An algorithm and metric for network decomposition from similarity matrices: Application to positional analysis

We present an algorithm for decomposing a social network into an optimal number of structurally equivalent classes. The k-means method is used to determine the best decomposition of the social network for various numbers of subgroups. The best number of subgroups into which to decompose a network is determined by minimizing the intra-cluster variance of similarity subject to the constraint that the improvement in going to more subgroups is better than a random network would achieve. We also describe a decomposability metric that assesses how closely the derived decomposition approaches an ideal network having only structurally equivalent classes.     

What makes a network go round? Exploring the structure of a strong component with exponential random graph models

A strong component is a subgraph in a directed network where, following the direction of ties, all nodes in the graph are reachable from one another. Mutual reachability implies that every node in the graph is theoretically able to send materials to and/or influence every other node suggesting that strong components are amongst the more egalitarian network structures. Despite this intriguing feature, they remain understudied. Using exponential random graph models (ERGM) for directed networks, we investigate the social and structural processes underlying the generation of strong components. We illustrate our argument using a network of 301 nodes and 703 personal lending ties from Renaissance Florence. ERGM shows that our strong component arises from triadic clustering alongside an absence of higher-order star structures. We contend that these processes produce a strong component with a hierarchical, rather than an egalitarian structure: while some nodes are deeply embedded in a dense network of exchange, the involvement of others is more tenuous. More generally, we argue that such tiered core-periphery strong components will predominate in networks where the social context creates conditions for an absence of preferential attachment alongside the presence of localized closure. Although disparate social processes can give rise to hierarchical strong components linked to these two structural mechanisms, in Florence they are associated with the presence of multiple dimensions of social status and the connectedness of participants across disparate network domains.     

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.     

A mixed membership model-based measure for subgroup integration in social networks

Social networks analysis often involves quantifying subgroup structure in which tie density is greater among nodes in the same subgroup than between subgroups. One such measure, subgroup insularity or segregation, is the extent that subgroups are separate from each other. We introduce a new measure, γ, which is a parameter from the mixed membership stochastic blockmodel (MMSBM; Airoldi et al., 2008), and differs from many existing measures in that γ does not depend on node membership. We compare this measure to several well-known measures and use simulation studies and real data analysis to provide insight into how this measure can be used in practice.     

Missing data in networks: exponential random graph (p1) models for networks with non-respondents

Survey studies of complete social networks often involve non-respondents, whereby certain people within the “boundary” of a network do not complete a sociometric questionnaire—either by their own choice or by the design of the study—yet are still nominated by other respondents as network partners. We develop exponential random graph (p¹) models for network data with non-respondents. We model respondents and non-respondents as two different types of nodes, distinguishing ties between respondents from ties that link respondents to non-respondents. Moreover, if we assume that the non-respondents are missing at random, we invoke homogeneity across certain network configurations to infer effects as applicable to the entire set of network actors. Using an example from a well-known network dataset, we show that treating a sizeable proportion of nodes as non-respondents may still result in estimates, and inferences about structural effects, consistent with those for the entire network.If, on the other hand, the principal research focus is on the respondent-only structure, with non-respondents clearly not missing at random, we incorporate the information about ties to non-respondents as exogenous. We illustrate this model with an example of a network within and between organizational departments. Because in this second class of models the number of non-respondents may be large, values of parameter estimates may not be directly comparable to those for models that exclude non-respondents. In the context of discussing recent technical developments in exponential random graph models, we present a heuristic method based on pseudo-likelihood estimation to infer whether certain structural effects may contribute substantially to the predictive capacity of a model, thereby enabling comparisons of important effects between models with differently sized node sets.     

