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.

The redundancy of a system denotes the amount of duplicate components or mechanisms in it. For a network, especially one in which mass or information is being transferred from an origin to a destination, redundancy is related to the robustness of the system. Existing network measures of redundancy rely on local connectivity (e.g. clustering coefficients) or the existence of multiple paths. As in many systems there are functional dependencies between components and paths, a measure that not only characterizes the topology of a network, but also takes into account these functional dependencies, becomes most desirable.

We propose a network redundancy measure in a prototypical model that contains functionally dependent directed paths: a Boolean model of a signal transduction network. The functional dependencies are made explicit by using an expanded network and the concept of elementary signaling modes (ESMs). We define the redundancy of a Boolean signal transduction network as the maximum number of node-independent ESMs and develop a methodology for identifying all maximal node-independent ESM combinations. We apply our measure to a number of signal transduction network models and show that it successfully distills known properties of the systems and offers new functional insights. The concept can be easily extended to similar related forms, e.g. edge-independent ESMs.

One of the fundamental principles driving diversity or homogeneity in domains such as cultural differentiation, political affiliation, and product adoption is the tension between two forces: influence (the tendency of people to become similar to others they interact with) and selection (the tendency to be affected most by the behavior of others who are already similar). Influence tends to promote homogeneity within a society, while selection frequently causes fragmentation. When both forces act simultaneously, it becomes an interesting question to analyze which societal outcomes should be expected.

To study this issue more formally, we analyze a natural stylized model built upon active lines of work in political opinion formation, cultural diversity, and language evolution. We assume that the population is partitioned into “types” according to some traits (such as language spoken or political affiliation). While all types of people interact with one another, only people with sufficiently similar types can possibly influence one another. The “similarity” is captured by a graph on types in which individuals of the same or adjacent types can influence one another. We achieve an essentially complete characterization of (stable) equilibrium outcomes and prove convergence from all starting states. We also consider generalizations of this model.

The dynamic drivers of interfirm interactions across space have rarely been explored in the context of disaster recovery; therefore, the mechanism through which shocks propagate is unclear. This paper uses stochastic actor-oriented modeling to examine how trade networks among the 500 largest Japanese companies evolved during 2010 and 2011, i.e. before and after the Great East Japan Earthquake to identify sources of vulnerability in the system. In contrast to previous reports on broken supply chains, the network displayed only modest change even in the directly affected areas. Controlling for distance and for firm size, we find that when firms changed their partners, they preferred firms that were popular among other firms, that had partners in common with them and that also bought some products or services from them. These findings concur with a criticism that Japanese firms avoid external actors and exhibit inflexibility in reorganizing their networks in times of need, which contrasts with the non-cliquish network structures observed in high-performing economic sectors. The results also highlight the role of energy firms in disaster resilience. Unlike other large Japanese companies that cluster in major urban centers, energy firms are distributed across Japan. However, despite their peripheral physical locations, energy firms are centrally located in trade networks. Thus, while a disaster in any region may affect some energy firms and lead to large-scale temporary shocks, the entire network is unlikely to be disconnected by any region-specific disaster because of the spatial distribution of the topological network core formed by energy companies.

Models of network evolution are based on the implicit assumption that network growth is continuous, uniform, and steady. Using the data collected from a large online-blogging platform, we show that the addition and removal of network ties by users do not occur sporadically at isolated nodes spread all over the network, as assumed by the vast majority of stochastic network models, but rather occur in brief bursts of intense local activity.

These bursts of network growth and attrition (addition and removal of network ties) are highly localized around focal nodes. Such network changes coincide with nearly instantaneous densification of the ties between the affected nodes, resulting in an increase of local clustering. Furthermore, we find that these network changes are tightly coupled to the dynamics of individual attributes, particularly the increase in homology between neighboring nodes (homophily) within the scope of the burst. Coincidence of the localized network change with the increase in homophily suggests a strong coupling between the selection and influence processes that lead to simultaneous elevation of assortativity and clustering.

Traveling homeless youth (i.e., youth who migrate from city to city, after a short period of time) exhibit elevated levels of risky substance use behaviors compared to non-traveling youth. Using sociometric data, we examined the characteristics of homeless youth who are connected to at least one traveler and how at least one connection to travelers was associated with substance use of travelers and non-travelers. Data were collected from 386 homeless youth at two drop-in centers in Los Angeles, CA. Results indicated that, comparing to what would be expected by chance, youth who reported at least one connection to travelers were more likely to use substances, more travelers reported at least one tie to another traveler, and more substance-using travelers reported at least one tie to other substance-using travelers. Multivariate analyses revealed that binge drinking and marijuana use were associated with homeless youth reporting connections to at least one substance-using traveler. Injection drug use was associated with reporting at least one connection to an injection drug user, regardless of traveler status, once we controlled for demographic characteristics and traveler status. We suggest the need for substance-using traveler-specific services using brief one-on-one interventions to prevent further engagement in risky substance use.

