Decremental All-Pairs ALL Shortest Paths and Betweenness Centrality

We consider the all pairs all shortest paths (APASP) problem, which maintains the shortest path dag rooted at every vertex in a directed graph G=(V,E) with positive edge weights. For this problem we present a decremental algorithm (that supports the deletion of a vertex, or weight increases on edges incident to a vertex). Our algorithm runs in amortized O(\vstar^2 \cdot \log n) time per update, where n=|V|, and \vstar bounds the number of edges that lie on shortest paths through any given vertex. Our APASP algorithm can be used for the decremental computation of betweenness centrality (BC), a graph parameter that is widely used in the analysis of large complex networks. No nontrivial decremental algorithm for either problem was known prior to our work. Our method is a generalization of the decremental algorithm of Demetrescu and Italiano [DI04] for unique shortest paths, and for graphs with \vstar =O(n), we match the bound in [DI04]. Thus for graphs with a constant number of shortest paths between any pair of vertices, our algorithm maintains APASP and BC scores in amortized time O(n^2 \log n) under decremental updates, regardless of the number of edges in the graph.Comment: An extended abstract of this paper will appear in Proc. ISAAC 201

Betweenness Centrality – Incremental and Faster

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The interplay of the physical landscape and social dynamics in shaping movement of African savanna elephants (loxodonta africana)

Free ranging African savanna elephants (Loxodonta africana) are increasingly impacted by human-induced habitat loss and poaching for ivory. Because elephants live in tightly knit groups, this combination of threats not only reduces the size of their populations but also degrades their social interactions. Long-term relationships with socially competent individuals, such as experienced seniors, benefit the ability of other group members to access limiting resources and avoid danger. Understanding how anthropogenic pressure may affect persistence of elephant populations is important, because elephants are an economically important keystone species. This doctoral thesis characterizes how individual elephants influence the movement of their social partners, and how the social network properties of elephant groups related to information sharing may change when socially competent members are killed by poachers. To that end, two techniques commonly used to study movement of individuals in their habitat, and one used to study the consequences of repeated social interactions, are modified and extended to incorporate a number of the social processes typically found in groups of elephants. First, an established, choice-based statistical framework for movement analysis is modified and validated using synthetic and empirical data. It allows for simultaneous modeling of the effects of the habitat quality and social interactions on individual movement choices. Next, this new model is applied to a unique set of remotely sensed tracks from five male elephants navigating across the same habitat in southern Africa. A key result is that known dominance relationships observed at water points and other gathering places are determined to persist even when elephants are ranging more widely across the landscape. Lastly, an existing \u27social network and poaching\u27 simulation model is parameterized with data from wild elephants. It reveals debilitating effects of poaching on various network metrics thought to correlate with group communication efficiency. The modeling and simulation tools developed over the course of this doctoral research may be generalized to include the influence of \u27dynamic points\u27 other than social conspecifics, such as predators or poachers, on long-term movement patterns, and thus may provide a tool to both understand and mitigate human-wildlife conflict. In addition, they may aid hypothesis testing about disturbance of social dynamics in animal systems subject to exploitation by humans or lethal management