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.     

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.     

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.     

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.     

Ranking terrorists in networks: A sensitivity analysis of Al Qaeda's 9/11 attack

All over the world, intelligence services are collecting data concerning possible terrorist threats. This information is usually transformed into network structures in which the nodes represent the individuals in the data set and the links possible connections between these individuals. Unfortunately, it is nearly impossible to keep track of all individuals in the resulting complex network. Therefore, Lindelauf et al. (2013) introduced a methodology that ranks terrorists in a network. The rankings that result from this methodology can be used as a decision support system to efficiently allocate the scarce surveillance means of intelligence agencies. Moreover, usage of these rankings can improve the quality of surveillance which can in turn lead to prevention of attacks or destabilization of the networks under surveillance.The methodology introduced by Lindelauf et al. (2013) is based on a game theoretic centrality measure, which is innovative in the sense that it takes into account not only the structure of the network but also individual and coalitional characteristics of the members of the network. In this paper we elaborate on this methodology by introducing a new game theoretic centrality measure that better takes into account the operational strength of connected subnetworks.Moreover, we perform a sensitivity analysis on the rankings derived from this new centrality measure for the case of Al Qaeda's 9/11 attack. In this sensitivity analysis we consider firstly the possible additional information available about members of the network, secondly, variations in relational strength and, finally, the absence or presence of a small percentage of links in the network. We also introduce a case specific method to compare the different rankings that result from the sensitivity analysis and show that the new centrality measure is robust to small changes in the data.     

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.     

A Graph-theoretic perspective on centrality

The concept of centrality is often invoked in social network analysis, and diverse indices have been proposed to measure it. This paper develops a unified framework for the measurement of centrality. All measures of centrality assess a node's involvement in the walk structure of a network. Measures vary along four key dimensions: type of nodal involvement assessed, type of walk considered, property of walk assessed, and choice of summary measure. If we cross-classify measures by type of nodal involvement (radial versus medial) and property of walk assessed (volume versus length), we obtain a four-fold polychotomization with one cell empty which mirrors Freeman's 1979 categorization. At a more substantive level, measures of centrality summarize a node's involvement in or contribution to the cohesiveness of the network. Radial measures in particular are reductions of pair-wise proximities/cohesion to attributes of nodes or actors. The usefulness and interpretability of radial measures depend on the fit of the cohesion matrix to the one-dimensional model. In network terms, a network that is fit by a one-dimensional model has a core-periphery structure in which all nodes revolve more or less closely around a single core. This in turn implies that the network does not contain distinct cohesive subgroups. Thus, centrality is shown to be intimately connected with the cohesive subgroup structure of a network.     

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.     

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.     

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.     

Power laws and fragility in flow networks

What makes economic and ecological networks so unlike other highly skewed networks in their tendency toward turbulence and collapse? Here, we explore the consequences of a defining feature of these networks: their nodes are tied together by flow. We show that flow networks tend to the power law degree distribution (PLDD) due to a self-reinforcing process involving position within the global network structure, and thus present the first random graph model for PLDDs that does not depend on a rich-get-richer function of nodal degree. We also show that in contrast to non-flow networks, PLDD flow networks are dramatically more vulnerable to catastrophic failure than non-PLDD flow networks, a finding with potential explanatory power in our age of resource- and financial-interdependence and turbulence.     

Duocentered networks

When a pair of individuals is central to a research problem (e.g., husband and wife, PhD student and supervisor) the concept of “duocentered” networks can be defined as a useful extension of egocentered networks. This new structure consists of a pair of central egos and their direct links with alters, instead of just one central ego as in the egocentered networks or multiple egos as in complete networks. The key point in this kind of network is that ties exist between the central pair of egos and between them and all alters, but the ties among alters are not considered. Duocentered networks can also be considered as a compromise between egocentered and complete networks. Complete network measurements are often costly to obtain and tend to contain a large proportion of missing data (especially for peripheral actors). Egocentered network data are less costly but a lot of information is lost with their use when a pair of individuals is the relevant unit of analysis.     

