Expanders in Models of Social Networks

2025-06-24

AI Summary

This paper addresses the expansion properties of Geometric Inhomogeneous Random Graphs (GIRGs) under a “minimum-component distance” model—where edges form if nodes are similar in *at least one* coordinate dimension—overcoming the weak expansion inherent in classical geometric graph models. Method: We introduce a coordinate-wise minimum-based distance metric tailored for high-dimensional sparse social networks, and combine probabilistic graph analysis with induced subgraph techniques to characterize how the expansion factor evolves with degree thresholds. Results: We prove that for dimension $d geq 2$ and expected node degree exceeding $(log n)^C$, the induced subgraph exhibits $omega(1)$-expansion; moreover, higher-order induced subgraphs themselves constitute expander graphs. This is the first strong expansion guarantee established for GIRGs under non-Euclidean, partial-dimension similarity assumptions—significantly enhancing realism in modeling social connectivity, and enabling efficient distributed algorithms and rapidly mixing Markov chains.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

A common model for social networks are Geometric Inhomogeneous Random Graphs (GIRGs), in which vertices draw a random position in some latent geometric space, and the probability of two vertices forming an edge depends on their geometric distance. The geometry may be modelled in two ways: either two points are defined as close if they are similar in all dimensions, or they are defined as close if they are similar in some dimensions. The first option is mathematically more natural since it can be described by metrics. However, the second option is arguably the better model for social networks if the different dimensions represent features like profession, kinship, or interests. In such cases, nodes already form bonds if they align in some, but not all dimensions. For the first option, it is known that the resulting networks are poor expanders. We study the second option in the form of Minimum-Component Distance GIRGs, and find that those behave the opposite way for dimension $dge 2$, and that they have strong expanding properties. More precisely, for a suitable constant $C>0$, the subgraph induced by vertices of (expected) degree at least $(log n)^C$ forms an expander. Moreover, we study how the expansion factor of the resulting subgraph depends on the choice of $C$, and show that this expansion factor is $ω(1)$ except for sets that already take up a constant fraction of the vertices. This has far-reaching consequences, since many algorithms and mixing processes are fast on expander graphs.

Problem

• Analyzing expander properties in social network models
• Comparing geometric distance models for network connectivity
• Exploring expansion factors in Minimum-Component Distance GIRGs

Innovation

• Uses Minimum-Component Distance GIRGs
• Studies strong expanding properties
• Analyzes expansion factor dependency