Modularity of regular and treelike graphs

TL;DR

This paper studies the modularity of regular and treelike graphs, providing bounds and typical values for different classes, which aids in statistical analysis of network structures.

Contribution

It offers new bounds and typical values for the modularity of regular and treelike graphs, extending known results and establishing baselines for network analysis.

Findings

Random cubic graphs have modularity typically between 0.666 and 0.804.

For large r-regular graphs, modularity is usually of order 1/√r.

Extends modularity bounds to treelike graphs with low treewidth and degree.

Abstract

Clustering algorithms for large networks typically use modularity values to test which partitions of the vertex set better represent structure in the data. The modularity of a graph is the maximum modularity of a partition. We consider the modularity of two kinds of graphs. For $r$-regular graphs with a given number of vertices, we investigate the minimum possible modularity, the typical modularity, and the maximum possible modularity. In particular, we see that for random cubic graphs the modularity is usually in the interval $(0.666, 0.804)$, and for random $r$-regular graphs with large $r$ it usually is of order $1/\sqrt{r}$. These results help to establish baselines for statistical tests on regular graphs. The modularity of cycles and low degree trees is known to be close to 1: we extend these results to `treelike' graphs, where the product of treewidth and maximum degree is much less than the number of edges. This yields for example the (deterministic) lower bound $0.666$ mentioned above on the modularity of random cubic graphs.