Coverage centralities for temporal networks

Abstract

Abstract. Structure of real networked systems, such as social relationship, can be modeled as temporal networks in which each edge appears only at the prescribed time. Understanding the structure of temporal networks requires quantifying the importance of a temporal vertex, which is a pair of vertex index and time. In this paper, we define two centrality measures of a temporal vertex based on the fastest temporal paths which use the temporal vertex. The definition is free from parameters and robust against the change in time scale on which we focus. In addition, we can efficiently compute these centrality values for all temporal vertices. Using the two centrality measures, we reveal that distributions of these centrality values of realworld temporal networks are heterogeneous. For various datasets, we also demonstrate that a majority of the highly central temporal vertices are located within a narrow time window around a particular time. In other words, there is a bottleneck time at which most information sent in the temporal network passes through a small number of temporal vertices, which suggests an important role of these temporal vertices in spreading phenomena.

Ego-betweenness centrality in link streams

“…However, a major drawback of these approaches is the computational cost of this path enumeration, which can be as large as O(T 3 N 3 ), T being the number of time intervals, N the number of nodes, in the case of [15] and O(T N 3 ) in [26]. Similarly, the authors of [25] introduce the temporal coverage centrality, which can be seen as an adaptation of betweenness centrality to temporal networks, as it evaluates the importance of a node through its capacity to relay information from a vertex to another. However, it slightly differs from betweenness centrality, as coverage counts the fraction of pairs of nodes, which have a fastest path going through a given node at a given time, without normalizing it to the overall number of such paths.…”

“…In [21], the authors make the choice of calling temporal betweenness of v the number of shortest time-respecting paths going through v. They also do not consider a fraction of the number of shortest paths, and refer to this quantity as "unnormalized betweenness centrality". Thus, this quantity is not a direct generalization of betweenness centrality to a dynamical network, but as in the case of [25], it is cheaper in terms of computational complexity.…”

“…In others, such as [15], [21], links go strictly forward in time, therefore, a shortest path cannot contain simultaneous links. In yet other cases [25], a notion of delay is integrated to the link. Depending on the application and data under study, all these choices are legitimate.…”

“…Note that this measurement is parametrized by the length of the snapshot. We also use the temporal coverage centrality [25], which aims to describe the importance of a node in a link stream with similar intentions as our definition, that is to say taking into account the dynamics without resorting to any timescale. Other dynamical centrality measurements have been considered for this comparison, in particular the one proposed by Tang et al [26], however our implementation is not efficient enough to process the datasets described in the following in a reasonable time.…”

“…In recent works, many definitions have been proposed to extend Freeman's graph-based definition to the dynamical context - [26], [15], [25], [21] among others. As we shall see, these definitions are often close to each other, although exhibiting subtle differences.…”