Robust Distributed Averaging: When are Potential-Theoretic Strategies Optimal?

TL;DR

This paper analyzes a strategic interaction between a network designer and an adversary in a distributed averaging network, proposing potential-theoretic strategies and conditions for equilibrium.

Contribution

It introduces a novel game-theoretic framework for network defense and attack, linking optimal strategies to potential theory and establishing equilibrium conditions.

Findings

Potential-theoretic strategies are optimal under certain conditions.

A sufficient condition for saddle-point equilibrium existence is provided.

Alternative characterization of optimal strategies simplifies complex equations.

Abstract

We study the interaction between a network designer and an adversary over a dynamical network. The network consists of nodes performing continuous-time distributed averaging. The adversary strategically disconnects a set of links to prevent the nodes from reaching consensus. Meanwhile, the network designer assists the nodes in reaching consensus by changing the weights of a limited number of links in the network. We formulate two Stackelberg games to describe this competition where the order in which the players act is reversed in the two problems. Although the canonical equations provided by the Pontryagin's maximum principle seem to be intractable, we provide an alternative characterization for the optimal strategies that makes connection to potential theory. Finally, we provide a sufficient condition for the existence of a saddle-point equilibrium for the underlying zero-sum game.