Network centralization with the Gil Schmidt power centrality index

Network centralization is a network index that measures the degree of dispersion of all node centrality scores in a network from the maximum centrality score obtained in the network. The Gil Schmidt power centrality index was developed for use in describing the political networks of Mexico, Gil and Schmidt [Gil, J., Schmidt, S., 1996a. The origin of the Mexican network of power. In: International Social Network Conference, Charleston, SC, USA, pp. 22–25; Gil, J., Schmidt, S., 1996b. The political network in Mexico. Social Networks 18, 355–381]. Upper bounds for network centralization, using the Gil Schmidt power centrality index, are derived for networks of fixed order and for when the network is bipartite, such as can arise from two mode data. In each case the networks that have maximum network centralization are described.     

Egocentric and sociocentric measures of network centrality

Egocentric centrality measures (for data on a node’s first-order zone) parallel to Freeman’s [Social Networks 1 (1979) 215] centrality measures for complete (sociocentric) network data are considered. Degree-based centrality is in principle identical for egocentric and sociocentric network data. A closeness measure is uninformative for egocentric data, since all geodesic distances from ego to other nodes in the first-order zone are 1 by definition. The extent to which egocentric and sociocentric versions of Freeman’s betweenness centrality measure correspond is explored empirically. Across seventeen diverse networks, that correspondence is found to be relatively close—though variations in egocentric network composition do lead to some notable differences in egocentric and sociocentric betweennness. The findings suggest that research design has a relatively modest impact on assessing the relative betweenness of nodes, and that a betweenness measure based on egocentric network data could be a reliable substitute for Freeman’s betweenness measure when it is not practical to collect complete network data. However, differences in the research methods used in sociocentric and egocentric studies could lead to additional differences in the respective betweenness centrality measures.     

On the robustness of centrality measures under conditions of imperfect data

An analysis is conducted on the robustness of measures of centrality in the face of random error in the network data. We use random networks of varying sizes and densities and subject them (separately) to four kinds of random error in varying amounts. The types of error are edge deletion, node deletion, edge addition, and node addition. The results show that the accuracy of centrality measures declines smoothly and predictably with the amount of error. This suggests that, for random networks and random error, we shall be able to construct confidence intervals around centrality scores. In addition, centrality measures were highly similar in their response to error. Dense networks were the most robust in the face of all kinds of error except edge deletion. For edge deletion, sparse networks were more accurately measured.     

Categorical attribute based centrality: E–I and G–F centrality

In a paper examining informal networks and organizational crisis, Krackhardt and Stern (1988) proposed a measure assessing the extent to which relations in a network were internal to a group as opposed to external. They called their measure the E–I index. The measure is now in wide use and is implemented in standard network packages such as UCINET ( Borgatti et al., 2002). The measure is based on a partition-based degree centrality measure and as such can be extended to other centrality measures and group level data. We explore extensions to closeness, betweenness and eigenvector centrality, and show how to apply the technique to sets of subgroups that do not form a partition. In addition, the extension to betweenness suggests a linkage to the Gould and Fernandez brokerage measures, which we explore.     

Legislative cosponsorship networks in the US House and Senate

In the US House and Senate, each piece of legislation is sponsored by a unique legislator. In addition, legislators can publicly express support for a piece of legislation by cosponsoring it. The network of sponsors and cosponsors provides information about the underlying social networks among legislators. I use a number of statistics to describe the cosponsorship network in order to show that it behaves much differently than other large social networks that have been recently studied. In particular, the cosponsorship network is much denser than other networks and aggregate features of the network appear to be influenced by institutional arrangements and strategic incentives. I also demonstrate that a weighted closeness centrality measure that I call ‘connectedness’ can be used to identify influential legislators.     

Bridging: Locating critical connectors in a network

This paper proposes several measures for bridging in networks derived from Granovetter's (1973) insight that links which reduce distances in a network are important structural bridges. Bridging is calculated by systematically deleting links and calculating the resultant changes in network cohesion (measured as the inverse average path length). The average change for each node's links provides an individual level measure of bridging. We also present a normalized version which controls for network size and a network-level bridging index. Bridging properties are demonstrated on hypothetical networks, empirical networks, and a set of 100 randomly generated networks to show how the bridging measure correlates with existing network measures such as degree, personal network density, constraint, closeness centrality, betweenness centrality, and vitality. Bridging and the accompanying methodology provide a family of new network measures useful for studying network structure, network dynamics, and network effects on substantive behavioral phenomenon.     

