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.
Some authors also considered bounded (or truncated) walks @cite_13 @cite_5 @cite_33 and obtained promising results on large graphs. This approach could also be considered in our framework in order to tackle large networks; this will be investigated in further work.
@cite_5 @cite_13 @cite_33
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Recently there has been much interest in graph-based learning, with applications in collaborative filtering for recommender networks, link prediction for social networks and fraud detection. These networks can consist of millions of entities, and so it is very important to develop highly efficient techniques. We are especially interested in accelerating random walk approaches to compute some very interesting proximity measures of these kinds of graphs. These measures have been shown to do well empirically (Liben-Nowell & Kleinberg, 2003; Brand, 2005). We introduce a truncated variation on a well-known measure, namely commute times arising from random walks on graphs. We present a very novel algorithm to compute all interesting pairs of approximate nearest neighbors in truncated commute times, without computing it between all pairs. We show results on both simulated and real graphs of size up to 100; 000 entities, which indicate near-linear scaling in computation time. In this note, we demonstrate an instance of bounded-degree graphs of size @math , for which the total variation mixing time for the random walk is decreased by a factor of @math if we multiply the edge-conductances by bounded factors in a certain way. Motivated by the immense success of random walk and Markov chain methods in the design of classical algorithms, we consider quantum walks on graphs. We analyse in detail the behaviour of unbiased quantum walk on the line, with the example of a typical walk, the Hadamard walk''''. In particular, we show that after @math time steps, the probability distribution on the line induced by the Hadamard walk is almost uniformly distributed over the interval @math . This implies that the same walk defined on the circle mixes in linear time. This is in direct contrast with the quadratic mixing time for the corresponding classical walk. We conclude by indicating how our techniques may be applied to more general graphs. We investigate network exploration by random walks defined via stationary and adaptive transition probabilities on large graphs. We derive an exact formula valid for arbitrary graphs and arbitrary walks with stationary transition probabilities (STP), for the average number of discovered edges as a function of time. We show that for STP walks site and edge exploration obey the same scaling nλ as a function of time n. Therefore, edge exploration on graphs with many loops is always lagging compared to site exploration, the revealed graph being sparse until almost all nodes have been discovered. We then introduce the edge explorer model (EEM), which presents a novel class of adaptive walks, that perform faithful network discovery even on dense networks.