Abstract
The harmonic closeness centrality measure associates, to each node of a graph, the average of the inverse of its distances from all the other nodes (by assuming that unreachable nodes are at infinite distance). This notion has been adapted to temporal graphs (that is, graphs in which edges can appear and disappear during time) and in this paper we address the question of finding the top-k nodes for this metric. Computing the temporal closeness for one node can be done in O(m) time, where m is the number of temporal edges. Therefore computing exactly the closeness for all nodes, in order to find the ones with top closeness, would require O(nm) time, where n is the number of nodes. This time complexity is intractable for large temporal graphs. Instead, we show how this measure can be efficiently approximated by using a “backward” temporal breadth-first search algorithm and a classical sampling technique. Our experimental results show that the approximation is excellent for nodes with high closeness, allowing us to detect them in practice in a fraction of the time needed for computing the exact closeness of all nodes. We validate our approach with an extensive set of experiments.
Highlights
Determining indices capable of capturing the importance of a node in a complex network has been an active research area since the end of the forties, especially in the field of social network analysis where the ultimate goal has always been to develop theories “to explain the human behavior” [1].After observing “that centrality is an important structural attribute of social networks”, and that there “is certainly no unanimity on exactly what centrality is or on its conceptual foundations”, in [2] the author proposed such a conceptual foundation of centrality by making use of graph theory concepts.The node indices proposed in that paper have become quite standard notions in complex network analysis
The authors used an algorithm for computing the t-closeness of a node of a temporal graph, whose time complexity is linear in the number m of temporal edges and whose space complexity is linear in the number n of nodes
Our first contribution is the design and analysis of an algorithm for computing the temporal closeness of a node of a temporal graph in a given time interval, whose time complexity is linear in the number m of temporal edges and whose space complexity is linear in the number n of nodes
Summary
Determining indices capable of capturing the importance of a node in a complex network has been an active research area since the end of the forties, especially in the field of social network analysis where the ultimate goal has always been to develop theories “to explain the human behavior” [1]. In the case of real-world complex networks, the number of nodes and of edges is typically so large that this algorithm is practically useless For this reason, several approaches have been followed in order to deal with huge graphs, such as computing an approximation of the closeness centrality (see, for example, [4,12]) or limiting ourselves to find the top-k nodes with respect to the closeness centrality [10]. This is not a contradiction, since, in general, a temporal edge may contribute to the t-closeness of a node for all t preceding the appearing time of the temporal edge itself).
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