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Probabilistic Paths and Centrality in Time
, 2010
"... Traditionally, graph centrality measures such as betweenness centrality are applied to discrete, static graphs, where binary edges represent the ‘presence’ or ‘absence’ of a relationship. However, when considering the evolution of networks over time, it is more natural to consider interactions at pa ..."
Abstract

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Traditionally, graph centrality measures such as betweenness centrality are applied to discrete, static graphs, where binary edges represent the ‘presence’ or ‘absence’ of a relationship. However, when considering the evolution of networks over time, it is more natural to consider interactions at particular timesteps as observational evidence of the latent (i.e., hidden) relationships among entities. In this formulation, there is inherent uncertainty about the strength of the underlying relationships and/or whether they are still active at a particular point in time. For example, if we observe an email communication between two people at time t, that indicates they have an active relationship at t, but at time t + k we are less certain the relationship still holds. In this work, we develop a framework to capture this uncertainty, centered around the notion of probabilistic paths. In order to model the effect of relationship uncertainty on network connectivity and its change over time, we formulate a measure of centrality based on most probable paths of communication, rather than shortest paths. In addition to the notion of the relationship strength, we also incorporate uncertainty with regard to the transmission of information using a binomial prior. We show that shortest paths in a unweighted, discrete graph can be formulated using probabilistic paths with a prior and we develop an algorithm to compute the most likely paths in O(VE + V² log V). We demonstrate the effectiveness of our approach by computing probabilistic betweenness centrality over time in the the Enron email dataset.