Cutoff phenomena for random walks on random regular graphs

We present analytical results for the distribution of cover times of random walks (RWs) on random regular graphs consisting of N nodes of degree c (c ⩾ 3). Starting from a random initial node at time t = 1, at each time step t ⩾ 2 an RW hops into a random neighbor of its previous node. In some of the time steps the RW may visit a new, yet-unvisited node, while in other time steps it may revisit a node that has already been visited before. The cover time T C is the number of time steps required for the RW to visit every single node in the network at least once. We derive a master equation for the distribution P t (S = s) of the number of distinct nodes s visited by an RW up to time t and solve it analytically. Inserting s = N we obtain the cumulative distribution of cover times, namely the probability P(T C ⩽ t) = P t (S = N) that up to time t an RW will visit all the N nodes in the network. Taking the large network limit, we show that P(T C ⩽ t) converges to a Gumbel distribution. We calculate the distribution of partial cover (PC) times P(T PC,k = t), which is the probability that at time t an RW will complete visiting k distinct nodes. We also calculate the distribution of random cover (RC) times P(T RC,k = t), which is the probability that at time t an RW will complete visiting all the nodes in a subgraph of k randomly pre-selected nodes at least once. The analytical results for the distributions of cover times are found to be in very good agreement with the results obtained from computer simulations.

Recent results establish for 2-spin antiferromagnetic systems that the computational complexity of approximating the partition function on graphs of maximum degree D undergoes a phase transition that coincides with the uniqueness phase transition on the infinite D-regular tree. For the ferromagnetic Potts model we investigate whether analogous hardness results hold. Goldberg and Jerrum showed that approximating the partition function of the ferromagnetic Potts model is at least as hard as approximating the number of independent sets in bipartite graphs (#BIS-hardness). We improve this hardness result by establishing it for bipartite graphs of maximum degree D. We first present a detailed picture for the phase diagram for the infinite D-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and ordered phases. We then prove for all temperatures below this critical temperature that it is #BIS-hard to approximate the partition function on bipartite graphs of maximum degree D. As a corollary, it is #BIS-hard to approximate the number of k-colorings on bipartite graphs of maximum degree D when k <= D/(2 ln D). The #BIS-hardness result for the ferromagnetic Potts model uses random bipartite regular graphs as a gadget in the reduction. The analysis of these random graphs relies on recent connections between the maxima of the expectation of their partition function, attractive fixpoints of the associated tree recursions, and induced matrix norms. We extend these connections to random regular graphs for all ferromagnetic models and establish the Bethe prediction for every ferromagnetic spin system on random regular graphs. We also prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm is torpidly mixing on random D-regular graphs at the critical temperature for large q.

We present a new approach to showing that random graphs are nearly optimal expanders. This approach is based on recent deep results in combinatorial group theory. It applies to both regular and irregular random graphs. Let G be a random d-regular graph on n vertices, and let \lambda be the largest absolute value of a non-trivial eigenvalue of its adjacency matrix. It was conjectured by Alon [86'] that a random d-regular graph is almost Ramanujan, in the following sense: for every e>0, \lambda<2\sqrt{d-1} + e asymptotically almost surely. Friedman famously presented a proof of this conjecture in [08']. Here we suggest a new, substantially simpler proof of a nearly-optimal result: we show that a random d-regular graph satisfies \lambda < 2\sqrt{d-1} + 1 a.a.s. A main advantage of our approach is that it is applicable to a generalized conjecture: For d even, a d-regular graph on n vertices is an n-covering space of a bouquet of d/2 loops. More generally, fixing an arbitrary base graph H, we study the spectrum of G, a random n-covering of H. Let \lambda be the largest absolute value of a non-trivial eigenvalue of G. Extending Alon's conjecture to this more general model, Friedman [03'] conjectured that for every e>0, a.a.s. \lambda < \rho+e, where \rho is the spectral radius of the universal cover of H. When H is regular we get a bound of \rho+0.84, and for an arbitrary H, we prove a nearly optimal upper bound of \sqrt{3}\rho. This is a substantial improvement upon all known results (by Friedman, Linial-Puder, Lubetzky-Sudakov-Vu and Addario-Berry-Griffiths).

Several popular problems in information networking and social networking such as search, distribution, and dissemination can be modeled by random walk of single or multiple agents on a graph. As an approach to remedy the redundancy in multiple random walks, Branching Random Walk (BRW) has been proposed, in which the number of agents of random walks is dynamically changed by the branching-and-aggregation mechanism. An agent starting its walk from an originating node randomly selects b nodes among its neighbor nodes, and it activates those neighbor nodes (branching). If multiple agents simultaneously visit the same node, those agents are unified into a single agent (aggregation). BRW is a relatively new mobility model on a graph. Hence, to the best of our knowledge, understanding of its dynamical properties is quite limited. In this paper, we describe the dynamics of BRW on a random regular graph, and clarify the relation between the branching parameter b and the mean and the distribution of hitting times and the mean cover time.

