Social Networks 32 (2010) 245–251 Contents lists available at ScienceDirect Social Networks journal homepage: www.elsevier.com/locate/socnet Node centrality in weighted networks: Generalizing degree and shortest paths Tore Opsahl a,∗ , Filip Agneessens b , John Skvoretz c a Imperial College Business School, Imperial College London, London SW7 2AZ, UK b Department of Organization Sciences, VU University Amsterdam, 1081 HV Amsterdam, The Netherlands c College of Arts & Sciences, University of South Florida, Tampa, FL 33620, USA article info Degree Closeness Betweenness Weighted networks abstract Ties often have a strength naturally associated with them that differentiate them from each other. Tie strength has been operationalized as weights. A few network measures have been proposed for weighted networks, including three common measures of node centrality: degree, closeness, and betweenness. However, these generalizations have solely focused on tie weights, and not on the number of ties, which was the central component of the original measures. This paper proposes generalizations that combine both these aspects. We illustrate the benefits of this approach by applying one of them to Freeman’s EIES dataset. © 2010 Elsevier B.V. All rights reserved. 1. Introduction Social network scholars are increasingly interested in trying to capture more complex relational states between nodes. One of these avenues of research has focused on the issue of tie strength, and a number of studies from a wide range of fields have begun to explore this issue (Barrat et al., 2004; Brandes, 2001; Doreian et al., 2005; Freeman et al., 1991; Granovetter, 1973; Newman, 2001; Opsahl and Panzarasa, 2009; Yang and Knoke, 2001). Whether the nodes represent individuals, organizations, or even countries, and the ties refer to communication, cooperation, friendship, or trade, ties can be differentiated in most settings. These differences can be analyzed by defining a weighted network, in which ties are not just either present or absent, but have some form of weight attached to them. In a social network, the weight of a tie is gen- erally a function of duration, emotional intensity, intimacy, and exchange of services (Granovetter, 1973). For non-social networks, the weight often quantifies the capacity or capability of the tie (e.g., the number of seats among airports; Colizza et al., 2007; Opsahl et al., 2008) or the number of synapses and gap junctions in a neural network (Watts and Strogatz, 1998). Nevertheless, most social net- work measures are solely defined for binary situations and, thus, unable to deal with weighted networks directly (Freeman, 2004; Wasserman and Faust, 1994). By dichotomizing the network, much of the information contained in a weighted network datasets is lost, and consequently, the complexity of the network topology cannot be described to the same extent or as richly. As a result, there has ∗ Corresponding author. Tel.: +44 20 7594 3035. E-mail addresses: t.opsahl@imperial.ac.uk (T. Opsahl), f.agneessens@fsw.vu.nl (F. Agneessens), skvoretz@cas.usf.edu (J. Skvoretz). been a growing need for network measures that directly account for tie weights. The centrality of nodes, or the identification of which nodes are more “central” than others, has been a key issue in network analy- sis (Freeman, 1978; Bonacich, 1987; Borgatti, 2005; Borgatti et al., 2006). Freeman (1978) argued that central nodes were those “in the thick of things” or focal points. To exemplify his idea, he used a network consisting of 5 nodes (see Fig. 1). The middle node has three advantages over the other nodes: it has more ties, it can reach all the others more quickly, and it controls the flow between the others. Based on these three features, Freeman (1978) formalized three different measures of node centrality: degree, closeness, and betweenness. Degree is the number of nodes that a focal node is connected to, and measures the involvement of the node in the net- work. Its simplicity is an advantage: only the local structure around a node must be known for it to be calculated (e.g., when using data from the General Social Survey; McPherson et al., 2001). However, there are limitations: the measure does not take into consideration the global structure of the network. For example, although a node might be connected to many others, it might not be in a position to reach others quickly to access resources, such as information or knowledge (Borgatti, 2005; Brass, 1984). To capture this feature, closeness centrality was defined as the inverse sum of shortest dis- tances to all other nodes from a focal node. A main limitation of closeness is the lack of applicability to networks with disconnected components: two nodes that belong to different components do not have a finite distance between them. Thus, closeness is generally restricted to nodes within the largest component of a network 1 . 1 A possible method for overcoming this limitation is to sum the inversed dis- tances instead of the inverse sum of distances as the limit of 1 over infinity is 0. 0378-8733/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.socnet.2010.03.006