Tailored graph ensembles as proxies or null models for real networks I: tools for quantifying structure

Abstract

We study the tailoring of structured random graph ensembles to real networks, with the objective of generating precise and practical mathematical tools for quantifying and comparing network topologies macroscopically, beyond the level of degree statistics. Our family of ensembles can produce graphs with any prescribed degree distribution and any degree-degree correlation function, its control parameters can be calculated fully analytically, and as a result we can calculate (asymptotically) formulae for entropies and complexities, and for information-theoretic distances between networks, expressed directly and explicitly in terms of their measured degree distribution and degree correlations.

Statistical mechanics of multiedge networks

“…where we have used the properties of the multinomial distribution presented in (11). Although the theoretical basis for the generation of graphs in different ensembles is introduced in this paper (see Appendix B), the challenges for the exact and efficient generation of such ensembles will be shortly tackled and presented in future work.…”

“…In some cases, groups of these events connect the same pair of nodes, hence forming an edge with multiple connections, which has a different nature from a weighted one where no quantization is imposed. If one aims to apply the well-known ensemble theory used in statistical physics to networks [11,12], one needs to take into account this subtle yet important difference, which has profound implications on the associated statistics. The choice of one or the other representation will thus depend on the problem at hand and makes a big difference in terms of collective behavior of the whole network as will be shown along this paper.…”

Tailored Random Graph Ensembles

“…π(k) is the Poissonian distribution with the same average degree, and W (k) indicates the marginal. In this case [1] find the Shannon entropy per bond of this ensemble to be…”

“…Quantifying the topology of a network via tailored random graph ensembles [1] and [2] derive closed form solutions for the Shannon entropy per bond…”

“…Structural dissimilarity of graphs c A and c B , based on information-theoretic distance between associated ensembles [1] uses the same methods as were used to derive equation 2 to determine a Kullback-Leibler distance between ensemble A (defined through p A and W A ) and ensemble B (defined through p B and W B ) This is evaluated to be…”