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

Graph sampling by lagged random walk

"This is the peer reviewed version of the following article: Zhang, L.-C. (2022). Graph sampling by lagged random walk. Stat, 11( 1), e444, which has been published in final form at https://doi.org/10.1002/sta4.444 This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. This article may not be enhanced, enriched or otherwise transformed into a derivative work, without express permission from Wiley or by statutory rights under applicable legislation. Copyright notices must not be removed, obscured or modified. The article must be linked to Wiley’s version of record on Wiley Online Library and any embedding, framing or otherwise making available the article or pages thereof by third parties from platforms, services and websites other than Wiley Online Library must be prohibited."We propose a family of lagged random walk sampling methods in simple undirected graphs, where transition to the next state (i.e., node) depends on both the current and previous states—hence, lagged. The existing random walk sampling methods can be incorporated as special cases. We develop a novel approach to estimation based on lagged random walks at equilibrium, where the target parameter can be any function of values associated with finite-order subgraphs, such as edge, triangle, 4-cycle and others.acceptedVersio

Multiplex Decomposition of Non-Markovian Dynamics and the Hidden Layer Reconstruction Problem

Elements composing complex systems usually interact in several different ways and as such the interaction architecture is well modelled by a multiplex network. However often this architecture is hidden, as one usually only has experimental access to an aggregated projection. A fundamental challenge is thus to determine whether the hidden underlying architecture of complex systems is better modelled as a single interaction layer or results from the aggregation and interplay of multiple layers. Here we show that using local information provided by a random walker navigating the aggregated network one can decide in a robust way if the underlying structure is a multiplex or not and, in the former case, to determine the most probable number of hidden layers. As a byproduct, we show that the mathematical formalism also provides a principled solution for the optimal decomposition and projection of complex, non-Markovian dynamics into a Markov switching combination of diffusive modes. We validate the proposed methodology with numerical simulations of both (i) random walks navigating hidden multiplex networks (thereby reconstructing the true hidden architecture) and (ii) Markovian and non-Markovian continuous stochastic processes (thereby reconstructing an effective multiplex decomposition where each layer accounts for a different diffusive mode). We also state and prove two existence theorems guaranteeing that an exact reconstruction of the dynamics in terms of these hidden jump-Markov models is always possible for arbitrary finite-order Markovian and fully non-Markovian processes. Finally, we showcase the applicability of the method to experimental recordings from (i) the mobility dynamics of human players in an online multiplayer game and (ii) the dynamics of RNA polymerases at the single-molecule level.Comment: 40 pages, 24 figure

Sampling Online Social Networks via Heterogeneous Statistics

Most sampling techniques for online social networks (OSNs) are based on a particular sampling method on a single graph, which is referred to as a statistics. However, various realizing methods on different graphs could possibly be used in the same OSN, and they may lead to different sampling efficiencies, i.e., asymptotic variances. To utilize multiple statistics for accurate measurements, we formulate a mixture sampling problem, through which we construct a mixture unbiased estimator which minimizes asymptotic variance. Given fixed sampling budgets for different statistics, we derive the optimal weights to combine the individual estimators; given fixed total budget, we show that a greedy allocation towards the most efficient statistics is optimal. In practice, the sampling efficiencies of statistics can be quite different for various targets and are unknown before sampling. To solve this problem, we design a two-stage framework which adaptively spends a partial budget to test different statistics and allocates the remaining budget to the inferred best statistics. We show that our two-stage framework is a generalization of 1) randomly choosing a statistics and 2) evenly allocating the total budget among all available statistics, and our adaptive algorithm achieves higher efficiency than these benchmark strategies in theory and experiment

Learning attribute and homophily measures through random walks

We investigate the statistical learning of nodal attribute functionals in homophily networks using random walks. Attributes can be discrete or continuous. A generalization of various existing canonical models, based on preferential attachment is studied (model class P), where new nodes form connections dependent on both their attribute values and popularity as measured by degree. An associated model class U is described, which is amenable to theoretical analysis and gives access to asymptotics of a host of functionals of interest. Settings where asymptotics for model class U transfer over to model class P through the phenomenon of resolvability are analyzed. For the statistical learning, we consider several canonical attribute agnostic sampling schemes such as Metropolis-Hasting random walk, versions of node2vec (Grover and Leskovec, 2016) that incorporate both classical random walk and non-backtracking propensities and propose new variants which use attribute information in addition to topological information to explore the network. Estimators for learning the attribute distribution, degree distribution for an attribute type and homophily measures are proposed. The performance of such statistical learning framework is studied on both synthetic networks (model class P) and real world systems, and its dependence on the network topology, degree of homophily or absence thereof, (un)balanced attributes, is assessed.S. Banerjee is partially supported by the NSF CAREER award DMS-2141621. S. Bhamidi and V. Pipiras are partially supported by NSF DMS-2113662. S. Banerjee, S. Bhamidi and V.Pipiras are partially supported by NSF RTG grant DMS-2134107info:eu-repo/semantics/publishedVersio