Statistical Complexity of Heterogeneous Geometric Networks

TL;DR

A parsimonious normalised measure of statistical complexity for networks is introduced and it is proved that this measure tends to 0 in Erdös-Rényi random graphs in the thermodynamic limit.

Abstract

Degree heterogeneity and latent geometry, also referred to as popularity and similarity, are key explanatory components underlying the structure of real-world networks. The relationship between these components and the statistical complexity of networks is not well understood. We introduce a parsimonious normalised measure of statistical complexity for networks. The measure is trivially 0 in regular graphs and we prove that this measure tends to 0 in Erdös-Rényi random graphs in the thermodynamic limit. We go on to demonstrate that greater complexity arises from the combination of heterogeneous and geometric components to the network structure than either on their own. Further, the levels of complexity achieved are similar to those found in many real-world networks. However, we also find that real-world networks establish connections in a way which increases complexity and which our null models fail to explain. We study this using ten link growth mechanisms and find that only one mechanism successfully and consistently replicates this phenomenon -- probabilities proportional to the exponential of the number of common neighbours between two nodes. Common neighbours is a mechanism which implicitly accounts for degree heterogeneity and latent geometry. This explains how a simple mechanism facilitates the growth of statistical complexity in real-world networks.

Figures (6)

• Figure 1: On the left we see a geometric graph with a regular structure. Node shapes indicate distinct degrees while colours indicate distinct, repeating neighbourhood degree sequences. On the right, nodes are randomly assigned different numbers of connections. These connections are made to the closest nodes, maintaining a geometric nature, but now we see the diversity of structure this opens up. Again, different shapes indicate distinct degrees, but now there are many unique neighbourhood degree sequences which remain colourless. This diversity reflects a higher hierarchical complexity.
• Figure 2: Results show measurements for random realisations ($n\sim U[50,10000]$, $d\sim U[0,1]$) of different random graphs as denoted in the legend, against size $n$ and density $d$.
• Figure 3: The normalised hierarchical complexity of cylinders of increasing sizes of the Allen Brain V1 mouse model vs ER graphs of the same size and density.
• Figure 4: Top, average normalised hierarchical complexity per degree of heterogeneous geometric graphs with with $n=1000$, $d\sim U[0,1]$, and $\sigma_{h}= 0.01,0.02,\dots,2$. One hundred realisations are created for each $\sigma$. Bottom, global normalised hierarchical complexity plotted against heterogeneity of these networks.
• Figure 5: Scatterplots visualising the positive association between density and hierarchical complexity in real-world networks. Spearman's correlation coefficient and associated $p$-value shown inset. Bottom row shows average results of the values of NHC as we increase density of networks according to the link growth mechanisms as described in the legend.

Theorems & Definitions (8)

• Definition 1
• Theorem 1
• proof
• Example 1
• Conjecture 1
• Theorem 2
• proof
• Conjecture 2