Analysing Information Flows and Key Mediators through Temporal Centrality Metrics ∗ John Tang Mirco Musolesi Cecilia Mascolo University of Cambridge University of St. Andrews University of Cambridge Vito Latora Vincenzo Nicosia University of Catania University of Catania ABSTRACT Keywords The study of influential members of human networks is an Temporal Graphs, Temporal Metrics, Temporal Centrality, important research question in social network analysis. How- Key Mediators, Social Networks, Complex Networks, Infor- ever, the current state-of-the-art is based on static or ag- mation Diffusion gregated representation of the network topology. We argue that dynamically evolving network topologies are inherent in many systems, including real online social and techno- 1. INTRODUCTION logical networks: fortunately the nature of these systems is Identifying key nodes has become an essential part of such that they allow the gathering of large quantities of fine- analysing and understanding networked systems with ap- grained temporal data on interactions amongst the network plication to a wide range of fields including finding the best members. person to target in a viral marketing campaign [15, 22], lo- In this paper we propose novel temporal centrality metrics cating key neurons in cortical networks [5], protecting im- which take into account such dynamic interactions over time. portant species in ecological systems [14] and finding bottle- Using a real corporate email dataset we evaluate the impor- necks in traffic networks [13]. The position of a node with tant individuals selected by means of static and temporal respect to other nodes can be classified and exploited: one analysis taking two perspectives: firstly, from a semantic could argue that people with the most friends are popular level, we investigate their corporate role in the organisation; and hence important; a node with high geodesic locality to and secondly, from a dynamic process point of view, we mea- other nodes could spread information quickly to high num- sure information dissemination and the role of information bers of nodes; and a person who lies between the most paths mediators. We find that temporal analysis provides a better of communication could act as a mediator among groups of understanding of dynamic processes and a more accurate people. These concepts are more commonly known as de- identification of important people compared to traditional gree, closeness and betweenness centrality [22, 4]. static methods. Existing centrality metrics for the study of real online so- cial networks (OSN) are based on a static network model where edges that appear (and disappear over time) are ag- Categories and Subject Descriptors gregated into a single static graph [1, 9, 10]. In particu- C.2.1 [Network Architecture and Design]: Network Topol- lar, if we consider a graph of interactions over time where ogy; C.2.0 [General]: Data communications each edge corresponds to an interaction (such as posting of a message) between two users represented by the nodes, we observe a continuous change of the graphs corresponding to General Terms the dynamic user activity. Examples are instant messenger systems and email where also the interactions often happen Measurement, Algorithms, Theory in bursts [17]. With this in mind, the key contribution of this paper is the ∗This work was carried out in part when Mirco Musolesi was introduction of temporal centrality metrics for the identifica- at the Computer Laboratory, University of Cambridge. tion of key nodes in OSNs based on temporal shortest paths. Tang et al. [20] proposed a temporal network model to study real dynamic dataset. The idea is that the behaviour of dy- namic networks can be more accurately captured by a se- Permission to make digital or hard copies of all or part of this work for quence of snapshots of the network topology as it changes personal or classroom use is granted without fee provided that copies are over time (Figure 1). Using this model, a temporal version of not made or distributed for profit or commercial advantage and that copies shortest path was also defined. Since static closeness and be- bear this notice and the full citation on the first page. To copy otherwise, to tweenness centrality metrics are derived from static shortest republish, to post on servers or to redistribute to lists, requires prior specific paths, we extended these metrics to temporal graphs by in- permission and/or a fee. SNS’10, April 13, 2010, Paris, France. troducing the definitions of temporal closeness and temporal Copyright 2010 ACM 978-1-4503-0080-3 ...