Subtyping Non-treatment-seeking Problem Gamblers Using the Pathways Model

This study examined whether distinct subgroups could be identified among a sample of non-treatment-seeking problem and pathological/disordered gamblers (PG) using Blaszczynski and Nower’s (Addiction 97:487–499, 2002) pathways model (N = 150, 50% female). We examined coping motives for gambling, childhood trauma, boredom proneness, risk-taking, impulsivity, attention-deficit/hyperactivity disorder (ADHD), and antisocial personality disorder as defining variables in a hierarchical cluster analysis to identify subgroups. Subgroup differences in gambling, psychiatric, and demographic variables were also assessed to establish concurrent validity. Consistent with the pathways model, our analyses identified three gambling subgroups: (1) behaviorally conditioned (BC), (2) emotionally vulnerable (EV), and (3) antisocial-impulsivist (AI) gamblers. BC gamblers (n = 47) reported the lowest levels of lifetime depression, anxiety, gambling severity, and interest in problem gambling treatment. EV gamblers (n = 53) reported the highest levels of childhood trauma, motivation to gamble to cope with negative emotions, gambling-related suicidal ideation, and family history of gambling problems. AI gamblers (n = 50) reported the highest levels of antisocial personality disorder and ADHD symptoms, as well as higher rates of impulsivity and risk-taking than EV gamblers. The findings provide evidence for the validity of the pathways model as a framework for conceptualizing PG subtypes in a non-treatment-seeking sample, and underscore the importance of tailoring treatment approaches to meet the respective clinical needs of these subtypes.     

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.     

A measure of distance between judgment sets

In the literature on judgment aggregation, an important open question is how to measure the distance between any two judgment sets. This is relevant for issues of social choice: if two individuals hold different beliefs then we might want to find a compromise that lies somewhere between them. We propose a set of axioms that determine a measure of distance uniquely. This measure differs from the widely used Hamming metric. The difference between Hamming’s metric and ours boils down to one axiom. Given judgment sets A and B, this axiom says that if the propositions in ${A \cap B}$ jointly imply that the propositions in A?B share the same truth value, then the disagreement between A and B over those propositions in A?B should be counted as a single disagreement. We consider the application of our metric to judgment aggregation, and also use the metric to measure the distance between preference rankings.     

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.     

Where Lies the Harm in Lottery Gambling? A Portrait of Gambling Practices and Associated Problems

Lotteries are one of the most prevalent forms of gambling and generate substantial state revenues. They are also argued to be one of the least harmful forms of gambling. This paper is one of the first to examine exclusive lottery gamblers and compares their gambling patterns and problems as well other associated risky behaviours to those who are not exclusive lottery gamblers. Data were derived from two large surveys conducted with representative adult samples in France (n?=?15,635) and Québec (n?=?23,896). Participants were separated into two groups: exclusive lottery gamblers (ELGs) and non-exclusive lottery gamblers. Using multivariate analysis, study results reveal that ELGs, who represent two thirds of gamblers, generally exhibit less intensive gambling patterns and are less likely to report other risky behaviours. However, harms associated with moderate risk and problem gambling are found to be concentrated in specific subpopulations for both groups, primarily males, older individuals, and those who report lower income and education level. Given widespread participation in lotteries and concentration of harm within specific subgroups, these findings point to the need for prevention efforts despite the lower levels of harm associated with lottery gambling.     

Network unfolding

Networt unfolding is a measurement model for representing relational dta by a connected and weighted graph. If the data — partial or complete rank orders — can be represented by such a graph then the complete graph yields a representation. However, our aim is to minimize the number of lines in the representation and to find a maximally reduced graph. The maximally reduced graph for a specific set of a data may not be a tree but may contain one or more cycles. THe scale level of the weights is at least that of an ordered metric scale.Four examples are provided to illustrate the model and the algorithm to find the reduce graph. The first example serves to introduce the terms and notations and represents the similarity of Apachean languages. The communication network of the second example on the network structure of exchange of positive messages as a directed graph. In the third example on the network structure of human associative memory we show by means of Monte Carlo study that the obtained reduction of the graph is larger than to be expected by chance, and infer that the structure is different from that assumed by Anderson and Bower. In the fourth example on popularity status we conceive status as a social agreement structure.We consider network unfolding to be an alternative to other models of strucuture as, e.g. multidimensional scaling, cluster, and factor analysis. Substantial theory should guid the selection among these models.     

Detecting large cohesive subgroups with high clustering coefficients in social networks

Clique relaxations are used in classical models of cohesive subgroups in social network analysis. Clustering coefficient was introduced more recently as a structural feature characterizing small-world networks. Noting that cohesive subgroups tend to have high clustering coefficients, this paper introduces a new clique relaxation, α-cluster, defined by enforcing a lower bound α on the clustering coefficient in the corresponding induced subgraph. Two variations of the clustering coefficient are considered, namely, the local and global clustering coefficient. Certain structural properties of α-clusters are analyzed and mathematical optimization models for determining α-clusters of the largest size in a network are developed and validated using several real-life social networks. In addition, a network clustering algorithm based on local α-clusters is proposed and successfully tested.     

