Large Deviation Principle for the Greedy Exploration Algorithm over Erdös-Rényi Graphs

We prove a large deviation principle for a greedy exploration process on an Erdös-Rényi (ER) graph when the number of nodes goes to infinity. To prove our main result, we usethe general strategy to study large deviations of processes proposed by Feng and Kurtz (2006),based on the convergence of non-linear semigroups. The rate function can be expressed in a closed-form formula, and associated optimization problems can be solved explicitly, providing the largedeviation trajectory. Also, we derive large deviation results for the size of the maximum independentset discovered by such an algorithm and analyse the probability that it exceeds known bounds forthe maximal independent set. We also analyse the link between these results and the landscapecomplexity of the independent set and the exploration dynamic.