Exploring Network Structure with the Density of States

TL;DR

The density of states is introduced as a tool for studying the space of all possible network partitions and it is shown that, even using modularity to measure quality, the density of states can still rule out spurious structure in random networks and overcome resolution limits.

Abstract

Community detection, as well as the identification of other structures like core periphery and disassortative patterns, is an important topic in network analysis. While most methods seek to find the best partition of the network according to some criteria, there is a body of results that suggest that a single network can have many good but distinct partitions. In this paper we introduce the density of states as a tool for studying the space of all possible network partitions. We demonstrate how to use the well known Wang-Landau method to compute a network's density of states. We show that, even using modularity to measure quality, the density of states can still rule out spurious structure in random networks and overcome resolution limits. We demonstrate how these methods can be used to find `building blocks', groups of nodes which are consistently found together in detected communities. This suggests an approach to partitioning based on exploration of the network's structure landscape rather than optimisation.

Figures (4)

• Figure 1: Left to right: karate club, dolphin social nework, ER grandom graph. Showing density of states under label swapping (blue) and label swapping configuration model (orange). Inset figure shows observed network and the 'best' partition found during the Wang Landau sampling.
• Figure 2: Left: DOS ratios for assortative and disassortative structures in the karate club network. Inset shows the highest modularity partitions for assortative (top) and disassortative case (bottom). Right: DOS ratio, where structure $B$ can also vary. Top shows the highest modularity partition (the same one found in arthur2023discovering), centre shows a partition with highest DOS ratio.
• Figure 3: Building blocks (shapes) composed to create groups (grey rectangles). The left partition is perfectly complete, $c=1$, with respect to the blocks, but not homogeneous. The middle is neither homogeneous nor complete. The right partition is more homogeneous than the first, but less complete.
• Figure 4: Left: completeness (blue) and number of blocks (orange) as a function of threshold $\theta$. Middle: the matrix $W$ sorted into blocks for the threshold indicated by the dashed red line. Right: the observed network with the building blocks coloured. Top to bottom: connected cavemen network, dolphin social network, Les Misérables character interaction network.