Temporal Network Creation Games: The Impact of Non-Locality and Terminals

TL;DR

This work explores the impact of two novel conceptual features: agents are no longer restricted to creating incident edges, called the global setting, and agents might only want to ensure that they can reach a subset of the other nodes, called the terminal model, and studies the existence, structure, and quality of equilibrium networks.

Abstract

We live in a world full of networks where our economy, our communication, and even our social life crucially depends on them. These networks typically emerge from the interaction of many entities, which is why researchers study agent-based models of network formation. While traditionally static networks with a fixed set of links were considered, a recent stream of works focuses on networks whose behavior may change over time. In particular, Bilò et al. (IJCAI 2023) recently introduced a game-theoretic network formation model that embeds temporal aspects in networks. More precisely, a network is formed by selfish agents corresponding to nodes in a given host network with edges having labels denoting their availability over time. Each agent strategically selects local, i.e., incident, edges to ensure temporal reachability towards everyone at low cost. In this work we set out to explore the impact of two novel conceptual features: agents are no longer restricted to creating incident edges, called the global setting, and agents might only want to ensure that they can reach a subset of the other nodes, called the terminal model. For both, we study the existence, structure, and quality of equilibrium networks. For the terminal model, we prove that many core properties crucially depend on the number of terminals. We also develop a novel tool that allows translating equilibrium constructions from the non-terminal model to the terminal model. For the global setting, we show the surprising result that equilibria in the global and the local model are incomparable and we establish a high lower bound on the Price of Anarchy of the global setting that matches the upper bound of the local model. This shows the counter-intuitive fact that allowing agents more flexibility in edge creation does not improve the quality of equilibrium networks.

Figures (6)

• Figure 1: This figure shows two host graphs $H_1$ and $H_2$ (dotted and solid lines) and two respective strategy profiles $s^1$ and $s^2$ (solid lines) on the left. Yellow nodes are terminals and all edges are bought by the nodes where they originate. On the right, you can see the resulting graph product according to \ref{['def:graph_product']}. For clarity, all edges with label 6 are not displayed.
• Figure 2: This figure illustrates the equilibrium constructions for the proof of \ref{['thm:NE_two_terminals']}. On the left, we have the case where $M\neq\varnothing$ and $N\neq\varnothing$. The middle shows the case where $N=\varnothing$ and $\min\lambda(m_1,t_1)>\min\lambda(t_1,t_2)$ and the right illustrates the case where $N=\varnothing$ and $\min\lambda(m_1,t_1)<\min\lambda(t_1,t_2)$.
• Figure 3: A forbidden structure in a strategy profile. The node $z$ has two neighbors $u_1$ and $u_2$ that both buy two distinct edges that they need to reach the nodes $x$ and $y$ respectively. For both of them, the two needed edges have at least a label as high as their edge to $z$.
• Figure 5: Simple temporal clique with a given minimal temporal spanner (blue edges). The numbers represent the labels of the edges. For blue edges it is also given which nodes could not reach all other nodes anymore when the edge is removed from the spanner.
• Figure 6: Simple temporal cliques with given global or local equilibrium respectively. The numbers on the edges indicate the time labels and all non depicted edges have label 3. In the left graph the blue edges form a global equilibrium when bought by the indicated nodes. In the right graph the directions of the edges indicate that the source buys this edge in a local Nash equilibrium.

Theorems & Definitions (29)

• definition 1: graph product
• theorem 1
• corollary 1
• lemma 1
• corollary 2
• theorem 2
• definition 2: necessary edge
• lemma 2
• lemma 3
• theorem 3