On the radius of centrality in evolving communication networks

In this article, we investigate how the choice of the attenuation factor in an extended version of Katz centrality influences the centrality of the nodes in evolving communication networks. For given snapshots of a network, observed over a period of time, recently developed communicability indices aim to identify the best broadcasters and listeners (receivers) in the network. Here we explore the attenuation factor constraint, in relation to the spectral radius (the largest eigenvalue) of the network at any point in time and its computation in the case of large networks. We compare three different communicability measures: standard, exponential, and relaxed (where the spectral radius bound on the attenuation factor is relaxed and the adjacency matrix is normalised, in order to maintain the convergence of the measure). Furthermore, using a vitality-based measure of both standard and relaxed communicability indices, we look at the ways of establishing the most important individuals for broadcasting and receiving of messages related to community bridging roles. We compare those measures with the scores produced by an iterative version of the PageRank algorithm and illustrate our findings with two examples of real-life evolving networks: the MIT reality mining data set, consisting of daily communications between 106 individuals over the period of one year, a UK Twitter mentions network, constructed from the direct \emph{tweets} between 12.4k individuals during one week, and a subset the Enron email data set

Centrality and spectral radius in dynamic communication networks

We explore the influence of the choice of attenuation factor on Katz centrality indices for evolving communication networks. For given snapshots of a network observed over a period of time, recently developed communicability indices aim to identify best broadcasters and listeners in the network. In this article, we looked into the sensitivity of communicability indices on the attenuation factor constraint, in relation to spectral radius (the largest eigenvalue) of the network at any point in time and its computation in the case of large networks. We proposed relaxed communicability measures where the spectral radius bound on attenuation factor is relaxed and the adjacency matrix is normalised in order to maintain the convergence of the measure. Using a vitality based measure of both standard and relaxed communicability indices we looked at the ways of establishing the most important individuals for broadcasting and receiving of messages related to community bridging roles. We illustrated our findings with two examples of real-life networks, MIT reality mining data set of daily communications between 106 individuals during one year and UK Twitter mentions network, direct messages on Twitter between 12.4k individuals during one week

Communicability across evolving networks

Many natural and technological applications generate time ordered sequences of networks, defined over a fixed set of nodes; for example time-stamped information about ‘who phoned who’ or ‘who came into contact with who’ arise naturally in studies of communication and the spread of disease. Concepts and algorithms for static networks do not immediately carry through to this dynamic setting. For example, suppose A and B interact in the morning, and then B and C interact in the afternoon. Information, or disease, may then pass from A to C, but not vice versa. This subtlety is lost if we simply summarize using the daily aggregate network given by the chain A-B-C. However, using a natural definition of a walk on an evolving network, we show that classic centrality measures from the static setting can be extended in a computationally convenient manner. In particular, communicability indices can be computed to summarize the ability of each node to broadcast and receive information. The computations involve basic operations in linear algebra, and the asymmetry caused by time’s arrow is captured naturally through the non-mutativity of matrix-matrix multiplication. Illustrative examples are given for both synthetic and real-world communication data sets. We also discuss the use of the new centrality measures for real-time monitoring and prediction

A survey on Human Mobility and its applications

Human Mobility has attracted attentions from different fields of studies such as epidemic modeling, traffic engineering, traffic prediction and urban planning. In this survey we review major characteristics of human mobility studies including from trajectory-based studies to studies using graph and network theory. In trajectory-based studies statistical measures such as jump length distribution and radius of gyration are analyzed in order to investigate how people move in their daily life, and if it is possible to model this individual movements and make prediction based on them. Using graph in mobility studies, helps to investigate the dynamic behavior of the system, such as diffusion and flow in the network and makes it easier to estimate how much one part of the network influences another by using metrics like centrality measures. We aim to study population flow in transportation networks using mobility data to derive models and patterns, and to develop new applications in predicting phenomena such as congestion. Human Mobility studies with the new generation of mobility data provided by cellular phone networks, arise new challenges such as data storing, data representation, data analysis and computation complexity. A comparative review of different data types used in current tools and applications of Human Mobility studies leads us to new approaches for dealing with mentioned challenges

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

Dynamic communicability and epidemic spread: a case study on an empirical dynamic contact network

We analyze a recently proposed temporal centrality measure applied to an empirical network based on person-to-person contacts in an emergency department of a busy urban hospital. We show that temporal centrality identifies a distinct set of top-spreaders than centrality based on the time-aggregated binarized contact matrix, so that taken together, the accuracy of capturing top-spreaders improves significantly. However, with respect to predicting epidemic outcome, the temporal measure does not necessarily outperform less complex measures. Our results also show that other temporal markers such as duration observed and the time of first appearance in the the network can be used in a simple predictive model to generate predictions that capture the trend of the observed data remarkably well.Comment: 31 pages, 15 figures, 11 tables; typos corrected; references added; Figure 3 added; some changes to the conclusion and introductio

Decentralized algorithms for evaluating centrality in complex networks

Im vorliegenden Bericht beschreiben wir eine neue Familie von dezentralen Algorithmen, mit denen autonome Knoten in einem komplexen Netzwerk ihre Zentralität berechnen können. Insbesondere gehen wir auf die Betweenness Centrality - Berechnung eines Knotens ein. Diese kann in einem Kommunikationsnetzwerk als Maß für die zu erwartende Vermittlungstätigkeit eines Knotens genommen werden. Wir beschreiben weiterhin, wie eine solche Analyse zur Verbesserung von Kommunikationsnetzwerken verwendet werden kann.Centrality indeices are often used to analyze the functionality of nodes in a communication network. Up to date most analyses are done on static networks where some entity has global knowledge of the networks properties. To expand the scope of these analyzing methods to decentral networks we propose a general framework for decentral algorithms that calculate different centralities, with emphasis on the algorithm of betwenness centrality. The betweenness centrality is the most complex measure and best suited for describing network communication based on shortest paths and predicting the congestion sensitivity of a network. The communication complexity of this latter algorithm is asymptotically optimal and the time complexity scales with the diameter of the network. The calculated centrality index can be used to adapt the communication network to given constraints and changing demands such that the relevant properties like the diameter of the network or uniform distribution of energy consumption is optimized

Structure of Heterogeneous Networks

Heterogeneous networks play a key role in the evolution of communities and the decisions individuals make. These networks link different types of entities, for example, people and the events they attend. Network analysis algorithms usually project such networks unto simple graphs composed of entities of a single type. In the process, they conflate relations between entities of different types and loose important structural information. We develop a mathematical framework that can be used to compactly represent and analyze heterogeneous networks that combine multiple entity and link types. We generalize Bonacich centrality, which measures connectivity between nodes by the number of paths between them, to heterogeneous networks and use this measure to study network structure. Specifically, we extend the popular modularity-maximization method for community detection to use this centrality metric. We also rank nodes based on their connectivity to other nodes. One advantage of this centrality metric is that it has a tunable parameter we can use to set the length scale of interactions. By studying how rankings change with this parameter allows us to identify important nodes in the network. We apply the proposed method to analyze the structure of several heterogeneous networks. We show that exploiting additional sources of evidence corresponding to links between, as well as among, different entity types yields new insights into network structure