In this paper we examine natural generalizations of four widely used centrality measures to subgroups of nodes in a network. This allows for a division into local and global influence. As an example, we analyze a classic network and discuss previously hidden features made visible by these new techniques. Network-wide measures and centralization formulae are derived.

Real-world networks are often compared to random graphs to assess whether their topological structure could be a result of random processes. However, a simple random graph in large scale often lacks social structure beyond the dyadic level. As a result we need to generate clustered random graph to compare the local structure at higher network levels. In this paper a generalized version of Gleeson's algorithm G(VS, VT, ES, ET, S, T) is advanced to generate a clustered random graph in large-scale which persists the number of vertices |V|, the number of edges |E|, and the global clustering coefficient CΔ as in the real network and it works successfully for nine large-scale networks. Our new algorithm also has advantages in randomness evaluation and computation efficiency when compared with the existing algorithms.

Network science gathers methods coming from various disciplines which sometimes hardly cross the boundaries between these disciplines. Widely used in molecular biology in the study of protein interaction networks, the enumeration, in a network, of all possible subgraphs of a limited size (usually around four or five nodes), often called graphlets, can only be found in a few works dealing with social networks. In the present work, we apply this approach to an original corpus of about 10,000 non-overlapping Facebook ego networks gathered from voluntary participants by a survey application. To deal with so many similar networks, we adapt the relative graphlet frequency to a measure that we call graphlet representativity, which we show to be more effective to classify random networks having slight structural differences. From our data, we produce two clusterings, one of graphlets (paths, star-like, holes, light triangles, and dense), one of networks. The latter is presented with a visualization scheme using our representativity measure. We describe the distinct structural characteristics of the five clusters of Facebook ego networks so obtained and discuss the empirical differences between results obtained with 4-node and 5-node graphlets. We also provide suggestions of follow-ups of this work, both in sociology and in network science.

Latent stochastic blockmodels are flexible statistical models that are widely used in social network analysis. In recent years, efforts have been made to extend these models to temporal dynamic networks, whereby the connections between nodes are observed at a number of different times. In this paper, we propose a new Bayesian framework to characterize the construction of connections. We rely on a Markovian property to describe the evolution of nodes' cluster memberships over time. We recast the problem of clustering the nodes of the network into a model-based context, showing that the integrated completed likelihood can be evaluated analytically for a number of likelihood models. Then, we propose a scalable greedy algorithm to maximize this quantity, thereby estimating both the optimal partition and the ideal number of groups in a single inferential framework. Finally, we propose applications of our methodology to both real and artificial datasets.

Network representations of systems from various scientific and societal domains are neither completely random nor fully regular, but instead appear to contain recurring structural features. These features tend to be shared by networks belonging to the same broad class, such as the class of social networks or the class of biological networks. Within each such class, networks describing similar systems tend to have similar features. This occurs presumably because networks representing similar systems would be expected to be generated by a shared set of domain-specific mechanisms, and it should therefore be possible to classify networks based on their features at various structural levels. Here we describe and demonstrate a new hybrid approach that combines manual selection of network features of potential interest with existing automated classification methods. In particular, selecting well-known network features that have been studied extensively in social network analysis and network science literature, and then classifying networks on the basis of these features using methods such as random forest, which is known to handle the type of feature collinearity that arises in this setting, we find that our approach is able to achieve both higher accuracy and greater interpretability in shorter computation time than other methods.

Social networks influence children and adolescents' physical activity. The focus of this paper is to examine the differences in the effects of physical activity on friendship selection, with eye to the implications on physical activity interventions for young children. Prior to implementing a network intervention in the field, it is important to understand potential heterogeneities in the effects that activity level have on network structure. In this study, the associations between activity level and cross-sectional network structure, and activity level and change in network structure are assessed. We studied a real-world friendship network among 81 children (average age 7.96 years) who lived in low SES neighborhoods, attended public schools, and attended one of two structured aftercare programs, of which one has existed and the other was new. We modeled network selection effects and cross-sectional properties, while accounting for potential heterogeneities between networks. There was heterogeneity in the effect of physical activity on both cross-sectional network structure and the formation and dissolution processes, both across time and between networks. This suggests that if peer selection processes are changing within a network, a static network intervention strategy for childhood physical activity could become inefficient as the network evolves.

