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Markov chains are convenient means of generating realizations of networks with a given (joint or otherwise) degree distribution, since they simply require a procedure for rewiring edges. The major challenge is to find the right number of…
Multiscale community detection can be viewed from a dynamical perspective within the Markov Stability framework, which uses the diffusion of a Markov process on the graph to uncover intrinsic network substructures across all scales. Here we…
This paper presents a simple algorithm to check whether reachability probabilities in parametric Markov chains are monotonic in (some of) the parameters. The idea is to construct - only using the graph structure of the Markov chain and…
This monograph is an exposition on an exciting new technique known as spectral independence, which has been instrumental in analyzing the convergence rate of Markov Chain Monte Carlo (MCMC) algorithms. For a high-dimensional distribution…
Recently there is a large amount of work devoted to the study of Markov chain stochastic gradient methods (MC-SGMs) which mainly focus on their convergence analysis for solving minimization problems. In this paper, we provide a…
Most methods proposed to uncover communities in complex networks rely on combinatorial graph properties. Usually an edge-counting quality function, such as modularity, is optimized over all partitions of the graph compared against a null…
A particular type of random dynamical processes is considered, in which the stochasticity is introduced through randomly fluctuating parameters. A method of local multipliers is developed for treating the local stability of such dynamical…
We consider the Ising, and more generally, $q$-state Potts Glauber dynamics on random $d$-regular graphs on $n$ vertices at low temperatures $\beta \gtrsim \frac{\log d}{d}$. The mixing time is exponential in $n$ due to a bottleneck between…
Locally-biased graph algorithms are algorithms that attempt to find local or small-scale structure in a large data graph. In some cases, this can be accomplished by adding some sort of locality constraint and calling a traditional graph…
Local search is a powerful heuristic in optimization and computer science, the complexity of which has been studied in the white box and black box models. In the black box model, we are given a graph $G = (V,E)$ and oracle access to a…
As sample sizes grow, scalability has become a central concern in the development of Markov chain Monte Carlo (MCMC) methods. One general approach to this problem, exemplified by the popular stochastic gradient Langevin dynamics (SGLD)…
The present paper studies local distributed graph problems in highly dynamic networks. Communication and changes of the graph happen in synchronous rounds and our algorithms always, i.e., in every round, satisfy non-trivial guarantees, no…
We consider graphs with two communities and analyze an algorithm for learning the community labels when the edges of the graph and only a small fraction of the labels are known in advance. The algorithm is based on the Glauber dynamics for…
Stochastic gradient methods are central to large-scale learning, yet their generalization theory typically relies on independent sampling assumptions. In many practical applications, data are generated by Markov chains and learning is…
Classical graphical modeling of multivariate random vectors uses graphs to encode conditional independence. In graphical modeling of multivariate stochastic processes, graphs may encode so-called local independence analogously. If some…
We study planar "vertex" models, which are probability measures on edge subsets of a planar graph, satisfying certain constraints at each vertex, examples including dimer model, and 1-2 model, which we will define. We express the local…
I show how Markov chain sampling with the Metropolis-Hastings algorithm can be modified so as to take bigger steps when the distribution being sampled from has the characteristic that its density can be quickly recomputed for a new point if…
We study irreducible time-homogenous Markov chains with finite state space in discrete time. We obtain results on the sensitivity of the stationary distribution and other statistical quantities with respect to perturbations of the…
The mixing time of a graph is an important metric, which is not only useful in analyzing connectivity and expansion properties of the network, but also serves as a key parameter in designing efficient algorithms. We introduce a new notion…
In order to study the stochastic Markov processes conditioned on a specific value of a time-integrated observable, the concept of ensembles of trajectories has been recently used extensively. In this paper, we consider a generic…