Fastest mixing reversible Markov chain on friendship graph: Trade-off between transition probabilities among friends and convergence rate

TL;DR

This paper investigates the trade-off between transition probabilities among friends and the convergence rate in social networks, identifying Pareto optimal points and showing the star topology as optimal for certain friendship graphs.

Contribution

It formulates the FMMC problem on friendship graphs, derives the Pareto frontier, and proves the star topology as the optimal structure for graphs with at least three blades.

Findings

Pareto frontier reduces to a single point for graphs with three or more blades.

Star topology (minimum spanning tree) is optimal for convergence.

Transition probability lower limit beyond which no improvement occurs.

Abstract

A long-standing goal of social network research has been to alter the properties of network to achieve the desired outcome. In doing so, DeGroot's consensus model has served as the popular choice for modeling the information diffusion and opinion formation in social networks. Achieving a trade-off between the cost associated with modifications made to the network and the speed of convergence to the desired state has shown to be a critical factor. This has been treated as the Fastest Mixing Markov Chain (FMMC) problem over a graph with given transition probabilities over a subset of edges. Addressing this multi-objective optimization problem over the friendship graph, this paper has provided the corresponding Pareto optimal points or the Pareto frontier. In the case of friendship graph with at least three blades, it is shown that the Pareto frontier is reduced to a global minimum point which is same as the optimal point corresponding to the minimum spanning tree of the friendship graph, i.e., the star topology. Furthermore, a lower limit for transition probabilities among friends has been provided, where values higher than this limit do not have any impact on the convergence rate.