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

In this paper, we study a network formation game in which agents seek to maximize their influence by allocating constrained resources to connections with other agents. 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 allocation by each agent determines the weights of its outgoing edges, and the allocations of all agents thereby induce a network. This defines 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 and show that it converges to the set of Nash equilibria under bounded budgets and the assumption that every agent takes a best response infinitely often. 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.