Chemotherapy is often administered in openly designed hospital wards, where the possibility of patient–patient social influence on health exists. Previous research found that social relationships influence cancer patient's health; however, we have yet to understand social influence among patients receiving chemotherapy in the hospital. We investigate the influence of co-presence in a chemotherapy ward. We use data on 4,691 cancer patients undergoing chemotherapy in Oxfordshire, United Kingdom who average 59.8 years of age, and 44% are Male. We construct a network of patients where edges exist when patients are co-present in the ward, weighted by both patients' time in the ward. Social influence is based on total weighted co-presence with focal patients' immediate neighbors, considering neighbors' 5-year mortality. Generalized estimating equations evaluated the effect of neighbors' 5-year mortality on focal patient's 5-year mortality. Each 1,000-unit increase in weighted co-presence with a patient who dies within 5 years increases a patient's mortality odds by 42% (β = 0.357, CI:0.204,0.510). Each 1,000-unit increase in co-presence with a patient surviving 5 years reduces a patient's odds of dying by 30% (β =−0.344, CI:−0.538,0.149). Our results suggest that social influence occurs in chemotherapy wards, and thus may need to be considered in chemotherapy delivery.

Using the direct current power flow model, we study cascading failures and their spatial and temporal properties in the U.S. Western Interconnection (USWI) power grid. We show that yield (the fraction of demand satisfied after the cascade) has a bimodal distribution typical of a first-order transition. The single line failure leads either to an insignificant power loss or to a cascade which causes a major blackout with yield less than 0.8. The former occurs with high probability if line tolerance α (the ratio of the maximal load a line can carry to its initial load) is greater than 2, while a major blackout occurs with high probability in a broad range of 1 < α < 2. We also show that major blackouts begin with a latent period (with duration proportional to α) during which few lines overload and yield remains high. The existence of the latent period suggests that intervention during early stages of a cascade can significantly reduce the risk of a major blackout. Finally, we introduce the preferential Degree And Distance Attachment model to generate random networks with similar degree, resistance, and flow distributions to the USWI. Moreover, we show that the Degree And Distance Attachment model behaves similarly to the USWI against failures.

We introduce a weighted configuration model graph, where edge weights correspond to the probability of infection in an epidemic on the graph. On these graphs, we study the development of a Susceptible–Infectious–Recovered epidemic using both Reed–Frost and Markovian settings. For the special case of having two different edge types, we determine the basic reproduction numberR0, the probability of a major outbreak, and the relative final size of a major outbreak. Results are compared with those for a calibrated unweighted graph. The degree distributions are based on both theoretical constructs and empirical network data. In addition, bivariate standard normal copulas are used to model the dependence between the degrees of the two edge types, allowing for modeling the correlation between edge types over a wide range. Among the results are that the weighted graph produces much richer results than the unweighted graph. Also, while R0 always increases with increasing correlation between the two degrees, this is not necessarily true for the probability of a major outbreak nor for the relative final size of a major outbreak. When using copulas we see that these can produce results that are similar to those of the empirical degree distributions, indicating that in some cases a copula is a viable alternative to using the full empirical data.

We discuss our outreach efforts to introduce school students to network science and explain why researchers who study networks should be involved in such outreach activities. We provide overviews of modules that we have designed for these efforts, comment on our successes and failures, and illustrate the potentially enormous impact of such outreach efforts.

