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。这一观察促使近年来对时序图开展了严谨研究。在时序图中，节点集合固定，节点间的连接仅在特定时间步可用。这引发了此类图中大量算法问题的研究，其中最突出的是时序生成子图问题，即计算能保证所有节点对通过时序路径可达的子图。据我们所知，目前仅存在解决该问题的中心化方法。然而，许多现实网络并非由中心设计者塑造，而是通过多个策略性智能体的交互涌现并演化。这一观察正是近期博弈论网络形成模型密集研究的驱动力。本文融合了这两个新兴研究方向：时序图与博弈论网络形成。作为该新领域的第一步，我们聚焦于简化场景：给定完整时序宿主图，对应于其节点的智能体自利地创建邻接边，以确保在生成的网络中可通过时序路径到达所有其他节点。这使时序生成子图成为我们博弈的均衡。我们证明了均衡网络的收敛性与存在性、寻找最优智能体策略的复杂性问题，以及均衡质量的相关结论。通过迈出这些重要第一步，我们揭示了需要深入探索策略性智能体创建时序图的开放性挑战问题。