Influence Spread in Social Networks: A Study via a Fluid Limit of the Linear Threshold Model

TL;DR

This paper models influence spread in social networks using a fluid limit of the linear threshold model, deriving differential equations that describe influence dynamics and connecting them to epidemiological models.

Contribution

It generalizes the linear threshold model to arbitrary threshold distributions and derives a fluid limit as an ODE, linking influence dynamics to hazard functions and epidemic models.

Findings

Fluid limit accurately approximates influence evolution.

Explicit solutions for uniform threshold distribution.

Connection established between influence spread and SIR epidemic model.

Abstract

Threshold based models have been widely used in characterizing collective behavior on social networks. An individual's threshold indicates the minimum level of influence that must be exerted, by other members of the population engaged in some activity, before the individual will join the activity. In this work, we begin with a homogeneous version of the Linear Threshold model proposed by Kempe et al. in the context of viral marketing, and generalize this model to arbitrary threshold distributions. We show that the evolution can be modeled as a discrete time Markov chain, and, by using a certain scaling, we obtain a fluid limit that provides an ordinary differential equation model (o.d.e.). We find that the threshold distribution appears in the o.d.e. via its hazard rate function. We demonstrate the accuracy of the o.d.e. approximation and derive explicit expressions for the trajectory of influence under the uniform threshold distribution. Also, for an exponentially distributed threshold, we show that the fluid dynamics are equivalent to the well-known SIR model in epidemiology. We also numerically study how other hazard functions (obtained from the Weibull and loglogistic distributions) provide qualitative different characteristics of the influence evolution, compared to traditional epidemic models, even in a homogeneous setting. We finally show how the model can be extended to a setting with multiple communities and conclude with possible future directions.