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

Mapping networks of influence: tracking Twitter conversations through time and space

The increasing use of social media around global news events, such as the London Olympics in 2012, raises questions for international broadcasters about how to engage with users via social media in order to best achieve their individual missions. Twitter is a highly diverse social network whose conversations are multi-directional involving individual users, political and cultural actors, athletes and a range of media professionals. In so doing, users form networks of influence via their interactions affecting the ways that information is shared about specific global events. This article attempts to understand how networks of influence are formed among Twitter users, and the relative influence of global news media organisations and information providers in the Twittersphere during such global news events. We build an analysis around a set of tweets collected during the 2012 London Olympics. To understand how different users influence the conversations across Twitter, we compare three types of accounts: those belonging to a number of well-known athletes, those belonging to some well-known commentators employed by the BBC, and a number of corporate accounts belonging to the BBC World Service and the official London Twitter account. We look at the data from two perspectives. First, to understand the structure of the social groupings formed among Twitter users, we use a network analysis to model social groupings in the Twittersphere across time and space. Second, to assess the influence of individual tweets, we investigate the ageing factor of tweets, which measures how long users continue to interact with a particular tweet after it is originally posted. We consider what the profile of particular tweets from corporate and athletes’ accounts can tell us about how networks of influence are forged and maintained. We use these analyses to answer the questions: How do different types of accounts help shape the social networks? and, What determines the level and type of influence of a particular account

Understanding and modeling the small-world phenomenon in dynamic networks

The small-world phenomenon first introduced in the context of static graphs consists of graphs with high clustering coefficient and low shortest path length. This is an intrinsic property of many real complex static networks. Recent research has shown that this structure is also observable in dynamic networks but how it emerges remains an open problem. In this paper, we propose a model capable of capturing the small-world behavior observed in various real traces. We then study information diffusion in such small-world networks. Analytical and simulation results with epidemic model show that the small-world structure increases dramatically the information spreading speed in dynamic networks

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

A model for dynamic communicators

We develop and test an intuitively simple dynamic network model to describe the type of time-varying connectivity structure present in many technological settings. The model assumes that nodes have an inherent hierarchy governing the emergence of new connections. This idea draws on newly established concepts in online human behaviour concerning the existence of discussion catalysts, who initiate long threads, and online leaders, who trigger feedback. We show that the model captures an important property found in e-mail and voice call data – ‘dynamic communicators’ with sufficient foresight or impact to generate effective links and having an influence that is grossly underestimated by static measures based on snaphots or aggregated data

Emerging Network-Based Tools in Movement Ecology

New technologies have vastly increased the available data on animal movement and behaviour. Consequently, new methods deciphering the spatial and temporal interactions between individuals and their environments are vital. Network analyses offer a powerful suite of tools to disentangle the complexity within these dynamic systems, and we review these tools, their application, and how they have generated new ecological and behavioural insights. We suggest that network theory can be used to model and predict the influence of ecological and environmental parameters on animal movement, focusing on spatial and social connectivity, with fundamental implications for conservation. Refining how we construct and randomise spatial networks at different temporal scales will help to establish network theory as a prominent, hypothesis-generating tool in movement ecology

Evolving Networks and Social Network Analysis Methods and Techniques

Evolving networks by definition are networks that change as a function of time. They are a natural extension of network science since almost all real-world networks evolve over time, either by adding or by removing nodes or links over time: elementary actor-level network measures like network centrality change as a function of time, popularity and influence of individuals grow or fade depending on processes, and events occur in networks during time intervals. Other problems such as network-level statistics computation, link prediction, community detection, and visualization gain additional research importance when applied to dynamic online social networks (OSNs). Due to their temporal dimension, rapid growth of users, velocity of changes in networks, and amount of data that these OSNs generate, effective and efficient methods and techniques for small static networks are now required to scale and deal with the temporal dimension in case of streaming settings. This chapter reviews the state of the art in selected aspects of evolving social networks presenting open research challenges related to OSNs. The challenges suggest that significant further research is required in evolving social networks, i.e., existent methods, techniques, and algorithms must be rethought and designed toward incremental and dynamic versions that allow the efficient analysis of evolving networks