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.     

On the core of characteristic function games associated with exchange networks

An exchange network is a social system in which the actors gain valued resources from bilateral transactions, but an opportunity to negotiate a deal is given only to those pairs of actors whose positions are tied with each other in a fixed communication network. A transaction consists in a mutually agreed-on division of a resource pool assigned to a network line. An additional constraint imposed on such a network restricts the range of transaction sets which may happen in a single negotiation round to those consistent with a given “exchange regime.” Under the one-exchange regime every actor is permitted to make no more than one deal per round. Bienenstock and Bonacich [Bienenstock, E.J., Bonacich, P., 1992. The core as a solution to exclusionary networks. Social Networks 14, 231–243] proposed to represent a one-exchange network with an n-person game in characteristic function form. The aim of this paper is to develop a mathematical theory of games associated with homogenous one-exchange networks (network homogeneity means that all lines are assigned resource pools of the same size). The focus is on the core, the type of solution considered most important in game theory. In particular, all earlier results obtained by Bonacich are re-examined and there is given a new graph-theoretic necessary and sufficient condition for the existence of nonempty core for the game representing a homogenous one-exchange network.     

Models,methods and network topology: Experimental design for the study of interference

How should a network experiment be designed to achieve high statistical power? Experimental treatments on networks may spread. Randomizing assignment of treatment to nodes enhances learning about the counterfactual causal effects of a social network experiment and also requires new methodology (ex. Aronow and Samii, 2017a, Bowers et al., 2013, Toulis and Kao, 2013). In this paper we show that the way in which a treatment propagates across a social network affects the statistical power of an experimental design. As such, prior information regarding treatment propagation should be incorporated into the experimental design. Our findings justify reconsideration of standard practice in circumstances where units are presumed to be independent even in simple experiments: information about treatment effects is not maximized when we assign half the units to treatment and half to control. We also present an example in which statistical power depends on the extent to which the network degree of nodes is correlated with treatment assignment probability. We recommend that researchers think carefully about the underlying treatment propagation model motivating their study in designing an experiment on a network.     

Peer influence groups: identifying dense clusters in large networks

Sociologists have seen a dramatic increase in the size and availability of social network data. This represents a poverty of riches, however, since many of our analysis techniques cannot handle the resulting large (tens to hundreds of thousands of nodes) networks. In this paper, I provide a method for identifying dense regions within large networks based on a peer influence model. Using software familiar to most sociologists, the method reduces the network to a set of m position variables that can then be used in fast cluster analysis programs. The method is tested against simulated networks with a known small-world structure showing that the underlying clusters can be accurately recovered. I then compare the performance of the procedure with other subgroup detection algorithms on the MacRea and Gagnon prison friendship data and a larger adolescent friendship network, showing that the algorithm replicates other procedures for small networks and outperforms them on the larger friendship network.     

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.     

Using egocentered networks in survey research. A methodological preview on an application of social network analysis in the area of family research

Can questions about families be adequately answered with household survey data? This question leads us to the problem to defining the family. To help us answer this question we use network analysis as a tool to examine the family without a preconceived definition. To get information about the population of family configurations in Federal Republic of Germany, we choose a survey technique that measured ego-centered networks (10–12 network generators, 4–6 network interpreters). Based on two pretests (n₁ = 98, n₂ = 534) and the main survey (n = 10,000), we discuss some problems encountered during data collection. We also examine the validity and reliability of the data. We used variations of the measurement instrument and asked a group of 99 respondents twice to test reliability. Then we compare our results with census data and compatible US and German studies to test validity. The results show: (1) It is possible to get reliable network data by using survey technique. (2) Size of networks, named persons, named relations are reasonably stable. (3) The instrument to measure networks is constant against minor changes in formulation of questionnaire and against variation of collecting techniques. These findings hold both on the macro level (aggregate data), and also on the micro level (person related data).