A bag-of-paths framework for network data analysis

Abstract

This work develops a generic framework, called the bag-of-paths (BoP), for link and network data analysis. The central idea is to assign a probability distribution on the set of all paths in a network. More precisely, a Gibbs-Boltzmann distribution is defined over a bag of paths in a network, that is, on a representation that considers all paths independently. We show that, under this distribution, the probability of drawing a path connecting two nodes can easily be computed in closed form by simple matrix inversion. This probability captures a notion of relatedness between nodes of the graph: two nodes are considered as highly related when they are connected by many, preferably low-cost, paths. As an application, two families of distances between nodes are derived from the BoP probabilities. Interestingly, the second distance family interpolates between the shortest path distance and the resistance distance. In addition, it extends the Bellman-Ford formula for computing the shortest path distance in order to integrate sub-optimal paths by simply replacing the minimum operator by the soft minimum operator. Experimental results on semi-supervised classification show that both of the new distance families are competitive with other state-ofthe-art approaches. In addition to the distance measures studied in this paper, the bag-of-paths framework enables straightforward computation of many other relevant network measures.

Absorbing random walks interpolating between centrality measures on complex networks

“…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.…”

Community detection and unveiling of hierarchy in networks: a density-based clustering approach

“…This step provides a fundamental starting point for interpreting the network and a powerful tool for further exploration of its characteristics using standard multivariate statistics or machine learning methods. To explore the structure and dynamics of the network, we start by modeling the interactions among nodes based on the concept of randomized shortest path (RSP) dissimilarity (Kivimäki et al, 2014;Yen et al, 2008;Saerens et al, 2009;Francoisse et al, 2017). The calculation involves the search for the optimal path that minimizes the expected cost obtained by imposing the constraint that the relative entropy has a constant value spread throughout the network.…”

“…The parameter β, which controls the distribution, plays the role of the inversetemperature in thermodynamics. It is shown (Francoisse et al, 2017) that, under the Gibbs-Boltzmann distribution, the probability of drawing a path connecting two nodes can easily be computed in closed form by simple matrix inversion. Moreover, in contrast to common distance measures, such as the Shortest Path (SP) (the length of the shortest paths between nodes), and the Commute Time (CT) distance (Akamatsu, 1996) (the expected length of paths that a random walker moving along the edges of the graph takes from one node to the other and back (Kivimäki et al, 2014)), RSP captures the global structure of the network.…”

“…These distances can be expressed in terms of path lengths, namely the smallest number of links or edges needed to connect the nodes within the network. Here, we model the interactions among nodes based on the concept of randomized shortest path (RSP) dissimilarity (Kivimäki et al, 2014;Yen et al, 2008;Saerens et al, 2009;Francoisse et al, 2017), which has its foundation in statistical physics and is derived with the constraint of fixed relative "distance" entropy. It was originally inspired by the work of Akamatsu (Akamatsu, 1996) on the decomposition of path choice entropy in g\eneral transport networks, and motivated by the fact that, generally common distances do not take into account the global structure of the graph (Francoisse et al, 2017).…”

“…The process of randomization assigns a probability distribution over the set of paths to be followed by each node, so that chance is favored over choice with the advantage of exploration versus efficiency. This reveals the degree of connectedness between nodes where two nodes are considered highly connected or related when they are linked by many, preferably low-cost, paths (Francoisse et al, 2017). The calculation involves the search for the optimal path that minimizes the expected cost obtained by imposing the constraint that the relative entropy has a constant value spread throughout the network.…”