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.

Where do new markets come from? I construct a network model in which national markets are nodes and flows of recorded music between them are links and conduct a longitudinal analysis of the global pattern of trade in the period 1976–2010. I hypothesize that new export markets are developed through a process of transitive closure in the network of international trade. When two countries' markets experience the same social influences, it brings them close enough together for new homophilous ties to be formed. The implication is that consumption of foreign products helps, not hurts, home-market producers develop overseas markets, but only in those countries that have a history of consuming the same foreign products that were consumed in the home market. Selling in a market changes what is valued in that market, and new market formation is a consequence of having social influences in common.

In this paper, we attempt to investigate the attack tolerance of human mobility networks where the mobility is restricted to some extent, for instance, in a hospital, one is not allowed to access all locations. Similar situations also arise in schools. In such a network, we will show that people need to rely upon some intermediate agents, popularly known as the brokers to disseminate information. In order to establish this fact, we have followed the approach of attack in a network which in turn helps to identify important nodes in the network in order to maintain the overall connectivity. In this direction, we have proposed, a new temporal metric, brokerage frequency which significantly outperforms all other state-of-the-art attack strategies reported in Trajanovski et al. (2012), Sur et al. (2015).

Collaboration networks provide a method for examining the highly heterogeneous structure of collaborative communities. However, we still have limited theoretical understanding of how individual heterogeneity relates to network heterogeneity. The model presented here provides a framework linking an individual's skill set to her position in the collaboration network, and the distribution of skills in the population to the structure of the collaboration network as a whole. This model suggests that there is a non-trivial relationship between skills and network position: individuals with a useful combination of skills will have a disproportionate number of links in the network. Indeed, in some cases, an individual's degree is non-monotonic in the number of skills she has—an individual with very few skills may outperform an individual with many. Special cases of the model suggest that the degree distribution of the network will be skewed, even when the distribution of skills is uniform in the population. The degree distribution becomes more skewed as problems become more difficult, leading to a community dominated by a few high-degree superstars. This has striking implications for labor market outcomes in industries where production is largely the result of collaborative effort.

Social network analysis is the study of how links between a set of actors are formed. Typically, it is believed that links are formed in a structured manner, which may be due to, for example, political or material incentives, and which often may not be directly observable. The stochastic blockmodel represents this structure using latent groups which exhibit different connective properties, so that conditional on the group membership of two actors, the probability of a link being formed between them is represented by a connectivity matrix. The mixed membership stochastic blockmodel extends this model to allow actors membership to different groups, depending on the interaction in question, providing further flexibility.

Attribute information can also play an important role in explaining network formation. Network models which do not explicitly incorporate covariate information require the analyst to compare fitted network models to additional attributes in a post-hoc manner. We introduce the mixed membership of experts stochastic blockmodel, an extension to the mixed membership stochastic blockmodel which incorporates covariate actor information into the existing model. The method is illustrated with application to the Lazega Lawyers dataset. Model and variable selection methods are also discussed.

Social networks, particularly those defined by friendships, influence many childhood and adolescent health behaviors such as the use of alcohol, tobacco, and other drugs, as well as diet and physical activity. Few, if any, studies have examined the concordance between friendship networks and sun exposure/safety behaviors. This study examines the friendship networks and sun safety behaviors for a group of fourth and fifth grade students taking part in a larger sun safety intervention, “SunSmart” (n = 128). Intra-class correlation, homophily hypothesis testing, and exponential random graph models were used to test friendship homophily based on sun safety behaviors. Peer Leaders were identified through social network popularity, and sun safety change scores were compared between Peer Leaders and non-leaders. Results show that students cluster based on shared demographic characteristics and some sun safety behaviors, and that there was a trend for Peer Leaders to respond better to the SunSmart intervention than non-leaders. Implications for future sun safety interventions using Peer Leaders as champions for sun safety behavior change are discussed.

We investigate the role of manipulation in boundedly rational opinion dynamics. Agents are subject to persuasion bias and repeatedly communicate with their neighbors in a social network. They can exert effort to manipulate trust in the opinions of others in their favor and update their opinions about some issue of common interest by taking weighted averages of neighbors' opinions. We show that manipulation can connect a segregated society and thus lead to mutual consensus. Second, we show that manipulation fosters opinion leadership; and surprisingly agents with low trust in their own opinion might get more influential even by being manipulated. Finally, comparative simulations reveal that manipulation is beneficial to information aggregation when preferences and abilities for manipulation are homogeneous, but detrimental in case abilities are concentrated at few powerful agents.

