Link Fraction Mixed Membership Reveals Community Diversity in Aggregated Social Networks

TL;DR

The paper tackles ecological fallacy and MAUP in aggregated social networks by introducing Link Fraction Mixed Membership (LFMM), which defines node-community membership as the fraction of a node's link volume to each community and proves aggregation-consistency. LFMM computes $m_i(k)=\frac{M_i(k)}{\sum_k M_i(k)}$ with $M_i(k)=\sum_{j\in C_k} w_{ij}(1-\frac{\delta_{ij}}{2})$, and extends to an aggregate-level matrix formulation $M'=A'C$, enabling a single-matrix computation and diffusion extensions. Validation on synthetic SBMs shows perfect aggregation-consistency in raw LFMM and near-perfect consistency in normalized forms, while an application to the population-scale Netherlands network uncovers urban centers as mixing hubs and tracks decade-long evolution, supported by a gravity-null model to separate geographic from social diversity. The study demonstrates LFMM as a robust tool for analyzing mixed membership and diversity in large, aggregated networks and outlines avenues for benchmarking against MMSBM and extending to more complex network types.

Abstract

Community detection is a critical tool for understanding the mesoscopic structure of large-scale networks. However, when applied to aggregated or coarse-grained social networks, disjoint community partitions cannot capture the diverse composition of community memberships within aggregated nodes. While existing mixed membership methods alleviate this issue, they may detected communities that are highly sensitive to the aggregation resolution, not reliably reflecting the underlying community structure of the underlying individual-level network. This paper presents the Link Fraction Mixed Membership (LFMM) method, which computes the mixed memberships of nodes in aggregated networks. Unlike existing mixed membership methods, LFMM is consistent under aggregation. Specifically, we show that it conserves community membership sums at different scales. The method is utilized to study a population-scale social network of the Netherlands, aggregated at different resolutions. Experiments reveal variation in community membership across different geographical regions and evolution over the last decade. In particular, we show how our method identifies large urban hubs that act as the melting pots of diverse, spatially remote communities.

Figures (8)

• Figure 1: Link Fraction Mixed Membership (LFMM) method and its consistency under aggregation. (a) Computing the link fraction of a node as fraction of connections to nodes within different community partitions. (b) Starting from an aggregated network with colored detected communities (top left): LFMM can be applied directly (red path) and be guaranteed to be equivalently computed by disaggregating communities to the individual level, computing LFMM, and aggregating back (black path). The dotted rectangles denote the aggregation partitioning of the individual-level network.
• Figure 2: LFMM consistency under aggregation and community affinity.(a) Sum of mixed memberships computed on the synthetic individual-level network ($m=0.2$) versus the mixed memberships computed directly on the aggregated network. Blue: LFMM values ($M$) following disaggregation then reaggregation. Orange: Normalized LFMM values ($m$). Green: LFMM of community detection performed at the individual level. (b) Isoline contour plot of mean LFMM value for different values of affinity ($\mu$) and aggregation intermixedness $m$.
• Figure 3: Disjoint community structure of the population-scale network of the Netherlands(a) Community partition of the municipality-aggregated network. Node sizes are proportional to population, colors represent community membership. Communities are named after the provinces they most closely match. (b) Community-aggregated adjacency matrix showing the density of connections within (diagonal) and between (off-diagonal) communities. Block sizes are proportional to community population. The matrix is ordered to minimize off-diagonal density values.
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