Constraints on Meso-Scale Structure in Complex Networks

TL;DR

The paper investigates when meso-scale network structures such as core-periphery, community-hierarchy, and nestedness can be detected using block modularity under the configuration model. It derives general inequalities for positive block contributions, demonstrates that the configuration null severely restricts detectable patterns (notably excluding 2x2 core-periphery and nested configurations) and generalizes the resolution limit to these structures; it also connects block modularity to the degree-corrected SBM, contrasting descriptive modularity with inferential likelihood-based approaches. Through a three-group CP example and a dc-SBM analysis, it shows that modularity and inference can yield different conclusions about network structure under different null models. The work provides a principled framework to understand when modularity-based meso-scale detection is informative and how to choose between descriptive and inferential methods in practice. Overall, it clarifies the limitations of modularity under the configuration model and highlights the value of generative-model inference for uncovering certain meso-scale structures.

Abstract

A key topic in network science is the detection of intermediate or meso-scale structures. Community, core-periphery, disassortative and other partitions allow us to understand the organisation and function of large networks. In this work we study under what conditions certain common meso-scale structures are detectable using the idea of block modularity. We find that the configuration model imposes strong restrictions on core-periphery and related structures in directed networks. We derive inequalities expressing when such structures can be detected under the configuration model. Nestedness is closely related to core-periphery and is similarly restricted to only be detectable under certain conditions. We show that these conditions are a generalisation of the resolution limit to structures other than assortative communities. We show how block modularity is related to the degree corrected Stochastic Block Model and that optimisation of one can be made equivalent to the other in general. Finally, we discuss these issues in inferential versus descriptive approaches to meso-scale structure detection.

Figures (5)

• Figure 1: All 2x2 block patterns for undirected networks and an example network realising that block pattern.
• Figure 2: All 2x2 block patterns for directed networks and an example network realising that block pattern.
• Figure 3: Color is $Q_{CP} - Q_{Bipartite}$ for fixed $p_m$ and variable $p_c, p_p$. Blue means CP can be detected under the configuration model, red means a bipartite structure is always preferred. The dashed line shows where $Q_{CP} = Q_{Bipartite}$.
• Figure 4: Left: $Q_{CP} - Q_{Bipartite}$ networks constructed as described in the text. Blue means CP can be detected under the configuration model, red means a bipartite structure is always preferred. The dotted line shows where $Q_{CP} = Q_{Bipartite}$. Right: NODF metric for the same networks. High values (near 1) imply a strongly nested network. The average of 20 random networks used for every $p_{cp}, p_{cc}$ pair.
• Figure 5: Left: SBM network, partition inferred by maximising the likelihood, Equation \ref{['eqn:genmod']}. Centre: Proportion of three structures found in 1000 samples from the configuration model of the left hand network. Right: Label assignment inferred for a configuration model network with CP structure.