Social Distancing, Gathering, Search Games: Mobile Agents on Simple Networks

TL;DR

This paper models social distancing and gathering as optimization problems involving mobile agents performing lazy random walks on simple networks, analyzing their expected times to achieve or avoid certain configurations.

Contribution

It introduces a new Markov chain-based framework for modeling social distancing and gathering, extending to multi-agent search games with different agent behaviors on simple networks.

Findings

Optimal laziness parameter minimizes expected time to achieve social distancing.

Markov chain models unify social distancing and gathering problems.

Extension to multiple searchers with different behaviors broadens existing search game literature.

Abstract

During epidemics, the population is asked to Socially Distance, with pairs of individuals keeping two meters apart. We model this as a new optimization problem by considering a team of agents placed on the nodes of a network. Their common aim is to achieve pairwise graph distances of at least D, a state we call socially distanced. (If D=1, they want to be at distinct nodes; if D=2 they want to be non-adjacent.) We allow only a simple type of motion called a Lazy Random Walk: with probability p (called the laziness parameter), they remain at their current node next period; with complementary probability 1-p , they move to a random adjacent node. The team seeks the common value of p which achieves social distance in the least expected time, which is the absorption time of a Markov chain. We observe that the same Markov chain, with different goals (absorbing states), models the gathering, or multi-rendezvous problem (all agents at the same node). Allowing distinct laziness for two types of agents (searchers and hider), extends the existing literature on predator-prey search games to multiple searchers. We consider only special networks: line, cycle and grid. Keywords: epidemic, random walk, dispersion, rendezvous search