Finding Planted Partitions in Random Graphs with General Degree Distributions

Source abstract

We consider the problem of recovering a planted partition such as a coloring, a small bisection, or a large cut in an (apart from that) random graph. In the last 30 years many algorithms for this problem have been developed that work provably well on various random graph models resembling the Erdős–Rényi model $G_{n,m}$. In these random graph models edges are distributed uniformly, and thus the degree distribution is very regular. By contrast, the recent theory of large networks shows that real-world networks frequently have a significantly different distribution of the edges and hence also a different degree distribution. Therefore, a variety of new types of random graphs have been introduced to capture these specific properties. One of the most popular models is characterized by a prescribed expected degree sequence. We study a natural variant of this model that features a planted partition. Our main result is that there is a polynomial time algorithm for recovering (a large part of) the planted partition in this model even in the sparse case, where the average degree is constant. In contrast to prior work, the input of the algorithm consists only of the graph, i.e., no further parameters of the model (such as the expected degree sequence) are revealed to the algorithm.