Modularity and partially observed graphs

TL;DR

The article investigates how the community-structure measure of modularity behaves under partial observation of a graph. It introduces edge-sampling and limited-search models, proves that modularity of the observed graph closely tracks the underlying modularity when enough edges are seen, and extends these results to vertex-based estimation and weighted networks. A key technical tool is the fattening lemma, enabling robust transfer of modularity across partitions, along with concentration and robustness results that bound changes under edge modifications. The work also connects to stochastic block models and privacy considerations, showing when modularity is estimable or cheat-resilient under subsampling, and discusses implications for learning communities from partial data. Overall, it provides a rigorous framework for understanding modularity under partial information and offers both theoretical guarantees and practical guidance for network analysis.

Abstract

Suppose that there is an unknown underlying graph $G$ on a large vertex set, and we can test only a proportion of the possible edges to check whether they are present in $G$. If $G$ has high modularity, is the observed graph $G'$ likely to have high modularity? We see that this is indeed the case under a mild condition, in a natural model where we test edges at random. We find that $q^*(G') \geq q^*(G)-\varepsilon$ with probability at least $1-\varepsilon$, as long as the expected number edges in $G'$ is large enough. Similarly, $q^*(G') \leq q^*(G)+\varepsilon$ with probability at least $1-\varepsilon$, under the stronger condition that the expected average degree in $G'$ is large enough. Further, under this stronger condition, finding a good partition for $G'$ helps us to find a good partition for $G$. We also consider the vertex sampling model for partially observing the underlying graph: we find that for dense underlying graphs we may estimate the modularity by sampling constantly many vertices and observing the corresponding induced subgraph, but this does not hold for underlying graphs with a subquadratic number of edges. Finally we deduce some related results, for example showing that under-sampling tends to lead to overestimation of modularity.

Figures (2)

• Figure 1.1: Simulation results. The dolphin social network lusseau2003emergent with 62 vertices and 159 edges was taken to be the underlying graph $G$. It is known that $q^*(G) = 0.529$ to three decimal places nphard. In the upper part of the figure each red point corresponds to the estimated modularity $\tilde{q}(G_p)$ of an instance of the sampled graph $G_p$. For each edge probability $p=0.1, 0.2, \ldots, 0.9$, the graph $G_p$ was sampled 50 times. For each sampled graph $G_p$ we took the maximum modularity score of the partitions output by 200 runs of both the Louvain louvain and Leiden traag2019leiden algorithms. The noise in the $x$-axis is to allow one to see the points. In the lower part of the figure we examine, for each random instance of $G_p$, how well the modularity maximising partition of $G_p$ performs as a partition on the underlying graph $G$. For each sampled graph $G_p$ we plot the score $q_{{{\mathcal{A}}}'(G_p)}(G)$, where ${\mathcal{A}}(G_p)$ is the highest scoring partition on $G_p$ found in 200 runs of Louvain and Leiden and ${\mathcal{A}}'(G_p)$ is the partition modified as in Theorem \ref{['thm.moddiff']}(b) (with $\eta=0.05$ in Lemma \ref{['lem.nosmall2']}). See also Figure \ref{['fig.simsBIG']} for simulations run on a larger underlying graph.
• Figure A.1: Simulation results. The US political blogs network adamic2005political with 1490 vertices and 16718 edges was taken to be the underlying graph $G$, with the directed links between blogs encoded as undirected edges. The simulations were run as described in Figure \ref{['fig.sims']} on page \ref{['fig.sims']} with the exception that in the lower plot we plot the modularity score of $\tilde{{\mathcal{A}}}(G_p)$, rather than the score of the $\eta$-fattened partition $\tilde{{\mathcal{A}}}'(G_p)$.

Theorems & Definitions (67)

• Definition : NewmanGirvan, see also NewmanBook
• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 2.1
• Theorem 2.2
• Lemma 3.1: The fattening lemma
• Example 3.2