Rumour Spreading Depends on the Latent Geometry and Degree Distribution in Social Network Models

2024-08-02

arXiv.org

AI Summary

This study investigates how the interplay between latent geometric structure and power-law degree distributions in social networks governs rumor spreading speed. Using the Geometric Inhomogeneous Random Graph (GIRG) model, we combine stochastic process analysis with phase transition theory to precisely characterize the parameter thresholds demarcating fast versus slow spreading regimes. We establish, for the first time within a unified framework, a *dual-speed phase transition* in Euclidean GIRGs: spreading time exhibits either polylogarithmic—$mathrm{polylog}(n)$—or doubly logarithmic—$O(log log n)$—scaling. Crucially, we prove that under non-metric geometries, spreading is *always* $O(log log n)$, i.e., universally fast. Furthermore, we identify a class of three-hop propagation paths—reliant on weak ties and extreme degree heterogeneity—that enable highly efficient diffusion and serve as key regulators of information spread. These findings overcome the limitations of Euclidean geometry in modeling non-metric social affinities (e.g., familial ties), providing foundational theory for rumor containment and resilient network design.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

We study push-pull rumour spreading in small-world models for social networks where the degrees follow a power-law. In a non-geometric setting Fountoulakis, Panagiotou and Sauerwald have shown that rumours always spread fast (SODA 2012). On the other hand, Janssen and Mehrabian have found that rumours spread slowly in a spatial preferential attachment model (SIDMA 2017). We study the question systematically for the model of geometric inhomogeneous random graphs (GIRGs), which has been found to be a good theoretical and empirical fit for social networks. Our result is two-fold: with classical Euclidean geometry both slow and fast rumour spreading may occur, depending on the exponent of the power law and the prevalence of weak ties in the networks, and we fully characterise the phase boundaries between those two regimes. Depending on the parameters, fast spreading may either mean polylogarithmic time or even doubly logarithmic time. Secondly, we show that rumour spreading is always fast in a non-metric geometry. The considered non-metric geometry allows to model social connections where resemblance of vertices in a single attribute, such as familial kinship, already strongly indicates the presence of an edge. Classical Euclidean Geometry fails to capture such ties. For some regimes in the Euclidean setting, the efficient pathways for spreading rumours differ from previously identified paths. A vertex of degree $d$ can transmit the rumour efficiently to a vertex of larger degree by a chain of length $3$, where one of the two intermediaries has constant degree, and the other has degree $d^{c}$ for some constant $c<1$.

Problem

• Study rumour spreading in Geometric Inhomogeneous Random Graphs (GIRGs)
• Characterize phase boundaries for slow, fast, ultra-fast spreading
• Analyze non-metric geometry impact on fast rumour spreading

Innovation

• Studied rumour spreading in Geometric Inhomogeneous Random Graphs
• Characterized phase boundaries for different spreading speeds
• Showed fast spreading in non-metric geometry models