## Saturday, November 12, 2016

### Notes 12 and 13: Evolution of complex networks

 Networks evolve under different constraints and forces, in which ; network science is especially interested. This figure is under CC:BY with a reference to Prof. Dr. Katharina A. Zweig or to this blogpost.

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.

"Note 12. Network science describes the topology of a network as the eﬀects of forces on either the entities or
the whole system.
" (Zweig, 2016)

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.

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 'criticality' 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.

"Note 13. Network science is interested in equilibrium
structures as they can be used to understand the forces,
constraints, or incentives under which a network is built.
" (Zweig, 2016)

Presumed constraints are, for example, Dunbar's number 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.

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.

Reference:

(Zweig2016) Katharina A. Zweig: Network Analysis Literacy, ISBN 978-3-7091-0740-9, Springer Vienna, 2016

All figures are under CC:BY with a reference to Prof. Dr. Katharina A. Zweig or to this blogpost, if not mentioned otherwise.