tag:blogger.com,1999:blog-67796253440391537932024-03-05T23:00:57.630-08:00Network Analysis LiteracyWhen what where2Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.comBlogger33125tag:blogger.com,1999:blog-6779625344039153793.post-16868823268366687892016-12-04T13:44:00.003-08:002016-12-11T10:46:12.418-08:00Note 18 - A quote by Barabási<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEimsrPOVXIrXuZ9esja19wNoOR3oO97vrYaumR8Imxb1Jh4-L6vgLkkEcIFLOCQ8tkaee0cBJqHTyEV5z9qhxnfPnQtRqaoE24-7KYkCOX18qXm-QvlYYrP9mMQjftjlb34gAGRfeAKgUk/s1600/smallBarabasi.jpg" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEimsrPOVXIrXuZ9esja19wNoOR3oO97vrYaumR8Imxb1Jh4-L6vgLkkEcIFLOCQ8tkaee0cBJqHTyEV5z9qhxnfPnQtRqaoE24-7KYkCOX18qXm-QvlYYrP9mMQjftjlb34gAGRfeAKgUk/s320/smallBarabasi.jpg" width="253" /></a></td></tr>
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Albert-Laszlo Barabási at the World Economic Forum 2012</div>
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CC BY-SA 2.0: By World Economic Forum from Cologny, </div>
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Switzerland - Mastering Complexity, </div>
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https://commons.wikimedia.org/w/index.php?curid=21275699</div>
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Note 18 is simply a quote from an early article by <a href="http://barabasi.com/" target="_blank">Albert-László Barabási</a> on how complex network analysis helps to tame complexity from 2005. I believe that we are still at the beginning of the journey, while---of course---the field has made serious progress on the analysis of, e.g., dynamic and multiplex networks:<br />
<br />
<blockquote class="tr_bq">
Yet, the road to a fundamental and comprehensive understanding of networks is still rather rocky. (Barabási, 2005)</blockquote>
Reference:<br />
<br />
Albert-László Barabási: "Taming complexity", Nature Physics, 1:68–70, 2005 2Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.com0tag:blogger.com,1999:blog-6779625344039153793.post-5024929574555673012016-12-04T13:31:00.002-08:002016-12-11T10:45:17.851-08:00Note 17: Universal features vs. contextual interpretationThe last point already made the point that some edge weights representing real-world concepts such as probabilities or friendship do not allow a meaningful interpretation of graph theoretic distances. Such an interpretation is depending on the <b>context</b>, the meaning of the relationships and weights in the real-world. However, hip new <i>"network science"</i> was in part so very hot in contrast to lame, old <i>"social network analysis"</i>, because it just applied any kind of measure to all kinds of complex networks to identify structures common to all of them. This was the case for Watts' and Strogatz' seminal paper on Small-Worlds (Watts, 1998) or for Barabási and Albert's paper on Scale-Free Networks (Barabási, 1999).<br />
<br />
<a name='more'></a><br />
<br />
As one example, consider one of the data sets used by Watts and Strogatz, the neural network of a very small worm. In general, in a neural network, the interpretation of a small average graph theoretic distance in a contextually meaningful way is difficult. While one could say that it is a lower bound on the average number of firing events necessary to bring a signal from any one cell to any other cell, we know that the wiring of neural cells is so complex as to not allow all cells to talk to all other cells. However, in the new physics' way of looking at networks, the value did not have to `mean' anything much. They were looking at general patterns that would occur in all kinds of complex networks such as to identify universal laws of network generation.<br />
<br />
For this goal, all kinds of network analytic measures can be applied to all kinds of network representation. However, without matching a network measure and the context of a research question, an interpretation is in general not possible.<br />
<br />
<blockquote class="tr_bq">
Note 17. Applying network analytic measures to com-<br />
plex networks in order to find universal structures does<br />
not require a careful choice of the measure but also does<br />
not lend itself to contextual interpretation. (Zweig, 2016) </blockquote>
<br />
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<br />
Reference:<br />
<br />
(Zweig2016) Katharina A. Zweig: Network Analysis Literacy, ISBN
978-3-7091-0740-9, Springer Vienna, 2016<br />
<br />2Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.com0tag:blogger.com,1999:blog-6779625344039153793.post-57473408321485821712016-11-12T03:03:00.005-08:002016-12-04T13:08:38.606-08:00Note 16: On distances, edge weights, and other modeling decisions<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgV9MIP8iHtlhSJH-84Hz-QxznVV00eTij7Z5Grp07a9wZ4OsbimypHZrIUfQCAz6fqoqBlTaw3JUoZ3PgChVFKrkrZYWxBVl7nWbZV5B6tOOlaqXfM8oGkTdZiZkzZCIj9dKBi-YZjZnc/s1600/Note15.png" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="270" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgV9MIP8iHtlhSJH-84Hz-QxznVV00eTij7Z5Grp07a9wZ4OsbimypHZrIUfQCAz6fqoqBlTaw3JUoZ3PgChVFKrkrZYWxBVl7nWbZV5B6tOOlaqXfM8oGkTdZiZkzZCIj9dKBi-YZjZnc/s400/Note15.png" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">(Graph theoretic) distance is certainly one of the best understood <br />
concepts in network analysis. However, not every weighted graph <br />
should be used to compute the distance between all pairs of nodes. <br />
This figure is under CC:BY with a reference to <br />
Prof. Dr. Katharina A. Zweig or to this blogpost.
</td></tr>
</tbody></table>
The distance between any two nodes in an undirected, unweighted graph is defined as the minimal number of edges one has to traverse, to get from one node to the other. A natural generalization of this concept to weighted graphs defines the distance as the minimal sum of the weights on any sequence of edges between the two nodes.<br />
<br />
For weights that represents the length of streets, this makes perfectly sense. But of course, weights in complex networks can represent almost anything. Let's consider the number of hours two people called each other in the last two weeks. Or the probability to surf from one webpage through another by a direct link from the first to the second page.<br />
<br />
The distance between two nodes in a complex network is used for many things, foremost so-called centrality indices like the betweenness centrality or the closeness centrality. In general, for most centrality indices, a low distance to many other nodes will make a node more central. However, the length of calls between two persons is rather a measure of their closeness, not their distance. Thus, summing up these values will actually favor those pairs of nodes, who are linked by paths with people that do not call each other for a long time. It can help here, to invert the weights to make the meaning of distance more intuitive. However, even in this case: what does it actually mean if I am connected to another person by two other persons who talk to each other for two hours each? Then my distance to that guy is 1/2 + 1/2 = 1. Does that make me closer to that guy than being directly connected to another person, which I only call for 10 minutes, i.e., with a "distance" of 6?<br />
<br />
With the probabilities, a summation obviously makes not much sense. Here, a multiplication of the weights might be most meaningful to yield interpretable results.<br />
<blockquote class="tr_bq">
<br />
<b>Note 16. If network representation and network ana-<br />lytic measure are well matched, the measure’s value can<br />be interpreted with respect to the functionality of the net-<br />work for the complex system of interest. </b>(Zweig, 2016)</blockquote>
<br />
Read more on this on the general keyword of "trilemma of complex network analysis" and in Chapter 5 "Network representations of complex systems" of my book. <br />
<br />
<br />
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<br />
<b><u>Reference:</u></b><br />
<b><u>
</u></b><br />
(Zweig2016) Katharina A. Zweig: Network Analysis Literacy, ISBN
978-3-7091-0740-9, Springer Vienna, 20162Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.com0tag:blogger.com,1999:blog-6779625344039153793.post-53769344867142579322016-11-12T03:03:00.004-08:002016-11-12T03:54:32.046-08:00Note 14: Network analysis and statistics<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOhlEi9CClF-_I9Ray3ZuP46JQR8Tn5_m3tyuIzavjeiwE9CDHH7D8SkA8_tCJBgnEXQSuU-FuRYvjQ6dt18WxgksnlrK5e2jP-hsfvXy4ZrlgQBhBFBAcYdd4OaXJ_EHPa5gSvQyHNzA/s1600/Note14_Equation.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="132" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOhlEi9CClF-_I9Ray3ZuP46JQR8Tn5_m3tyuIzavjeiwE9CDHH7D8SkA8_tCJBgnEXQSuU-FuRYvjQ6dt18WxgksnlrK5e2jP-hsfvXy4ZrlgQBhBFBAcYdd4OaXJ_EHPa5gSvQyHNzA/s400/Note14_Equation.png" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Network Analysis has made heavy use of statistics - and it seems that statistics is not humankind's strength. That does not make network analysis any easier. <br />
This figure is under CC:BY with a reference to
Prof. Dr. Katharina A. Zweig or to this blogpost.
