Counting Strict Gridlock on Graphs

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.

Figures (8)

• Figure 1: The edge-minimal connected graph with a specified $\mathop{\mathrm{LO}}\nolimits$-polynomial degree $d$.
• Figure 2: Demonstration of computing the locally-optimal polynomial for a graph with a low maximum degree. Bivalent vertices are represented by squares. First, we subdivide the edges connected to leaves, and then we apply \ref{['thm:trivalent']} on the central vertex. All graphs in the bottom row are bivalent dense and therefore have easily accessible LO polynomials of $k^2$ or $k$.
• Figure 3: Demonstration of \ref{['thm:any-degree']} where $v$ is a degree four vertex. Each symbol shows the four edges connected to $v$ and represents the LO-polynomial of the graph. Initially, we have no information about which neighbors $v$ shares a color with, but \ref{['thm:any-degree']} tells us how we can express this LO-polynomial in terms of other LO-polynomials where certain edges are subdivided, represented by blue.
• Figure 4: Demonstration of \ref{['ex:degree-four-step-two']} with degree 4. Like \ref{['fig:degree_four_recur_1']}, blue edges represent subdivided edges. An additional leaf in blue can be seen in the seven graphs in the top two rows. Non-voting edges, shown in green, connect vertices that have the same color but do not contribute towards local-optimality.
• Figure 5: The complete recursion on four edges.

Theorems & Definitions (33)

• Theorem 1
• Proposition 2
• proof
• Proposition 3
• Corollary 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof