We consider the specification of effects of numerical actor attributes, having an interval level of measurement, in statistical models for directed social networks. A fundamental mechanism is homophily or assortativity, where actors have a higher likelihood to be tied with others having similar values of the variable under study. But there are other mechanisms that may also play a role in how the attribute values of two actors influence the likelihood of a tie between them. We discuss three additional mechanisms: aspiration, the tendency to send more ties to others having high values; attachment conformity, sending more ties to others whose values are close to the “social norm”; and sociability, where those having higher values will tend to send more ties generally. These mechanisms may operate jointly, and then their effects will be confounded. We present a specification representing these effects simultaneously by a four-parameter quadratic function of the values of sender and receiver. Flexibility can be increased by a five-parameter extension. We argue that for numerical actor attributes having important effects on directed networks, these specifications may provide an improvement. An illustration is given of dependence of advice ties on academic grades, analyzed by the Stochastic Actor-oriented Model.

When collecting egocentric network data, visual representations of networks can function as a cognitive aid for depicting relationships, helping to maintain an overview of the relationships, and keeping the attention of the interviewees. Additionally, network maps can serve as a narration generator in qualitative and in mixed-methods studies. While varying visual instruments are used for collecting egocentric network data, little is known about differences among visual tools concerning the influence on the resulting network data, the usability for interviewees, and data validity. The article provides an overview of existing visually oriented tools that are used to collect egocentric networks and discusses their functions, advantages, and limitations. Then, we present results of an experimental study where we compare four different visual tools with regard to networks elicited, manageability, and the impact of follow-up questions. In order to assess the manageability of the four tools, we used the thinking aloud method. The results provide evidence that the decision in favor of a specific visual tool (structured vs. unstructured) can affect the size and composition of the elicited networks. Follow-up questions greatly affect the elicited networks and follow-up cues can level out differences among tools. Respondents tend to prefer the concentric circles tool, with some differences in preferences and manageability of tools between participants with low and those with high socioeconomic status. Finally, assets and drawbacks of the four instruments are discussed with regard to data quality and crucial aspects of the data collection process when using visual tools.

We derive properties of latent variable models for networks, a broad class of models that includes the widely used latent position models. We characterize several features of interest, with particular focus on the degree distribution, clustering coefficient, average path length, and degree correlations. We introduce the Gaussian latent position model, and derive analytic expressions and asymptotic approximations for its network properties. We pay particular attention to one special case, the Gaussian latent position model with random effects, and show that it can represent the heavy-tailed degree distributions, positive asymptotic clustering coefficients, and small-world behaviors that often occur in observed social networks. Finally, we illustrate the ability of the models to capture important features of real networks through several well-known datasets.

We assess the reliability of relational event model (REM) parameters estimated under two sampling schemes: (1) uniform sampling from the observed events and (2) case–control sampling which samples nonevents, or null dyads (“controls”), from a suitably defined risk set. We experimentally determine the variability of estimated parameters as a function of the number of sampled events and controls per event, respectively. Results suggest that REMs can be reliably fitted to networks with more than 12 million nodes connected by more than 360 million dyadic events by analyzing a sample of some tens of thousands of events and a small number of controls per event. Using the data that we collected on the Wikipedia editing network, we illustrate how network effects commonly included in empirical studies based on REMs need widely different sample sizes to be reliably estimated. For our analysis we use an open-source software which implements the two sampling schemes, allowing analysts to fit and analyze REMs to the same or other data that may be collected in different empirical settings, varying sample parameters or model specification.

We sketch the history of spectral ranking—a general umbrella name for techniques that apply the theory of linear maps (in particular, eigenvalues and eigenvectors) to matrices that do not represent geometric transformations, but rather some kind of relationship between entities. Albeit recently made famous by the ample press coverage of Google's PageRank algorithm, spectral ranking was devised more than 60 years ago, almost exactly in the same terms, and has been studied in psychology, social sciences, bibliometrics, economy, and choice theory. We describe the contribution given by previous scholars in precise and modern mathematical terms: Along the way, we show how to express in a general way damped rankings, such as Katz's index, as dominant eigenvectors of perturbed matrices, and then use results on the Drazin inverse to go back to the dominant eigenvectors by a limit process. The result suggests a regularized definition of spectral ranking that yields for a general matrix a unique vector depending on a boundary condition.

A measure of relative importance of network effects in the stochastic actor-oriented model (SAOM) is proposed. The SAOM is a parametric model for statistical inference in longitudinal social networks. The complexity of the model makes the interpretation of inferred results difficult. So far, the focus is on significance tests while the relative importance of effects is usually ignored. Indeed, there is no established measure to determine the relative importance of an effect in a SAOM. We introduce such a measure based on the influence effects have on decisions of individual actors in the network. We demonstrate its utility on empirical data by analyzing an evolving friendship network of university freshmen.

In dynamic networks, the presence of ties are subject both to endogenous network dependencies and spatial dependencies. Current statistical models for change over time are typically defined relative to some initial condition, thus skirting the issue of where the first network came from. Additionally, while these longitudinal network models may explain the dynamics of change in the network over time, they do not explain the change in those dynamics. We propose an extension to the longitudinal exponential random graph model that allows for simultaneous inference of the changes over time and the initial conditions, as well as relaxing assumptions of time-homogeneity. Estimation draws on recent Bayesian approaches for cross-sectional exponential random graph models and Bayesian hierarchical models. This is developed in the context of foreign direct investment relations in the global electricity industry in 1995–2003. International investment relations are known to be affected by factors related to: (i) the initial conditions determined by the geographical locations; (ii) time-dependent fluctuations in the global intensity of investment flows; and (iii) endogenous network dependencies. We rely on the well-known gravity model used in research on international trade to represent how spatial embedding and endogenous network dependencies jointly shape the dynamics of investment relations.

A wide variety of networked systems in human societies are composed of repeated communications between actors. A dyadic relationship made up of repeated interactions may be reciprocal (both actors have the same probability of directing a communication attempt to the other) or non-reciprocal (one actor has a higher probability of initiating a communication attempt than the other). In this paper we propose a theoretically motivated index of reciprocity appropriate for networks formed from repeated interactions based on these probabilities. We go on to examine the distribution of reciprocity in a large-scale social network built from trace-logs of over a billion cell-phone communication events across millions of actors in a large industrialized country. We find that while most relationships tend toward reciprocity, a substantial minority of relationships exhibit large levels of non-reciprocity. This is puzzling because behavioral theories in social science predict that persons will selectively terminate non-reciprocal relationships, keeping only those that approach reciprocity. We point to two structural features of human communication behavior and relationship formation—the division of contacts into strong and weak ties and degree-based assortativity—that either help or hinder the ability of persons to obtain communicative balance in their relationships. We examine the extent to which deviations from reciprocity in the observed network are partially traceable to the operation of these countervailing tendencies.

Most of the current complex networks that are of interest to practitioners possess a certain community structure that plays an important role in understanding the properties of these networks. For instance, a closely connected social communities exhibit faster rate of transmission of information in comparison to loosely connected communities. Moreover, many machine learning algorithms and tools that are developed for complex networks try to take advantage of the existence of communities to improve their performance or speed. As a result, there are many competing algorithms for detecting communities in large networks. Unfortunately, these algorithms are often quite sensitive and so they cannot be fine-tuned for a given, but a constantly changing, real-world network at hand. It is therefore important to test these algorithms for various scenarios that can only be done using synthetic graphs that have built-in community structure, power law degree distribution, and other typical properties observed in complex networks. The standard and extensively used method for generating artificial networks is the LFR graph generator. Unfortunately, this model has some scalability limitations and it is challenging to analyze it theoretically. Finally, the mixing parameter μ, the main parameter of the model guiding the strength of the communities, has a non-obvious interpretation and so can lead to unnaturally defined networks. In this paper, we provide an alternative random graph model with community structure and power law distribution for both degrees and community sizes, the Artificial Benchmark for Community Detection (ABCD graph). The model generates graphs with similar properties as the LFR one, and its main parameter ξ can be tuned to mimic its counterpart in the LFR model, the mixing parameter μ. We show that the new model solves the three issues identified above and more. In particular, we test the speed of our algorithm and do a number of experiments comparing basic properties of both ABCD and LFR. The conclusion is that these models produce graphs with comparable properties but ABCD is fast, simple, and can be easily tuned to allow the user to make a smooth transition between the two extremes: pure (independent) communities and random graph with no community structure.

A recurrent finding in personal network research is that individual and social outcomes are influenced not just by the kind of people one knows, but also by how those people are connected to each other. Personal network structure – the way in which one’s personal contacts know and interact with each other – reflects broader trends in social organization and personal communities, and shapes patterns of social capital, support, and isolation. This article proposes a method to identify typologies of structure in large collections of personal networks. The method is applied to six datasets collected in widely different circumstances and using various survey instruments. It is then compared with another recently introduced method to extract typologies of egocentric network structure. Findings show that personal network structure can be effectively summarized using just three measures of cohesive subgroup characteristics. Structural typologies can then be identified by applying standard cluster analysis techniques to the three variables. Both methods considered in the article capture significant variation in network structures, but they also show substantial levels of disagreement and cross-classification. I discuss similarities and differences between the methods, and potential applications of the proposed typologies to substantive research on personal communities, social support, and social capital.

Face-to-face social contacts are potentially important transmission routes for acute respiratory infections, and understanding the contact network can improve our ability to predict, contain, and control epidemics. Although workplaces are important settings for infectious disease transmission, few studies have collected workplace contact data and estimated workplace contact networks. We use contact diaries, architectural distance measures, and institutional structures to estimate social contact networks within a Swiss research institute. Some contact reports were inconsistent, indicating reporting errors. We adjust for this with a latent variable model, jointly estimating the true (unobserved) network of contacts and duration-specific reporting probabilities. We find that contact probability decreases with distance, and that research group membership, role, and shared projects are strongly predictive of contact patterns. Estimated reporting probabilities were low only for 0–5 min contacts. Adjusting for reporting error changed the estimate of the duration distribution, but did not change the estimates of covariate effects and had little effect on epidemic predictions. Our epidemic simulation study indicates that inclusion of network structure based on architectural and organizational structure data can improve the accuracy of epidemic forecasting models.

Homeless men are frequently unsheltered and isolated, disconnected from supportive organizations and individuals. However, little research has investigated these men's social networks. We investigate the structure and composition of homeless men's social networks, vis-a-vis short- and long-term homelessness with a sample of men drawn randomly from meal lines on Skid Row in Los Angeles. Men continuously homeless for the past six months display networks composed of riskier members when compared to men intermittently homeless during that time. Men who report chronic, long-term homelessness display greater social network fragmentation when compared to non-chronically homeless men. While intermittent homelessness affects network composition in ways that may be addressable with existing interventions, chronic homelessness fragments networks, which may be more difficult to address with those interventions. These findings have implications for access to social support from network members which, in turn, impacts the resources homeless men require from other sources such as the government or non-governmental organizations.

Many studies that gather social network data use survey methods that lead to censored, missing, or otherwise incomplete information. For example, the popular fixed rank nomination (FRN) scheme, often used in studies of schools and businesses, asks study participants to nominate and rank at most a small number of contacts or friends, leaving the existence of other relations uncertain. However, most statistical models are formulated in terms of completely observed binary networks. Statistical analyses of FRN data with such models ignore the censored and ranked nature of the data and could potentially result in misleading statistical inference. To investigate this possibility, we compare Bayesian parameter estimates obtained from a likelihood for complete binary networks with those obtained from likelihoods that are derived from the FRN scheme, and therefore accommodate the ranked and censored nature of the data. We show analytically and via simulation that the binary likelihood can provide misleading inference, particularly for certain model parameters that relate network ties to characteristics of individuals and pairs of individuals. We also compare these different likelihoods in a data analysis of several adolescent social networks. For some of these networks, the parameter estimates from the binary and FRN likelihoods lead to different conclusions, indicating the importance of analyzing FRN data with a method that accounts for the FRN survey design.

To expand the toolbox available to network science, we study the isomorphism between distance and Fuzzy (proximity or strength) graphs. Distinct transitive closures in Fuzzy graphs lead to closures of their isomorphic distance graphs with widely different structural properties. For instance, the All Pairs Shortest Paths (APSP) problem, based on the Dijkstra algorithm, is equivalent to a metric closure, which is only one of the possible ways to calculate shortest paths in weighted graphs. We show that different closures lead to different distortions of the original topology of weighted graphs. Therefore, complex network analyses that depend on the calculation of shortest paths on weighted graphs should take into account the closure choice and associated topological distortion. We characterize the isomorphism using the max-min and Dombi disjunction/conjunction pairs. This allows us to: (1) study alternative distance closures, such as those based on diffusion, metric, and ultra-metric distances; (2) identify the operators closest to the metric closure of distance graphs (the APSP), but which are logically consistent; and (3) propose a simple method to compute alternative path length measures and corresponding distance closures using existing algorithms for the APSP. In particular, we show that a specific diffusion distance is promising for community detection in complex networks, and is based on desirable axioms for logical inference or approximate reasoning on networks; it also provides a simple algebraic means to compute diffusion processes on networks. Based on these results, we argue that choosing different distance closures can lead to different conclusions about indirect associations on network data, as well as the structure of complex networks, and are thus important to consider.

