Researchers have designed many algorithms to measure the distances between graph nodes, such as average hitting times of random walks, cosine distances from DeepWalk, personalized PageRank, etc. Successful although these algorithms are, still they are either underperforming or too time-consuming to be applicable to huge graphs that we encounter daily in this big data era. To address these issues, here we propose a faster algorithm based on an improved version of random walks that can beat DeepWalk results with more than ten times acceleration. The reason for this significant acceleration is that we can derive an analytical formula to calculate the expected hitting times of this random walk quickly. There is only one parameter (the power expansion order) in our algorithm, and the results are robust with respect to its changes. Therefore, we can directly find the optimal solution without fine-tuning of model parameters. Our method can be widely used for fraud detection, targeted ads, recommendation systems, topic-sensitive search, etc.
We consider a graph observability problem: how many edge colors are needed for an unlabeled graph so that an agent, walking from node to node, can uniquely determine its location from just the observed color sequence of the walk? Specifically, let G(n,d) be an edge-colored subgraph of d-dimensional (directed or undirected) lattice of size n^d = n * n * ... * n. We say that G(n,d) is t-observable if an agent can uniquely determine its current position in the graph from the color sequence of any t-dimensional walk, where the dimension is the number of different directions spanned by the edges of…
We present a new algorithm for estimating the Personalized PageRank (PPR) between a source and target node on undirected graphs, with sublinear running-time guarantees over the worst-case choice of source and target nodes. Our work builds on a recent line of work on bidirectional estimators for PPR, which obtained sublinear running-time guarantees but in an average-case sense, for a uniformly random choice of target node. Crucially, we show how the reversibility of random walks on undirected networks can be exploited to convert average-case to worst-case guarantees. While past bidirectional me…
In this paper we derive an analytical expression for the mean load at each node of an arbitrary undirected graph for the uniform multicommodity flow problem under random walk routing. We show the mean load is linearly dependent on the nodal degree with a common multiplier equal to the sum of the inverses of the non-zero eigenvalue of the graph Laplacian. Even though some aspects of the mean load value, such as linear dependence on the nodal degree, are intuitive and may be derived from the equilibrium distribution of the random walk on the undirected graph, the exact expression for the mean lo…
This paper studies the problem of proper-walk connection number: given an undirected connected graph, our aim is to colour its edges with as few colours as possible so that there exists a properly coloured walk between every pair of vertices of the graph i.e. a walk that does not use consecutively two edges of the same colour. The problem was already solved on several classes of graphs but still open in the general case. We establish that the problem can always be solved in polynomial time in the size of the graph and we provide a characterization of the graphs that can be properly connected w…
In a coalescing random walk, a set of particles make independent random walks on a graph. Whenever one or more particles meet at a vertex, they unite to form a single particle, which then continues the random walk through the graph. Coalescing random walks can be used to achieve consensus in distributed networks, and is the basis of the self-stabilizing mutual exclusion algorithm of Israeli and Jalfon. Let G=(V,E), be an undirected, connected n vertex graph with m edges. Let C(n) be the expected time for all particles to coalesce, when initially one particle is located at each vertex of an n v…
We address the problem of determining a natural local neighbourhood or "cluster" associated to a given seed vertex in an undirected graph. We formulate the task in terms of absorption times of random walks from other vertices to the vertex of interest, and observe that these times are well approximated by the components of the principal eigenvector of the corresponding fundamental matrix of the graph's adjacency matrix. We further present a locally computable gradient-descent method to estimate this Dirichlet-Fiedler vector, based on minimising the respective Rayleigh quotient. Experimental ev…
Coalescing random walks is a fundamental stochastic process, where a set of particles perform independent discrete-time random walks on an undirected graph. Whenever two or more particles meet at a given node, they merge and continue as a single random walk. The coalescence time is defined as the expected time until only one particle remains, starting from one particle at every node. Despite recent progress the coalescence time for graphs such as binary trees, d-dimensional tori, hypercubes and more generally, vertex-transitive graphs, remains unresolved. We provide a powerful toolkit that res…
In this dissertation we demonstrate that the continuous-time quantum walk models remain powerful for nontrivial graph structures. We consider two aspects of this problem. First, it is known that the standard Continuous-Time Quantum Walk (CTQW), proposed by Childs and Goldstone, can propagate quickly on the infinite path graph. However, the Schrödinger equation requires the Hamiltonian to be symmetric, and thus only undirected graphs can be implemented. In this thesis, we address the question, whether it is possible to construct a continuous-time quantum walk on general directed graphs, preserv…
In a model of network communication based on a random walk in an undirected graph, what subset of nodes (subject to constraints on the set size), enables the fastest spread of information? In this paper, we assume the dynamics of spread is described by a network consensus process, but to find the most effective seeds we consider the target set of a random walk--the process dual to network consensus spread. Thus an optimal set $A$ minimizes the sum of the expected first hitting times $F(A)$ of random walks that start at nodes outside the set. We introduce a submodular, non-decreasing rank funct…
We study three mixing properties of a graph: large algebraic connectivity, large Cheeger constant (isoperimetric number) and large spectral gap from 1 for the second largest eigenvalue of the transition probability matrix of the random walk on the graph. We prove equivalence of this properties (in some sense). We give estimates for the probability for a random graph to satisfy these properties. In addition, we present asymptotic formulas for the numbers of Eulerian orientations and Eulerian circuits in an undirected simple graph.
