Note 4 and Note 6: What does a measure measure?

Network analysis promises to give insight into the inner
workings of a complex system by representing
it as a complex network, by analyzing this representation,
and by interpreting the results of the analysis.
But: what does a measure actually measure?
How is the result mapped to the inner workings of the system?
This figure is under CC:BY with a reference to
Prof. Dr. Katharina A. Zweig  or to this blogpost.

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.

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. However, when we apply network analysis, we hope for the following:

"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)

 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:

1. 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.
2. 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:
1. The message is sent at the same time to all neighbors.
2. They send it to all their neighbors within one time step.
3. There is such a thing as a time step, which all nodes know.
3. 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).

 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.

"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)

Reference:

(Borgatti2005) Borgatti, S. P.: Centrality and Network Flow, Social Networks, 2005, 27, 55-71

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