</td></tr>
</tbody></table>
<br />
<br />
A main part of network analysis is computing and interpreting statistical numbers. Unfortunately, statistics is not the most intuitive part of mathematics and it is well-known that even trained scientists have problem in correctly interpreting statistical results. Consider the following test:<br />
<br />
A group of young students without any symptoms of sickness make a blood donation. Routinely, their blood is checked for an HIV infection. The test is very sensitive. For simplicity, assume that it detects 99.99% of all infected persons and that non-infected persons will get a negative test with 99.99%. If now a person's first test returns a positive result, what is her likelihood to actually be infected?<br />
<br />
If you think it is 99.99%, you are in very good company (but wrong):<br />
<br />
<blockquote class="tr_bq">
<b>Note 14. Gigerenzer and his team showed in various<br />studies that almost none of the experts was able to give<br />the correct answer . Most answered that, as the test is<br />so specific and so sensitive, the probability that a person<br />is infected if the test says so is 99.99%. </b>(Zweig, 2016)</blockquote>
<br />
Actually, the question cannot really be answered without knowing the chance that a person without any symptoms is infected. This probability can be approximated by the so-called <i>incidence rate</i>, the number of new infections per year. For Germany, this is around 3,000 in a nation with about 80 million inhabitants (for young people, it might actually be higher than for the general population, but as an approximation that is fine).<br />
<br />
We now want to know the probability that a person is infected if her test turns out to be positive. There are two ways for a positive test: the person is infected and detected or the person is not-infected but falsely flagged. If we would test whole Germany, we would in essence find all of the 3,000 newly infected persons. However, from the remaining (still roughly) 80 million people, we would flag 0,01%, i.e., 1 person in 1 in 10,000. Thus, we additionally flag 8,000 people as positive. From all 11,000 people with a positive test, 8,000 would actually not be infected. I.e., the probability that a person with a positive test is infected is less than 50%, namely around 27%. Surprised?<br />
<br />
This computation has a very important consequence. Let our 'null-hypothesis' be that any given person is not infected. Now, we know that the probability that a person is not infected and gets a positive test is very small - this value is called her <i>p-value</i> (probability to observe the data given the assumption in the null-hypothesis). Especially, it is smaller than p=0.05, the classic threshold value to 'reject the null-hypothesis'. However, as we have seen, we need to compute the probability that the person is infected given a positive test result. And this probability can differ strongly from the other one when the ratio of the two classes (infected vs not infected) is not around 0.5. Thus, rejecting a null-hypothesis, just because given the assumption ("not infected") the observation of the data ("positive test") is unlikely, is the wrong way.<br />
<blockquote class="tr_bq">
<br />
<b>Note 15. The only correct verbal descriptions of a p-value need to<br />contain the words <i>given that the null-hypothesis is true</i> as<br />the p-value conditions on that. As the p-value does not<br />say anything about the probability of the hypothesis to<br />be true, given the observed data, it cannot be used as a<br />basis for rejecting the null-hypothesis.</b> (Zweig, 2016) </blockquote>
<br />
It is just the first step to update our probability of the assumption, given the observed data. This will be important, e.g., to identify <a href="https://en.wikipedia.org/wiki/Network_motif" target="_blank">network motif</a>s.<br />
<br />
If statistics is already hard, then this makes network analysis no way easier! Read more about <a href="https://en.wikipedia.org/wiki/Statistical_hypothesis_testing" target="_blank">statistical hypotheses testing on Wikipedia</a>. Or join my Mendely group on "<a href="https://www.mendeley.com/groups/758201/good-statistics-papers-for-non-statisticians/" target="_blank">Good statistics papers for non-statisticians</a>".<br />
<br />
<br />
<br />
<b><u> References:</u></b><br />
<br />
(Gigerenzer, 2007) Gerd Gigerenzer, Wolfgang Gaissmaier, Elke Kurz-Milcke, Lisa M. Schwartz, and Steven Woloshin. <a href="http://www. psychologicalscience.org/journals/pspi/pspi_8_2_article.pdf" target="_blank">Helping doctors and patients make sense of health statistics</a>. Psychological science in the public interest, 8(2), 2007.<br />
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" Priority="39" Name="Table Grid"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" Priority="61" Name="Light List"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2"/>
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<w:LsdException Locked="false" SemiHidden="true" Name="Revision"/>
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<w:LsdException Locked="false" Priority="29" QFormat="true" Name="Quote"/>
<w:LsdException Locked="false" Priority="30" QFormat="true"
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<w:LsdException Locked="false" Priority="33" QFormat="true" Name="Book Title"/>
<w:LsdException Locked="false" Priority="37" SemiHidden="true"
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<w:LsdException Locked="false" Priority="39" SemiHidden="true"
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<w:LsdException Locked="false" Priority="46"
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Name="List Table 6 Colorful Accent 5"/>
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Name="List Table 6 Colorful Accent 6"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 6"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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</w:LatentStyles>
</xml><![endif]--><!--[if gte mso 10]>
<style>
/* Style Definitions */
table.MsoNormalTable
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mso-tstyle-rowband-size:0;
mso-tstyle-colband-size:0;
mso-style-noshow:yes;
mso-style-priority:99;
mso-style-parent:"";
mso-padding-alt:0in 5.4pt 0in 5.4pt;
mso-para-margin-top:0in;
mso-para-margin-right:0in;
mso-para-margin-bottom:8.0pt;
mso-para-margin-left:0in;
line-height:107%;
mso-pagination:widow-orphan;
font-size:11.0pt;
font-family:"Calibri",sans-serif;
mso-ascii-font-family:Calibri;
mso-ascii-theme-font:minor-latin;
mso-hansi-font-family:Calibri;
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<br />
(Zweig2016) Katharina A. Zweig: Network Analysis Literacy, ISBN
978-3-7091-0740-9, Springer Vienna, 20162Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.com0tag:blogger.com,1999:blog-6779625344039153793.post-65423228777067960552016-11-12T03:03:00.002-08:002016-11-12T03:06:03.832-08:00Notes 12 and 13: Evolution of complex networks<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjS6mH_MH-1RM07g-BH0i_ScqSq3BnWsDeIK4Ij-iTwLfbaiI5Fw6SuMiwaiVKSbRQ843mXuHg-tJVVgCSORe76zvH-Wfwxcw4rNNolFngX1U8NC2F95pPcdyFyzQQWbNIc6UiirOKWZoY/s1600/note12_13.png" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="227" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjS6mH_MH-1RM07g-BH0i_ScqSq3BnWsDeIK4Ij-iTwLfbaiI5Fw6SuMiwaiVKSbRQ843mXuHg-tJVVgCSORe76zvH-Wfwxcw4rNNolFngX1U8NC2F95pPcdyFyzQQWbNIc6UiirOKWZoY/s320/note12_13.png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Networks evolve under different constraints and forces, in which <br />
;
network science is especially interested.<br />
This figure is under CC:BY with a reference to <br />
Prof. Dr. Katharina A. Zweig or to this blogpost.
</td><td class="tr-caption" style="text-align: center;"><br /></td><td class="tr-caption" style="text-align: center;"><br /></td></tr>
</tbody></table>
<br />
<br />
The perspective of statistical physics is always on finding universal forces that form and determine a given phenomenon. From this perspective, network science originating in statistical physics was also always searching for the forces on either the entities or the whole system that govern the evolution of a network's structure.<br />
<br />
"<b>Note 12. Network science describes the topology of a network as the effects of forces on either the entities or<br />the whole system.</b>" (Zweig, 2016)<br />
<br />
For example, the "smallness" of small-worlds can be explained by minimizing cost (energy) and the diameter at the same time, under the assumption that an edge between more distance nodes is more expensive. Putting some energy in long-distance edges is the optimal way to reduce the diameter of the network with the smallest total cost involved.<br />
<br />
Looking at a dynamic system, statistical physis is then most interested in two phenomena: phase transitions and equilibria. Phase transitions indicate the point, where a small shift in one parameter radically changes the way the system behaves. As such, this is not a phase in which it is easy to understand the forces and constraints under which the networks are built. It is more a state of <a href="https://en.wikipedia.org/wiki/Critical_phenomena" target="_blank">'criticality'</a> of which scale-free distributions are a tell-tale sign. When the "scale-free" degree distribution was observed (which actually is in most cases not directly scale-free) it seemed as if complex networks were in a state of 'self-organized criticality', i.e., a system that keeps itself in this state. This would have been very interesting because it points to a certain equilibrium of forces. In any case, the other most interesting state is the equilibrium, which can be either a stable one or an instable one. In the first case, small perturbations will be quickly diminished and the system returns back to the equilibrium state. In an instable one, a small perturbation will lead to a totally different state. In equilibria it is often easier to better understand the forces and constraints that form the system's behavior.<br />
<br />
<b>"Note 13. Network science is interested in equilibrium<br />structures as they can be used to understand the forces,<br />constraints, or incentives under which a network is built.</b>" (Zweig, 2016)<br />
<br />
Presumed constraints are, for example, <a href="https://en.wikipedia.org/wiki/Dunbar%27s_number" target="_blank">Dunbar's number</a> in social networks, i.e., the observation that people might not be able to manage more than 150 close acquaintances. Reasonable forces are energy/cost minimization for most networks, i.e., that the number and length of edges to be built is limited or should be minimized while - in most cases - building connected and maybe even locally dense networks.<br />
<br />
You can find more on the perspective of statistical physics in the 2nd chapter "Universal structures vs. individual featuers" of my book. There I compare it with the perspective of social science and why I believe that Network Science is really something different than (Social) Network Analysis. <br />
<br />
<br />
Reference:<br />
<br />
(Zweig2016)
Katharina A. Zweig: Network Analysis Literacy, ISBN 978-3-7091-0740-9, Springer
Vienna, 2016<br />
<br />
All figures are under <a href="https://creativecommons.org/licenses/by/4.0/">CC:BY </a>with
a reference to Prof. Dr. Katharina A. Zweig or to this blogpost, if not mentioned otherwise. 2Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.com0tag:blogger.com,1999:blog-6779625344039153793.post-51324636729847470672016-09-27T14:51:00.000-07:002016-09-27T14:51:13.725-07:00Drawing graphs for publications or presentationsThe one software I would not like to miss for drawing and editing graphs is <a href="https://www.yworks.com/products/yed">yEd</a>, a free software provided by yWorks. It is an intuitive tool to draw small graphs and networks where - in contrast to most normal vector graphics programs - moving around the nodes will automatically include the edges as well. Furthermore it is possible to mark a subgraph and to turn it or to mirror its position - without mirroring or turning the node labels as well.<br />
<br />
I've created almost all drawings for my publications with it, especially those for my book and this blog. My normal design pipeline is to draw the graph in yEd and then either to save it directly as eps or pdf for inclusion in LaTeX documents or to first save it as an svg. Then I would give some finishing touches to the drawing in inkscape, from which it can then again be exported as eps and pdf.<br />
<br />
Despite the fact that I've been using this tool now for more than 12 years, I just learned that it is also the best way to create an image for Power Point presentations. Instead of exporting it to bmp which scales badly, try exporting it to emf. Scaling this format leads to much better results. 2Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.com0tag:blogger.com,1999:blog-6779625344039153793.post-21888220952761804212016-09-26T08:52:00.001-07:002016-09-26T08:52:16.718-07:00Foreword by Steve Borgatti<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzyvZ8TUCnKErsP1WS6ykZcNgxusI19tLKQUljmiiQSXRh-VwfskNyVDV4PlRKFb29eNdEVjXdUpQRzzcgA6i7qo1kVF5-YYQQRyL2QV1DySqbp7-1axmW_lwyb-Nuu2zEWKWygFVL_Ns/s1600/WordCloud_Borgatti.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="185" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzyvZ8TUCnKErsP1WS6ykZcNgxusI19tLKQUljmiiQSXRh-VwfskNyVDV4PlRKFb29eNdEVjXdUpQRzzcgA6i7qo1kVF5-YYQQRyL2QV1DySqbp7-1axmW_lwyb-Nuu2zEWKWygFVL_Ns/s320/WordCloud_Borgatti.png" width="320" /></a></div>
I was very happy to hear that <a href="http://www.steveborgatti.com/">Steve Borgatti</a> agreed to write a foreword for my book. As I describe it in my introduction, his research was the starting point for my book. So, I'll hand over to him and copy his foreword here:<br />
<br />
<br />
<blockquote class="tr_bq">
This is a delightful book. It’s so easy to read, you can almost accidentally learn quite a bit of network science without even noticing it. Written in a playful manner, it tends to enliven the brain rather than put it to sleep – quite a change from the usual pedantic tome. It’s a quirky book that does not try to be systematic. For example, it does not cover “community detection” (that’s cluster analysis to you social scientists). As a result, the book has a great deal of personality.<br />But what I really like about the book is the subtext. What it’s actually about, in my opinion, is how to think, and here, that means how to think with models. Most academics are very gullible when it comes to concepts outside their disciplines. Within their area, any new idea or phrasing is treated with withering skepticism, but outside their area, they adopt ideas with the speed of teenagers adopting slang or fashion. Thus, a management scholar hears about small worlds and clustering coefficients and immediately shoehorns them into their next study. A physicist learns about betweenness centrality and suddenly there are 500 papers that reference the idea. If the first paper associates betweenness with influential spreaders in the spread of a disease, all of the following papers do the same. If you internalize this book, you won’t make that mistake. You will realize that, although there is a sense in which network measures are tools like hammers, there is much more to them. Hammers work pretty much the way they work in any setting, but using a network measure implicitly entails fitting a model of how things work. And if the model doesn’t fit, the measure doesn’t either. <br />Curiously, although I associate model-based thinking with the physical sciences, my experience is that both physical and social scientists are equally likely to have this mindless, “pluginski” attitude about network concepts. Therefore, I think this book would be useful for both audiences. But since the content of the book is mostly drawn from what Katharina calls the “network science” field (as opposed to the “social network analysis” field), I’m guessing it will appeal mostly to budding physical scientists. Too bad, because if there was ever an introduction to network science that was especially suitable for social scientists, this is it. <br />I look forward to seeing this in print. <br />Lexington,KY<br />Steve Borgatti</blockquote>
