Graph Metrics for Temporal Networks

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.Comment: 26 pages, 5 figures, Chapter in Temporal Networks (Petter Holme and Jari Saram\"aki editors). Springer. Berlin, Heidelberg 201

The reachability of contagion in temporal contact networks: how disease latency can exploit the rhythm of human behavior

The symptoms of many infectious diseases influence their host to withdraw from social activity limiting their own potential to spread. Successful transmission therefore requires the onset of infectiousness to coincide with a time when its host is socially active. Since social activity and infectiousness are both temporal phenomena, we hypothesize that diseases are most pervasive when these two processes are synchronized. We consider disease dynamics that incorporate a behavioral response that effectively shortens the infectious period of the disease. We apply this model to data collected from face-to-face social interactions and look specifically at how the duration of the latent period effects the reachability of the disease. We then simulate the spread of the model disease on the network to test the robustness of our results. Diseases with latent periods that synchronize with the temporal social behavior of people, i.e. latent periods of 24 hours or 7 days, correspond to peaks in the number of individuals who are potentially at risk of becoming infected. The effect of this synchronization is present for a range of disease models with realistic parameters. The relationship between the latent period of an infectious disease and its pervasiveness is non-linear and depends strongly on the social context in which the disease is spreading.Comment: 9 Pages, 5 figure

Spatio-temporal networks: reachability, centrality and robustness

Recent advances in spatial and temporal networks have enabled researchers to more-accurately describe many real-world systems such as urban transport networks. In this paper, we study the response of real-world spatio-temporal networks to random error and systematic attack, taking a unified view of their spatial and temporal performance. We propose a model of spatio-temporal paths in time-varying spatially embedded networks which captures the property that, as in many real-world systems, interaction between nodes is non-instantaneous and governed by the space in which they are embedded. Through numerical experiments on three real-world urban transport systems, we study the effect of node failure on a network's topological, temporal and spatial structure. We also demonstrate the broader applicability of this framework to three other classes of network. To identify weaknesses specific to the behaviour of a spatio-temporal system, we introduce centrality measures that evaluate the importance of a node as a structural bridge and its role in supporting spatio-temporally efficient flows through the network. This exposes the complex nature of fragility in a spatio-temporal system, showing that there is a variety of failure modes when a network is subject to systematic attacks

Temporal Networks

A great variety of systems in nature, society and technology -- from the web of sexual contacts to the Internet, from the nervous system to power grids -- can be modeled as graphs of vertices coupled by edges. The network structure, describing how the graph is wired, helps us understand, predict and optimize the behavior of dynamical systems. In many cases, however, the edges are not continuously active. As an example, in networks of communication via email, text messages, or phone calls, edges represent sequences of instantaneous or practically instantaneous contacts. In some cases, edges are active for non-negligible periods of time: e.g., the proximity patterns of inpatients at hospitals can be represented by a graph where an edge between two individuals is on throughout the time they are at the same ward. Like network topology, the temporal structure of edge activations can affect dynamics of systems interacting through the network, from disease contagion on the network of patients to information diffusion over an e-mail network. In this review, we present the emergent field of temporal networks, and discuss methods for analyzing topological and temporal structure and models for elucidating their relation to the behavior of dynamical systems. In the light of traditional network theory, one can see this framework as moving the information of when things happen from the dynamical system on the network, to the network itself. Since fundamental properties, such as the transitivity of edges, do not necessarily hold in temporal networks, many of these methods need to be quite different from those for static networks

Discovering and Predicting Temporal Patterns of WiFi-interactive Social Populations

Extensive efforts have been devoted to characterizing the rich connectivity patterns among the nodes (components) of such complex networks (systems), and in the course of development of research in this area, people have been prompted to address on a fundamental question: How does the fascinating yet complex topological features of a network affect or determine the collective behavior and performance of the networked system? While elegant attempts to address this core issue have been made, for example, from the viewpoints of synchronization, epidemics, evolutionary cooperation, and the control of complex networks, theoretically or empirically, this widely concerned key question still remains open in the newly emergent field of network science. Such fruitful advances also push the desire to understand (mobile) social networks and characterize human social populations with the interdependent collective dynamics as well as the behavioral patterns. Nowadays, a great deal of digital technologies are unobtrusively embedded into the physical world of human daily activities, which offer unparalleled opportunities to explosively digitize human physical interactions, who is contacting with whom at what time. Such powerful technologies include the Bluetooth, the active Radio Frequency Identification (RFID) technology, wireless sensors and, more close to our interest in this paper, the WiFi technology. As a snapshot of the modern society, a university is in the coverage of WiFi signals, where the WiFi system records the digital access logs of the authorized WiFi users when they access the campus wireless services. Such WiFi access records, as the indirect proxy data, work as the effective proxy of a large-scale population's social interactions.Comment: 11 pages, 10 page

Activity clocks: spreading dynamics on temporal networks of human contact

Dynamical processes on time-varying complex networks are key to understanding and modeling a broad variety of processes in socio-technical systems. Here we focus on empirical temporal networks of human proximity and we aim at understanding the factors that, in simulation, shape the arrival time distribution of simple spreading processes. Abandoning the notion of wall-clock time in favour of node-specific clocks based on activity exposes robust statistical patterns in the arrival times across different social contexts. Using randomization strategies and generative models constrained by data, we show that these patterns can be understood in terms of heterogeneous inter-event time distributions coupled with heterogeneous numbers of events per edge. We also show, both empirically and by using a synthetic dataset, that significant deviations from the above behavior can be caused by the presence of edge classes with strong activity correlations