  This paper introduces a novel, 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 semi-supervised 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 bag-of-paths 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 a n x 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, therefore defining a group betweenness
for each class. Unlabeled nodes are then classified according to the class
showing the highest group betweenness. Experiments on various real-world data
sets show that BoP group betweenness outperforms 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.

Machine Learning;Machine Learning