<br /><br />
<br />
Foreword by Steve Borgatti for Katharina A. Zweig's book: "Network Analysis Literacy", in print; (c) by Springer Wien, used with permission2Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.com0tag:blogger.com,1999:blog-6779625344039153793.post-66214569704806363082016-06-26T00:28:00.001-07:002016-11-12T02:34:01.719-08:00Notes 8-11: What is the difference between social network analysis and network science?The following four notes are from my book "Network Analysis Literacy" (Zweig2016).<br />
<br />
<blockquote class="tr_bq">
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZ61I55rfaCPhNFAsyUOgHijlf2xnxX1LorkwHOu9OwLCCkuyoDfVIo7P03ECd3h5jLd3nTT9IqfI6RMUhMy4wUoLUGIfcc94c__yrnDTx5c6w4Cw44gQDUjGxg4bmI5ugco_HV709Pno/s1600/Note_8_11_Table.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="231" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZ61I55rfaCPhNFAsyUOgHijlf2xnxX1LorkwHOu9OwLCCkuyoDfVIo7P03ECd3h5jLd3nTT9IqfI6RMUhMy4wUoLUGIfcc94c__yrnDTx5c6w4Cw44gQDUjGxg4bmI5ugco_HV709Pno/s400/Note_8_11_Table.png" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Summary of the differences between social network analysis and network science. Of course, this is a generalization and will not apply to every single network analytic project from either field.</td></tr>
</tbody></table>
"<b>Note 8. The first big difference between social and complex network analysis as a part of network science, however, is that the underlying data is not restricted to social systems but comprises all relationships between any kind of entities in any given complex system.</b>" </blockquote>
<blockquote class="tr_bq">
"<b>Note 9. A second important difference between network science and social network analysis is that (in general) the first induces micro-behavior from observed macro-behavior while (in general) the second predicts macro-behavior from hypothesized micro-behavior.</b>"</blockquote>
<blockquote class="tr_bq">
"<b>Note 10. Social network analysis tries to capture many details from the social system of interest. Often, additional parameters of the persons under observation are requested and used for the analysis. The approach is thus a contextual approach that takes the context into account. In network science, the abstraction level is in most cases much higher and individual properties of the entities are much less often taken into account. The approach can be characterized as being largely context-free.</b>"</blockquote>
<blockquote class="tr_bq">
"<b>Note 11. In summary (and a bit bold), social network analysis is a theory-driven, bottom-up approach that carefully models additional social information where available and takes it into account when interpreting the results. Network science follows a data-driven, top-down approach that tries to clean the data from all detail to compare the core structure of different complex networks.</b>"</blockquote>
<br />
Reference:<br />
<br />
(Zweig2016)
Katharina A. Zweig: Network Analysis Literacy, ISBN 978-3-7091-0740-9, Springer
Vienna, 2016 2Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.com0tag:blogger.com,1999:blog-6779625344039153793.post-32920345573023378472016-06-26T00:03:00.000-07:002016-06-26T00:05:27.525-07:00Note 7: What is the difference between graph theory and network analysis?<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3nBD1PLwATDTiRntBlnkodHidML7jwgW4V9tYiyYr_gpFvuJuhCyw32wZtTvyHHX5wkkkt8J8hq1dn7qmNpX9fPcfOeFBKzCTgMJm7D8i9LuW-nE_BsaVbtz6qdY-2QNRCLI_LfJvtj4/s1600/note7.png" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="149" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3nBD1PLwATDTiRntBlnkodHidML7jwgW4V9tYiyYr_gpFvuJuhCyw32wZtTvyHHX5wkkkt8J8hq1dn7qmNpX9fPcfOeFBKzCTgMJm7D8i9LuW-nE_BsaVbtz6qdY-2QNRCLI_LfJvtj4/s320/note7.png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">What is the difference between graph theory and network analysis?<br />
This
figure is under <a href="https://creativecommons.org/licenses/by/4.0/">CC:BY </a>with
a reference to<br />
Prof. Dr. Katharina A. Zweig or to this blogpost. </td></tr>
</tbody></table>
Graph theory is often seen as one building block of network analysis. But what exactly are the differences between the two fields? Graph theory is a very abstract science that defines different graph classes and tries to understand their specific properties. Furthermore, it is concerned with so-called graph problems. Let me give you the most classic example of such a graph problem:<br />
<br />
<blockquote class="tr_bq">
<b>Given:</b> A graph G<br />
<b>Wanted:</b> The maximum clique C in G, i.e., the largest subset of nodes in G such that any two nodes in C are connected by an edge (complete subgraph)</blockquote>
<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLElFICxIMhS8WV8cMQESrzWJDQMnZ8IfwNUYi11MqhjQF5JhDl4Gfz6SnkO7oPn02AQjJ6hg143wvjsVLu96FkDpQubnWsP5TBV_Ovc_wuB8ib2jN0ZyuRXMI_TZNreBugbsz_sIwCLE/s1600/Note7_Interval.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLElFICxIMhS8WV8cMQESrzWJDQMnZ8IfwNUYi11MqhjQF5JhDl4Gfz6SnkO7oPn02AQjJ6hg143wvjsVLu96FkDpQubnWsP5TBV_Ovc_wuB8ib2jN0ZyuRXMI_TZNreBugbsz_sIwCLE/s320/Note7_Interval.png" width="315" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Figure 1: A set of intervals (above) and the corresponding interval graph (below). This figure is under <a href="https://creativecommons.org/licenses/by/4.0/">CC:BY</a> with a reference to Prof. Dr. Katharina A. Zweig
or to this blogpost.</td></tr>
</tbody></table>
A graph problem is always described by an input ("Given: XX") and a definition of the wanted output ("Wanted: YY") with resepct to the input. Finding a maximum clique in a given graph is problem that takes a long time to solve. The problem is that, intuitively, all possible subsets of nodes have to be tested of whether they constitute a clique. Until now, there is no other solution process that is substantially faster for all possible inputs. However, for some graph classes, it can be done much faster. One of those graph classes is a so-called interval graph (see Figure 1). Given a set I of intervals, the interval graph represents the intervals as node and connects any two nodes with each other, if the corresponding intervals overlap. In such a graph, we just need to find a spot where the maximum number of intervals overlap. It can be shown that we just have to look at all the endpoints of the intervals to find that spot. This can be done very fast, even if the network is extremely large.<br />
<br />
Thus, graph theory is concerned with different graph classes and their relationship to graph problems and the runtime necessary to find the solutions to the graph problems. <br />
<br />
<b>"Note 7. In general, graph theory is concerned with the relationships
between different graph classes and the relationship between certain
graph structures, a graph problem, and its algorithmic solution."
(Zweig2016)</b><br />
<br />
In contrast, network analysis is interested in the connection between a certain graph structure and the function this graph structure has in the complex system of interest. For this, we use graph theoretic insights. For example, in random graph theory, we know how likely it is that a certain random graph contains a clique of order k (order = number of nodes in it). If a clique of order k is found in a complex network but it would also be very likely to find one of those in a random graph of a corresponding size and order, network analysis would deduce that the clique is not likely to be functional for the complex system of interest.<br />
<b> </b><br />
In summary: network analysis builds on (some of the) insights of graph theory, but is not interested so much in the abstract behavior of graph classes, but rather in the relationship between structure and function of<b> </b>a graph and the complex system it represents.<b><br /></b><br />
<br />
<br />2Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.com0tag:blogger.com,1999:blog-6779625344039153793.post-2266437737460371292016-06-25T22:35:00.004-07:002016-11-12T02:32:16.211-08:00Note 4 and Note 6: What does a measure measure?<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh9BB9GgC8GA4krysgCOfvzqogvTGYBFmG8VgyL5cVHWuZLvpT7cCpFiPBsEHwk5MPtBcgsEk0i860lACn2aPQeAX8SvcRCYMgb6fVKSqiTrhb8xBI03nJerlfMTrAPQ1k42_aE4-2xYRY/s1600/Note4.png" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="239" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh9BB9GgC8GA4krysgCOfvzqogvTGYBFmG8VgyL5cVHWuZLvpT7cCpFiPBsEHwk5MPtBcgsEk0i860lACn2aPQeAX8SvcRCYMgb6fVKSqiTrhb8xBI03nJerlfMTrAPQ1k42_aE4-2xYRY/s320/Note4.png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Network analysis promises to give insight into the inner<br />
workings of a complex system by representing<br />
it as a complex network, by analyzing this representation,<br />
and by interpreting the results of the analysis.<br />
But: what does a measure actually measure?<br />
How is the result mapped to the inner workings of the system? <br />
This figure is under
<a href="https://creativecommons.org/licenses/by/4.0/">CC:BY </a>with a reference to<br />
Prof. Dr. Katharina A. Zweig
or to this blogpost.
</td></tr>
</tbody></table>
Network analysis as a framework is mainly seen as a set of functions applied to the network's structure. These functions are often called "measures" but from a mathematical standpoint, a measure is a very specific function, that would need to fulfill a set of properties. However, since it is common to call them "measures", I will stick to that term.<br />
<br />
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<div class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;">
<span style="font-family: "times new roman" , serif; font-size: 12.0pt;">Now, what
does a network analytic measure like the eccentricity measure? Mathematically,
the eccentricity of a node is defined as its maximal distance to any other node
in the graph. So, this is what the eccentricity measures.</span><span style="font-family: "times new roman" , serif; font-size: 12.0pt;"> However,
when we apply network analysis, we hope for the following:</span>
</div>
<br />
<blockquote class="tr_bq">
<div class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;">
<b><span style="font-family: "times new roman" , serif; font-size: 12.0pt;">"Note
4. The promise of network analysis is that the abstraction of a complex system
as represented by a complex network and its underlying graph still allows to
infer something about the complex system of interest. That is actually a strong
assumption [and there are] preconditions to enable this transfer."
(Zweig2016)</span></b><span style="font-family: "times new roman" , serif; font-size: 12.0pt;"></span></div>
</blockquote>
<br />
<div class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;">
<span style="font-family: "times new roman" , serif; font-size: 12.0pt;"> So,
what is the insight that a measure like the eccentrictiy can give? It kind of
determines the centrality of a node, but under very strong assumptions:</span></div>
<ol start="1" type="1">
<li class="MsoNormal" style="line-height: normal; mso-list: l0 level1 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span style="font-family: "times new roman" , serif; font-size: 12.0pt;">The graph theoretic distance
between v and w, i.e., the minimal number of edges to be traversed to get
from v to w, is of interest in the complex system to be investigated. For
example, if the complex network represents a street network as an
unweighted graph, then the graph theoretic distance which only counts the
number of edges (i.e., streets) to be traversed, is hardly of interest.</span></li>
<li class="MsoNormal" style="line-height: normal; mso-list: l0 level1 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span style="font-family: "times new roman" , serif; font-size: 12.0pt;">The eccentricity looks at the
maximal distance. If we identify the node with the lowest eccentricity,
this is the node that can, in principle, send a message to all nodes and
it will reach even the farthest node in the minimal time possible. This
again under some assumptions:</span></li>
<ol start="1" type="1">
<li class="MsoNormal" style="line-height: normal; mso-list: l0 level2 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 72.0pt;"><span style="font-family: "times new roman" , serif; font-size: 12.0pt;">The message is sent at the
same time to all neighbors.</span></li>
<li class="MsoNormal" style="line-height: normal; mso-list: l0 level2 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 72.0pt;"><span style="font-family: "times new roman" , serif; font-size: 12.0pt;">They send it to all their
neighbors within one time step.</span></li>
<li class="MsoNormal" style="line-height: normal; mso-list: l0 level2 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 72.0pt;"><span style="font-family: "times new roman" , serif; font-size: 12.0pt;">There is such a thing as a
time step, which all nodes know.</span></li>
</ol>
<li class="MsoNormal" style="line-height: normal; mso-list: l0 level1 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span style="font-family: "times new roman" , serif; font-size: 12.0pt;">In other words: there is a
process in the real-world complex system which uses the relationship
represented by the complex network modeling the system. This process needs
to be well described by the implicit assumptions of the network measure
(cf. Borgatti2005). </span></li>
</ol>
<div class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;">
<span style="font-family: "times new roman" , serif; font-size: 12.0pt;"> This
is a first indication, that network analytic measures implicitly contain a
model of a network flow process, and that the measure needs to match the
network flow process of interest.</span></div>
<div class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;">
<br /></div>
<br />
<blockquote class="tr_bq">
<div class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;">
<b><span style="font-family: "times new roman" , serif; font-size: 12.0pt;">"Note
6. While it is absolutely true that the result of a formula is never wrong in
the sense of 'different than what it is supposed to be', the application of the
formula might be a mismatch with the intention of what is to be measured."