Equicentrality and network centralization: A micro–macro linkage

This paper investigates a linkage between micro- and macrostructures as an intrinsic property of social networks. In particular, it examines the linkage between equicentrality [Kang, S.M., 2007. A note on measures of similarity based on centrality. Social Networks 29, 137–142] as a conceptualization of a microstructural process (i.e., the likelihood of social actors to be connected with similarly central others) and network centralization as a macrostructural construct, and shows that they have a negative linear association. In other words, when actors are connected with similarly central alters (i.e., high equicentrality), the overall network centralization is low. Conversely, when highly central actors are connected with low-centrality actors (i.e., low equicentrality), the overall network centralization is high. The relationship between degree equicentrality and degree centralization is more significant in observed networks, especially those evolving over time, as compared to random networks. An application of this property is given by venture capital co-investment networks.     

Measurement error in network data: A re-classification

Research on measurement error in network data has typically focused on missing data. We embed missing data, which we term false negative nodes and edges, in a broader classification of error scenarios. This includes false positive nodes and edges and falsely aggregated and disaggregated nodes. We simulate these six measurement errors using an online social network and a publication citation network, reporting their effects on four node-level measures – degree centrality, clustering coefficient, network constraint, and eigenvector centrality. Our results suggest that in networks with more positively-skewed degree distributions and higher average clustering, these measures tend to be less resistant to most forms of measurement error. In addition, we argue that the sensitivity of a given measure to an error scenario depends on the idiosyncracies of the measure's calculation, thus revising the general claim from past research that the more ‘global’ a measure, the less resistant it is to measurement error. Finally, we anchor our discussion to commonly-used networks in past research that suffer from these different forms of measurement error and make recommendations for correction strategies.     

Centrality in social networks conceptual clarification

The intuitive background for measures of structural centrality in social networks is reviewed and existing measures are evaluated in terms of their consistency with intuitions and their interpretability.Three distinct intuitive conceptions of centrality are uncovered and existing measures are refined to embody these conceptions. Three measures are developed for each concept, one absolute and one relative measure of the centrality of positions in a network, and one reflecting the degree of centralization of the entire network. The implications of these measures for the experimental study of small groups is examined.     

Structural effects of network sampling coverage I: Nodes missing at random

Network measures assume a census of a well-bounded population. This level of coverage is rarely achieved in practice, however, and we have only limited information on the robustness of network measures to incomplete coverage. This paper examines the effect of node-level missingness on 4 classes of network measures: centrality, centralization, topology and homophily across a diverse sample of 12 empirical networks. We use a Monte Carlo simulation process to generate data with known levels of missingness and compare the resulting network scores to their known starting values. As with past studies (0035 and 0135), we find that measurement bias generally increases with more missing data. The exact rate and nature of this increase, however, varies systematically across network measures. For example, betweenness and Bonacich centralization are quite sensitive to missing data while closeness and in-degree are robust. Similarly, while the tau statistic and distance are difficult to capture with missing data, transitivity shows little bias even with very high levels of missingness. The results are also clearly dependent on the features of the network. Larger, more centralized networks are generally more robust to missing data, but this is especially true for centrality and centralization measures. More cohesive networks are robust to missing data when measuring topological features but not when measuring centralization. Overall, the results suggest that missing data may have quite large or quite small effects on network measurement, depending on the type of network and the question being posed.     

Public sphere as assemblage: the cultural politics of roadside memorialization

This paper investigates contemporary academic accounts of the public sphere. In particular, it takes stock of post‐Habermasian public sphere scholarship, and acknowledges a lively and variegated debate concerning the multiple ways in which individuals engage in contemporary political affairs. A critical eye is cast over a range of key insights which have come to establish the parameters of what ‘counts’ as a/the public sphere, who can be involved, and where and how communicative networks are established. This opens up the conceptual space for re‐imagining a/the public sphere as an assemblage. Making use of recent developments in Deleuzian‐inspired assemblage theory – most especially drawn from DeLanda's (2006) ‘new philosophy of society’ – the paper sets out an alternative perspective on the notion of the public sphere, and regards it as a space of connectivity brought into being through a contingent and heterogeneous assemblage of discursive, visual and performative practices. This is mapped out with reference to the cultural politics of roadside memorialization. However, a/the public sphere as an assemblage is not simply a ‘social construction’ brought into being through a logic of connectivity, but is an emergent and ephemeral space which reflexively nurtures and assembles the cultural politics (and political cultures) of which it is an integral part. The discussion concludes, then, with a consideration of the contribution of assemblage theory to public sphere studies. (Also see Campbell 2009a)     

Induced, endogenous and exogenous centrality

Centrality measures are based upon the structural position an actor has within the network. Induced centrality, sometimes called vitality measures, take graph invariants as an overall measure and derive vertex level measures by deleting individual nodes or edges and examining the overall change. By taking the sum of standard centrality measures as the graph invariant we can obtain measures which examine how much centrality an individual node contributes to the centrality of the other nodes in the network, we call this exogenous centrality. We look at exogenous measures of degree, closeness and betweenness.     