Most social networks present complex structures. They can be both multi-modal and multi-relational. In addition, each relationship can be observed across time occasions. Relational data observed in such conditions can be organized into multidimensional arrays and statistical methods from the theory of multiway data analysis may be exploited to reveal the underlying data structure. In this paper, we adopt an exploratory data analysis point of view, and we present a procedure based on multiple factor analysis and multiple correspondence analysis to deal with time-varying two-mode networks. This procedure allows us to create static displays in order to explore network evolutions and to visually analyze the degree of similarity of actor/event network profiles over time while preserving the different statuses of the two modes.

We study the interplay between the diffusion of a harmful state in a network of contacts and the possibility of individual agents to undertake costly investment to protect themselves against infection. We characterize how equilibrium diffusion outcomes, such as the immunization rate, total prevalence and welfare, respond to changes in the architecture of the network, and show that these responses depend on the details of the diffusion process.

Dynamic network models describe many important scientific processes, from cell biology and epidemiology to sociology and finance. Estimating dynamic networks from noisy time series data is a difficult task since the number of components involved in the system is very large. As a result, the number of parameters to be estimated is typically larger than the number of observations. However, a characteristic of many real life networks is that they are sparse. For example, the molecular structure of genes make interactions with other components a highly-structured and, therefore, a sparse process. Until now, the literature has focused on static networks, which lack specific temporal interpretations.

We propose a flexible collection of ANOVA-like dynamic network models, where the user can select specific time dynamics, known presence or absence of links, and a particular autoregressive structure. We use undirected graphical models with block equality constraints on the parameters. This reduces the number of parameters, increases the accuracy of the estimates and makes interpretation of the results more relevant. We illustrate the flexibility of the method on both synthetic and real data.

Values—the motivational goals that define what is important to us—guide our decisions and actions every day. Their importance is established in a long line of research investigating their universality across countries and their evolution from childhood to adulthood. In adolescence, value structures are subject to substantial change, as life becomes increasingly social. Value change has thus far been understood to operate independently within each person. However, being embedded in various social systems, adolescents are constantly subject to social influence from peers. Thus, we introduce a framework investigating the emergence and evolution of value priorities in the dynamic context of friendship networks. Drawing on stochastic actor-oriented network models, we analyze 73 friendship networks of adolescents. Regarding the evolution of values, we find that adolescents’ value systems evolve in a continuous cycle of internal validation through the selection and enactment of goals—thereby experiencing both congruence and conflicts—and external validation through social comparison among their friends. Regarding the evolution of friendship networks, we find that demographics are more salient for the initiation of new friendships, whereas values are more relevant for the maintenance of existing friendships.

Relational event models (REMs) for the analysis of social interaction were first introduced 15 years ago. Since then, a number of important substantive and methodological contributions have produced their progressive refinement and hence facilitated their increased adoption in studies of social and other networks. Today REMs represent a well-established class of statistical models for relational processes. This special issue of Network Science demonstrates the standing and recognition that REMs have achieved within the network analysis and networks science communities. We wrote this brief introductory editorial essay with four main objectives in mind: (i) positioning relational event data and models in the larger context of contemporary network science and social network research; (ii) reviewing some of the most important recent developments; (iii) presenting the innovative studies collected in this special issue as evidence of the empirical value of REMs, and (iv) identifying open questions and future research directions.

Triadic closure has been conceptualized and measured in a variety of ways, most famously the clustering coefficient. Existing extensions to affiliation networks, however, are sensitive to repeat group attendance, which does not reflect common interpersonal interpretations of triadic closure. This paper proposes a measure of triadic closure in affiliation networks designed to control for this factor, which manifests in bipartite models as biclique proliferation. To avoid arbitrariness, the paper introduces a triadic framework for affiliation networks, within which a range of measures can be defined; it then presents a set of basic axioms that suffice to narrow this range to the one measure. An instrumental assessment compares the proposed and two existing measures for reliability, validity, redundancy, and practicality. All three measures then take part in an investigation of three empirical social networks, which illustrates their differences.

