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.     

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.     

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.     

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.     

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.     

Adolescent network positions and memory performance in adulthood: Evidence from sibling fixed effects models with sociometric network data

Although research has explored social factors influencing memory performance during adolescence, the impact of adolescent social network positions remains largely unknown. This study examines whether adolescent network position is associated with memory performance in adulthood, while also considering potential gender differences. The study used a sibling sample from the National Longitudinal Study of Adolescent to Adult Health (N = 2462) and employed sibling fixed effects models to account for unobserved family background factors, such as genetics, parental characteristics, family environment, and childhood neighborhood. Four dimensions of adolescent network position—i.e., popularity, sociality, degree centrality, and closeness centrality—were sociometrically assessed in schools. Memory performance in adulthood was measured using the Rey Auditory Verbal Learning Test. The sibling fixed effects estimates indicate that sociality, degree centrality, and closeness centrality are significantly associated with increased memory performance in adulthood, even after controlling for unobserved family heterogeneity as well as a set of individual-level covariates. In contrast, controlling for unobserved family heterogeneity attenuated the association for popularity, making it statistically insignificant. This study also provides evidence of gender differences in the association between social network position and memory performance. The associations for popularity, sociality, and degree centrality are more pronounced among men than women. This study’s findings highlight the importance of adolescent network positions as social determinants in shaping cognitive outcomes over the life course. Interventions that encourage positive peer interactions and reduce social isolation during adolescence may help improve cognitive health in the population.     

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.     

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.

     

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.     

Closeness centrality via the Condorcet principle

We provide a characterization of closeness centrality in the class of distance-based centralities. To this end, we introduce a natural property, called majority comparison, that states that out of two adjacent nodes the one closer to more nodes is more central. We prove that any distance-based centrality that satisfies this property gives the same ranking in every graph as closeness centrality. The axiom is inspired by the interpretation of the graph as an election in which nodes are both voters and candidates and their preferences are determined by the distances to the other nodes.     

Embedding time in positions: Temporal measures of centrality for social network analysis

Digital data enable researchers to obtain fine-grained temporal information about social interactions. However, positional measures used in social network analysis (e.g., degree centrality, reachability, betweenness) are not well suited to these time-stamped interaction data because they ignore sequence and time of interactions. While new temporal measures have been developed, they consider time and sequence separately. Building on formal algebra, we propose three temporal equivalents to positional network measures that incorporate time and sequence. We demonstrate how these temporal equivalents can be applied to an empirical context and compare the results with their static counterparts. We show that, compared to their temporal counterparts, static measures applied to interaction networks obscure meaningful differences in the way in which individuals accumulate alters over time, conceal potential disconnections in the network by overestimating reachability, and bias the distribution of betweenness centrality, which can affect the identification of key individuals in the network.     

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.     

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.     

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.     

Resistance distance,closeness, and betweenness

In a seminal paper Stephenson and Zelen (1989) rethought centrality in networks proposing an information-theoretic distance measure among nodes in a network. The suggested information distance diverges from the classical geodesic metric since it is sensible to all paths (not just to the shortest ones) and it diminishes as soon as there are more routes between a pair of nodes. Interestingly, information distance has a clear interpretation in electrical network theory that was missed by the proposing authors. When a fixed resistor is imagined on each edge of the graph, information distance, known as resistance distance in this context, corresponds to the effective resistance between two nodes when a battery is connected across them. Here, we review resistance distance, showing once again, with a simple proof, that it matches information distance. Hence, we interpret both current-flow closeness and current-flow betweenness centrality in terms of resistance distance. We show that this interpretation has semantic, theoretical, and computational benefits.     

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.     

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.     

Interorganizational network findings from a nationwide cardiovascular disease prevention initiative

ObjectivesTo use network analysis in order to evaluate the effectiveness of interorganizational networks in implementing policy, systems, and environmental interventions for cardiovascular disease prevention throughout the United States.MethodsEvaluators conducted an interorganizational network (ION) survey to examine information sharing and joint planning within organizational relationships in 15 community-based cardiovascular disease prevention partnership networks. Density and betweenness centrality scores at the node- and network-level were calculated for each partnership network using UCINET© network analysis software. Common data patterns were then extracted using a multiple case study format.ResultsNetwork density scores ranged from 0.50 to 1.00 (M = 0.84, SD = 0.14) for information sharing and 0.43–1.00 (M = 0.77, SD = 0.15) for joint planning. Centralization indices ranged from 0.00 to 0.11 (M = 0.04, SD = 0.03), and 0.00-0.17 (M = 0.06, SD = 0.05), respectively. Overall, 73.33 % of communities were successful in meeting their partnership goals.ConclusionsWhen planning and implementing interorganizational networks, high betweenness centrality and more hierarchically structured networks were identified as the most salient partnership characteristics to programmatic success. The network findings were triangulated with previously published qualitative data to provide context. These findings provide valuable insight on how national networks can be designed and leveraged to implement systematic community health projects.