Graphs and Path Equilibria

TL;DR

This paper explores equilibrium conditions for optimal or stable paths in arbitrary arc-labelled directed graphs, providing necessary and sufficient conditions and applying findings to network routing.

Contribution

It introduces a general equilibrium framework for graphs, establishing conditions for equilibrium existence and clarifying their logical relationship.

Findings

A sufficient condition for equilibrium existence in all graphs.

A necessary condition for equilibrium existence in all graphs.

Conditions coincide under total order optimality, ensuring full equivalence.

Abstract

The quest for optimal/stable paths in graphs has gained attention in a few practical or theoretical areas. To take part in this quest this chapter adopts an equilibrium-oriented approach that is abstract and general: it works with (quasi-arbitrary) arc-labelled digraphs, and it assumes very little about the structure of the sought paths and the definition of equilibrium, \textit{i.e.} optimality/stability. In this setting, this chapter presents a sufficient condition for equilibrium existence for every graph; it also presents a necessary condition for equilibrium existence for every graph. The necessary condition does not imply the sufficient condition a priori. However, the chapter pinpoints their logical difference and thus identifies what work remains to be done. Moreover, the necessary and the sufficient conditions coincide when the definition of optimality relates to a total order, which provides a full-equivalence property. These results are applied to network routing.