Network sampling coverage II: The effect of non-random missing data on network measurement

Missing data is an important, but often ignored, aspect of a network study. Measurement validity is affected by missing data, but the level of bias can be difficult to gauge. Here, we describe the effect of missing data on network measurement across widely different circumstances. In Part I of this study (Smith and Moody, 2013), we explored the effect of measurement bias due to randomly missing nodes. Here, we drop the assumption that data are missing at random: what happens to estimates of key network statistics when central nodes are more/less likely to be missing? We answer this question using a wide range of empirical networks and network measures. We find that bias is worse when more central nodes are missing. With respect to network measures, Bonacich centrality is highly sensitive to the loss of central nodes, while closeness centrality is not; distance and bicomponent size are more affected than triad summary measures and behavioral homophily is more robust than degree-homophily. With respect to types of networks, larger, directed networks tend to be more robust, but the relation is weak. We end the paper with a practical application, showing how researchers can use our results (translated into a publically available java application) to gauge the bias in their own data.     

A representation for the Mexican political networks

A compressed graph representation for use with the Mexican political networks is introduced. Properties of these graphs are investigated. It is also explained how the Jorge–Schmidt power centrality index can be used to index the centrality of nodes in the original network from the compressed graph representation.     

Node centrality in weighted networks: Generalizing degree and shortest paths

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.     

A network centrality bias: Central individuals in workplace networks have more supportive coworkers

Combining the results of two empirical studies, we investigate the role of alters’ motivation in explaining change in ego’s network position over time. People high in communal motives, who are prone to supportive and altruistic behavior in their interactions with others as a way to gain social acceptance, prefer to establish ties with co-workers occupying central positions in organizational social networks. This effect results in a systematic network centrality bias: The personal network of central individuals (individuals with many incoming ties from colleagues) is more likely to contain more supportive and altruistic people than the personal network of individuals who are less central (individuals with fewer incoming ties). This result opens the door to the possibility that the effects of centrality so frequently documented in empirical studies may be due, at least in part, to characteristics of the alters in an ego’s personal community, rather than to egos themselves. Our findings invite further empirical research on how alters’ motives affect the returns that people can reap from their personal networks in organizations.     

Networks of Contention: The Shape of Online Transnationalism in Early Twenty-First Century Social Movement Coalitions

The study of new media use by transnational social movements is central to contemporary investigations of social contention. In order to shed light on the terrain in which the most recent examples of online mobilization have grown and developed, this paper combines the interest in the transnational dynamics of social contention and the exploration of the use of new information and communication technologies (ICTs) for protest action. In specific terms, the study investigates how early twenty-first century social movement coalitions used Internet tools to build symbolically transnational collective identities. By applying a hyperlink network analysis approach, the study focuses on a website network generated by local chapters of the World Social Forum (WSF), one of the earliest social movement coalitions for global justice. The sample network, selected through snowball sampling, is composed of 222 social forum websites from around the world. The study specifically looks at hyperlinks among social forum websites as signs of belonging and potential means of alliance. The analysis uses network measures, namely of cohesion, centrality, structural equivalence and homophily, to test dynamics of symbolic collective identification underlying the WSF coalition. The findings show that in early twenty-first century transnational contention, culture and place still played a central role in the emergence of transnational movement networks.     

The structure of global centrality measures

The aim of this article is to identify and analyse the logic and structure of centrality measures applied to social networks. On the basis of the article by Borgatti and Everett, identifying the latent functions of centrality, we first use a survey of personal networks with 450 cases to perform an empirical study of the differences and correspondences between degree, closeness and betweenness centrality in personal networks. Then, we examine the correspondences between the three global indicators in each type of centrality: the maximum value, the mean value and the hierarchy or centralization. The results provide a better understanding of the centrality indicators of networks and the reality that they express in an empirical context.     

A New Model for Information Diffusion in Heterogeneous Social Networks

This paper discusses a new model for the diffusion of information through heterogeneous social networks. In earlier models, when information was given by one actor to another the transmitter did not retain the information. The new model is an improvement on earlier ones because it allows a transmitter of information to retain that information after telling it to somebody else. Consequently, the new model allows more actors to have information during the information diffusion process. The model provides predictions of diffusion times in a given network at the global, dyadic, and individual levels. This leads to straightforward generalizations of network measures, such as closeness centrality and betweenness centrality, for research problems that focus on the efficiency of information transfer in a network. We analyze in detail how information diffusion times and centrality measures depend on a series of network measures, such as degrees and bridges. One important finding is that predictions about the time actors need to spread information in the network differ considerably between the new and old models, while the predictions about the time needed to receive information hardly differ. Finally, some cautionary remarks are made about using the model in empirical research.