Geodesic based centrality: Unifying the local and the global

A variety of node-level centrality measures, including purely structural measures (such as degree and closeness centrality) and measures incorporating characteristics of actors (such as the Blau's measure of heterogeneity) have been developed to measure a person's access to resources held by others. Each of these node-level measures can be placed on a continuum depending on whether they focus only on ego's direct contacts (e.g. degree centrality and Blau's measure of heterogeneity), or whether they also incorporate connections to others at longer distances in the network (e.g. closeness centrality or betweenness centrality). In this paper we propose generalized measures, where a tuning parameter δ regulates the relative impact of resources held by more close versus more distant others. We first show how, when a specific δ is chosen degree-centrality and reciprocal closeness centrality are two specific instances of this more general measure. We then demonstrate how a similar approach can be applied to node-level measures that incorporate attributes. When more or less weight is given to other nodes at longer distances with specific characteristics, a generalized measure of resource-richness and a generalized measure for diversity among one's connections can be obtained (following Blau's measure of heterogeneity). Finally, we show how this approach can also be applied to betweenness centrality to focus on more local (ego) betweenness or global (Freeman) betweenness. The importance of the choice of δ is illustrated on some classic network datasets.     

You are who remembers you. Detecting leadership through accuracy of recall

We measured interpersonal perception accuracy by focusing on the relationship between actors’ centrality and their ability to accurately report their social interactions. We used the network measures of actors’ betweenness centrality and degree centrality to identify the most prominent members by correlating ego-perception and alter-perception in a “non-reciprocity” type of misalignment. We found a positive correlation between actors’ centrality and their centrality as assessed by senior management, and a negative correlation between actors’ centrality and their accuracy in recalling interactions. Underreporting social interactions may represent a third way of measuring the importance of members and finding the most influential actors.     

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.     

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.     

Reliability of measures of centrality and prominence

This paper evaluates the reliability of measures of centrality and prominence of social networks among high school students. The authors present and discuss results from eight experiments. Four types of social support: (1) instrumental support, (2) informational support, (3) social companionship, and (4) emotional support—were measured three times within each class. Four measurement scales: (1) binary, (2) categorical, (3) categorical with labels and (4) line production—were applied. Reliability of in- and out-degree, in- and out-closeness, betweenness and flow betweenness was estimated by the Pearson correlation coefficient. Meta analysis of factors affecting the test-retest reliability of measures of centrality and prominence was done by multiple classification analysis. Results show that,

• -Global measures (considering direct and indirect choices) are more sensitive to measurement errors than local measures (considering only direct choices).
• -In-measures are more stable than out-measures.
• -Among types of social support, emotional support gives the least stable measures of centrality and prominence, whereas social companionship gives the most stable results.
• -The reliability of centrality and prominence measures is higher when the network is denser.

     

Learning in social networks: Selecting profitable choices among alternatives of uncertain profitability in various networks

Social capital theory assumes that information is valuable. However, only rarely is this value explicitly modeled, and there are few examples of empirical tests of mechanisms that connect social network structure to valuable information. We model an individual decision problem in which individuals make choices that yield uncertain outcomes. The individuals can learn about the profitability of options from their own choices and from the network. We generate computer-simulated data to derive hypotheses about the effect of network characteristics on making profitable choices. We conduct a laboratory experiment to empirically test these hypotheses and find that, at the individual level, degree centrality has a positive effect on making profitable choices whereas betweenness centrality has no effect. At the network level, density has a positive effect on making profitable choices, whereas centralization does not have an effect.     

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.     

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.     

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.     

Measuring centrality in film narratives using dynamic character interaction networks

The tools of social network analysis offer a promising framework for studying fictional texts and the relational activity of the characters therein. The goal of this paper is to offer both a conceptual refinement of the project of measuring the centrality of characters within narratives using network tools, as well as the proposal of a novel measure with which to do so. Conceptually, we argue that as questions of time, order and sequence are central in narratives, measures of characters’ narrative importance should be based on dynamic network representations which respect the time-ordering of narrative events. We suggest a directed dynamic measure of relative character importance based on character interactions and illustrate it through an examination of gender in the 2015 film Star Wars: The Force Awakens. We find that the measure helps illuminate important narrative dynamics which cannot be captured by static measures, and presents a platform on which future character network research can productively build.     

