Characterizing weak convergence error of sparse graph coloring process

This paper is an ongoing study of weak convergence properties of local coloring strategies which is a fundamental question about local coloring processes on large networks. Consider a piece of information spreading on a social network. Suppose 90% of the nodes receive the information. Is it possible that the consequence is by accident? i.e., is it possible that the next time the same information is only received by less than 10% of the nodes? The classic result of Sullivan says no. However, another question, remains. Is the consequence determined by the local structure of the graph ? The question, can. be phrased as whether the empirical process induced by an. in dependent -jump finite range process (local coloring process) converges if the graph structure converges. Here the local structure of a graph, G, refers to the limit of the empirical distribution, of all sub graphs of G. An independent jump finite range process (local coloring process) refers to a jump process with each node jumping independently of each other (independent jump), and the intensity measure of a node, c, depends on. the current state of nodes within a fixed distance to c (finite range). We give a condition for the local structure of the graph that garantees the independent jump finite range process (local coloring process) converges provided the local graph structure converges. 'The result improves our previous result. © 2016 ICIC INTERNATIONAL.