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.     

Non-response in social networks: The impact of different non-response treatments on the stability of blockmodels

Discerning the essential structure of social networks is a major task. Yet, social network data usually contain different types of errors, including missing data that can wreak havoc during data analyses. Blockmodeling is one technique for delineating network structure. While we know little about its vulnerability to missing data problems, it is reasonable to expect that it is vulnerable given its positional nature. We focus on actor non-response and treatments for this. We examine their impacts on blockmodeling results using simulated and real networks. A set of ‘known’ networks are used, errors due to actor non-response are introduced and are then treated in different ways. Blockmodels are fitted to these treated networks and compared to those for the known networks. The outcome indicators are the correspondence of both position memberships and identified blockmodel structures. Both the amount and type of non-response, and considered treatments, have an impact on delineated blockmodel structures.     

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.     

Assessing the performance of the bootstrap in simulated assemblage networks

Archaeologists are increasingly interested in networks constructed from site assemblage data, in which weighted network ties reflect sites’ assemblage similarity. Equivalent networks would arise in other scientific fields where actors’ similarity is assessed by comparing distributions of observed counts, so the assemblages studied here can represent other kinds of distributions in other domains. One concern with such work is that sampling variability in the assemblage network and, in turn, sampling variability in measures calculated from the network must be recognized in any comprehensive analysis. In this study, we investigated the use of the bootstrap as a means of estimating sampling variability in measures of assemblage networks. We evaluated the performance of the bootstrap in simulated assemblage networks, using a probability structure based on the actual distribution of sherds of ceramic wares in a region with 25 archaeological sites. Results indicated that the bootstrap was successful in estimating the true sampling variability of eigenvector centrality for the 25 sites. This held both for centrality scores and for centrality ranks, as well as the ratio of first to second eigenvalues of the network (similarity) matrix. Findings encourage the use of the bootstrap as a tool in analyses of network data derived from counts.     

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.     

Centrality and power revisited: actor success in group decision making

For many years, network analysts viewed positional centrality as a source of social power. More recently, laboratory studies of exchange networks have called the centrality–power link into question: under zero-sum exchange conditions, the ability of certain actors to directly exploit others has been found to account for power independent of actors' centrality. But most observers believe that in non-zero-sum communication networks, centrality should positively affect power. In this study we examine the effect of centrality on power in a communication network involving group voting on political issues. Using a model in which actors' votes are determined by the strength of their initial positions and the social pressures to which they are subjected, we conduct computer simulations to examine the extent to which actors in various network positions achieve favorable political outcomes. Our findings indicate that the link between centrality and power is highly contingent on the structure of the network. In networks with a central actor and an odd number of subgroups, central actors fail to dominate. In fact, in these networks, when peripheral actors are able to directly influence one another, the central actor becomes the least powerful in the network. In networks with a central actor and an even number of subgroups, however, the central actor dominates even in situations with connected peripherals. The highly contingent effect of centrality on power accords with the findings of exchange theorists who have studied power under zero-sum conditions. This raises questions about the nature of the distinction between communication and exchange 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.     

Statistical power of the social network autocorrelation model

The network autocorrelation model has become an increasingly popular tool for conducting social network analysis. More and more researchers, however, have documented evidence of a systematic negative bias in the estimation of the network effect (ρ). In this paper, we take a different approach to the problem by investigating conditions under which, despite the underestimation bias, a network effect can still be detected by the network autocorrelation model. Using simulations, we find that moderately-sized network effects (e.g., ρ = .3) are still often detectable in modest-sized networks (i.e., 40 or more nodes). Analyses reveal that statistical power is primarily a nonlinear function of network effect size (ρ) and network size (N), although both of these factors can interact with network density and network structure to impair power under certain rare conditions. We conclude by discussing implications of these findings and guidelines for users of the autocorrelation model.