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.     

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.     

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.     

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.     

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.     

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.     

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.     

Ecological Influences on School Achievement in a Diverse Youth Sample: The Mediating Role of Substance Use

This study's purpose was to examine the extent to which closeness to family, peers, and school was associated with substance use and school achievement, based on the integrative model of ecological theory, social attachment theory, and social learning theory. A secondary data analysis was conducted on the first wave of the National Longitudinal Study of Adolescent Health (Add Health). The final sample yielded 3,147 boys and 3,356 girls.

A structural equation model was employed to test a hypothesized model. School closeness was found to be a primary ecological factor that significantly influenced school achievement while substance use emerged as a critical mediator of this relationship. Family closeness was negatively associated with school achievement. Also, substance use did not have a mediating function in the relationship between peer closeness and school achievement. No gender difference was found, except the relationship between family closeness and school achievement, in that family closeness had a significant, direct effect on school achievement among only boys but not girls.

The findings suggest that schools should strengthen adolescents' tie to school to promote their better academic success and to prevent them from substance use.     

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.     

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.     

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.     

Are respondents more likely to list alters with certain characteristics?: Implications for name generator data

Analyses of egocentric networks make the implicit assumption that the list of alters elicited by name generators is a complete list or representative sample of relevant alters. Based on the literature on free recall tasks and the organization of people in memory, I hypothesize that respondents presented with a name generator are more likely to name alters with whom they share stronger ties, alters who are more connected within the network, and alters with whom they interact in more settings. I conduct a survey that presents respondents with the GSS name generator and then prompts them to remember other relevant alters whom they have not yet listed. By comparing the alters elicited before and after prompts I find support for the first two hypotheses. I then go on to compare network-level measures calculated with the alters elicited by the name generator to the same measures calculated with data from all alters. These measures are not well correlated. Furthermore, the degree of underestimation of network size is related to the networks’ mean closeness, density, and mean duration of relationships. Higher values on these variables result in more accurate estimation of network size. This suggests that measures of egocentric network properties based on data collected using a single name generator may have high levels of measurement error, possibly resulting in misestimation of how these network properties relate to other variables.     

PERCEIVED GAME UNCERTAINTY,SUSPENSE AND THE DEMAND FOR SPORT

This study tries bridging between different behavioral economic explanations for the lack of support of the uncertainty of outcome hypothesis in spectator sports. We test a measure of perceived game uncertainty that is comparable to objective measures frequently tested in the literature. Econometric results suggest that fans do not perceive closeness of a game differently than how economists have tended to measure it. However, fans' perceptions of suspensefulness are distinct from their perceptions of game uncertainty. Moreover, the finding that fans' preferences for game uncertainty are dominated by loss aversion also emerges—independently of fanship status—in our stated‐preference setting. (JEL L83, D12, Z2)     

Adding valued data to social network measures: Does it add to associations with adolescent substance use?

This research examines whether adding information about adolescent friendships (feeling of closeness to friend, visits to friend's home, interaction outside school, and parent involvement in the friendship) to social network measures increases associations with adolescent substance use. In 2002, friendship nominations were obtained from more than 4700 students in grades 6 to 8 in 13 schools. Social network measures weighted and unweighted for the valued information were created from those data and compared. The findings suggest that, although non-valued network measures and ego substance use were associated, the valued network measures did not add to those associations.     

Power in politically charged networks

We offer a theory and measure for determining powerful nodal positions based on potential inter-actor control in “politically charged” networks, which contain both allies and adversaries. Power is derived from actors that are dependent on the focal actor and sociometrically weak, either due to a lack of alternative allies or from being threatened by others. We create a new Political Independence Index (PII), compare it to other established measures, and illustrate its use in the setting of an international network of alliances and military conflicts from 1946 to 2000. Results show that politically independent nations as measured by PII have smaller increases in military personnel than others over time.     

Exploring variation in active network size: Constraints and ego characteristics

Studies of active personal networks have primarily focused on providing reliable estimates of the size of the network. In this study, we examine how compositional properties of the network and ego characteristics are related to variation in network size. There was a negative relationship between mean emotional closeness and network size, for both related and unrelated networks. Further, there was a distinct upper bound on total network size. These results suggest that there are constraints both on the absolute number of individuals that ego can maintain in the network, and also on the emotional intensity of the relationships that ego can maintain with those individuals.     

Continuity and Change in Grandchildren's Closeness to Grandparents: Consequences of Changing Intergenerational Ties

Drawing on data from Waves 2 and 3 of the National Survey of Families and Households, this study examines whether grandchildren's (N = 496) previous patterns of closeness to grandparents is associated with their current closeness to grandparents and whether changes in parents' intergenerational ties make a difference in the development of grandchildren's closeness to grandparents when grandchildren experience young adulthood. The findings suggest that there is a possibility for both continuity and change in grandchildren's bond to grandparents. Grandchildren's closeness to grandparents was associated not only with their earlier patterns of closeness to grandparents and with parents' concurrent relations with the grandchild and grandparent generations, but also with changes in parents' intergenerational ties over time. Also, the grandchild gender moderated linkages between certain intergenerational ties in the family.     

CLOSENESS WITH PARENTS AND PERCEIVED CONSEQUENCES OF PREGNANCY AMONG MALE AND FEMALE ADOLESCENTS

Using the 1994–1995 National Longitudinal Study of Adolescent Health, we examine the association between closeness with parents and teen beliefs regarding the consequences of pregnancy. Unlike previous studies in this area, we include measures of closeness to both mother and father and examine effects for both male and female adolescents by family type (two-biological- or two-adoptive-parent, stepfather, and single-mother families). Our findings indicate that parental closeness is most salient for boys living with two biological or adoptive parents. For girls living with two biological or adoptive parents and boys and girls living in stepfather families, controlling for other aspects of the parent-child relationship, as well as adolescent, socioeconomic, and demographic characteristics, eliminates the protective influence of closeness to parents. For teens living in single-mother families, closeness with mother has no effect on beliefs about the consequences of pregnancy. Overall, our findings indicate that it is teen beliefs regarding parental viewpoints on educational achievement and engagement in sexual activity, rather than closeness per se, that has the link to adolescent beliefs about the consequences of pregnancy.     

Some properties of structural equivalence measures derived from sociometric choice data

I discuss and illustrate the extent to which different relation measures and pattern similarity measures can be expected to generate different structural equivalence results. Measures of network relations and pattern similarity are reviewed to establish clear comparisons between structural equivalence measures. Using Monte Carlo sociometric choice data drawn from four strategically designed study populations, alternative relation and pattern similarity measures are combined in a factorial design generating six measures of structural equivalence within each study population. I report the magnitudes of differences between structural equivalence measures within populations, compared across populations. Three conclusions are drawn: (1) There is significant reliability across alternative measures. (2) This reliability increases with the clarity of boundaries between statuses in a study population. (3) The noticeable differences between structural equivalence measures that exist under conditions at all weaker than strong equivalence are principally a function of how relations are measured rather than how relation pattern similarities are measured. Two inferences are drawn for applied network analysis: (1) Structural equivalence should be computed from path distance measures of network relations (however normalized) rather than being computed directly from patterns of binary choice data. (2) Renewed methodological attention should shift from how we measure pattern similarity to how we measure relationships.