The validity of network observations is sometimes of concern in empirical studies, since observed networks are prone to error and may not represent the population of interest. This lack of validity is not just a result of random measurement error, but often due to systematic bias that can lead to the misinterpretation of actors’ preferences of network selections. These issues in network observations could bias the estimation of common network models (such as those pertaining to influence and selection) and lead to erroneous statistical inferences. In this study, we proposed a simulation-based sensitivity analysis method that can evaluate the robustness of inferences made in social network analysis to six forms of selection mechanisms that can cause biases in network observations—random, homophily, anti-homophily, transitivity, reciprocity, and preferential attachment. We then applied this sensitivity analysis to test the robustness of inferences for social influence effects, and we derived two sets of analytical solutions that can account for biases in network observations due to random, homophily, and anti-homophily selection.

In the context of a motivating study of dynamic network flow data on a large-scale e-commerce website, we develop Bayesian models for online/sequential analysis for monitoring and adapting to changes reflected in node–node traffic. For large-scale networks, we customize core Bayesian time series analysis methods using dynamic generalized linear models (DGLMs). These are integrated into the context of multivariate networks using the concept of decouple/recouple that was recently introduced in multivariate time series. This method enables flexible dynamic modeling of flows on large-scale networks and exploitation of partial parallelization of analysis while maintaining coherence with an over-arching multivariate dynamic flow model. This approach is anchored in a case study on Internet data, with flows of visitors to a commercial news website defining a long time series of node–node counts on over 56,000 node pairs. Central questions include characterizing inherent stochasticity in traffic patterns, understanding node–node interactions, adapting to dynamic changes in flows and allowing for sensitive monitoring to flag anomalies. The methodology of dynamic network DGLMs applies to many dynamic network flow studies.

Population analyses of functional connectivity have provided a rich understanding of how brain function differs across time, individual, and cognitive task. An important but challenging task in such population analyses is the identification of reliable features that describe the function of the brain, while accounting for individual heterogeneity. Our work is motivated by two particularly important challenges in this area: first, how can one analyze functional connectivity data over populations of individuals, and second, how can one use these analyses to infer group similarities and differences. Motivated by these challenges, we model population connectivity data as a multilayer network and develop the multi-node2vec algorithm, an efficient and scalable embedding method that automatically learns continuous node feature representations from multilayer networks. We use multi-node2vec to analyze resting state fMRI scans over a group of 74 healthy individuals and 60 patients with schizophrenia. We demonstrate how multilayer network embeddings can be used to visualize, cluster, and classify functional regions of the brain for these individuals. We furthermore compare the multilayer network embeddings of the two groups. We identify significant differences between the groups in the default mode network and salience network—findings that are supported by the triple network model theory of cognitive organization. Our findings reveal that multi-node2vec is a powerful and reliable method for analyzing multilayer networks. Data and publicly available code are available at https://github.com/jdwilson4/multi-node2vec.

Social relationships are important among persons experiencing homelessness, but there is little research on changes in social networks among persons moving into permanent supportive housing (PSH). Using data collected as part of a longitudinal study of 405 adults (aged 39+) moving into PSH, this study describes network upheaval during this critical time of transition. Interviews conducted prior to and after three months of living in PSH assessed individual-level (demographics, homelessness history, health, and mental health) and social network characteristics, including network size and composition (demographics, relationship type, and social support). Interviewers utilized network member characteristics to assess whether network members were new or sustained between baseline and three months post-housing. Multilevel logistic regression models assessed characteristics of network members associated with being newly gained or persisting in networks three months after PSH move-in. Results show only one-third of social networks were retained during the transition to PSH, and veterans, African Americans, and other racial/ethnic minorities, and those living in scattered site housing, were more likely to experience network disruption. Relatives, romantic partners, and service providers were most likely to be retained after move-in. Some network change was moderated by tie strength, including the retention of street-met persons. Implications are discussed.