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This paper introduces a novel and well-founded betweenness measure, called the bag-of-paths (BoP) betweenness, as well as its extension, the BoP group betweenness, to tackle semisupervised classification problems on weighted directed graphs. The objective of semisupervised classification is to assign a label to unlabeled nodes using the whole topology of the graph and the labeled nodes at our disposal. The BoP betweenness relies on a BoP framework, assigning a Boltzmann distribution on the set of all possible paths through the network such that long (high-cost) paths have a low probability of being picked from the bag, while short (low-cost) paths have a high probability of being picked. Within that context, the BoP betweenness of node j is defined as the sum of the a posteriori probabilities that node j lies in between two arbitrary nodes (i, k) when picking a path starting in i and ending in k. Intuitively, a node typically receives a high betweenness if it has a large probability of appearing on paths connecting two arbitrary nodes of the network. This quantity can be computed in closed form by inverting an n&#x00D7;n matrix where n is the number of nodes. For the group betweenness, the paths are constrained to start and end in nodes within the same class, thereby defining a within-class group betweenness for each class. Unlabeled nodes are then classified according to the class showing the highest group betweenness. Experiments on various real-world datasets show that the BoP group betweenness performs competitively compared to all the tested state-of-the-art methods. The benefit of the BoP betweenness is particularly noticeable when only a few labeled nodes are available.

Semisupervised Classification Through the Bag-of-Paths Group Betweenness