Group betweenness and co-betweenness: Inter-related notions of coalition centrality

Vertex betweenness centrality is a metric that seeks to quantify a sense of the importance of a vertex in a network in terms of its ‘control’ on the flow of information along geodesic paths throughout the network. Two natural ways to extend vertex betweenness centrality to sets of vertices are (i) in terms of geodesic paths that pass through at least one of the vertices in the set, and (ii) in terms of geodesic paths that pass through all vertices in the set. The former was introduced by Everett and Borgatti [Everett, M., Borgatti, S., 1999. The centrality of groups and classes. Journal of Mathematical Sociology 23 (3), 181–201], and called group betweenness centrality. The latter, which we call co-betweenness centrality here, has not been considered formally in the literature until now, to the best of our knowledge. In this paper, we show that these two notions of centrality are in fact intimately related and, furthermore, that this relationship may be exploited to obtain deeper insight into both. In particular, we provide an expansion for group betweenness in terms of increasingly higher orders of co-betweenness, in a manner analogous to the Taylor series expansion of a mathematical function in calculus. We then demonstrate the utility of this expansion by using it to construct analytic lower and upper bounds for group betweenness that involve only simple combinations of (i) the betweenness of individual vertices in the group, and (ii) the co-betweenness of pairs of these vertices. Accordingly, we argue that the latter quantity, i.e., pairwise co-betweenness, is itself a fundamental quantity of some independent interest, and we present a computationally efficient algorithm for its calculation, which extends the algorithm of Brandes [Brandes, U., 2001. A faster algorithm for betweenness centrality. Journal of Mathematical Sociology 25, 163] in a natural manner. Applications are provided throughout, using a handful of different communication networks, which serve to illustrate the way in which our mathematical contributions allow for insight to be gained into the interaction of network structure, coalitions, and information flow in social networks.     

Prevention of violence against children and psychosocial models of intervention: Network analysis of an inter-institutional service

This study describes an Italian service, the Inter-institutional Groups Operating against child Abuse and Maltreatment (GOIAM), which specializes in the treatment of child abuse. The GOIAM service is composed of 53 professionals divided into four different categories: social workers, psychologists, child neuropsychiatrists, and psychopedagogists. With the aim of detecting strengths and weaknesses of the operative model adopted by GOIAM, the whole service was analyzed through the application of social network analysis (SNA). The analysis revealed a low-density network, while the study of centrality yielded different distributions of power indicators (degree, closeness, and betweenness): prestige and pervasiveness were balanced, while influence, reachability, and betweenness were quite unbalanced. The search for subgroups (N-clans, lambda sets, community structure) suggested the presence of many overlapping triads, leading to the formation of large clusters. The inspection of such groups revealed that professionals belonging to the same district tended to cluster. This condition did not promote the diffusion of opinions and expertise, and could be an important factor of weakness in an interdistrict service. The analysis of inter-groups relationships (brokerage) and the study of social roles (REGular Equivalence) showed that most professionals had “diffuse” roles through which they try to perform every task, exchanging assignments, activities, and positions. The flattening of the roles inside the organization might at first appear as facilitation, although this could represent a limit for a complex organization in the long term because it generates a lack of specialization. Finally, the potential evolution of the GOIAM network (P1 model) seems to be characterized by a further decrease in density, more isolation of peripheral nodes, an increase in centralization, and, consequently, a higher level of hierarchy in the network.     

The social networks of academic performance in a student context of poverty in Mexico

Purpose: To describe the structure of the social network of junior high school students from a low socioeconomic status and assess the association between centrality measurements and academic performance. In the centrality positions (eigenvector, closeness, degree, and betweenness), the female gender and “only study” were significant predictors of high academic performance. The density of the student social network was presented by homophilic associations that hint at the existence of subcultures at school.     

Some unique properties of eigenvector centrality

Eigenvectors, and the related centrality measure Bonacich's c(β), have advantages over graph-theoretic measures like degree, betweenness, and closeness centrality: they can be used in signed and valued graphs and the beta parameter in c(β) permits the calculation of power measures for a wider variety of types of exchange. Degree, betweenness, and closeness centralities are defined only for classically simple graphs—those with strictly binary relations between vertices. Looking only at these classical graphs, where eigenvectors and graph–theoretic measures are competitors, eigenvector centrality is designed to be distinctively different from mere degree centrality when there are some high degree positions connected to many low degree others or some low degree positions are connected to a few high degree others. Therefore, it will not be distinctively different from degree when positions are all equal in degree (regular graphs) or in core-periphery structures in which high degree positions tend to be connected to each other.     

Ego network betweenness

In this paper, we look at the betweenness centrality of ego in an ego network. We discuss the issue of normalization and develop an efficient and simple algorithm for calculating the betweenness score. We then examine the relationship between the ego betweenness and the betweenness of the actor in the whole network. Whereas, we can show that there is no theoretical link between the two we undertake a simulation study, which indicates that the local ego betweenness is highly correlated with the betweenness of the actor in the complete network.     

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.     

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.     

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.     

Bridging,brokerage and betweenness

Valente and Fujimoto (2010) proposed a measure of brokerage in networks based on Granovetter's classic work on the strength of weak ties. Their paper identified the need for finding node-based measures of brokerage that consider the entire network structure, not just a node's local environment. The measures they propose, aggregating the average change in cohesion for a node's links, has several limitations. In this paper we review their method and show how the idea can be modified by using betweenness centrality as an underpinning concept. We explore the properties of the new method and provide point, normalized, and network level variations. This new approach has two advantages, first it provides a more robust means to normalize the measure to control for network size, and second, the modified measure is computationally less demanding making it applicable to larger networks.