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.     

Reinterpreting network measures for models of disease transmission

In the wake of AIDS and HIV, a better framework is needed to model the pattern of contacts between infectious and susceptible individuals. Several network measures have been defined elsewhere which quantify the distance between 2 nodes, and the centrality of a node, in a network. Stephenson and Zelen (S-Z), however, have recently presented a new measure based on statistical estimation theory and applied it to a network of AIDS cases. This paper shows that the closeness measure proposed by S-Z is equivalent to the effective conductance in an electrical network, fits the measure into the existing theory of percolation, and provides a more efficient algorithm for computing S-Z closeness. The S-Z methodology is compared with the closeness measures of maximal flow, first passage time, and random hitting time. Computational problems associated with the measure are discussed in a closing section.     

The VL control measure for symmetric networks

In this paper we measure “control” of nodes in a network by solving an associated optimisation problem. We motivate this so-called VL control measure by giving an interpretation in terms of allocating resources optimally to the nodes in order to maximise some search probability. We determine the VL control measure for various classes of networks. Furthermore, we provide two game theoretic interpretations of this measure. First it turns out that the VL control measure is a particular proper Shapley value of the associated cooperative network game. Secondly, we relate the measure to optimal strategies in an associated matrix search game.     

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.     

A game theoretic approach to measuring degree of centrality in social networks

We present a new measure of degree of centrality in a social network which is based on a natural extension of the Banzhaf (1965) index of power in an N-person game.     

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.     

The Cohesiveness of Blocks In Social Networks: Node Connectivity and Conditional Density

This study shows several ways that formal graph theoretic statements map patterns of network ties into substantive hypotheses about social cohesion. If network cohesion is enhanced by multiple connections between members of a group, for example, then the higher the global minimum of the number of independent paths that connect every pair of nodes in the network, the higher the social cohesion. The cohesiveness of a group is also measured by the extent to which it is not disconnected by removal of 1, 2, 3,..., k actors. Menger's Theorem proves that these two measures are equivalent. Within this graph theoretic framework, we evaluate various concepts of cohesion and establish the validity of a pair of related measures: 1. Connectivity—the minimum number k of its actors whose removal would not allow the group to remain connected or would reduce the group to but a single member—measures the social cohesion of a group at a general level. 2. Conditional density measures cohesion on a finer scale as a proportion of ties beyond that required for connectivity k over the number of ties that would force it to k + 1.
Calibrated for successive values of k, these two measures combine into an aggregate measure of social cohesion, suitable for both small- and large-scale network studies. Using these measures to define the core of a new methodology of cohesive blocking, we offer hypotheses about the consequences of cohesive blocks for social groups and their members, and explore empirical examples that illustrate the significance, theoretical relevance, and predictiveness of cohesive blocking in a variety of substantively important applications in sociology.     

Entropy as a measure of centrality in networks characterized by path-transfer flow

Recently, Borgatti [Borgatti, S.P., 2005. Centrality and network flow. Social Networks 27, 55–71] proposed a taxonomy of centrality measures based on the way that traffic flows through the network—whether over path, geodesic, trail, or walk, and whether by means of transfer, serial duplication, or parallel duplication. Most of the extant centrality measures assume that traffic propagates via parallel duplication or, alternatively, that it travels over geodesics. Few of the other flow possibilities have centrality measures associated with them. This article proposes an entropy-based measure of centrality appropriate for traffic that propagates by transfer and flows along paths. The proposed measure can be applied to most network types, whether binary or weighted, directed or undirected, connected or disconnected. The measure is illustrated on the gang alliance network of Kennedy et al. [Kennedy, D.M., Braga, A. A., Piehl, A.M., 1998. The (un)known universe: mapping gangs and gang violence in Boston. Crime Prevention Studies 8, 219–262].     

