The Decay of Impact with Network Distance in Linear Diffusion Processes

TL;DR

This paper derives an approximate mathematical model showing that influence impact in social networks declines exponentially with network distance, validated by numerical studies on educational networks.

Contribution

It provides a first-order eigenvector-based approximation for total impact in linear diffusion models, linking impact decay to spectral properties of the network.

Findings

Impact declines exponentially with network distance.

First-order eigenvector approximation closely matches exact impact calculations.

Model applicable to social influence and status processes in various settings.

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.