On coalescence time in graphs: When is coalescing as fast as meeting?

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 results in tight bounds for various topologies including the aforementioned ones. The meeting time is defined as the worst-case expected time required for two random walks to arrive at the same node at the same time. As a general result, we establish that for graphs whose meeting time is only marginally larger than the mixing time (a factor of log^2 n), the coalescence time of n random walks equals the meeting time up to constant factors. This upper bound is complemented by the construction of a graph family demonstrating that this result is the best possible up to constant factors. For almost-regular graphs, we bound the coalescence time by the hitting time, resolving the discrete-time variant of a conjecture by Aldous for this class of graphs. Finally, we prove that for any graph the coalescence time is bounded by O(n^3) (which is tight for the Barbell graph); surprisingly even such a basic question about the coalescing time was not answered before this work. By duality, our results give bounds on the voter model and therefore give bounds on the consensus time in arbitrary undirected graphs. We also establish a new bound on the hitting time and cover time of regular graphs, improving and tightening previous results by Broder and Karlin, as well as those by Aldous and Fill

On coalescence time in graphs: When is coalescing as fast as meeting?

Coalescing random walks is a fundamental stochastic process, where a set of particles perform independent discretetime 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 such as by Cooper, Elsasser, Ono, Radzik [13] and Cooper, Frieze and Radzik [12], 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 results in tight bounds for various topologies including the aforementioned ones. The meeting time is defined as the worst-case expected time required for two random walks to arrive at the same node at the same time. As a general result, we establish that for graphs whose meeting time is only marginally larger than the mixing time (a factor of log2 n), the coalescence time of n random walks equals the meeting time up to constant factors. This upper bound is complemented by the construction of a graph family demonstrating that this result is the best possible up to constant factors. For almostregular graphs, we bound the coalescence time by the hitting time, resolving the discrete-time variant of a conjecture by Aldous for this class of graphs. Finally, we prove that for any graph the coalescence time is bounded by O(n3) (which is tight for the Barbell graph); surprisingly even such a basic question about the coalescing time was not answered before this work. By duality, our results give bounds on the voter model and therefore give bounds on the consensus time in arbitrary undirected graphs. We also establish a new bound on the hitting time and cover time of regular graphs, improving and tightening previous results by Broder and Karlin [10], as well as those by Aldous and Fill [1].</p

Brief Announcement: How large is your graph?

We consider the problem of estimating the graph size, where one is given only local access to the graph. We formally define a query model in which one starts with a seed node and is allowed to make queries about neighbours of nodes that have already been seen. In the case of undirected graphs, an estimator of Katzir et al. (2014) based on a sample from the stationary distribution π uses O 1 kπk2 +davg queries; we prove that this is tight. In addition, we establish this as a lower bound even when the algorithm is allowed to crawl the graph arbitrarily; the results of Katzir et al. give an upper bound that is worse by a multiplicative factor tmix·log(n). The picture becomes significantly different in the case of directed graphs. We show that without strong assumptions on the graph structure, the number of nodes cannot be predicted to within a constant multiplicative factor without using a number of queries that are at least linear in the number of nodes; in particular, rapid mixing and small diameter, properties that most real-world networks exhibit, do not suffice. The question of interest is whether any algorithm can beat breadth-first search. We introduce a new parameter, generalising the well-studied conductance, such that if a suitable bound on it exists and is known to the algorithm, the number of queries required is sublinear in the number of edges; we show that this is tight

How large is your graph?

We consider the problem of estimating the graph size, where one is given only local access to the graph. We formally define a query model in which one starts with a seed node and is allowed to make queries about neighbours of nodes that have already been seen. In the case of undirected graphs, an estimator of Katzir et al. (2014) based on a sample from the stationary distribution π uses O 1 kπk2 +davg queries; we prove that this is tight. In addition, we establish this as a lower bound even when the algorithm is allowed to crawl the graph arbitrarily; the results of Katzir et al. give an upper bound that is worse by a multiplicative factor tmix·log(n). The picture becomes significantly different in the case of directed graphs. We show that without strong assumptions on the graph structure, the number of nodes cannot be predicted to within a constant multiplicative factor without using a number of queries that are at least linear in the number of nodes; in particular, rapid mixing and small diameter, properties that most real-world networks exhibit, do not suffice. The question of interest is whether any algorithm can beat breadth-first search. We introduce a new parameter, generalising the well-studied conductance, such that if a suitable bound on it exists and is known to the algorithm, the number of queries required is sublinear in the number of edges; we show that this is tight