(Zweig2016)</span></b></div>
</blockquote>
<br />
<div class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;">
<span style="font-family: "times new roman" , serif; font-size: 12.0pt;">Reference:</span></div>
<div class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;">
<span style="font-family: "times new roman" , serif; font-size: 12.0pt;">(Borgatti2005)
Borgatti, S. P.: Centrality and Network Flow, Social Networks, 2005, 27, 55-71 </span></div>
<div class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;">
<span style="font-family: "times new roman" , serif; font-size: 12.0pt;">(Zweig2016)
Katharina A. Zweig: Network Analysis Literacy, ISBN 978-3-7091-0740-9, Springer
Vienna, 2016 </span></div>
<br />
<br />2Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.com0tag:blogger.com,1999:blog-6779625344039153793.post-72932688331800861762016-06-23T13:33:00.002-07:002016-06-25T22:45:49.205-07:00Note 5: Can visualization replace analysis?<div class="separator" style="clear: both; text-align: center;">
</div>
<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZlNNgNP-mt1txXCJlMpXx10uEuEg9AVBpkshnGUEWbrYNT3dwruzenI6n6Ffr5FEsaTG7zk5Ppr7UniZJTFxQYUCgJzXq_pSP3NCJCXQSPY8T4HTinFv4pBYfRLNJGp_dPPFu6Ujl9so/s1600/Note5.png" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="137" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZlNNgNP-mt1txXCJlMpXx10uEuEg9AVBpkshnGUEWbrYNT3dwruzenI6n6Ffr5FEsaTG7zk5Ppr7UniZJTFxQYUCgJzXq_pSP3NCJCXQSPY8T4HTinFv4pBYfRLNJGp_dPPFu6Ujl9so/s320/Note5.png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">What is the most central node in the graph on the left,<br />
what is the most central node in the graph on the right?</td><td class="tr-caption" style="text-align: center;"></td><td class="tr-caption" style="text-align: center;"><br /></td><td class="tr-caption" style="text-align: center;"><br /></td></tr>
</tbody></table>
<br />
I always show this figure in my lecture. Then I ask: "What is the most central node in the left graph? What is the most central node in the right graph?" For the first question, it is almost impossible to suppress the urge to shout out: "The one in the middle is!". I think that this is because we humans are used to put the most important thing in the middle, where our (physical) focus is. We believe that if someone puts a thing in the middle, there must be meaning assigned to this. However, in a network visualization, it does not need to be. However, looking closely at the two visualizations, you will find that they display the same <b>graph, i.e., the connections are absolutely the same</b>. And on the right hand visualization, it becomes obvious that every node is interchangeable with every other node (in graph theoretical terms: the nodes are in an <a href="https://en.wikipedia.org/wiki/Automorphism">automorphism class</a>). All nodes thus have the same centrality in the network.<br />
<br />
Of course, most algorithms try to put nodes in the middle of their neighbors. This already indicates that a node which is at the center of a network (in the sense of smallest closeness) might also be in the center of the visualization of the network. However, there are also other aesthetic considerations, such as the overall ratio of the resulting figure, or edge crossing minimization.<br />
<br />
Looking at a visualization of a network is always a good idea. Being inspired by the visualization and creating a new hypothesis about the structure of it, is absolutely helpful in the first stages of any network analytic project. However, the hypothesis needs to be tested by a quantifiable method - not by another visualization.<br />
<br />
<blockquote class="tr_bq">
<b>"Note 5. A visualization of a network can be both revealing and deceiving. This is why Gephi, <a href="https://www.yworks.com/products/yed">yEd</a>,
and other visualization tools are perfect for exploration and
hypothesis building; it is also the reason why statistical software
packages or self-tailored applications are needed to collect
quantifiable evidence that a given hypothesis is true." (Zweig2016)</b></blockquote>
<br />
Reference:<br />
<br />
(Zweig2016) Katharina A. Zweig: Network Analysis Literacy, ISBN 978-3-7091-0740-9, Springer Vienna, publication expected Dec 2016 2Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.com0tag:blogger.com,1999:blog-6779625344039153793.post-26652980366309265932016-06-05T02:07:00.001-07:002016-06-25T22:47:01.700-07:00Note 3: What is actually the differece between a graph and a complex network?<br />
<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhR5ON_Ao_YExedt1eZIxxytjhUqlOqq491fbGRuk-z6EFnEewF3CS7BZJVg9d03YGDK27ByiNOg8VXaINejNP_Au7g12KKOFCibOBnYPscj3lcqrRY8YLoSi1aDhSJ0UOeNTfkNZuWe78/s1600/Note3_CN_vs_Graph.png" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="296" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhR5ON_Ao_YExedt1eZIxxytjhUqlOqq491fbGRuk-z6EFnEewF3CS7BZJVg9d03YGDK27ByiNOg8VXaINejNP_Au7g12KKOFCibOBnYPscj3lcqrRY8YLoSi1aDhSJ0UOeNTfkNZuWe78/s320/Note3_CN_vs_Graph.png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">A graph is a mathematical structure that is composed of a set of<br />
elements and a relation defined on that set. It does not know about<br />
the set of real-world entities and their relationship it <br />
(might) represent. It is important to note that a graph does, <br />
of course, not need to represent any real-world situation.<br />
The 'complex network' defines the relationship between<br />
the set of entities in the real-world and their representation<br />
in the graph.This figure is under
<a href="https://creativecommons.org/licenses/by/4.0/">CC:BY </a>
with a<br />
reference to
Prof. Dr. Katharina A. Zweig or to this blogpost.</td></tr>
</tbody></table>
When I started to do network analysis, I was confused by the terms 'graph' and 'complex network', and I wondered whether they can be used totally interchangeably or not.<br />
<blockquote class="tr_bq">
<br />
<b>"Note 3. What is the difference between a (complex) network and a graph? The quick answer is that a graph is the abstract representation of a relation between entities while a network combines the graph with additional information about the entities and their relationship represented by the graph."</b><br />
<b>(Zweig2016)</b></blockquote>
<br />
A <i><a href="https://en.wikipedia.org/wiki/Graph_(discrete_mathematics)">graph</a></i> is, mathematically, just the combination of any set of elements V and a <a href="https://en.wikipedia.org/wiki/Binary_relation">relation</a> E defined on it. A relation is just a subset of all the possible pairs of elements in the set. By definition, these pairs have an order, i.e., it makes a difference whether the pair $(x,y)$ or the air $(y,x)$ is included. If for all pairs both directions are included in the relation, we speak of a <i>symmetric relation</i>. If there is at least one pair that is only included in one direction, it is an <i>asymmetric relation.</i> The elements of the relation are called <i>edges</i>, when they are part of a graph, and the elements of V are called <i>nodes</i> or <i>vertices.</i><br />
A graph can be associated with functions, that assign values to nodes or edges. You see, on this mathematical level, everything is pretty abstract.<br />
<br />
A complex network fills these things with meaning: the nodes suddenly represents a set of real-world entities, e.g., persons. The edges represent a relationship between the nodes. The mathematical property of the relation called 'symmetry' suddenly represents an undirected relationship, while an asymmetric relation represents a directed relationship. Functions associated with the edges are <i>weights</i> that capture an important aspect of the relationship, and functions associated with the nodes capture important <i>properties</i> of the nodes.<br />
<br />
In most network analytic publications, you will see an identification of the nodes with their entities and the edges with the relationship they represent. This is in most cases unproblematic and saves a lot of text. Instead of writing "Two nodes are connected by an edge if the corresponding street corners are connected by a street", it is much faster to write: "In the network, street corners are the nodes and streets are the edges.".<br />
<br />
However, such a formulation also indicates there would be a clear one-to-one-mapping. As will be seen in later blog posts, this is almost never the case: there are multiple modeling decisions to be made to come from a heap of raw data to a network representation. Here is how Brandes et al. phrase the problem in their <a href="http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8888786&fulltextType=ED&fileId=S2050124213000027">editorial of the first issue</a> of their journal "<a href="http://journals.cambridge.org/action/displayJournal?jid=NWS">Network Science</a>":<br />
<br />
<blockquote class="tr_bq">
"As representation is usually defined via an isomorphism, i.e., a one-to-one mapping<br />
between structures preserving relations, a phenomenon cannot be represented<br />
directly but needs to be conceptualized first.<br />
Of course, this is by no means an unusual division in science or other areas of<br />
knowledge. Possibly because of the graphic and metaphoric connotations of the<br />
term network, the implications of a preceding abstraction step are often overlooked<br />
or blurred. Sometimes this may be on purpose for terminological convenience.<br />
More often, there appears to be a lack of awareness. We feel, however, that this<br />
distinction is crucially important for serious applications of network science to the<br />
understanding of substantive phenomena as it points to the delicacy of interpreting<br />
the results of network data analysis.<br />
Interpretation essentially reverses the process of abstraction and representation<br />
to get back to the phenomenon so that substantive theory is required to secure<br />
conclusions."(Brandes2013)</blockquote>
Defining the <i><a href="http://mathworld.wolfram.com/Isomorphism.html">isomorphism</a></i> between the elements in the real-world and their counterpart in the graph is a step that is often also called <i><a href="https://en.wikipedia.org/wiki/Operationalization">operationalization</a></i> and later blog post will have a lot to say about this step.<br />
<br />
In summary: there is an important distinction between a complex network and the graph it contains. And at least in the description of how the real-world phenomenon is turned into a network representation, it is good practice to differentiate between the two different layers---the real-world and the graph representation of it. In later parts of the text, however, it might be cumbersome to differentiate between the elements of the graph and the real-world entities they represent.<br />
<br />
<b>Reference:</b><br />
<br />
(Brandes2013) Brandes, U.; Robins, G.; McCranie, A. & Wasserman, S.: "What is Network Science?", Network Science, 2013, 1, Editorial<br />
(Zweig2016)
Katharina A. Zweig: Network Analysis Literacy, ISBN 978-3-7091-0740-9, Springer
Vienna, publication expected Dec 2016 2Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.com0tag:blogger.com,1999:blog-6779625344039153793.post-32727396326988979952016-06-04T14:14:00.000-07:002016-06-23T12:39:01.661-07:00Note 2: Minimal requirements to represent data as a "complex network"<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiG1QU4HnrEB596BIdA-MIMvutR0LgUFTxDw-9ne044lXJknhpUnnCMhks_A75WHUC2bvLYyr7PDiQ_pHNI6dSsJ40B0xCBCJrxQg1QzuxgmnWMnDSYHU15GBo987DJcID5LYSzbQldMRI/s1600/Spiegelei.png" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiG1QU4HnrEB596BIdA-MIMvutR0LgUFTxDw-9ne044lXJknhpUnnCMhks_A75WHUC2bvLYyr7PDiQ_pHNI6dSsJ40B0xCBCJrxQg1QzuxgmnWMnDSYHU15GBo987DJcID5LYSzbQldMRI/s320/Spiegelei.png" width="182" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Not all well-defined relations lend <br />
themselves to a meaningful representation <br />
