Most networks are not static objects, but instead they change over time. This observation has sparked rigorous research on temporal graphs within the last years. In temporal graphs, we have a fixed set of nodes and the connections between them are only available at certain time steps. This gives rise to a plethora of algorithmic problems on such graphs, most prominently the problem of finding temporal spanners, i.e., the computation of subgraphs that guarantee all pairs reachability via temporal paths. To the best of our knowledge, only centralized approaches for the solution of this problem are known. However, many real-world networks are not shaped by a central designer but instead they emerge and evolve by the interaction of many strategic agents. This observation is the driving force of the recent intensive research on game-theoretic network formation models. In this work we bring together these two recent research directions: temporal graphs and game-theoretic network formation. As a first step into this new realm, we focus on a simplified setting where a complete temporal host graph is given and the agents, corresponding to its nodes, selfishly create incident edges to ensure that they can reach all other nodes via temporal paths in the created network. This yields temporal spanners as equilibria of our game. We prove results on the convergence to and the existence of equilibrium networks, on the complexity of finding best agent strategies, and on the quality of the equilibria. By taking these first important steps, we uncover challenging open problems that call for an in-depth exploration of the creation of temporal graphs by strategic agents.

翻译：大多数网络并非静态对象，而是随时间动态变化。这一观察推动了近年来对时态图（temporal graphs）的严谨研究。在时态图中，节点集固定，节点间的连接仅在特定时间步可用，这引发了该类图上的诸多算法问题，其中最突出的便是时态生成子图（temporal spanners）的求解，即计算能通过时态路径保证所有节点对可达的子图。据我们所知，目前该问题的求解方法均为集中式方法。然而，许多现实网络并非由中心设计者塑造，而是由众多策略性智能体的交互涌现并演化而成。这一观察正是近期博弈论网络形成模型研究的驱动力。在本工作中，我们将这两个前沿研究方向——时态图与博弈论网络形成——结合起来。作为进入这一新领域的初步探索，我们聚焦于一个简化设定：给定一个完全时态宿主图，对应于其节点的智能体自私地创建邻接边，以确保在生成网络中能够通过时态路径到达所有其他节点。这使时态生成子图成为我们博弈的纳什均衡。我们证明了均衡网络收敛性及存在性、寻找最优智能体策略的复杂性，以及均衡质量的相关结果。通过这些重要的初步步骤，我们揭示了挑战性的开放问题，亟需对策略性智能体创建时态图的过程进行深入探索。