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.

翻译：本文研究了一种网络形成博弈，其中每个代理通过分配有限资源来选择与其他代理的连接，以最大化自身影响力。具体而言，我们采用卡兹中心性来刻画代理在网络中的影响力。分配约束被限定在给定无权网络中的邻居节点内，该网络编码了拓扑约束条件。每个代理的分配对应于其出边权重，所有代理的分配共同构成一个完整的网络。该模型是一个策略式博弈，其中代理的效用由其卡兹中心性定义。我们刻画了该博弈的纳什均衡网络并分析了其性质。提出了一种序贯最优反应动态（BRD）来模拟网络形成过程，并证明在非常宽松的假设下该动态收敛于纳什均衡集。对于完全底层拓扑结构，我们发现纳什均衡下代理的卡兹中心性与其预算成正比。对于每个代理均存在自环的一般底层拓扑结构，我们证明纳什均衡会形成层次化网络。最后，仿真实验验证了我们的发现。