We consider the task of simultaneous clustering of the two node sets involved in a bipartite network. The approach we adopt is based on use of the exact integrated complete likelihood for the latent blockmodel. Using this allows one to infer the number of clusters as well as cluster memberships using a greedy search. This gives a model-based clustering of the node sets. Experiments on simulated bipartite network data show that the greedy search approach is vastly more scalable than competing Markov chain Monte Carlo-based methods. Application to a number of real observed bipartite networks demonstrate the algorithms discussed.

This paper proposes a model of network interactions in the interbank market. Our innovation is to model systemic risk in the interbank network as the propagation of incentives or strategic behavior rather than the propagation of losses after default. Transmission in our model is not based on default. Instead, we explain bank profitability based on competition incentives and the outcome of a strategic game. As competitors' lending decisions change, banks adjust their own decisions as a result: generating a “transmission” of shocks through the system. We provide a unique equilibrium characterization of a static model, and embed this model into a full dynamic model of network formation. We also determine the key bank, which is the bank that is crucial for the stability of the financial network.

Nowadays, scientific challenges usually require approaches that cross traditional boundaries between academic disciplines, driving many researchers towards interdisciplinarity. Despite its obvious importance, there is a lack of studies on how to quantify the influence of interdisciplinarity on the research impact, posing uncertainty in a proper evaluation for hiring and funding purposes. Here, we propose a method based on the analysis of bipartite interconnected multilayer networks of citations and disciplines, to assess scholars, institutions, and countries interdisciplinary importance. Using data about physics publications and US patents, we show that our method allows to reward, using a quantitative approach, scholars and institutions that have carried out interdisciplinary work and have had an impact in different scientific areas. The proposed method could be used by funding agencies, universities and scientific policy decision makers for hiring and funding purposes, and to complement existing methods to rank universities and countries.

This study examines changes in individual social capital during adult life within a 19-year period. Social capital theory and life course theory are combined, and it is argued that changes in social networks do not necessarily go together with changes in social capital: while personal networks are known to decline in size with age, social capital can be expected to accumulate, in particular for those who had a better starting position and therefore more resources to share. Panel data from the survey of the social networks of the Dutch (SSND) (1999–2018) at four points of measurement are employed to inquire into this argument. Social capital is measured by the position generator instrument, and three indicators, that is, resource extensity, mean prestige access, and resource range are analyzed. Results of fixed effect models show that, on average, people maintain access to social capital, and that men and higher educated gain social capital through their life as opposed to women and lower educated. Implications for the understanding of the reproduction of social inequality are discussed. The paper concludes with a reflection upon the value of ego-centered network analysis in the era of big data and data science.

In this paper, we seek to advance an updated concept of social space that integrates the multilayer and dynamic statistical network methods currently at the disposal of social network researchers. We demonstrate the analytic value of the new concept of social space that we propose with the help of an illustrative analysis of an organizational field involving organizations' external and internal decisions that congeal into a multilevel system of action that shapes the space of possibilities for other participants in the field. Through these internal and external decisions, organizations seek certain positions in their social space while simultaneously modifying that social space over time. We conclude by arguing that network researchers' choices of goodness-of-fit statistics should reflect a consideration about the dimensions of social space of most interest to the nodes involved.

The clustering coefficient is typically used as a measure of the prevalence of node clusters in a network. Various definitions for this measure have been proposed for the cases of networks having weighted edges which may or not be directed. However, these techniques consistently assume that only a subset of all possible edges is present in the network, whereas there are weighted networks of interest in which all possible edges are present, that is, complete weighted networks. For this situation, the concept of clustering is redefined, and computational techniques are presented for computing an associated clustering coefficient for complete weighted undirected or directed networks. The performance of this new definition is compared with that of current clustering definitions when extended to complete weighted networks.