Network Coloring and Colored Coin Games

Abstract

Kearns et al. introduced the Graph Coloring Problem to model dynamic conflict resolution in social networks. Players, represented by the nodes of a graph, consecutively update their color from a fixed set of colors with the prospect of finally choosing a color that differs from all neighbors choices. The players only react on local information (the colors of their neighbors) and do not communicate. The reader might think of radio stations searching for transmission frequencies which are not subject to interference from other stations. While Kearns et al. (see [10]) empirically examined how human players deal with such a situation, Chaudury et al. performed a theoretical study and showed that, under a simple, greedy and selfish strategy, the players find a proper coloring of the graph within time $O\left (\log \left (\frac{n} {\delta } \right )\right )$ with probability ≥ 1 − δ, where n is the number of nodes in the network and δ is arbitrarily small. In other words, the graph is properly colored within τ steps and $\tau < c\log \left (\frac{n} {\delta } \right )$ with high probability for some constant c. Previous estimates on the constant c are very large. In this chapter we substantially improve the analysis and upper time bound for the proper coloring, by combining ideas from search games and probability theory.