The influence of secrecy on the communication structure of covert networks

In order to be able to devise successful strategies for destabilizing covert organizations it is vital to recognize and understand their structural properties. Every covert organization faces the constant dilemma of staying secret and ensuring the necessary coordination between its members. Using elements from multi-objective optimization and bargaining game theory we analyze which communication structures are optimal in the sense of providing a balanced tradeoff between secrecy and operational efficiency. For several different secrecy and information scenarios this tradeoff is analyzed considering the set of connected graphs of given order as possible communication structures. Assuming uniform exposure probability of individuals in the network we show that the optimal communication structure corresponds to either a network with a central individual (the star graph) or an all-to-all network (the complete graph) depending on the link detection probability, which is the probability that communication between individuals will be detected. If the probability that an individual is exposed as member of the network depends on the information hierarchy determined by the structure of the graph, the optimal communication structure corresponds to a reinforced ring or wheel graph in case of an information measure based on average performance. In worst case performance with respect to information it can be seen that windmill wing graphs approximate optimal structures. Finally we give an example how optimal structures change when considering a non-balanced tradeoff between secrecy and operational efficiency.     

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.     

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.     

Detecting interaction links in a collaborating group using manually annotated data

Identification of network linkages through direct observation of human interaction has long been a staple of network analysis. It is, however, time consuming and labor intensive when undertaken by human observers. This paper describes the development and validation of a two-stage methodology for automating the identification of network links from direct observation of groups in which members are free to move around a space. The initial manual annotation stage utilizes a web-based interface to support manual coding of physical location, posture, and gaze direction of group members from snapshots taken from video recordings of groups. The second stage uses the manually annotated data as input for machine learning to automate the inference of links among group members. The manual codings were treated as observed variables and the theory of turn taking in conversation was used to model temporal dependencies among interaction links, forming a Dynamic Bayesian Network (DBN). The DBN was modeled using the Bayes Net Toolkit and parameters were learned using Expectation Maximization (EM) algorithm. The Viterbi algorithm was adapted to perform the inference in DBN. The result is a time series of linkages for arbitrarily long segments that utilizes statistical distributions to estimate linkages. The validity of the method was assessed through comparing the accuracy of automatically detected links to manually identified links. Results show adequate validity and suggest routes for improvement of the method.     

Sect,Church, and Organizational Change

Abstract

Advances in research on national and local influence structures suggest that comparative study of the relationships among the influential in the context of the centralization of power and decision-making is both possible and fruitful. Techniques based on graph theory may be used to operationalize the concept of centralization. We applied Nieminen's (1973) index of centrality to a sample of the elite of the Valle region of Columbia.

Positional, reputational and decisional methods were combined to select the sample, and relationships among respondents in a regionally active core of the elite were mapped from the organizational affiliations they had in common. The adjacency matrix produced by these data showed that direct links among the leaders were structured by organizational brokers in several of the seven cities in the region. The derivative distance matrix and index of centrality incorporate the indirect relationships that exist. All but one person in the sector are closely interrelated in a network, or one clique. An individual's centrality to the network appears to be associated with indicators of status rather than class, but the possibility that differences within this sector of the elite may arise from the interplay of class and status in an industrializing society disappears when the effects of the one deviant case are removed. Reinforced by this finding, we conclude instead that this subset of the elite forms a group of people likely to be in the kind of contact and communication with one another from which consensus emerges. Centrality to a network therefore becomes an indicator of centralization.     

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.     

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.     

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.     

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.     

An agent-based model of network effects on tax compliance and evasion

Agent-based models are flexible analytical tools suitable for exploring and understanding complex systems such as tax compliance and evasion. The agent-based model created in this research builds upon two other agent-based models of tax evasion, the Korobow et al., 2007, Hokamp and Pickhardt, 2010 models. The model utilizes their rules for taxpayer behavior and apprehension of tax evaders in order to test the effects of network topologies in the propagation of evasive behavior. Findings include that network structures have a significant impact on the dynamics of tax compliance, demonstrating that taxpayers are more likely to declare all their income in networks with higher levels of centrality across the agents, especially when faced with large penalties proportional to their incomes. These results suggest that network structures should be chosen selectively when modeling tax compliance, as different topologies yield different results. Additionally, this research analyzed the special case of a power law distribution and found that targeting highly interconnected individuals resulted in a lower mean gross tax rate than targeting disconnected individuals, due to the penalties inflating the mean gross tax rate in the latter case.