A Network Formation Game for Katz Centrality Maximization: A Resource Allocation Perspective

TL;DR

This paper models a strategic network formation game where agents allocate resources to maximize influence via Katz centrality, characterizing Nash equilibria and proposing a convergent best-response dynamics.

Contribution

It introduces a game-theoretic framework for influence maximization using Katz centrality, analyzing equilibrium structures and proposing a dynamic formation process.

Findings

Nash equilibria networks are characterized for different topologies.

Katz centralities are proportional to budgets in complete networks.

Hierarchical networks emerge at equilibria in general topologies.

Abstract

In this paper, we study a network formation game in which agents seek to maximize their influence by allocating constrained resources to choose connections with other agents. In particular, we use Katz centrality to model agents' influence in the network. Allocations are restricted to neighbors in a given unweighted network encoding topological constraints. The allocations by an agent correspond to the weights of its outgoing edges. Such allocation by all agents thereby induces a network. This models a strategic-form game in which agents' utilities are given by their Katz centralities. We characterize the Nash equilibrium networks of this game and analyze their properties. We propose a sequential best-response dynamics (BRD) to model the network formation process. We show that it converges to the set of Nash equilibria under very mild assumptions. For complete underlying topologies, we show that Katz centralities are proportional to agents' budgets at Nash equilibria. For general underlying topologies in which each agent has a self-loop, we show that hierarchical networks form at Nash equilibria. Finally, simulations illustrate our findings.