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

Bavaud, François; Guex, Guillaume

doi:10.1007/978-3-642-35386-4_6

Cited by 18 publications

(30 citation statements)

References 20 publications

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“…getting killed. Another way of interpreting this is by imagining an additional, absorbing "cemetery" node [28], where the walker can end up from each node of the graph with a non-zero probability.…”

Section: Algorithm For Faster Computation Of ∆ Rspmentioning

confidence: 99%

“…Then, the stationary distribution of the defined Markov chain is the free-energy rank score. In [28], the authors compute edge flows minimizing the free energy between two nodes. The resulting flows define some new edge and node betweenness measures, balancing exploration and exploitation through an adjustable temperature parameter.…”

Section: A New Generalized Distance Based On Helmholtz Free Energymentioning

confidence: 99%

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Developments in the theory of randomized shortest paths with a comparison of graph node distances

Kivimäki

Shimbo

Saerens

2014

Physica A: Statistical Mechanics and its Applications

202

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There have lately been several suggestions for parametrized distances on a graph that generalize the shortest path distance and the commute time or resistance distance. The need for developing such distances has risen from the observation that the above-mentioned common distances in many situations fail to take into account the global structure of the graph. In this article, we develop the theory of one family of graph node distances, known as the randomized shortest path dissimilarity, which has its foundation in statistical physics. We show that the randomized shortest path dissimilarity can be easily computed in closed form for all pairs of nodes of a graph. Moreover, we come up with a new definition of a distance measure that we call the free energy distance. The free energy distance can be seen as an upgrade of the randomized shortest path dissimilarity as it defines a metric, in addition to which it satisfies the graph-geodetic property. The derivation and computation of the free energy distance are also straightforward. We then make a comparison between a set of generalized distances that interpolate between the shortest path distance and the commute time, or resistance distance. This comparison focuses on the applicability of the distances in graph node clustering and classification. The comparison, in general, shows that the parametrized distances perform well in the tasks. In particular, we see that the results obtained with the free energy distance are among the best in all the experiments.

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Section: Algorithm For Faster Computation Of ∆ Rspmentioning

confidence: 99%

Section: A New Generalized Distance Based On Helmholtz Free Energymentioning

confidence: 99%

Developments in the theory of randomized shortest paths with a comparison of graph node distances

Kivimäki

Shimbo

Saerens

2014

Physica A: Statistical Mechanics and its Applications

202

View full text Add to dashboard Cite

show abstract

“…In [45], RSP is used to interpolate between graph distance and resistance distance, while in [46] it is used to interpolate between random-walk betweenness and a measure similar to standard betweenness centrality. In [47], Bavaud and Guex accomplish a weighting equivalent to RSP through the minimization of a free-energy functional. Françoisse et al .…”

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confidence: 99%

Absorbing random walks interpolating between centrality measures on complex networks

Gurfinkel

Rikvold

2020

Phys. Rev. E

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Centrality, which quantifies the importance of individual nodes, is among the most essential concepts in modern network theory. As there are many ways in which a node can be important, many different centrality measures are in use. Here, we concentrate on versions of the common betweenness and closeness centralities. The former measures the fraction of paths between pairs of nodes that go through a given node, while the latter measures an average inverse distance between a particular node and all other nodes. Both centralities only consider shortest paths (i.e., geodesics) between pairs of nodes. Here we develop a method, based on absorbing Markov chains, that enables us to continuously interpolate both of these centrality measures away from the geodesic limit and toward a limit where no restriction is placed on the length of the paths the walkers can explore. At this second limit, the interpolated betweenness and closeness centralities reduce, respectively, to the well-known current betweenness and resistance closeness (information)centralities. The method is tested numerically on four real networks, revealing complex changes in node centrality rankings with respect to the value of the interpolation parameter. Non-monotonic betweenness behaviors are found to characterize nodes that lie close to inter-community boundaries in the studied networks.ity's preference for geodesics. However, these do not precisely reduce to the betweenness. Kivimäki et al . [45,46] introduce the randomized-shortest-path (RSP) framework, which assigns Boltzmann weights to all paths in the network. Their inverse temperature parameter also tunes the preference for geodesics. In [45], RSP is used to interpolate between graph distance and resistance distance, while in [46] it is used to interpolate between random-walk betweenness and a measure similar to standard betweenness centrality. In [47], Bavaud and Guex accomplish a weighting equivalent to RSP through the minimization of a free-energy functional. Françoisse et al . also reach similar results with a different path-weighting scheme in [48]. Estrada, Higham, and Hatano [25] calculate a version of betweenness centrality by assigning lower weights to longer paths.The remainder of this paper is organized as follows. In Sec. II A, we introduce notations and conventions. In Sec. II B, we discuss well-known centrality measures. In Sec. II C, we develop two new parametrized centralities, based on a specific absorbing random walk, that interpolate between (a) closeness and resistance-closeness centralities and (b) betweenness and random-walk betweenness centralities. In Sec. III A, we report the behavior of these centralities on four example networks. In Sec. III B, we use the numerical examples to explain how the presence of similarly-long paths leads to specific centrality behaviors. In Sec. III C, we provide a model for the non-monotonicity encountered in some of the numerics. In

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“…On one hand, the feasible set X st defined by (1) is convex; on the other hand, F (X) is convex iff ϕ(x) is convex, in which case the solutionX st is unique and given by (Bavaud and Guex 2012) x…”

Section: Interpolating Random Walks and Shortest Pathsmentioning

confidence: 99%

“…In particular, the free energy path functional (Saerens et al (2009)) generates optimal flows interpolating between shortest paths and random walks (Section 2.4), where the edge resistances and transition matrix can be fixed independently (Bavaud and Guex (2012)). After reviewing the main definitions involved in the taxonomy of dissimilarities (Section 2.1), the geometric properties of the energy and free energy path functional dissimilarities (Section 2.5) are investigated.…”

Section: Introductionmentioning

confidence: 99%

Flow-Based Dissimilarities: Shortest Path, Commute Time, Max-Flow and Free Energy

Guex

Bavaud

2015

Data Science, Learning by Latent Structures, and Knowledge Discovery

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Random-walk based dissimilarities on weighted networks have demonstrated their efficiency in clustering algorithms. This contribution considers a few alternative network dissimilarities, among which a new max-flow dissimilarity, and more general flow-based dissimilarities, freely mixing shortest paths and random walks in function of a free parameter-the temperature. Their geometrical properties, and in particular their squared Euclidean nature are investigated through their power indices and multidimensional scaling properties. In particular, formal and numerical studies demonstrate the existence of critical temperatures, where flow-based dissimilarities cease to be squared Euclidean. The clustering potential of medium range temperatures is emphasised.

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

Cited by 18 publications

References 20 publications

Developments in the theory of randomized shortest paths with a comparison of graph node distances

Developments in the theory of randomized shortest paths with a comparison of graph node distances

Absorbing random walks interpolating between centrality measures on complex networks

Flow-Based Dissimilarities: Shortest Path, Commute Time, Max-Flow and Free Energy

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