as a complex network. This figure is under <a href="https://creativecommons.org/licenses/by/4.0/">CC:BY</a> <br />
with a reference to Prof. Dr. Katharina A. Zweig<br />
or to this blogpost.</td></tr>
</tbody></table>
Have you ever asked yourself what the minimal requirement is to turn something into a "complex network"? Well, mathematically seen, it is the following:<br />
<blockquote class="tr_bq">
<br />
Note 2. Mathematically, a relation R on a given set<br />
of entities or objects is just an arbitrary choice of pairs<br />
of these entities (objects), denoted by R ⊆ O × O. In<br />
principle, any relation can be represented as a graph. <br />
(Zweig2016)</blockquote>
<br />
So, the minimal requirements are actually - minimal. While mathematically possible, not all relations gain from being represented as a graph and by being treated as a "complex network". Look, for example at the set of all living humans that own at least one ID card and connect any two of them if their oldest ID-card's ID number shares the last digit. This is surely a relation, but it is also surely a <i>relationship </i>between humans that will not be any better analyzed by turning it into a complex network.<br />
<br />
Why is this so? The whole idea of complex network analysis is to understand the interaction structure of entities in a complex system. Complex systems are those with <a href="https://en.wikipedia.org/wiki/Emergence">emergent phenomena</a>. Emergence often - well - emerges, when interactions between pairs of entities change the interactions of other pairs of entities because of the interactions between the pairs, i.e., when indirect effects are transferred via the interactions. Brandes et al. express it like this:<br />
<br />
<blockquote class="tr_bq">
By postulating a friendship network in (say) a school class-<br />
room of 25 students, we have taken a theoretical step that<br />
is non-trivial. We have supposed that separate individ-<br />
uals are not an adequate representation, moreover that<br />
even separate dyads are insufficient; rather, that there<br />
is a unity within the classroom that makes it proper to<br />
talk of “a” network, not 25 children or 300 dyads. To con-<br />
ceptualize the classroom in network terms is an implicit<br />
(and strong) claim that connectedness across individual<br />
elements is fundamentally important, so that the class-<br />
room can be thought of as one “system”. (Brandes2013)</blockquote>
I believe that this feature that turns a set of pairs (or dyads) into "one system" is a network process that induces indirect effects via the relationship that binds the pairs together in one network. Thus, the answer to the question is: while mathematically any relation defined on a finite set of entities is good enough for a representation as a graph, semantically, not all relations make sense to be represented as a complex network. And since a relation can once represent a meaningful relationship and the very same relationship can also represent a meaningless relationship (like the one above), it is not the relation itself that decides about its "networkability". It is the relationship the relation represents.<br />
<br />
<b>Reference:</b><br />
<br />
(Brandes2013) Brandes, U.; Robins, G.; McCranie, A. & Wasserman, S.: "What is Network Science?", Network Science, 2013, 1, Editorial<br />
(Zweig2016) Katharina A. Zweig: Network Analysis Literacy, ISBN 978-3-7091-0740-9, Springer Vienna, publication expected Dec 2016 2Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.com0tag:blogger.com,1999:blog-6779625344039153793.post-4841586572038322682016-06-04T10:29:00.002-07:002016-06-23T12:21:36.785-07:00Note 1: Trilemma of Complex Network Analysis<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjLQTMvk_kuYaeoRFEZif0wWzhwi6pDxtLGhyqGWbh8OgWSgCkhGDrQVwE3fS-pZmfxkfNPrY-puoR-dknRRhjFqt-vnyf5MqO2xygxlKJlWotjhHpWeBRkeMYN9b2RiIoacNN77VsNKcg/s1600/Trilemma.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="242" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjLQTMvk_kuYaeoRFEZif0wWzhwi6pDxtLGhyqGWbh8OgWSgCkhGDrQVwE3fS-pZmfxkfNPrY-puoR-dknRRhjFqt-vnyf5MqO2xygxlKJlWotjhHpWeBRkeMYN9b2RiIoacNN77VsNKcg/s320/Trilemma.png" width="320" /></a></div>
I started my doctoral studies over 13 years ago - and it seemed that complex network analysis is the framework to use whenever you can define a meaningful relationship between a set of entities: proteins interacting with each other, airports connected by scheduled flights, people connected by pressing a 'retweet' button in their browser to express their opinion on someone else's tweet.<br />
<br />
The main hypothesis and first Note in my upcoming book "Network Analysis Literacy" (Zweig2016) condenses my findings of these more than a dozen years:<br />
<br />
<blockquote class="tr_bq">
<b>Note 1.</b> "To interpret the values of a distance-based measure, the way of calculating the distance must be matched to the process of interest. To interpret any walk-based measure, the set of walks used by the measure needs to be closely adapted to the process. (K.A. Zweig: Network Analysis Literacy, (c) by Springer Verlag, Heidelberg, to be published)</blockquote>
<br />
It refers to the so-called <b>trilemma of complex network analysis, </b>a term Isadory Dorn, Andreas Lindenblatt, and I developed in 2012 (Dorn2012). It summarizes the interdependencies between raw data, relationship of interest, network process of interest, research question, and methods used to analyze the latter. The <b>research question </b>determines a <b>network process</b> that uses a (set of) <b>relationships</b> to exert indirect effects on entities connected to each other by this relationship. By representing this relationship as a complex network, all classic network analytic methods can be applied---in principle.<br />
<a href="http://www.steveborgatti.com/">Stephen P. Borgatti</a> was the first to show that centrality indices, one of the most classic and widely used set of methods, have an inbuilt model of a network process they are associated with (Borgatti2005): they secretely determine the paths on which indirect effects are induced! Take your beloved <a href="https://en.wikipedia.org/wiki/Betweenness_centrality">betweenness centrality</a>, defined as follows:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhmCKHe4JsTE1G4GPWlS4AVcrqpvnxg1LFY4qzFtC4R_O8xANtO0oEslyzzhQMf72VJNwPpVemT6SHkQm_AI-TTmOQakR9d3EvYh_rpO6mxIzgAAL3V-gzW7hSlV6jnqxR_EM6HuzCX1wI/s1600/Betweenness.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="88" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhmCKHe4JsTE1G4GPWlS4AVcrqpvnxg1LFY4qzFtC4R_O8xANtO0oEslyzzhQMf72VJNwPpVemT6SHkQm_AI-TTmOQakR9d3EvYh_rpO6mxIzgAAL3V-gzW7hSlV6jnqxR_EM6HuzCX1wI/s320/Betweenness.png" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
where $\delta_v(s,t)$ refers to the number of shortest paths between s and t, containing v, and $\delta(s,t)$ refers to the number of all shortest paths between s and t. There will be another dedicated blog entry to the implicit assumptions the betweenness centrality makes, but here it suffices to say that it assumes the following: all entities want to interact with each other in the same intensity (all pairs of s and t are treated equally) and all of them interact on shortest paths.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
If your network process of interest does not follow these two assumptions, the betweenness centrality might not be the best centrality index to answer your research question. </div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
This is just the first example of how network analysis literacy, e.g., knowing the implicit models behind your favourite network analytic measure or the relationship between research question, network process, and relationship, may help you to make well-grounded choices, </div>
<br />
<u><b>References:</b></u><br />
<br />
(Borgatti2005) Borgatti, S. P.: "<a href="http://www.sciencedirect.com/science/article/pii/S0378873304000693">Centrality and Network Flow</a>", Social Networks, 2005, 27, 55-71<br />
<br />
(Dorn2012) Dorn, I.; Lindenblatt, A. & Zweig, K. A.: "<a href="http://dl.acm.org/citation.cfm?id=2457082&dl=ACM&coll=DL&CFID=625287760&CFTOKEN=98233934">The Trilemma of Network Analysis</a>", Proceedings of the 2012 IEEE/ACM international conference on Advances in Social Network Analysis and Mining, Istanbul, 2012<br />
<br />
(Zweig2016) Katharina A. Zweig: Network Analysis Literacy, ISBN 978-3-7091-0740-9, Springer Vienna, publication expected Dec 2016 <br />
<br />2Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.com0tag:blogger.com,1999:blog-6779625344039153793.post-80504664125988416612015-08-04T01:51:00.001-07:002015-08-04T01:51:31.549-07:00Books to readMy personal all-time-favourite list of books on the general topic of network analysis and everything related. Will be edited from time to time.<br />
<br />
<ol>
<li><a href="http://www.amazon.com/Simulation-Similarity-Understand-Studies-Philosophy-ebook/dp/B00PXTJF72/ref=sr_1_1?ie=UTF8&qid=1438678228&sr=8-1&keywords=weisberg+simulation">Michael Weisberg: "Simulation and Similarity - using models to understand the world", Oxford University Press, 2014</a> [Link to amazon.com - I am not an affiliate]<br /><br />An understandable book written by a philosopher - that, in itself, makes it an all-time-favourite! Weisberg gives a very understandable and helpful characterization of scientific models, especially including computational models. Even Watts and Strogatz' small-world network model is briefly mentioned (p. 29, as an example of a mathematical structure). </li>
</ol>
2Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.com0tag:blogger.com,1999:blog-6779625344039153793.post-67933314562717259532015-08-04T01:46:00.001-07:002016-06-04T09:35:51.251-07:00Best articles on the topic of "Network Analysis Literacy"This is a post that will be updated from time to time, whenever I see a good paper that makes us aware of possible pittfalls when doing network analysis.<br />
<br />
<ol>
<li>
<span style="font-family: "arial";"><a href="http://www.biocomp.unibo.it/gigi/CB/articoli/foxkellerBioessays.pdf">Keller, E. F.: "Revisiting ``Scale-Free'' networks", BioEssays, 2005, 27, 1060-1068</a> [Link to PDF]<br /><br />Evelyn Keller explains (mainly for the biologist community) why there was such a hype about scale-free networks. She summarizes the ideas in physics behind it and the history of statistics that shows that power-law distributions as such are not too surprising. Her article is a bit outdated because it goes against a phenomenon that has slowly ceased, namely the usage of the Barabási-Albert model (BA model) to understand various network flow processes. She rightfully states that (biological and other) complex networks are not only scale-free but have various other properties that need to be regarded and that the BA-model shows none of these other properties but a very peculiar structure in which the hub nodes are also strongly interconnected. Anyway, a readable summary, especially for people new to the field of statistical physics.</span>
</li>
<li><span style="font-family: "arial";"><a href="http://science.sciencemag.org/content/325/5939/414">Carter Butts: Revisiting the Foundations of Network Analysis, Science, 325(5939), 414--416, 2009</a><br /><br />My all-time favourite point to the problem of clearly defining nodes and edges based on raw data. He shows along which lines the questions of "When is a node a node" and "When is an edge an edge" can be answered. Great read!</span></li>
</ol>
2Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.com2tag:blogger.com,1999:blog-6779625344039153793.post-39839338306223705192015-05-01T14:42:00.000-07:002018-05-28T14:47:04.182-07:00ImpressumThis blog is written by<br />
<br />
Prof. Dr. Katharina A. Zweig<br />
<br />
Gottlieb-Daimler-Str. 48<br />
Department of Computer Science <br />
67663 Kaiserslautern<br />
<br />
