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Temporal Network Creation Games: The Impact of Flexible Labels

Hans Gawendowicz, Nicolas Klodt, Aleksandrs Morgensterns, George Skretas

TL;DR

Two reachability models and several cost functions based on the label an agent assigns to an edge are studied, as well as proofs of existence of Nash equilibria, as well as lower and upper bounds on the Price of Anarchy and Price of Stability.

Abstract

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.

Temporal Network Creation Games: The Impact of Flexible Labels

TL;DR

Two reachability models and several cost functions based on the label an agent assigns to an edge are studied, as well as proofs of existence of Nash equilibria, as well as lower and upper bounds on the Price of Anarchy and Price of Stability.

Abstract

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 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.
Paper Structure (19 sections, 20 theorems, 16 equations, 5 figures, 1 table)

This paper contains 19 sections, 20 theorems, 16 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Work
  4. Preliminaries
  5. Temporal Graphs
  6. Our Model
  7. Variants
  8. Uniform Label cost
  9. Non-strict Paths
  10. Strict Paths
  11. Monotone Label Cost Model
  12. Non-strict Paths
  13. Strict Paths
  14. Proper Label Model
  15. Strict
  16. ...and 4 more sections

Key Result

Proposition 3

In the non-strict uniform label cost model, $\text{PoS}_{> 0}^{\leq, 0} = 1$.

Figures (5)

  • Figure 1: Temporal graph with lifetime 6. $(\{a, d\}, \{d, b\})$ is a temporal path, while $(\{a,c\}, \{c, b\})$ is not.
  • Figure 2: NE for $k = 3$. Dashed edges are labeled 2 and solid edges are labeled 1
  • Figure 3: PoA in the any-label model for strict paths with $n = 6$
  • Figure 4: Flattened 3D hypercube, NE for $n = 8$.
  • Figure 5: Nash equilibria for $n$ between 1 and 6. Vertices that buy 2 edges are highlighted.

Theorems & Definitions (42)

  • Definition 1: $k$-label tree
  • Definition 2: Reachability Tree
  • Proposition 3
  • proof
  • Lemma 4
  • proof
  • Theorem 5
  • proof
  • Lemma 6
  • proof
  • ...and 32 more