Clustering redemption–beyond the impossibility of Kleinberg’s axioms

Kleinberg (2002) stated three axioms that any clustering procedure should satisfy and showed there is no clustering procedure that simultaneously satisfies all three. One of these, called the consistency axiom, requires that when the data is modified in a helpful way, i.e. if points in the same cluster are made more similar and those in different ones made less similar, the algorithm should output the same clustering. To circumvent this impossibility result, research has focused on considering clustering procedures that have a clustering quality measure (or a cost) and showing that a modification of Kleinberg’s axioms that takes cost into account lead to feasible clustering procedures. In this work, we take a different approach, based on the observation that the consistency axiom fails to be satisfied when the “correct” number of clusters changes. We modify this axiom by making use of cost functions to determine the correct number of clusters, and require that consistency holds only if the number of clusters remains unchanged. We show that single linkage satisfies the modified axioms, and if the input is well-clusterable, some popular procedures such as k-means also satisfy the axioms, taking a step towards explaining the success of these objective functions for guiding the design of algorithms

Hierarchical clustering beyond the worst-case

Hiererachical clustering, that is computing a recursive partitioning of a dataset to obtain clusters at increasingly finer granularity is a fundamental problem in data analysis. Although hierarchical clustering has mostly been studied through procedures such as linkage algorithms, or top-down heuristics, rather than as optimization problems, recently Dasgupta proposed an objective function for hierarchical clustering and initiated a line of work developing algorithms that explicitly optimize an objective. In this paper, we consider a fairly general random graph model for hierarchical clustering, called the hierarchical stochastic block model (HSBM), and show that in certain regimes the SVD approach of McSherry combined with specific linkage methods results in a clustering that give an Op1q approximation to Dasgupta’s cost function. We also show that an approach based on SDP relaxations for balanced cuts based on the work of Makarychev et al., combined with the recursive sparsest cut algorithm of Dasgupta, yields an Op1q approximation in slightly larger regimes and also in the semi-random setting, where an adversary may remove edges from the random graph generated according to an HSBM. Finally, we report empirical evaluation on synthetic and real-world data showing that our proposed SVD-based method does indeed achieve a better cost than other widely-used heurstics and also results in a better classification accuracy when the underlying problem was that of multi-class classification

Hierarchical clustering beyond the worst-case

Hiererachical clustering, that is computing a recursive partitioning of a dataset to obtain clusters at increasingly finer granularity is a fundamental problem in data analysis. Although hierarchical clustering has mostly been studied through procedures such as linkage algorithms, or top-down heuristics, rather than as optimization problems, recently Dasgupta proposed an objective function for hierarchical clustering and initiated a line of work developing algorithms that explicitly optimize an objective. In this paper, we consider a fairly general random graph model for hierarchical clustering, called the hierarchical stochastic block model (HSBM), and show that in certain regimes the SVD approach of McSherry combined with specific linkage methods results in a clustering that give an Op1q approximation to Dasgupta’s cost function. We also show that an approach based on SDP relaxations for balanced cuts based on the work of Makarychev et al., combined with the recursive sparsest cut algorithm of Dasgupta, yields an Op1q approximation in slightly larger regimes and also in the semi-random setting, where an adversary may remove edges from the random graph generated according to an HSBM. Finally, we report empirical evaluation on synthetic and real-world data showing that our proposed SVD-based method does indeed achieve a better cost than other widely-used heurstics and also results in a better classification accuracy when the underlying problem was that of multi-class classification

Books Received

Hierarchical clustering is a recursive partitioning of a dataset into clusters at an increasingly finer granularity. Motivated by the fact that most work on hierarchical clustering was based on providing algorithms, rather than optimizing a specific objective, Dasgupta (2016) framed similarity-based hierarchical clustering as a combinatorial optimization problem, where a `good' hierarchical clustering is one that minimizes some cost function. He showed that this cost function has certain desirable properties, such as in order to achieve optimal cost disconnected components must be separated first and that in `structureless' graphs, i.e., cliques, all clusterings achieve the same cost. We take an axiomatic approach to defining `good' objective functions for both similarity and dissimilarity-based hierarchical clustering. We characterize a set of admissible objective functions (that includes the one introduced by Dasgupta) that have the property that when the input admits a `natural' ground-truth hierarchical clustering, the ground-truth clustering has an optimal value. Equipped with a suitable objective function, we analyze the performance of practical algorithms, as well as develop better and faster algorithms for hierarchical clustering. For similarity-based hierarchical clustering, Dasgupta (2016) showed that a simple recursive sparsest-cut based approach achieves an O(log^3/2 n)-approximation on worst-case inputs. We give a more refined analysis of the algorithm and show that it in fact achieves an O(√log n)-approximation. This improves upon the LP-based O(log n)-approximation of Roy and Pokutta (2016). For dissimilarity-based hierarchical clustering, we show that the classic average-linkage algorithm gives a factor 2 approximation, and provide a simple and better algorithm that gives a factor 3/2 approximation. This aims at explaining the success of this heuristics in practice. Finally, we consider `beyond-worst-case' scenario through a generalisation of the stochastic block model for hierarchical clustering. We show that Dasgupta's cost function also has desirable properties for these inputs and we provide a simple algorithm that for graphs generated according to this model yields a 1 + o(1) factor approximation.</p