In this paper, we set forth a new algorithm for generating approximately uniformly random spanning trees in undirected graphs. We show how to sample from a distribution that is within a multiplicative $(1+δ)$ of uniform in expected time $\TO(m\sqrt{n}\log 1/δ)$. This improves the sparse graph case of the best previously known worst-case bound of $O(\min \{mn, n^{2.376}\})$, which has stood for twenty years. To achieve this goal, we exploit the connection between random walks on graphs and electrical networks, and we use this to introduce a new approach to the problem that integrates discrete r…
The famous Chinese Postman Problem (CPP) is polynomial time solvable on both undirected and directed graphs. Gutin et al. [Discrete Applied Math 217 (2016)] generalized these results by proving that CPP on $c$-edge-colored graphs is polynomial time solvable for every $c\geq 2$. In CPP on weighted edge-colored graphs $G$, we wish to find a minimum weight properly colored closed walk containing all edges of $G$ (a walk is properly colored if every two consecutive edges are of different color, including the last and first edges in a closed walk). In this paper, we consider CPP on arc-colored digr…
We give an $m^{1+o(1)}β^{o(1)}$-time algorithm for generating a uniformly random spanning tree in an undirected, weighted graph with max-to-min weight ratio $β$. We also give an $m^{1+o(1)}ε^{-o(1)}$-time algorithm for generating a random spanning tree with total variation distance $ε$ from the true uniform distribution. Our second algorithm's runtime does not depend on the edge weights. Our $m^{1+o(1)}β^{o(1)}$-time algorithm is the first almost-linear time algorithm for the problem --- even on unweighted graphs --- and is the first subquadratic time algorithm for sparse weighted graphs. Our …
Let G be an edge weighted undirected graph. For every pair of nodes consider the shortest cycle containing these nodes in G. The cycle diameter of G is the maximum length of a cycle in this set. Let H be a directed graph obtained by directing the edges of G. The cycle diameter of H is similarly defined except for that cycles are replaced by directed closed walks. Is there always an orientation H of G whose cycle diameter is bounded by a constant times the cycle diameter of G? We prove this property for planar graphs. These results have implications on the problem of approximating an orientatio…
We consider a fundamental algorithmic question in spectral graph theory: Compute a spectral sparsifier of random-walk matrix-polynomial $$L_α(G)=D-\sum_{r=1}^dα_rD(D^{-1}A)^r$$ where $A$ is the adjacency matrix of a weighted, undirected graph, $D$ is the diagonal matrix of weighted degrees, and $α=(α_1...α_d)$ are nonnegative coefficients with $\sum_{r=1}^dα_r=1$. Recall that $D^{-1}A$ is the transition matrix of random walks on the graph. The sparsification of $L_α(G)$ appears to be algorithmically challenging as the matrix power $(D^{-1}A)^r$ is defined by all paths of length $r$, whose prec…
Cheeger's inequality shows that any undirected graph $G$ with minimum nonzero normalized Laplacian eigenvalue $λ_G$ has a cut with conductance at most $O(\sqrt{λ_G})$. Qualitatively, Cheeger's inequality says that if the relaxation time of a graph is high, there is a cut that certifies this. However, there is a gap in this relationship, as cuts can have conductance as low as $Θ(λ_G)$. To better approximate the relaxation time of a graph, we consider a more general object. Instead of bounding the mixing time with cuts, we bound it with cuts in graphs obtained by Schur complementing out vertices…
We present a new algorithm for generating a uniformly random spanning tree in an undirected graph. Our algorithm samples such a tree in expected $\tilde{O}(m^{4/3})$ time. This improves over the best previously known bound of $\min(\tilde{O}(m\sqrt{n}),O(n^ω))$ -- that follows from the work of Kelner and Mądry [FOCS'09] and of Colbourn et al. [J. Algorithms'96] -- whenever the input graph is sufficiently sparse. At a high level, our result stems from carefully exploiting the interplay of random spanning trees, random walks, and the notion of effective resistance, as well as from devising a way…
