The topological structure of complex networks has fascinated researchers for several decades, resulting in the discovery of many universal properties and reoccurring characteristics of different kinds of networks. However, much less is known today about the network dynamics: indeed, complex networks in reality are not static, but rather dynamically evolve over time. Our paper is motivated by the empirical observation that network evolution patterns seem far from random, but exhibit structure. Moreover, the specific patterns appear to depend on the network type, contradicting the existence of a " one fits it all " model. However, we still lack observables to quantify these intuitions, as well as metrics to compare graph evolutions. Such observables and metrics are needed for extrapolating or predicting evolutions, as well as for interpolating graph evolutions. To explore the many faces of graph dynamics and to quantify temporal changes, this paper suggests to build upon the concept of centrality, a measure of node importance in a network. In particular, we introduce the notion of centrality distance, a natural similarity measure for two graphs which depends on a given centrality, characterizing the graph type. Intuitively, centrality distances reflect the extent to which (non-anonymous) node roles are different or, in case of dynamic graphs, have changed over time, between two graphs. We evaluate the centrality distance approach for five evolutionary models and seven real-world social and physical networks. Our results empirically show the usefulness of centrality distances for characterizing graph dynamics compared to a null-model of random evolution, and highlight the differences between the considered scenarios. Interestingly, our approach allows us to compare the dynamics of very different networks, in terms of scale and evolution speed.
Another line of research attempts to extend the concept of centralities to dynamic graphs @cite_20 @cite_5 @cite_21 @cite_8 . Some researchers study how the importance of nodes changes over time in dynamic networks @cite_8 . Others define temporal centralities which to rank nodes in dynamic networks and study their distribution over time @cite_5 @cite_21 . Time centralities which describe the relative importance of time instants in dynamic networks are proposed in @cite_20 . In contrast to this existing body of work, our goal is to facilitate the direct comparison of entire networks and their dynamics, not only parts thereof.
@cite_5 @cite_21 @cite_20 @cite_8
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For a long time now, researchers have worked on defining different metrics able to characterize the importance of nodes in networks. Among them, centrality measures have proved to be pertinent as they relate the position of a node in the structure to its ability to diffuse an information efficiently. The case of dynamic networks, in which nodes and links appear and disappear over time, led the community to propose extensions of those classical measures. Yet, they do not investigate the fact that the network structure evolves and that node importance may evolve accordingly. In the present paper, we propose temporal extensions of notions of centrality, which take into account the paths existing at any given time, in order to study the time evolution of nodes' importance in dynamic networks. We apply this to two datasets and show that the importance of nodes does indeed vary greatly with time. We also show that in some cases it might be meaningless to try to identify nodes that are consistently important over time, thus strengthening the interest of temporal extensions of centrality measures. Temporal networks, i.e., networks in which the interactions among a set of elementary units change over time, can be modelled in terms of time-varying graphs, which are time-ordered sequences of graphs over a set of nodes. In such graphs, the concepts of node adjacency and reachability crucially depend on the exact temporal ordering of the links. Consequently, all the concepts and metrics proposed and used for the characterisation of static complex networks have to be redefined or appropriately extended to time-varying graphs, in order to take into account the effects of time ordering on causality. In this chapter we discuss how to represent temporal networks and we review the definitions of walks, paths, connectedness and connected components valid for graphs in which the links fluctuate over time. We then focus on temporal node–node distance, and we discuss how to characterise link persistence and the temporal small-world behaviour in this class of networks. Finally, we discuss the extension of classic centrality measures, including closeness, betweenness and spectral centrality, to the case of time-varying graphs, and we review the work on temporal motifs analysis and the definition of modularity for temporal graphs. There is an ever-increasing interest in investigating dynamics in time-varying graphs (TVGs). Nevertheless, so far, the notion of centrality in TVG scenarios usually refers to metrics that assess the relative importance of nodes along the temporal evolution of the dynamic complex network. For some TVG scenarios, however, more important than identifying the central nodes under a given node centrality definition is identifying the key time instants for taking certain actions. In this paper, we thus introduce and investigate the notion of time centrality in TVGs. Analogously to node centrality, time centrality evaluates the relative importance of time instants in dynamic complex networks. In this context, we present two time centrality metrics related to diffusion processes. We evaluate the two defined metrics using both a real-world dataset representing an in-person contact dynamic network and a synthetically generated randomized TVG. We validate the concept of time centrality showing that diffusion starting at the best ranked time instants (i.e., the most central ones), according to our metrics, can perform a faster and more efficient diffusion process. 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 real-world 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.