The Decay of Impact with Network Distance in Linear Diffusion Processes

2026-04-24

AI Summary

In linear diffusion models, the precise decay pattern of total influence between nodes with respect to network distance remains unclear, as the superposition of exponentially decaying influences along multiple paths yields complex interactions. This work proposes the first analytical first-order approximation of total influence as a function of network distance, revealing that it is governed by the product of the nodes’ eigenvector centralities and an exponential decay term dictated by the dominant eigenvalue of the graph’s adjacency matrix. Higher-order eigenvectors can further refine this approximation. Combining spectral graph theory, eigen-decomposition of the adjacency matrix, and linear diffusion modeling, the approach is validated through numerical simulations and empirical analysis on interpersonal networks in educational settings, demonstrating that total influence typically decays exponentially and that the first-order approximation effectively captures the exact solution.

Technology Category

• Application Domains: Humanities & Computational Social Science
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

Many processes related to status, power, and influence within social networks have been modeled using forced linear diffusion models; examples include the highly successful Friedkin-Johnsen model of social influence, the status/power scores of Katz and Bonacich, and the widely used network autocorrelation model. While a basic assumption of such models is that the impact of one individual on another through any given path falls exponentially with path length, the total impact of the first individual on the second involves contributions from walks of all lengths; thus, while total impact is expected to decline with network distance, the relationship is not trivial. Here, we provide an approximate solution for the total impact of one node on another as a function of network distance, showing that the total impact is given to first order by a product of eigenvector centrality scores together with an expression in terms of the graph spectrum (eigenvalues of the adjacency matrix) that falls exponentially with distance. We also show how this solution can be refined using higher-order eigenvectors of the adjacency matrix. A numerical study on interpersonal networks drawn from educational settings verifies an average exponential decline in impact strength under the linear diffusion model, and shows that the first-order eigenvector approximation can often be a good proxy for total impact as obtained from the exact solution. This suggests a simple model that can be used to approximate total impact for social influence or status processes in a range of settings.

Problem

• linear diffusion
• network distance
• social influence
• impact decay
• eigenvector centrality

Innovation

• linear diffusion
• network distance
• eigenvector centrality
• graph spectrum
• exponential decay