Polysaccharides, such as cellulose, are often processed by dissolution in solvent mixtures, e.g. an ionic liquid (IL) combined with a dipolar aprotic co-solvent (CS) that the polymer does not dissolve in. A multi-walker, discrete-time, discrete-space 1-dimensional random walk can be applied to model solvation of a polymer in a multi-component solvent mixture. The number of IL pairs in a solvent mixture and the number of solvent shells formable, x, is associated with n, the model time-step, and N, the number of random walkers. The mean number of distinct sites visited is proportional to the amount of polymer soluble in a solution. By also fitting a polynomial regression model to the data, we can associate the random walk terms with chemical interactions between components and probe where the system deviates from a 1-D random walk. The 'frustration' between solvents shells is given as ln x in the random walk model and as a negative IL:IL interaction term in the regression model. This frustration appears in regime II of the random walk model (high volume fractions of IL) where walkers interfere with each other, and the system tends to its limiting behaviour. In the low concentration regime, (regime I) the solvent shells do not interact, and the system depends only on IL and CS terms. In both models (and both regimes), the system is almost entirely controlled by the volume available to solvation shells, and thus is a counting/space-filling problem, where the molar volume of the CS is important. Small deviations are observed when there is an IL-CS interaction. The use of two models, built on separate approaches, confirm these findings, demonstrating that this is a real effect and offering a route to identifying such systems. Specifically, the majority of CSs - such as dimethylformide - follow the random walk model, whilst 1-methylimidazole, dimethyl sulfoxide, 1,3-dimethyl-2-imidazolidinone and tetramethylurea offer a CS-mediated improvement and propylene carbonate results in a CS-mediated hindrance. It is shown here that systems, which are very complex at a molecular level, may, nonetheless, be effectively modelled as a simple random walk in phase-space. The 1-D random walk model allows prediction of the ability of solvent mixtures to dissolve cellulose based on only two dissolution measurements (one in neat IL) and molar volume.

OF THE DISSERTATION Random matrices and random boxes by Linh V. Tran Dissertation Director: Prof. Van Vu This thesis concerns two questions on random structures: the semi-circular law for adjacency matrix of regular random graph and the piercing number for random boxes. • Random matrices We proved in full generality the semi-circular law for random d-regular graph model in the case d tends to infinity as n does. Our result complements the McKay law [19], which applied for the case d is an absolute constant. • Random boxes. Take n random boxes with axis-parallel edges inside the unit cube [0, 1]d, the piercing number τd(n) is the minimum number of points needed to pierce all boxes. Using hypergraph setting, we was able to prove a near sharp estimation for τd(n): for any d ≥ 2 Ωd( √ n(log n)d/2−1) = τd(n) = Od( √ n(log n)d/2−1 log log n). This thesis is based on two papers by the author [31] and [30] (joint work with Van Vu and Ke Wang).

We study the robustness under perturbations of mixing times, by studying mixing times of random walks in percolation clusters inside boxes in Z d . We show that for d≥2 and p>p c (Z d ), the mixing time of simple random walk on the largest cluster inside is Θ(n 2 ) – thus the mixing time is robust up to a constant factor. The mixing time bound utilizes the Lovasz-Kannan average conductance method. This is the first non-trivial application of this method which yields a tight result.

Mathematical modelling of the movement of animals, micro-organisms and cells is of great relevance in the fields of biology, ecology and medicine. Movement models can take many different forms, but the most widely used are based on the extensions of simple random walk processes. In this review paper, our aim is twofold: to introduce the mathematics behind random walks in a straightforward manner and to explain how such models can be used to aid our understanding of biological processes. We introduce the mathematical theory behind the simple random walk and explain how this relates to Brownian motion and diffusive processes in general. We demonstrate how these simple models can be extended to include drift and waiting times or be used to calculate first passage times. We discuss biased random walks and show how hyperbolic models can be used to generate correlated random walks. We cover two main applications of the random walk model. Firstly, we review models and results relating to the movement, dispersal and population redistribution of animals and micro-organisms. This includes direct calculation of mean squared displacement, mean dispersal distance, tortuosity measures, as well as possible limitations of these model approaches. Secondly, oriented movement and chemotaxis models are reviewed. General hyperbolic models based on the linear transport equation are introduced and we show how a reinforced random walk can be used to model movement where the individual changes its environment. We discuss the applications of these models in the context of cell migration leading to blood vessel growth (angiogenesis). Finally, we discuss how the various random walk models and approaches are related and the connections that underpin many of the key processes involved.

Let Gn,d be a random d-regular graph with n vertices, where d = o(n). Given a fixed graph H, YH denotes the number of induced copies of H in Gn,d. In this paper, the authors determine the threshold of the event “YH > 0”, and also obtain the induced subgraph counts inside the threshold interval.

Simple random walk and Stirling's formula Simple ramdon walk in many dimensions Self-avoiding walk Brownian motion Shuffling and random permutations Seven shuffles are enough (sort of) Markov chains on finite sets Markov chain Monte Carlo Random walks and electrical networks Uniform spanning trees Random walk simulations Other simulations Simulations in finance Problems Bibliography.

Let G be a given graph (modelling a communication network) which we assume suffers from static edge faults: That is we let each edge of G be present independently with probability p (or absent with fault probability f = 1 - p). In particular we are interested in robustness results for the case that the graph G itself is a random member of the class of all regular graphs with given degree. Our result is: If the degree d is fixed then p = 1/(d - 1) is a threshold probability for the existence of a linear-sized component in a faulty version of almost all random regular graphs. We show: If each edge of an arbitrary graph G with maximum degree bounded above by d is present with, probability p = Λ/(d - 1) where Λ 1 then for almost all G the faulted version of G has a linear-sized component with high probability. Note that these results imply some kind of optimality of random regular graphs among the class of graphs with the same degree bound. The theme is: Use the known expansion properties of almost all random regular graphs to obtain strong robustness results. This has not been done systematically before.

We study the neighborhoods of a typical point $$Z_n$$ visited at n-th step of a random walk, determined by the condition that the transition probabilities stay close to $$\mu ^{*n}(Z_n)$$ . If such neighborhood contains a ball of radius $$C \sqrt{n}$$ , we say that the random walk has almost invariant transition probabilities. We prove that simple random walks on wreath products of $$\mathbb {Z}$$ with finite groups have almost invariant distributions. A weaker version of almost invariance implies a necessary condition of Ozawa’s criterion for the property $$H_\mathrm{FD}$$ . We define and study the radius of almost invariance. We estimate this radius for random walks on iterated wreath products and show this radius can be asymptotically strictly smaller than n / L(n), where L(n) denotes the drift function of the random walk. We show that the radius of individual almost invariance of a simple random walk on the wreath product of $$\mathbb {Z}^2$$ with a finite group is asymptotically strictly larger than n / L(n). Finally, we show the existence of groups such that the radius of almost invariance is smaller than a given function, but remains unbounded. We also discuss possible limiting distribution of ratios of transition probabilities on non almost invariant scales.

We establish conditions on sequences of graphs which ensure that the mixing times of the random walks on the graphs in the sequence converge. The main assumption is that the graphs, associated measures and heat kernels converge in a suitable Gromov-Hausdorff sense. With this result we are able to establish the convergence of the mixing times on the largest component of the Erdős-Renyi random graph in the critical window, sharpening previous results for this random graph model. Our results also enable us to establish convergence in a number of other examples, such as finitely ramified fractal graphs, Galton-Watson trees and the range of a high-dimensional random walk.

Une marche aleatoire simple $\mathcal{W}$ sur $\mathbb{Z}^{2}$ induit pour chaque sous-graphe couvrant $G\subset \mathbb{Z}^{2}$ une marche $\mathcal{W}_{G}$ : notamment, si on regarde $\mathcal{W}$ comme un mot infini uniforme sur l’alphabet $\{\mathbf{x},-\mathbf{x},\mathbf{y},-\mathbf{y}\}$, alors la marche $\mathcal{W}_{G}$ commence a l’origine et suit les directions definis par $\mathcal{W}$, en acceptant seulement les pas de $\mathcal{W}$ le long desquels $\mathcal{W}_{G}$ reste sur $G$. Pour chaque $G\subset \mathbb{Z}^{2}$ fixe, la marche $\mathcal{W}_{G}$ a la loi d’une marche aleatoire simple sur $G$ et alors $\mathcal{W}_{G}$ est presque surement recurrent dans le sens que $\mathcal{W}_{G}$ visite chaque sommet de $G$ connexe a l’origine un nombre infini de fois. Alors, une question naturelle surgit : est-ce que presque surement pour $\mathcal{W}$, les marches $\mathcal{W}_{G}$ sont recurrents pour tous $G\subset \mathbb{Z}^{2}$ a la fois? Dans cet article, on repond a cette question d’une maniere negative, en montrent l’existence des graphes exceptionnels sur lesquels la marche est transitoire. En fait, on montre que meme si on considere un nombre denombrable des marches aleatoires simples independantes, alors des graphes exceptionnels existent. D’autre cote, on montre que il n’existe pas des graphes exceptionnels pour la marche aleatoire branchante.

Benjamini, Lyons and Schramm (1999) considered properties of an infinite graph $G$, and the simple random walk on it, that are preserved by random perturbations. To address problems raised by those authors, we study simple random walk on the infinite percolation cluster in Cayley graphs of certain amenable groups known as "lamplighter groups''.We prove that zero speed for random walk on a lamplighter group implies zero speed for random walk on an infinite cluster, for any supercritical percolation parameter $p$. For $p$ large enough, we also establish the converse. We prove that if $G$ has a positive anchored expansion constant then so does every infinite cluster of independent percolation with parameter $p$ sufficiently close to 1; We also show that positivity of the anchored expansion constant is preserved under a random stretch if, and only if, the stretching law has an exponential tail.