$10.00. betweenness. Naturally, both these temporal extensions are Time Sun Sat Fri Thu Wed Tue Mon Figure 1: Temporal Graph showing a typical week of activity during Nov 2001 using 24-hour windows (left) and aggregated static graph (right). Nodes represent employees; a link between two employees exists if an email was sent by one of them to the other in that 24-hour window. associated to the identification of central nodes in the net- 2. TEMPORAL CENTRALITY METRICS work with application to dynamic processes over an OSN. In As discussed, the representation of a time-varying network particular, temporal closeness quantifies how fast a user can by means of the associated static graph can convey mislead- disseminate a piece of information. Therefore, applications ing information about the network itself. For instance, a of this metric include viral marketing and the study of ru- static aggregated network usually has far more links than mour spreading. On the other hand, temporal betweenness the network has at each time instant (or if the aggregation distinguishes individuals who act as key mediators between is performed over short time windows). Since the time or- the most communication paths over time. dering of edges is not considered, the number of paths be- Hence, to evaluate our proposed temporal centrality met- tween pairs of nodes is overestimated. This also leads to an rics, we apply them to separate dynamic processes. First, underestimation of the length of shortest paths [20]. an information dissemination process captures the notion of Following from this, since traditional static closeness and speed and reachability of a piece of information spreading betweenness centrality are based upon static shortest paths, through a network starting from a subset of nodes. Thus, the estimation provided by these metrics is largely inaccu- we can compare the important nodes selected by static and rate. In order to overcome these problems, in this section, temporal closeness centrality. Second, since we observe that we introduce the definitions of temporal closeness and tem- if certain individuals lie between the most communication poral betweenness centrality by employing temporal shortest paths, their removal from the network will impact the over- paths, which do take into account time information. First all communication efficiency. we will present the basic definitions of the temporal graph We provide a preliminary evaluation using the publicly model and temporal paths. available Enron email dataset [19], describing corporate com- munication over time between 151 known employees during 2.1 Preliminaries the height of the company's accounting scandal. A temporal graph can be thought of as an ordered sequence The main contributions of this paper are as follows: of graphs. A state of the network topology is calculated by • Based on temporal path lengths proposed by Tang et aggregating all the edges that appear inside a certain time al. [20] we define novel temporal centrality metrics for window. An example is given in Figure 1: the temporal the study of key nodes in OSNs (Section 2). graph shown in the left panel is a sequence of seven graphs, each of them representing the contacts among nodes in a • We evaluate these temporal metrics using two dynamic time window of 24-hours. The corresponding aggregated processes, applied on the Enron email dataset. The static graph (which reports all the links among nodes, with- temporal analysis consistently finds that employees who out any information about time) is shown in the right panel. worked as energy traders fulfil both these roles. This More formally, given a real network trace starting at t gives us an interesting insight not only into the En- min and ending at t , the temporal graph Gw(t ; t ) is ron scandal but also into the differences between static max t min max defined as the ordered sequence of graphs G , G , and temporal analysis (Section 3.3). We also find that tmin tmin+w :::, G , where w is the size of each time window, ex- compared to existing static analysis, temporal metrics tmax pressed in some time units (e.g., seconds or hours). The not only uncover important nodes that are better for number of graphs in the sequence is denoted as W = ((t − information spreading but also individuals who play a max t )=w) = jGw(t ; t )j. The contact function Rs be- vital role in mediating between the most communica- min t min max ij tween nodes i; j at time s is equal to 1 if and only if there tion channels (Section 3.4). exists a link between i and j in Gt; t ≤ s ≤ t + w, otherwise s • We provide initial insights into temporal dynamics which Rij is equal to zero. All the graphs in the temporal graph make temporal metrics more suitable for studying time- have the same set of nodes V , while each of them has, in varying OSNs compared to static analysis (Section 3.5). general, a different set of edges Et, where an edge between s i; j 2 V belongs to Et if and only if Rij = 1. From this time that a message forwarded from i has to wait on k be- model a temporal path between two nodes i and j can be fore being passed to j, the higher the chance of disruption w defined over Gt (tmin; tmax) as a sequence of k hops via a removing the message which was destined to j.