Telephone number: +49 631 205 3346<br />
Email address: lastnameOfBlogger AT cs DOT uni-kl.de<br />
<br />
<br />
The opinions reported in my blogs are my own opinions and not the ones of the university I am working. All scientific results reported here are carefully researched, however, there is always the possibility of mistakes and misunderstandings from my side. Please make up your own mind (as always). <br />
<br />2Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.comtag:blogger.com,1999:blog-6779625344039153793.post-33490260334912973552014-01-04T07:18:00.001-08:002014-01-04T07:49:23.986-08:00First chapter of my upcoming book is onlineAs some of you know, I am working on my book Network Analysis Literacy to get it finished as soon as possible. Holiday was a good time to get the first chapter polished. <a href="http://www.ninasnet.de/MyBook/Chapters/Chapter1_AFirstEncounter.pdf">Have a nice read</a> and come back to me with any comments you may have.2Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.com0tag:blogger.com,1999:blog-6779625344039153793.post-57493381159004986812014-01-02T08:52:00.004-08:002016-06-04T09:41:05.450-07:00Data sets for network analysis<ul>
<li><a href="http://moreno.ss.uci.edu/data.html">Freeman's network data sets (UCINET format)</a></li>
<li>Jure Leskovec's <a href="http://snap.stanford.edu/">database of large networks</a> (SNAP), a real treasure of a huge variety of networks. However, almost no information on the entities in the networks. Thus, mainly usable only for structural analysis. </li>
</ul>
2Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.com0tag:blogger.com,1999:blog-6779625344039153793.post-36402036495040255222012-02-05T07:20:00.000-08:002012-02-05T07:20:39.035-08:00A bibtex-batch scriptI currently write my book in Texlipse, a plug-in for Eclipse. I organized it in several \input-files with a separate chapter bib for all of the chapters. I could not find any help within Texlipse how to enforce a bibtex-run on each single input file, so the following batch script can help. Open a text editor like the WordPad or NotePad++ and copy the following line:<br />
<br />
for /f %%a IN ('dir /b *.aux') do bibtex %%a<br />
<br />
Save the file, e.g., as bibtexBatch.BAT. For convenience, save it to the same directory in which your .aux-files are located.<br />
Then run the 'CMD' command to get a DOS shell and change to the directory in which your .aux files are located. You run the batch file by calling it directly: bibtexBatch.BAT<br />
<br />
It cycles through the current directory and returns all files with the .aux-extension and hands them over to bibtex.2Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.com0tag:blogger.com,1999:blog-6779625344039153793.post-31385247418439292522011-11-07T14:31:00.001-08:002014-01-01T08:25:15.764-08:00Quotes from network analysis papersI'm currently doing a heavy literature research in network analysis and from time to time I find quotes that need some more audience.<br />
<br />
<h3>
<b>Reciprocity</b></h3>
"We use the term reciprocity to depict the degree of mutuality of a relationship. A re-<br />
lationship with a high reciprocity is one where both are equally interested in keeping up<br />
the relationship — a good example is the relationship of Romeo and Juliet in the famous<br />
play with the same name by William Shakespeare, where the two lovers strive to share each<br />
other’s company despite their families’ objections. On the other hand, in a relationship with<br />
a low reciprocity one person is significantly more active in maintaining the relationship than<br />
1the other. We judge reciprocity from actual communications taking place between people.<br />
<br />
In Shakespeare’s time this would have meant counting the number of poems passed and<br />
sonnets sung, but in our modern era it is easier to make use of the prevalence of electronic<br />
communication." :-) [Lauri Kovanen, Jari Saramäki, Kimmo Kaski: "Reciprocity of mobile phone calls", ArXiv:1002.0763v1 [physics.soc-ph], p. 1-2]<br />
<br />
<h3>
Centrality Indices</h3>
Sabidussi about the introduction of new centrality indices: <br />
"There is no doubt that an appeal to intuition must be made at some level, but it has to be made in a mathematically precise fashion." (Sabidussi, The centrality index of a graph, Psychometrika 31(4), 581-603, 1966)<br />
So, how good is your mathematical precision on your intuition? ;-) <br />
<br />
Sabidussi tried an axiomatic approach to centrality indices. He required for example that adding an edge to the most central vertex of a graph should always result in a graph in which the same vertex is still most central. Although this sounds intuitive, most centrality indices do not stand this test (e.g., eccentricity). Nonetheless, Sabidussi concludes:<br />
<br />
"One may, of course, blame this wholesale failure on our axiom system.There is little doubt, however, that the three indices [which he tested and which failed as well, among them a closeness-type of centrality] [...] would not survive even a more sophisticated system of axioms. In view of this, we strongly suggest that [these measures] be discarded and that centrality be measured by the trivial index [1/(n - degree of the node)] defined [above]. [It] is more easily calculated than any of the other indices, and, whatever its intuitive shortcomings, it has the decided advantage of satisfying a well-defined system of axioms." Sabidussi, 1966 (p. 20, see above)<br />
<br />
<h3>
Citation Failures</h3>
As everyone knows and thankfully remarked by Goh, Kahn and Kim <span style="font-size: x-small;">[Universal Load of Load Distribution in Scale-Free Networks, PRL, 87(27), 278701, 2001]</span>, Mark Newman introduced the <b>BFS</b>: "... and measure the load along the shortest path using the modified version of the breath-first [sic!] search algorithm introduced by Newman <span style="font-size: xx-small;">[ M. E. J. Newman, Phys. Rev. E 64, 016131 (2001); 64, 016132 (2001)]</span> "<br />
<br />
<h3>
Excuses and Justifications</h3>
<br />
"Edge weights in networks have, with some exceptions [...], received relatively little attention in<br />
the physics literature for the excellent reason that in any field one is well advised to look at the simple cases first (unweighted networks) before moving on to more complex ones (weighted networks). " M.E.J. Newman, "Analysis of weighted networks", ArXiv:cond-mat/0407503v1<br />
<br />
Excuse for not asking all participants of the study the same questions:<br />
"We only asked the last two informants this question because it didn't occur to us earlier." <span style="font-family: arial;"> </span><br />
<span style="font-family: arial;"> </span>[Bernard, H. R.; Shelley, G. A. & Killworth, P.: "How
much of a network does the GSS and RSW dredge up?<i> Social Networks, </i><b>1987</b><i>,
9</i>, 49-61]
<br />
In general I have the feeling that scientific articles were more personal in these days. This particular article is started with the following quote: "At my twenty-fifth high school reunion, last year, you would have been proud of me. They way i called those names up, with seldom a quick half glance at a tag ... their names came to me like the list of vowels, because I had learned them when i was fresh, back before I had met or heard of 375,000 other Americans. By the time anyone gets to be 43, if he has followed current events and been out of town a few times, two thirds of the names he hears sound vaguely, but only vaguely, familiar" (From "Not exactly what I had in Mind", Roy Blount Jr., The Atlantic Monthly Press, 1985)<br />
I wish we would read more of the persons behind research. It would remind us all that science is quantifiable but still conducted by humans which err or make subjective decisions. <br />
<br />
<br />
The following is a quote by two physicists about their engagement in economy, but the statement seems so general that it might fit to network analysis as well: <br />"Physicists are not, however, accustomed to waiting for a fully formed theory before reporting new results." Tobias Preis and H. Eugene Stanley: "Bubble trouble", Physics World May, p. 29-32, 2011. <br /><br /><br />
<h3>
Data quality: Protein-protein interaction networks</h3>
Mackay et al. wrote a paper about protein-interaction data and how bad they are. They have tried to re-produce around 20 published protein-protein interaction pairs but were only able to reproduce less than half. Their article starts with the following sentences:<br />
<br />
"When Othello concluded, on the basis of flimsy evidence, that Desdemona had indulged in inappropriate physical interactions, great tragedy ensued. We believe that many reported protein interactions are similarly based on insufficient data, and therefore risk skewing the literature." <br />
(<span style="font-family: arial;"><span style="font-size: x-small;">Mackay, J. P.; Sunde, M.; Lowry, J. A.; Crossley, M. & Matthews, J. M. Protein interactions: is seeing believing?<i> TRENDS in Biochemical Sciences, </i><b>2008</b><i>, 32</i>, 530-531</span>)</span><br />
<br />
<br />
I really wonder whether we want to base any type of network analysis where up to 50% of all edges are false-positive.<br />
<br />
<h3>
<b>Copy-pasting story lines</b></h3>
<b> </b> There are some story lines in network analysis I just have heard way too often and it seems they are just copy-pasted from one paper to the other. One of them is: "Until recently, network modeling often assumed the topology was either a random graph or a regular lattice." In this case, that is a citation from Goldberg and Roth, "Assessing experimentally derived interactions in a small world", PNAS 100(8), p. 4372-76, 2003, but actually it can be found in slight variations in hundreds of papers. It is a very interesting topic in itself, how science builds narratives that convey their beliefs in a model but I think it is important to stop from time to time and re-think a story line. Often, the narrative becomes oversimplified by little "mutations" along their copy-and-paste evolution that essentially hinders science more than helps it. Coming back to the example: It is unreasonable that any mature scientist would actually <b>assume </b>that a real world network was either a random graph or a regular lattice. We have <b>modeled it as</b> a random graph or a regular lattice in the hope that these models capture the essential properties of it - or just because a random graph model and a lattice comes in handy when trying to calculate things. The origins of these two models, especially in complex network analysis, is that atomic interactions in cristal lattices could be analyzed and solved in exactly two models: a lattice structure, which comes natural for cristals, or in a random graph fashion - which is not a natural choice but one in which some insight into the model can be gained. Thus, in the paper of Watts and Strogatz, which both come from statistical physics in which these interactions are an important research question, they focused on these two models - they were prevalent in statistical physics and well understood there. It is thus reasonable to transfer these models to real-world interaction between other things than atoms just to see how well they do. This has nothing to do with actually<b> assuming </b>that the models capture the real-world interactions in any meaningful way. <br />
<br />
<h3>
Mathematical modeling (in general and for network analysis)</h3>
In his article <a href="http://www.stat.cmu.edu/%7Efienberg/Stat36-835/Rothenberg-Connections-1995.pdf">Sampling in social networks</a>, Rothenberg cites Rapoportas follows: "Mathematical modeling is a vehicle for absolutely rigorous reasoning and therein lies its advantage. A disadvantage of mathematical modeling is that it necessitates a drastic paring down of the details of the phenomena modeled...these simplifications...can impair or altogether destroy the model's pragmatic relevance."<br />
<br />
<br />
