Statistical inference on graphs often proceeds via spectral methods involving low-dimensional embeddings of matrix-valued graph representations such as the graph Laplacian or adjacency matrix. In this paper, we analyze the asymptotic information-theoretic relative performance of Laplacian spectral embedding and adjacency spectral embedding for block assignment recovery in stochastic blockmodel graphs by way of Chernoff information. We investigate the relationship between spectral embedding performance and underlying network structure (e.g., homogeneity, affinity, core-periphery, and (un)balancedness) via a comprehensive treatment of the two-block stochastic blockmodel and the class of K-blockmodels exhibiting homogeneous balanced affinity structure. Our findings support the claim that, for a particular notion of sparsity, loosely speaking, “Laplacian spectral embedding favors relatively sparse graphs, whereas adjacency spectral embedding favors not-too-sparse graphs.” We also provide evidence in support of the claim that “adjacency spectral embedding favors core-periphery network structure.”

Balance theory has advanced with interdisciplinary contributions from social science, physical science, engineering, and mathematics. The common focus of attention is social networks in which every individual has either a positive or negative, cognitive or emotional, appraisal of every other individual. The current frontier of work on balance theory is a hunt for a dynamical model that predicts the temporal evolution of any such appraisal network to a particular structure in the complete set of balanced networks allowed by the theory. Finding such a model has proved to be a difficult problem. In this article, we contribute a parsimonious solution of the problem that explicates the conditions under which a network will evolve either to a set of mutually antagonistic cliques or to an asymmetric structure that allows agreement, cooperation, and compromise among cliques.

Recently there has been significant work in the social sciences involving ensembles of social networks, that is, multiple, independent, social networks such as students within schools or employees within organizations. There remains, however, very little methodological work on exploring these types of data structures. We present methods for clustering social networks with observed nodal class labels, based on statistics of walk counts between the nodal classes. We extend this method to consider only non-backtracking walks, and introduce a method for normalizing the counts of long walk sequences using those of shorter ones. We then present a method for clustering networks based on these statistics to explore similarities among networks. We demonstrate the utility of this method on simulated network data, as well as on advice-seeking networks in education.

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.

Social networks, wherein the edges represent nonbehavioral relations such as friendship, power, and influence, can be difficult to measure and model. A powerful tool to address this is cognitive social structures (Krackhardt, D. (1987). Cognitive social structures. Social Networks, 9(2), 109–134.), where the perception of the entire network is elicited from each actor. We provide a formal statistical framework to analyze informants’ perceptions of the network, implementing a latent space network model that can estimate, e.g., homophilic effects while accounting for informant error. Our model allows researchers to better understand why respondents’ perceptions differ. We also describe how to construct a meaningful single aggregated network that ameliorates potential respondent error. The proposed method provides a visualization method, an estimate of the informants’ biases and variances, and we describe a method for sidestepping forced-choice designs.

The temporal exponential random graph model (TERGM) and the stochastic actor-oriented model (SAOM, e.g., SIENA) are popular models for longitudinal network analysis. We compare these models theoretically, via simulation, and through a real-data example in order to assess their relative strengths and weaknesses. Though we do not aim to make a general claim about either being superior to the other across all specifications, we highlight several theoretical differences the analyst might consider and find that with some specifications, the two models behave very similarly, while each model out-predicts the other one the more the specific assumptions of the respective model are met.

We investigate the implications of different forms of multigroup connectivity. Four multigroup connectivity modalities are considered: co-memberships, edge bundles, bridges, and liaison hierarchies. We propose generative models to generate these four modalities. Our models are variants of planted partition or stochastic block models conditioned under certain topological constraints. We report findings of a comparative analysis in which we evaluate these structures, controlling for their edge densities and sizes, on mean rates of information propagation, convergence times to consensus, and steady-state deviations from the consensus value in the presence of noise as network size increases.

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.

We investigate the impact of misinformation about the contact structure on the ability to predict disease outbreaks. We base our study on 31 empirical temporal networks and tune the frequencies in errors in the node identities or time stamps of contacts. We find that for both these spreading scenarios, the maximal misprediction of both the outbreak size and time to extinction follows an stretched exponential convergence as a function of the error frequency. We furthermore determine the temporal-network structural factors influencing the parameters of this convergence.

Most network studies rely on a measured network that differs from the underlying network which is obfuscated by measurement errors. It is well known that such errors can have a severe impact on the reliability of network metrics, especially on centrality measures: a more central node in the observed network might be less central in the underlying network. Previous studies have dealt either with the general effects of measurement errors on centrality measures or with the treatment of erroneous network data. In this paper, we propose a method for estimating the impact of measurement errors on the reliability of a centrality measure, given the measured network and assumptions about the type and intensity of the measurement error. This method allows researchers to estimate the robustness of a centrality measure in a specific network and can, therefore, be used as a basis for decision-making. In our experiments, we apply this method to random graphs and real-world networks. We observe that our estimation is, in the vast majority of cases, a good approximation for the robustness of centrality measures. Beyond this, we propose a heuristic to decide whether the estimation procedure should be used. We analyze, for certain networks, why the eigenvector centrality is less robust than, among others, the pagerank. Finally, we give recommendations on how our findings can be applied to future network studies.

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.

The topic of anomaly detection in networks has attracted a lot of attention in recent years, especially with the rise of connected devices and social networks. Anomaly detection spans a wide range of applications, from detecting terrorist cells in counter-terrorism efforts to identifying unexpected mutations during ribonucleic acid transcription. Fittingly, numerous algorithmic techniques for anomaly detection have been introduced. However, to date, little work has been done to evaluate these algorithms from a statistical perspective. This work is aimed at addressing this gap in the literature by carrying out statistical evaluation of a suite of popular spectral methods for anomaly detection in networks. