Random walks and the effective resistance of networks

In this article we present an interpretation ofeffective resistance in electrical networks in terms of random walks on underlying graphs. Using this characterization we provide simple and elegant proofs for some known results in random walks and electrical networks. We also interpret the Reciprocity theorem of electrical networks in terms of traversals in random walks. The byproducts are (a) precise version of thetriangle inequality for effective resistances, and (b) an exact formula for the expectedone-way transit time between vertices.     

Random walks on highly symmetric graphs

We consider uniform random walks on finite graphs withn nodes. When the hitting times are symmetric, the expected covering time is at least 1/2n logn-O(n log logn) uniformly over all such graphs. We also obtain bounds for the covering times in terms of the eigenvalues of the transition matrix of the Markov chain. For distance-regular graphs, a general lower bound of (n-1) logn is obtained. For hypercubes and binomial coefficient graphs, the limit law of the covering time is obtained as well.     

Recurrent networks and a theorem of Nash-Williams

In this paper, we generalize a result of Nash-Williams concerning recurrence of locally finite networks, by extending his result to networks with possibly vertices of infinite degree.Work completed at the University of Waterloo and at the University of Milan.     

Random walks on random simple graphs

This paper looks at random regular simple graphs and considers nearest neighbor random walks on such graphs. This paper considers walks where the degree d of each vertex is around (log n)^a where a is a constant which is at least 2 and where n is the number of vertices. By extending techniques of Dou, this paper shows that for most such graphs, the position of the random walk becomes close to uniformly distributed after slightly more than log n/log d steps. This paper also gets similar results for the random graph G(n, p), where p = d/(n − 1). © 1996 John Wiley & Sons, Inc.     

Random walks in varying dimensions

We establish recurrence criteria for sums of independent random variables which take values in Euclidean lattices of varying dimension. In particular, we describe transient inhomogeneous random walks in the plane which interlace two symmetric step distributions of bounded support.Research partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.Research supported in part by National Science Foundation Grant No. DMS 9300191, by a Sloan Foundation Fellowship, and by a Presidential Faculty Fellowship.     

On mixing of certain random walks, cutoff phenomenon and sharp threshold of random matroid processes

In this paper we define and analyze convergence of the geometric random walks, which are certain random walks on vector spaces over finite fields. We show that the behavior of such walks is given by certain random matroid processes. In particular, the mixing time is given by the expected stopping time, and the cutoff is equivalent to sharp threshold. We also discuss some random geometric random walks as well as some examples and symmetric cases.     

Random walks systems on complete graphs

We study two versions of random walks systems on complete graphs. In the first one, the random walks have geometrically distributed lifetimes so we define and identify a non-trivial critical parameter related to the proportion of visited vertices before the process dies out. In the second version, the lifetimes depend on the past of the process in a non-Markovian setup. For that version, we present results obtained from computational analysis, simulations and a mean field approximation. These three approaches match.     

On the cover time of random walks on graphs

This article deals with random walks on arbitrary graphs. We consider the cover time of finite graphs. That is, we study the expected time needed for a random walk on a finite graph to visit every vertex at least once. We establish an upper bound ofO(n ²) for the expectation of the cover time for regular (or nearly regular) graphs. We prove a lower bound of (n logn) for the expected cover time for trees. We present examples showing all our bounds to be tight.Mike Saks was supported by NSF-DMS87-03541 and by AFOSR-0271. Jeff Kahn was supported by MCS-83-01867 and by AFOSR-0271.     

Random walks in two-dimensional complexes

A 2-dimensional complex is a union of a finite number of quarter planes ₊ ² having some boundaries in common. The most interesting example is the union of all 2-dimensional faces of ₊ ^N . We consider maximally homogeneous random walks on such complexes and obtain necessary and sufficient conditions for ergodicity, null recurrence and transience up to some non-zero assumptions which are of measure 1 in the parameter space.The problem we address in this paper is of theoretical range. However, the results can be applied to performance evaluation of some telecommunication systems (e.g. local area networks) viewed as interacting queues. To enforce this assertion, a detailed example of coupled queues in differentregimes is presented.     

Agent-based randomized broadcasting in large networks

Mobile agents are software abstractions that can migrate across the links of a network. They naturally extend the object oriented program style and nicely correspond to agents as examined in game theory. In this paper, we introduce a simple, robust, and efficient randomized broadcast protocol within this mobile agent programming paradigm. We show that by using this scheme, broadcasting enquiries in a random graph of certain density O(lnn) steps, where n denotes the number of nodes in the graph. Then, we consider bounded degree graphs and prove that we are able to distribute an information among all nodes in O(D) steps, where D denotes the diameter of the graph. We also show that, in contrast to traditional randomized broadcasting (TRB), graphs exist in which agent-based randomized broadcasting requires Ω(n²) steps. On the other hand, some graphs which require Ω(nlnn) steps to spread the information in the traditional broadcast model, allow very fast agent-based broadcasting. It should be noted that the previously mentioned results are guaranteed with probability 1-o(1/n).     

Counting trees with random walks

We give a simple proof of Tutte’s matrix-tree theorem, a well-known result providing a closed-form expression for the number of rooted spanning trees in a directed graph. Our proof stems from placing a random walk on a directed graph and then applying the Markov chain tree theorem to count trees. The connection between the two theorems is not new, but it appears that only one direction of the formal equivalence between them is readily available in the literature. The proof we now provide establishes the other direction. More generally, our approach is another example showing that random walks can serve as a powerful glue between graph theory and Markov chain theory, allowing formal statements from one side to be carried over to the other.     

Random coloring evolution on graphs

In this short note we study how two colors, red and blue, painted on a given graph are evolved randomly according to a transition rule which aims to simulate how people influence each other. We shall also calculate the probability that the evolution will be trapped eventually using the martingale method.     

Random walks on graphs with interval weights and precise marginals

We propose a model of random walks on weighted graphs where the weights are interval valued, and connect it to reversible imprecise Markov chains. While the theory of imprecise Markov chains is now well established, this is a first attempt to model reversible chains. In contrast with the existing theory, the probability models that have to be considered are now non-convex. This presents a difficulty in computational sense, since convexity is critical for the existence of efficient optimization algorithms used in the existing models. The second part of the paper therefore addresses the computational issues of the model. The goal is finding sets of weights which maximize or minimize expectations corresponding to multiple steps transition probabilities. In particular, we present a local optimization algorithm and numerically test its efficiency. We show that its application allows finding close approximations of the globally best solutions in reasonable time.     

关于随机环境中的马尔可夫过程的简介——纪念李国平院士吴新谋教授诞辰100周年

该文系统地介绍随机环境中的马尔可夫过程. 共4章, 第一章介绍依时的随机环境中的马尔可夫链(MCTRE), 包括MCTRE的存在性及等价描述; 状态分类; 遍历理论及不变测度; p-θ 链的中心极限定理和不变原理. 第二章介绍依时的随机环境中的马尔可夫过程(MPTRE), 包括MPTRE的基本概念; 随机环境中的q -过程存在唯一性; 时齐的q -过程;MPTRE的构造及等价性定理.第三章介绍依时的随机环境中的分枝链(MBCRE), 包括有限维的和无穷维的MBCRE的模型和基本概念; 它们的灭绝概念;两极分化; 增殖率等.第四章介绍依时依空的随机环境中的马尔可夫链(MCSTRE), 包括MCSTRE的基本概念、构造; 依时依空的随机环境中的随机徘徊(RWSTRE)的中心极限定理、不变原理.     

A maximum principle for node voltages in finitely structured,transfinite, electrical networks

Transfinite electrical networks have unique finite-powered voltage-current regimes given in terms of branch voltages and branch currents, but they do not in general possess unique node voltages. However, if their structures are sufficiently restricted, those node voltages will exist and will satisfy a maximum principle much like that which holds for ordinary infinite electrical networks. The structure that is imposed in order to establish these results generalized the idea of local-finiteness. Other properties that do not hold in general for transfinite networks but do hold under the imposed structure are Kirchhoff's current laws for nodes of any ranks and the permissibility of connecting pure voltage sources to such nodes. This work lays the foundation for a theory of transfinite random walks, which will be the subject of a subsequent work.This work was supported by the National Science Foundation under the grants DMS-9200738 and MIP-9200748.     

Random walks on graphs with a strong isoperimetric property

A random walk on a graph is a Markov chain whose state space consists of the vertices of the graph and where transitions are only allowed along the edges. We study (strongly) reversible random walks and characterize the class of graphs where then-step transition probabilities tend to zero exponentially fast (geometric ergodicity). These characterizations deal with an isoperimetric property, norm inequalities for certain associated operators, and eigenvalues of the Laplace operator. There is some (strong) similarity with the theory of (non)amenable groups.     

Random walks on simplicial complexes and harmonics

In this paper, we introduce a class of random walks with absorbing states on simplicial complexes. Given a simplicial complex of dimension d, a random walk with an absorbing state is defined which relates to the spectrum of the k‐dimensional Laplacian for 1 ≤ k ≤ d. We study an example of random walks on simplicial complexes in the context of a semi‐supervised learning problem. Specifically, we consider a label propagation algorithm on oriented edges, which applies to a generalization of the partially labelled classification problem on graphs. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 379–405, 2016     

Extremal first passage times for trees

For random walks associated with trees with probability zero of staying at any vertex, we develop explicit graph theoretic formulas for the mean first passage times between states, we give lower and upper bounds for the entries of the mean first passage matrix E, and we characterize the cases of equality in these bounds. We also consider the variance of the first return time to a state and we find those trees which maximize the variance and those trees which minimize the variance. As may be expected, the trees which provide extremal behavior are given by paths and stars.     

随机环境中两性分枝过程的马氏性和对偶性

证明了独立同分布环境中的两性分枝过程是时奇的马氏链,给出了过程灭绝一爆炸这一对偶性的一个新的证明。在随机环境情形下,证明了一类单调函数的存在性。     

Random walks,totally unimodular matrices,and a randomised dual simplex algorithm

We discuss the application of random walks to generating a random basis of a totally unimodular matrix and to solving a linear program with such a constraint matrix. We also derive polynomial upper bounds on the combinatorial diameter of an associated polyhedron.Supported by NATO grant RG0088/89.Corresponding author. Supported by NSF grants CCR-8900112, CCR-9024935 and NATO grant RG0088/89.