Mixing times of random walks on dynamic configuration models

The mixing time of a random walk, with or without backtracking, on a random graph generated according to the configuration model on $n$ vertices, is known to be of order $\log n$. In this paper we investigate what happens when the random graph becomes {\em dynamic}, namely, at each unit of time a fraction $\alpha_n$ of the edges is randomly rewired. Under mild conditions on the degree sequence, guaranteeing that the graph is locally tree-like, we show that for every $\varepsilon\in(0,1)$ the $\varepsilon$-mixing time of random walk without backtracking grows like $\sqrt{2\log(1/\varepsilon)/\log(1/(1-\alpha_n))}$ as $n \to \infty$, provided that $\lim_{n\to\infty} \alpha_n(\log n)^2=\infty$. The latter condition corresponds to a regime of fast enough graph dynamics. Our proof is based on a randomised stopping time argument, in combination with coupling techniques and combinatorial estimates. The stopping time of interest is the first time that the walk moves along an edge that was rewired before, which turns out to be close to a strong stationary time.Comment: 23 pages, 6 figures. Previous version contained a mistake in one of the proofs. In this version we look at nonbacktracking random walk instead of simple random wal

Rapid mixing of Swendsen-Wang dynamics in two dimensions

We prove comparison results for the Swendsen-Wang (SW) dynamics, the heat-bath (HB) dynamics for the Potts model and the single-bond (SB) dynamics for the random-cluster model on arbitrary graphs. In particular, we prove that rapid mixing of HB implies rapid mixing of SW on graphs with bounded maximum degree and that rapid mixing of SW and rapid mixing of SB are equivalent. Additionally, the spectral gap of SW and SB on planar graphs is bounded from above and from below by the spectral gap of these dynamics on the corresponding dual graph with suitably changed temperature. As a consequence we obtain rapid mixing of the Swendsen-Wang dynamics for the Potts model on the two-dimensional square lattice at all non-critical temperatures as well as rapid mixing for the two-dimensional Ising model at all temperatures. Furthermore, we obtain new results for general graphs at high or low enough temperatures.Comment: Ph.D. thesis, 66 page

2.5K-Graphs: from Sampling to Generation

Understanding network structure and having access to realistic graphs plays a central role in computer and social networks research. In this paper, we propose a complete, and practical methodology for generating graphs that resemble a real graph of interest. The metrics of the original topology we target to match are the joint degree distribution (JDD) and the degree-dependent average clustering coefficient ($\bar{c}(k)$). We start by developing efficient estimators for these two metrics based on a node sample collected via either independence sampling or random walks. Then, we process the output of the estimators to ensure that the target properties are realizable. Finally, we propose an efficient algorithm for generating topologies that have the exact target JDD and a $\bar{c}(k)$ close to the target. Extensive simulations using real-life graphs show that the graphs generated by our methodology are similar to the original graph with respect to, not only the two target metrics, but also a wide range of other topological metrics; furthermore, our generator is order of magnitudes faster than state-of-the-art techniques