Understanding the local structure of a graph provides valuable insights about the underlying phenomena from which the graph has originated. Sampling and examining k-subgraphs is a widely used approach to understand the local structure of a graph. In this paper, we study the problem of sampling uniformly k-subgraphs from a given graph. We analyse a few different Markov chain Monte Carlo (MCMC) approaches, and obtain analytical results on their mixing times, which improve significantly the state of the art. In particular, we improve the bound on the mixing times of the standard MCMC approach, and the state-of-the-art MCMC sampling method PSRW, using the canonical-paths argument. In addition, we propose a novel sampling method, which we call recursive subgraph sampling RSS, and its optimized variant RSS+. The proposed methods, RSS and RSS+, are provably faster than the existing approaches. We conduct experiments and verify the uniformity of samples and the time efficiency of RSS and RSS+. or equivalently, sampling k-subgraphs uniformly at random, which is a challenge by itself. As a result, the problem of uniform sampling k-subgraphs, has been extensively studied in data mining, statistics, and theoretical computer science [1,4,6,7,10,14,22,25,27].In this paper, we study the problem of sampling uniformly at random k-subgraphs from a given input graph. Among the different methodologies that have been proposed, we focus on the Markov chain Monte Carlo (MCMC) approach [20], and in particular on the Metropolis-Hastings algorithm (MH) [11]. The high-level idea is to sample from the stationary distribution of a Markov chain, whose set of states is the set of k-subgraphs, by performing a random walk. The MH algorithm [11] is used to ensure that the stationary distribution, and thus, the sampling, is uniform. An important theoretical question is to upper bound the mixing time of the random walk, which is the time needed for the empirical sampling distribution to be close enough to the stationary distribution.We present improved results for the mixing time of Markov chains designed for uniform sampling of k-subgraphs. Our starting point is the recent work of Bressan et al. [7], who analyze a MCMC method and show that it mixes in timeÕ((k!) 2 ∆ 2k |V | 2 ), where |V | is the number of nodes in the input graph, ∆ is a maximum node degree, andÕ(·) is used to suppress logarithmic and other lower-term factors.Our first result is to analyze the Markov chain with the MH algorithm using the technique of canonical paths [23] and obtain an upper bound on the mixing time ofÕ(k!∆ k (D + k)|V |), where D is a diameter of the graph. Our bound is a significant improvement of the bound of Bressan et al. [7].Next, we proceed to improve this bound even further, by introducing a novel Markov chain to perform the random walk. In particular, we propose a technique based on recursive subgraph sampling (RSS), and a further improvement called RSS+, which exploits the fact that we can easily sample 2-subgraph uniformly at random in time O(1): sam...