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2Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.com0tag:blogger.com,1999:blog-6779625344039153793.post-43814756370042110862011-08-18T13:53:00.000-07:002011-08-18T13:55:38.288-07:00How to lie with statisticsI very much like to give a short course on 'how to lie with statistics' (<a href="http://www.ninasnet.de/Projects/HowToLieWithStatistics_Public.pptx">Check out my slides</a>). My experience is that most people feel very uncomfortable in interpreting statistics and applying statistical methods to their data. The problem seems to be that most of the young scientists seem to think they are the only ones that do not understand statistics. But it rather seems to be the case that humans in general are not very good at thinking in probabilities. Gigerenzer showed that the following question is very hard to solve correctly, even for medical doctors:<br />
<br />
If a normal student donates some blood and an HIV test turns out positive, how likely is it that she is really infected? To answer that you need to know the sensitivity and specificity of the test, i.e., the probability that an infected person will be detected by the test and the probability that a non-infected person will have a negative test result. Both probabilities are very high, around 99.8%. Still, the probability that a student with a positive first test is really infected is only 40/2040 = 1/51. Astonished?<br />
<br />
80% of the medical doctors he interviewed gave the wrong numbers, another 7% got close but presumably by some obvious error, and only the remaining ones gave the correct answer. <br />
But Gigerenzer also <a href="http://www.psychologicalscience.org/journals/pspi/pspi_8_2_article.pdf">showed very nicely</a> how to remedy the problem: forget about probabilities and think in natural frequencies! Then you'll notice that you need another information: in the normal population how many people are infected per year and do not know it? According to the German Robert Koch Institute this incidence rate of new HIV infections is about 3,000 per year in the whole population. So, if 1,000,000 people (out of 80 million Germans) donate blood we can expect around 40 of them to be infected with HIV. The other close to 1,000,000 donors are not infected but 2 out of 1000 of them will be tested positive nonetheless. Thus, in total there will be 2000+40 positive tests of which only 40 really point to infected donors.<br />
<br />
<br />
Also in network analysis there were some articles regarding statistics and problems with it. Most of them warned about wrong random models or at least warned that we need to think hard about the right random graph model:<br />
<ol><li><a href="http://www.sciencemag.org/content/305/5687/1107.3.full">Artzy-Randrup et al.</a> showed that network motifs might be evaluted differently depending on the underlying random graph model. </li>
<li><a href="http://nwb.cns.iu.edu/papers/2006-colizza-natphys2.pdf">Colizza et al.</a> show that also the rich-club effect needs to be assessed against the appropriate random network model: observing only those nodes with degree at least k, the fraction of realized edges between these most connected nodes is called the rich-club coefficient. It was thought that a rich-club coefficient that increases with k is a sign of a rich-club effect in which those that are most connected build a dense core. Colizza et al show that even random graphs have this increasing rich-club coefficient since high-degree nodes just have a higher probability to be connected to anybody, also to other high-degree nodes. Thus, it is important to compare with a random graph model which maintains the degree sequence.</li>
<li>We have shown a <a href="http://www.ninasnet.de/Publications/PDFs/2010_ASONAM.pdf">similar effect</a> in assessing the similarity of two nodes in a bipartite graph. Say you have a bipartite graph between customers and films and make an edge between two films if the customer stated she liked that film. If now two films are co-liked by 23 persons, is this statistically significant? Or if they are co-liked by 1179 persons? By choosing the right random network model, it is easy to quantify the significance of this number. (The second pair of films was 'Pretty Woman' and 'Star Wars V' and you might have guessed that their number is significantly too low given their respective popularity). See the figure for more examples.</li>
<li>Even modularity, which already checks the observed edges within partitions against the expected number of these edges in a graph, has some statistical flaws (or say, unintuitive properties) which were thankfully pointed out by<a href="http://www.google.de/url?sa=t&source=web&cd=1&ved=0CB0QFjAA&url=http%3A%2F%2Fwww.pnas.org%2Fcontent%2F104%2F1%2F36.full.pdf&rct=j&q=%22Resolution%20limit%20in%20community%20detection%22%20filetype%3Apdf&ei=qxhNTrb1H_H74QT4rsWtBw&usg=AFQjCNHGJA_Dq5oX4erfAUW2jcAgu2lfGg&sig2=XqH5G2wTzuc1yyo2-am_nQ&cad=rja"> Fortunato and Barthelémy.</a> </li>
</ol><br />
<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; margin-left: 1em; text-align: right;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEji2kiYcXplMlEWZk2hfQ4x7AZk_R-gEfFKELN3dk2W8zzCJ9Ls5yppDWHEgQziGYmYhd0gY0tZ4J6McQpapWify0ELP5UmaRfjaJpgAMK6dFlg4llJdYqOseTfCr1VvqOtv5ukHQBJy1o/s1600/3000MostPopularFilms.png" imageanchor="1" style="clear: right; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEji2kiYcXplMlEWZk2hfQ4x7AZk_R-gEfFKELN3dk2W8zzCJ9Ls5yppDWHEgQziGYmYhd0gY0tZ4J6McQpapWify0ELP5UmaRfjaJpgAMK6dFlg4llJdYqOseTfCr1VvqOtv5ukHQBJy1o/s400/3000MostPopularFilms.png" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Pairwise "similarity" of the most popular films in the Netflix data set (popular = rated by at least 30% of all customers). We have sorted the films such they fall in natural groups like action films in group 1 and chick flicks in group 5. For each pair we counted the number of customers that liked both films (rating 4 or 5 out of 5). Blue fields say that the pair of film has been co-liked more often than expected and red fields denote pairs of films that were significantly less co-liked than expected. Of course, expectation depends on the chosen random model. The classic random model (lower right) thinks that all popular films are more often liked together than expected. This includes the film pair: "50 first dates" and "The Patriot". If you also think that both films are simply wonderful, you can stop here. By using a different random model inspired from network analysis, a much more differentiated picture emerges (upper left). Here, it can be clearly seen that films from the same group are still more often co-liked than expected while different groups (e.g., 1 and 5) are distinctly less co-liked than expected. </td></tr>
</tbody></table><br />
I would like to know whether there is a similar trick as to thinking in frequencies rather than in probabilities to get the choice of the appropriate random graph model right. If you know one, let me know!<br />
<br />
Feel free to use my slides for your own lecture on 'How to lie with statistics' (please give credit and note that all the images are under some GPL and so should be your modification of it). I also like to travel, so if I'm around and your interested in this short-course I would love to come to your institution and give the course. It takes three hours and is targeted at PhD students from non-mathematical studies in their first year. 2Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.com0tag:blogger.com,1999:blog-6779625344039153793.post-77964207245909968632011-08-09T14:39:00.000-07:002011-08-09T14:48:49.991-07:00Our Word-Morph Network<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgiFK0BwzXI4ytLwBBBn9eMK_KZ_lYMpEmMW0MCpOQ-w2GoXzzAvG6n_Yq96oSOun04PsKW3wuocRpa9XDS-ea-V9hsehsGDE_xPeOEid6Utl4smEAqZz-J7THodSsV5Crax0yEmguhh8o/s1600/Wordmorph.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgiFK0BwzXI4ytLwBBBn9eMK_KZ_lYMpEmMW0MCpOQ-w2GoXzzAvG6n_Yq96oSOun04PsKW3wuocRpa9XDS-ea-V9hsehsGDE_xPeOEid6Utl4smEAqZz-J7THodSsV5Crax0yEmguhh8o/s640/Wordmorph.png" width="481" /></a></div>Do you remember the old word game in which you are asked to change the word 'cat' into 'dog' by exchanging one letter at a time, only using real English words? So, with <b>cat</b> and <b>dog</b> it is easier as it may seem: cat-cot-dot-dog. It is not always that easy: for <b>ivy</b> and <b>dry</b> the shortest path is: dry-dey-dee-dye-aye-ace-ice-icy-ivy. So, here, it is not possible to directly come 'closer' to the goal with each exchanged letter. The graph above shows all correct English three-letter words (extracted from the Oxford dictionary), where two words are connected by an edge if they differ in exactly one letter. So, this is the graph in which the word-morph game essentially takes place.<br />
<br />
We asked ourselves how people learn to play this game: will they be slow at the beginning and get faster soon? Will they learn the shortest paths between all pairs or do something different? We found out that people learn 'landmark words' through which they tend to navigate. Thus, they only need to learn n many paths in a network with n nodes, instead of learning n*n different paths. Of course, the paths that navigate through a certain landmark word are a bit longer than the shortest paths (in this case by a factor of about 1.6). It thus seems that our human mind tries to optimize both: path length but also the effort to learn about the network. <a href="http://www.google.de/url?sa=t&source=web&cd=1&ved=0CBkQFjAA&url=http%3A%2F%2Fcsjarchive.cogsci.rpi.edu%2FProceedings%2F2011%2Fpapers%2F0648%2Fpaper0648.pdf&rct=j&q=human%20wayfinding%20network%20analysis&ei=_p1BTvnVAsbMsgbyk7i2Bw&usg=AFQjCNGqnt8XQ7hF79hV4ZGDZtkQvcUezw&sig2=kw01oV3YSDA0dMi8DpeGcg&cad=rja">Check out our pape</a>r if you want to learn more about this research; we got a <a href="http://cognitivesciencesociety.org/uploads/cogsci2011_awards.pdf">best paper award for it from CogSci 2011</a>. <br />
<br />
The graph was layouted in Gephi, and coloured by an automatic clustering algorithm. An obvious pattern emerged, consisting of five central groups of words (dark blue, green red, yellow, light blue). Can you guess what these five groups are? 2Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.com0tag:blogger.com,1999:blog-6779625344039153793.post-13753936223434001602011-07-26T08:38:00.000-07:002011-07-26T08:47:19.332-07:00Egonet visualization in igraphJust for fun I thought I should implement the egonet visualization from yesterday in the igraph package as well (yesterday I used the network package). Merely 2 hours later I came up with the right recipe... :-) Basically, the fun was motivated by the not so overwhelming visualizations of my last blog. So, I had the idea to export the resulting ego-net-subgraph into some Gephi-readable format.In short, we want to go from the first figure to the second figure. Yes, I'm sure it is really the same graph!!<br />
<br />
<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgG2CMEGMOlJNIHIHphtY-Is2QQLf1XBIJwkJqySQHfZhHIvIW1294dDRYVraGIAgEnF_2ncuoIasVIiUW1H_F9TYm6K1tKK3gdiEQ-fzmXrj_JsnN0miAs3Nqdy6WrbeKHKJBsfsFPq4M/s1600/EgoNetVisualization_igraph-PLOT2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="160" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgG2CMEGMOlJNIHIHphtY-Is2QQLf1XBIJwkJqySQHfZhHIvIW1294dDRYVraGIAgEnF_2ncuoIasVIiUW1H_F9TYm6K1tKK3gdiEQ-fzmXrj_JsnN0miAs3Nqdy6WrbeKHKJBsfsFPq4M/s200/EgoNetVisualization_igraph-PLOT2.png" width="200" /></a></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgkp_L25pI2BnBhfi0SKnTTq-xXfGLlhYap1oABFiAZe-yYSxziK1b7Uefb_ZtUwwD5icIw-Avo4okOFDwTXLQ3bkJWjTaWqEmoVJ1bXGGOr8aV-TiEJGnegaKizaG__8eBygddsmUM7xg/s1600/EmailArenas2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgkp_L25pI2BnBhfi0SKnTTq-xXfGLlhYap1oABFiAZe-yYSxziK1b7Uefb_ZtUwwD5icIw-Avo4okOFDwTXLQ3bkJWjTaWqEmoVJ1bXGGOr8aV-TiEJGnegaKizaG__8eBygddsmUM7xg/s200/EmailArenas2.png" width="195" /></a></div><br />
