Abstract. 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(ν * 2 · log n) time per update, where n = |V |, and ν * 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 [3] for unique shortest paths, and for graphs with ν * = O(n), we match the bound in [3]. 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.
Abstract. We consider the incremental computation of the betweenness centrality (BC) of all vertices in a graph G = (V, E), directed or undirected, with positive real edge-weights. The current widely used algorithm is the Brandes algorithm that runs in O(mn + n 2 log n) time, where n = |V | and m = |E|. We present an incremental algorithm that updates the BC score of all vertices in G when a new edge is added to G, or the weight of an existing edge is reduced. Our incremental algorithm runs in O(m n+n 2 ) time, where m is bounded by m * = |E * |, and E * is the set of edges that lie on a shortest path in G. We achieve the same bound for the more general incremental update of a vertex v, where the edge update can be performed on any subset of edges incident to v. Our incremental algorithm is the first algorithm that is asymptotically faster on sparse graphs than recomputing with the Brandes algorithm even for a single edge update. It is also likely to be much faster than the Brandes algorithm on dense graphs since m * is often close to linear in n. Our incremental algorithm is very simple, and we give an efficient cache-oblivious implementation that incurs O(scan(n 2 )+n·sort(m )) cache misses, where scan and sort are well-known measures for efficient caching. We also give a static BC algorithm that runs in time O(m * n + n 2 log n), which is faster than the Brandes algorithm on any graph with m = ω(n log n) and m * = o(m). IntroductionBetweenness centrality (BC) is a widely-used measure in the analysis of large complex networks. The BC of a node v in a network is the fraction of all shortest paths in the network that go through v, and this measure is often used as an index that determines the relative importance of v in the network. Some applications of BC include analyzing social interaction networks [12], identifying lethality in biological networks [21], and identifying key actors in terrorist networks [13,4]. Given the changing nature of the networks under consideration, it is desirable to have algorithms that compute BC faster than computing it from scratch after every change. Our main contribution is the first incremental algorithm for computing BC after an incremental update on an edge or on a vertex that is provably faster on sparse graphs than the widely used static algorithm by Brandes [3]. By an incremental update on an edge (u, v) we mean a decrease in the weight of an existing edge (u, v), or the addition of a new edge (u, v) with finite weight if (u, v) is not present in the graph; in an incremental vertex update, updates can occur on any subset of edges incident to v, including the addition of new edges.Let G = (V, E) be a graph with positive real edge weights. Let n = |V | and m = |E|. To state our result we need the following definitions. For a vertex x ∈ V , let m * x denote the number of edges that lie on shortest paths through x. Letm * denote the average over all m * x , i.e.,m * = 1 n x∈V m * x . Finally, let m * denote the total number of edges that lie on shortest paths in G. For our incr...
Exhaustive identification of all possible alternate pathways that exist in metabolic networks can provide valuable insights into cellular metabolism. With the growing number of metabolic reconstructions, there is a need for an efficient method to enumerate pathways, which can also scale well to large metabolic networks, such as those corresponding to microbial communities. We developed MetQuest, an efficient graph-theoretic algorithm to enumerate all possible pathways of a particular size between a given set of source and target molecules. Our algorithm employs a guided breadth-first search to identify all feasible reactions based on the availability of the precursor molecules, followed by a novel dynamic-programming based enumeration, which assembles these reactions into pathways of a specified size producing the target from the source. We demonstrate several interesting applications of our algorithm, ranging from identifying amino acid biosynthesis pathways to identifying the most diverse pathways involved in degradation of complex molecules. We also illustrate the scalability of our algorithm, by studying large graphs such as those corresponding to microbial communities, and identify several metabolic interactions happening therein. MetQuest is available as a Python package, and the source codes can be found at https://github.com/RamanLab/metquest.
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