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

Balaji R, Prashil Wankhede, Pavankumar Tallapragada

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.

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

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.

Paper Structure

This paper contains 6 sections, 11 theorems, 22 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. Modeling and Problem Setup
  3. Analysis of the Game and Sequential BRD
  4. About the Nash Equilibrium Networks
  5. Simulations
  6. Conclusion

Key Result

Lemma B.1

(Interdependence between agent centralities.) Consider the Katz centralities defined in eq:katz_score. Let $\mathbf{w} \in \mathcal{K}$ be a feasible allocation profile, satisfying eq:agent_constraint_set. Then, the Katz centrality of any agent $i \in \mathcal{V}$ in $\mathsf{G}( \mathbf{w} )$ satis $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure E1: Convergence of BRD. (Left) Underlying topology with self-loops at every node. (Right) Unweighted Nash equilibrium network attained by BRD with self loops at nodes $\{1,2,3\}$. The color bar represents the continuum of centrality values in $[0,10]$.
  • Figure E2: Evolution of $c_{i}$'s under BRD with $\mathsf{G}^\dagger$ as shown in Figure \ref{['fig:self_loops']}.
  • Figure E3: When $\mathsf{G}^\dagger$ is complete. (Left) Plot of centralities vs agents' budgets. (Right) Unweighted Nash equilibrium network reached under BRD. Self loops exist at nodes $\{4,5,6\}$. The color bar represents the continuum of centrality values in $[0,10]$.

Theorems & Definitions (12)

  • Lemma B.1
  • Lemma C.1
  • Lemma C.2
  • Lemma C.3
  • Theorem C.4
  • Corollary C.5
  • Theorem C.6
  • Remark C.7
  • Theorem D.1
  • Lemma D.2
  • ...and 2 more