I could not find any helpful write/export-function in the network package, but there is a flexible write.graph-function in the igraph package. Both packages are quite similar, thus a quick transformation would do the trick. So, let's look at the necessary transformations:<br />
<br />
<pre>tableEmail <- read.table("email_Arenas.txt")
emailNW <- graph.data.frame(tableEmail, directed=T)
randomSample <- sample(0:vcount(emailNW)-1, 10, replace=FALSE)
neighs <- vector()
for(x in randomSample){
neighs <- c(neighs,neighborhood(emailNW, 1, x, mode="all")[[1]])
}
subgraph <- subgraph(emailNW, neighs)
</pre><br />
<br />
The replacements are:<br />
<ol><li><b>graph.data.frame</b> instead of network. It creates the igraph object out of the data frame.</li>
<li><b>random sample</b> from the interval [0:n-1] where n is the number of nodes, explained later. </li>
<li><b>neighborhood</b> instead of get.neighborhood. It makes life a bit easier since it naturally includes all nodes in distance smaller than the given order, in this case 1. Thus, we do not need to explicitly include x itself. Careful, the result is a list, so make sure to only append the first entry of the list to the vector neighs</li>
<li><b>subgraph</b> instead of get.inducedSubgraph.</li>
</ol>The plot at this time point looks similar to the other one, all vertices are red. Now, the fun begins! How can we color the random seed nodes? In essence, what we did in the network package was to prepare a list of colors where we assigned a different color to the vertices from the random sample (as identified by their index). We then restricted this vector to those indices which are still present in the subgraph, as identified by the function network.vertex.names(subgraph). It can thus be seen that in the network package the induced subgraph keeps the old vertex IDs as vertex names:<br />
<br />
<pre>color <- rep(2, times=network.size(emailNW))
color[randomSample] = 3
plot(subgraph, vertex.col=color[network.vertex.names(subgraph)])</pre><pre></pre>However, igraph does not make it as simple for us. First of all, it re-assigns vertex IDs in the subgraph to make them subsequent. Second, these indices run from 0 to (number of nodes -1). This is already the case in the first igraph-object that we created, namely emailNW. This leads to the first surprising behavior. Let's look at the random sample:<br />
<br />
<pre>> randomSample
> [1] 670 97 352 346 465 53 37 1092 726 74
</pre><br />
Let's look at the vertices in the resulting induced subgraph with the function V(subgraph): <br />
<pre> </pre><pre>> V(subgraph)
[1] "2" "3" "5" "7" "18" "19" "21" "22" "23" "27"
[11] "31" "38" "40" "41" "45" "49" "51" "54" "69" "72"
[21] "74" "75" "76" "87" "98" "112" "124" "143" "148" "152"
[31] "183" "185" "187" "189" "191" "195" "231" "233" "237" "241"
[41] "254" "267" "268" "270" "275" "280" "290" "314" "316" "329"
[51] "330" "331" "333" "344" "345" "346" "347" "348" "349" "350"
[61] "351" "352" "353" "354" "355" "356" "362" "378" "392" "396"
[71] "454" "462" "463" "464" "465" "466" "467" "468" "501" "523"
[81] "538" "549" "556" "557" "558" "559" "560" "561" "568" "578"
[91] "590" "598" "627" "635" "636" "638" "671" "680" "711" "727"
[101] "743" "746" "748" "765" "778" "836" "841" "940" "941" "942"
[111] "943" "944" "945" "946" "954" "1030" "1031" "1092" "1093"
</pre><br />
There is not a single vertex from the random sample set but, suspiciously, for each of them there is a vertex with an ID increased by one. What happens is that igraph uses the random sample as indices to the vertices that are themselves labeled from 1 to n (number of nodes in the graph). I.e., if we address the first ten nodes of emailNW by [0:9] we will get vertices labeled 1 to 10:<br />
<br />
<pre>> V(emailNW)[0:9]
Vertex sequence:
[1] "1" "2" "3" "4" "5" "6" "7" "8" "9" "10"
</pre><br />
But (you knew there would be a but, didn't you?) all other attributes assigned to the node set of the graph are indexed by 1 to n. For example, the character with the names (labels) of the vertices is indexed 1 to n: <br />
<pre> </pre><pre>> V(emailNW)[0]
Vertex sequence:
[1] "1"
> V(emailNW)$name[0]
character(0)
> V(emailNW)$name[1:10]
[1] "1" "2" "3" "4" "5" "6" "7" "8" "9" "10"
> V(emailNW)[0:9]
Vertex sequence:
[1] "1" "2" "3" "4" "5" "6" "7" "8" "9" "10"
</pre><br />
Beautiful. With this it is now a piece of cake to do the coloring in igraph as well: <br />
<pre> </pre><pre>color <- rep(2, times=vcount(emailNW))
color[randomSample] <- 3
V(subgraph)$color <- color[as.numeric(V(subgraph)$name)-1]
V(subgraph)$layout <- layout.fruchterman.reingold(subgraph)
plot(subgraph)
</pre><br />
Now, the figure will have 10 green nodes as planned. <br />
<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYy_M0RKMuLL3EqDOsjAyJyggHC8bimIVoziHUsxdjcijHe5PMufGcysGVwOPc9OTu6_iZCCdaeYrg0FfcAg_jz4Y-X-DyhiQVbUpDe0vQygju6rZ2DRRYqUTMHw93L889pe-qDgF7uFA/s1600/EgoNetVisualization_igraph-PLOT2.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYy_M0RKMuLL3EqDOsjAyJyggHC8bimIVoziHUsxdjcijHe5PMufGcysGVwOPc9OTu6_iZCCdaeYrg0FfcAg_jz4Y-X-DyhiQVbUpDe0vQygju6rZ2DRRYqUTMHw93L889pe-qDgF7uFA/s400/EgoNetVisualization_igraph-PLOT2.png" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Yeah, I know, beautiful layout. That is WHY we need to export it to GEPHI and beautify it there.</td></tr>
</tbody></table><br />
Btw, the layout assignment will throw two warnings but please don't ask me why. I hope I will not have to enquire that as well... ;-)<br />
<br />
You might want to check that the right nodes got the right color: <br />
<pre> </pre><pre>V(subgraph)$name[V(subgraph)$color==3]
V(emailNW)[randomSample]
</pre><br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPELlMZUA66zJco4M6cw0gDDyYtwXyyyJJmFBmZ5bwSQu0KOZbKPRLz2kO0BINzlKbYfKJOkCiuKfBVQK_q3wxkrCgHgvhvx_ZIBJUTqub7UIDJxMiubyygUu2odb5BQkhmYOkAl-aFN8/s1600/EmailArenas2.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPELlMZUA66zJco4M6cw0gDDyYtwXyyyJJmFBmZ5bwSQu0KOZbKPRLz2kO0BINzlKbYfKJOkCiuKfBVQK_q3wxkrCgHgvhvx_ZIBJUTqub7UIDJxMiubyygUu2odb5BQkhmYOkAl-aFN8/s400/EmailArenas2.png" width="390" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Gephi's version</td></tr>
</tbody></table>So, here is the Gephi layout. Of course, a hairball is a hairball - but: did you see the isolated component beforehand?2Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.com0tag:blogger.com,1999:blog-6779625344039153793.post-28473906188604483652011-07-25T03:01:00.000-07:002011-07-25T03:12:09.832-07:00Visualizing and Coloring of Induced SubgraphsIf the network is too large to visualize it as a whole, it might be a good idea to choose some nodes at random and to visualize their direct surroundings, also called their ego network. The ego network comprises the direct neighbors of a node and the relationship between these neighbors. In this example we will combine all edges between 10 randomly chosen nodes and their direct neighbors. The data set we are working on is <a href="http://deim.urv.cat/%7Eaarenas/data/welcome.htm">freely available</a> from <a href="http://deim.urv.cat/%7Eaarenas/">Alex Arena'</a>s homepage. We use his data on the email contact network of the University Rovira y Virgili and we will assume that it is stored as 'email_Arenas.txt'. (For all readers who are fluent in latex, here is the according <a href="http://www.ninasnet.de/MyBook/Rnws/AFirstEncounter/EgoNetVisualization.Rnw">Sweave-file</a> which will create all figures in eps/pdf/png format. It assumes you have set the working directory to where itself and the data is located. )<br />
<br />
<br />
<pre>> library(network)
> setwd("path-to-wherever-you-stored-the-data")
> tableEmail <- read.table("email_Arenas.txt")
# create a network from the data frame
> emailNW <- network(tableEmail, directed = T)
# choose 10 random vertex IDs without replacement
> randomSample <- sample(1:network.size(emailNW), 10, replace = FALSE)
# show the randomly drawn vertex IDs
> randomSample
[1] 843 548 691 871 524 858 290 1059 485 593
#initialize a new vector
> neighs <- vector()
#for all vertex IDs in the random sample
> for (x in randomSample) {
# add themselves and their direct neighborhood to the vector 'neighs'
> neighs <- c(neighs, x, get.neighborhood(emailNW, x, type = "combined"))
> }
#create the induced subgraph
> igraph <- get.inducedSubgraph(emailNW, neighs)
</pre>Now, we would like to see this random sample from the graph: <br />
<pre>> plot(igraph)
</pre><br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQjOSe8-PERSqxWQECnLdecH3WWnDsvC9ML0eQjq5K3ec5g-Be_u-V1LQI72uVrBQUCkwAJsyrpb88dpbTjgTeqqiaL6yusXNTIIL05ZPGVLVjlo8KWtQRPUdAMhDWNnuJOQm4-oesdOo/s1600/EgoNetVisualization-CREATEEGONWS.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" heigth="600" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQjOSe8-PERSqxWQECnLdecH3WWnDsvC9ML0eQjq5K3ec5g-Be_u-V1LQI72uVrBQUCkwAJsyrpb88dpbTjgTeqqiaL6yusXNTIIL05ZPGVLVjlo8KWtQRPUdAMhDWNnuJOQm4-oesdOo/s1600/EgoNetVisualization-CREATEEGONWS.png" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">An induced subgraph based on 10 randomly drawn seed nodes and their direct neighborhoods.</td></tr>
</tbody></table><br />
<br />
<br />
The induced subgraphs gives an overall impression on how dense the local neighborhoods and the connections between 10 randomly drawn neighborhoods are. It would, however, be helpful to identify the 10 seeds. This can be done by coloring them accordingly. This was my first try: <br />
<pre>#make a vector of size n (=#number of nodes), assigning color 2 as default
> color <- rep(2, times = network.size(emailNW))
#for the seed nodes, assign color number 3
> color[network.vertex.names(emailNW) %in% randomSample] = 3
#plot the graph and assign the color vector
> plot(igraph, vertex.col = color[network.vertex.names(igraph)])
</pre><br />
Interestingly, this does not give the wanted results; it seems as if there were no seed vertices at all:<br />
<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHIuYu5Ed0eIi_JxkPKSZt0L6PmxnvcNRPuZ8N3X5mS5cOHoZ9FODtzmyQ3mIt53u_4Hw55uMX0jUDgAWXrdqI-f0_qndeBYzTYyhE-F-0ccJ85zmJHgYZoMnYTPlHuiXMuh20BYI5JX8/s1600/EgoNetVisualization-PLOT1.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHIuYu5Ed0eIi_JxkPKSZt0L6PmxnvcNRPuZ8N3X5mS5cOHoZ9FODtzmyQ3mIt53u_4Hw55uMX0jUDgAWXrdqI-f0_qndeBYzTYyhE-F-0ccJ85zmJHgYZoMnYTPlHuiXMuh20BYI5JX8/s1600/EgoNetVisualization-PLOT1.png" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">A test plot in which all seed nodes should have been colored in green. Did not work, obviously.</td></tr>
</tbody></table><br />
If you would run the same code again and again, you would sometimes see one or two or even more green vertices. So, what happens (or seems to happen) is the following: the induced subgraph creates a new graph with obviously fewer vertices than the original graph. The vertex names themselves are maintained, and their order is maintained as well - you can check that by typing network.vertex.names(igraph). But only the first n entries in vertex.col are used as an assigment to the colors. Thus, we need to reduce the color-vector to those entries which are contained in igraph:<br />
<br />
<pre>> plot(igraph ,vertex.col=color[network.vertex.names(igraph)])
</pre><br />
This will finally produce the graph with all 10 seed nodes colored in green:<br />
<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjVzVtUVsxO2CKEShZmX59c3gEsarlApkkyZfLos1M7c4vQYI1ceGI_Hzr_7xBYOOExOC7ubiW6PaH-2iyx9ayh2CBx5fCmBwyOLGlpSlLtfWcDTJCdP2Cn0GNE1sj3Pe3rhn7BUJoE8AU/s1600/EgoNetVisualization-PLOT2.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjVzVtUVsxO2CKEShZmX59c3gEsarlApkkyZfLos1M7c4vQYI1ceGI_Hzr_7xBYOOExOC7ubiW6PaH-2iyx9ayh2CBx5fCmBwyOLGlpSlLtfWcDTJCdP2Cn0GNE1sj3Pe3rhn7BUJoE8AU/s1600/EgoNetVisualization-PLOT2.png" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The final result with all 10 seed nodes colored in green.</td></tr>
</tbody></table>2Ghttp://www.blogger.com/profile/12395120175850181679noreply@blogger.com1