Counting Strict Gridlock on Graphs

TL;DR

This paper introduces a new framework for analyzing distributed graph coloring problems related to consensus formation, focusing on counting strict gridlock colorings that prevent groups from reaching agreement.

Contribution

It defines a novel type of graph coloring called strict gridlock colorings and provides a recurrence relation to count these colorings, linking graph structure to consensus difficulty.

Findings

Developed a recurrence relation for counting gridlock colorings

Established a mathematical measure of how graph structure hinders consensus

Linked gridlock colorings to social group dynamics and network analysis

Abstract

Graph colorings have been of interest to mathematicians for a long time, but relatively recently, social scientists have also found them to be interesting tools for studying group behavior. In the last 20 years, scientists have begun to study how coloring problems can be solved by groups of individuals on a graph, which has led to new insights into network structure, group dynamics, and individual human behavior. Despite this newfound utility, the exact nature of these distributed coloring problems is not well-understood, and established mathematical tools like the chromatic polynomial miss the unique challenges that arise in these social problem-solving situations with limited information. In this paper, we provide a new framework for understanding these distributed problems by defining a new kind of graph coloring with particular relevance to consensus formation on networks, in which all vertices are trying to agree on a common color. These strict gridlock colorings represent roadblocks to consensus where the group will not reach a uniform coloring using natural update processes. We describe a recurrence relation that provides an algorithm for counting these gridlocked colorings, which establishes a mathematical measure of how much a given graph hinders consensus in a group.