A family of dissimilarity measures between nodes generalizing both the shortest-path and the commute-time distances

Abstract

This work introduces a new family of link-based dissimilarity measures between nodes of a weighted directed graph. This measure, called the randomized shortest-path (RSP) dissimilarity, depends on a parameter θ and has the interesting property of reducing, on one end, to the standard shortest-path distance when θ is large and, on the other end, to the commute-time (or resistance) distance when θ is small (near zero). Intuitively, it corresponds to the expected cost incurred by a random walker in order to reach a destination node from a starting node while maintaining a constant entropy (related to θ) spread in the graph. The parameter θ is therefore biasing gradually the simple random walk on the graph towards the shortest-path policy. By adopting a statistical physics approach and computing a sum over all the possible paths (discrete path integral), it is shown that the RSP dissimilarity from every node to a particular node of interest can be computed efficiently by solving two linear systems of n equations, where n is the number of nodes. On the other hand, the dissimilarity between every couple of nodes is obtained by inverting an n × n matrix. The proposed measure can be used for various graph mining tasks such as computing betweenness centrality, finding dense communities, etc, as shown in the experimental section.

Interpolating between Random Walks and Shortest Paths: A Path Functional Approach

“…The construction is controlled by a temperature T and applies to any network endowed with a Markov transition matrix W and a resistance matrix R. The two matrices can be related, typically as (componentwise) inverses of each other (e.g. Yen et al 2008) or not, in which case continuity at T = 0 and T = ∞ however requires w ij > 0 whenever r ij < ∞.…”

“…Fouss et al 2007, and references therein; see also Yen et al (2008) and Chebotarev (2010) for further studies on resistance and shortest-path distances.…”

“…The conflict is arbitrered by a continuous parameter T ≥ 0, the temperature (or its inverse β := 1/T ), and results in an analytically solvable unique optimum continuously interpolating between shortest-paths and random walks. See Yen et al (2008) and Saerens and al. (2009) for a close proposal, yet distinct in its implementation.…”

K-Nearest Neighbor Search and Outlier Detection via Minimax Distances

“…The route-specific distance between objects i and j is sum of the edge weights on this route [19]. Their link-based distance is then obtained by summing up the route-specific measures of all different routes between them.…”

“…Such a distance measure is known to better capture the arbitrarily shaped structures compared to the basic measures such as Euclidean or Mahalanobis distances. Link-based measures are often obtained by inverting the Laplacian of the distance matrix, done in the context of regularized Laplacian kernel and Markov diffusion kernel [19,6]. However, computing all-pair link-based distances requires inverting a N × N matrix, which yields O(N 3 ) running time; thereby it is not applicable to large-scale datasets.…”