A crucial aspect of research is understanding how real-world networks, such as transportation and information networks, are formed. A prominent model for such networks was introduced by \cite{fabrikant_network_2003} and extended by \cite{bilo_temporal_2023}, incorporating temporal graphs to better represent real-world networks. In this model, there is a given host graph with $n$ agents (represented by nodes) and time labels on the edges. Each agent can establish connections by purchasing edges. This makes the edges present at the time steps given by the time labels of the host graph. The goal of each agent is to reach as many other agents as possible while minimizing the number of edges bought. However, this model makes the simplifying assumption that each edge comes with predetermined time steps. We address this deficiency by extending the model of Bilo et al. \cite{bilo_temporal_2023} to allow agents to purchase edges and to decide when they appear. To capture a variety of real-world applications, we study two reachability models and several cost functions based on the label an agent assigns to an edge. For these settings, we provide proofs of existence of Nash equilibria, as well as lower and upper bounds on the Price of Anarchy and Price of Stability.

翻译：理解现实世界网络（如交通网络和信息网络）如何形成是研究的关键方面。\cite{fabrikant_network_2003} 提出了此类网络的经典模型，并由 \cite{bilo_temporal_2023} 扩展引入时间图以更准确表征现实网络。该模型包含一个具有 $n$个智能体（表示为节点）和边时间标签的给定宿主图。每个智能体可通过购买边建立连接，这使得边在宿主图时间标签指定的时间步出现。每个智能体的目标是在最小化购买边数量的同时，尽可能接触更多其他智能体。然而，该模型存在简化假设：每条边均附带预设时间步。为弥补这一缺陷，我们扩展了 Bilo 等人 \cite{bilo_temporal_2023} 的模型，允许智能体购买边并自主决定边的出现时间。为涵盖多样化的现实应用场景，我们研究了两种可达性模型，以及基于智能体为边分配标签的多种成本函数。针对这些设定，我们证明了纳什均衡的存在性，并给出了无政府状态价格与稳定价格的下界和上界。