We study the persistence of network segregation in networks characterized by the co-evolution of vertex attributes and link structures, in particular where individual vertices form linkages on the basis of similarity with other network vertices (homophily), and where vertex attributes diffuse across linkages, making connected vertices more similar over time (influence). A general mathematical model of these processes is used to examine the relative influence of homophily and influence in the maintenance and decay of network segregation in self-organizing networks. While prior work has shown that homophily is capable of producing strong network segregation when attributes are fixed, we show that adding even minute levels of influence is sufficient to overcome the tendency towards segregation even in the presence of relatively strong homophily processes. This result is proven mathematically for all large networks and illustrated through a series of computational simulations that account for additional network evolution processes. This research contributes to a better theoretical understanding of the conditions under which network segregation and related phenomenon—such as community structure—may emerge, which has implications for the design of interventions that may promote more efficient network structures.

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.

Network centrality has been addressed for more than 30 years; however, few studies provided statistical tests for verifying this network characteristic. By applying the idea of focus test in spatial analysis, we propose a statistical method to test the centrality of a network. We consider not only the degree of node, but also give weights on other nodes with different lengths of the shortest paths, which are called “distances” in networks. According to the density of distance based on the hidden variable model and the property of the multinomial distribution, a test statistic called “focus centrality” is provided to evaluate a network centrality. Besides the theoretical construction, we verify that the proposed method is feasible and effective in the simulation studies. Further, two empirical network data are studied as demonstrations.

Despite the pivotal role that both power and interpersonal trust play in a multitude of social exchange situations, relatively little is known about their interplay. Moreover, previous theorizing makes competing claims. Do we consider our relatively more powerful exchange partners to be less trustworthy, as rational choice reasoning would suggest? Or do more complex psychological mechanisms lead us to trust them more, as motivated cognition reasoning implies? Extending the latter approach, we develop and empirically test three hypotheses on the interrelation between perceptions of interpersonal trust and power. According to the status value hypothesis, individuals are more likely to befriend those whom they or others perceive as powerful. The status signaling hypothesis states that the friends of people one perceives as powerful will also be seen as powerful. According to the self-monitoring hypothesis, high self-monitors are more likely than low self-monitors to befriend those they or others perceive as powerful. We use multiplex stochastic actor-based models to analyze the co-evolution of trust and power relations among n = 49 employees in a Dutch Youth Care organization. Data covers three waves of a longitudinal sociometric network survey collected over a period of 18 months in the years 2009–2010. In general, we find some support for all three hypotheses, though the effects are weak. Being one of the first organizational field studies on the co-evolution of power and trust, we conclude with discussing the implications of these findings for the study of social exchange processes.

We consider the problem of generating uniformly random graphs from a constrained distribution. A graph is valid if it obeys certain constraints such as a given number of nodes, edges, k-stars or degree sequence, and each graph must occur with equal probability. A typical application is to confirm the correctness of a model by repeated sampling and comparing statistical properties against empirical data. Markov Chain Monte Carlo (MCMC) algorithms are often used, but have certain difficulties such as the inability to search the space of all possible valid graphs. We propose an improved algorithm which overcomes these difficulties. Although each individual iteration of the MCMC algorithm takes longer, we obtain better coverage of the search space in the same amount of time. This leads to better estimates of various quantities such as the expected number of transitive triads given the constraints. The algorithm should be of general interest with many possible applications, including the world wide web, biological, and social networks.

We confirm a conjecture by Everett et al. (2004) regarding the problem of maximizing closeness centralization in two-mode data, where the number of data of each type is fixed. Intuitively, our result states that among all networks obtainable via two-mode data, the largest closeness is achieved by simply locally maximizing the closeness of a node. Mathematically, our study concerns bipartite graphs with fixed size bipartitions, and we show that the extremal configuration is a rooted tree of depth 2, where neighbors of the root have an equal or almost equal number of children.

Distributed information transfer is of paramount importance to the effectiveness of dynamic collective behaviors, especially when a swarm is confronted with complex environmental circumstances. Recently, the signaling network of interaction underlying such effective information transfers has been revealed in the particular case of bird flocks governed by a topological interaction. Such biological systems are known to be evolutionary optimized, but are also constrained by the very nature of the signaling mechanisms—owing to intrinsic limitations in sensory modalities—enabling communication among individuals. Here, we propose that artificial swarm design can be tackled from the angle of signaling network design. To this aim, we use different network models to investigate the impact of some network structural properties on the effectiveness of a specific emergent swarming behavior, namely global consensus. Two new network models are introduced, which together with the well-known Watts–Strogatz model form the basis for an analysis of the relationship between clustering, shortest path and speed to consensus. A network-theoretic approach combined with spectral graph theory tools are used to propose some signaling network design principles. Eventually, one key design principle—a concomitant reduction in clustering and connecting path—is successfully tested on simulations of swarms of self-propelled particles.