Deciding on the Type of the Degree Distribution of a Graph from Traceroute-like Measurements

Abstract

The degree distribution of the Internet topology is considered as one of its main properties. However, it is only known through a measure-ment procedure which gives a biased estimate. This measurement may in first approximation be modeled by a BFS (Breadth-First Search) tree. We explore here our ability to infer the type (Poisson or power-law) of the degree distribution from such a limited knowledge. We design procedures which estimate the degree distribution of a graph from a BFS or multi-BFS trees, and show experimentally (on models and real-world data) that our approaches succeed in making the differ-ence between Poisson and power-law degree distribution and in some cases can also estimate the number of links. In addition, we establish a method, which is a diminishing urn, to analyze the procedure of the queue. We analyze the profile of the BFS tree from a random graph with a given degree distribution. The expected number of nodes and the the expected number of invisible links at each level of BFS tree are two main results that we obtain. Using these informations, we propose two new methodologies to decide on the type of the underlying graph.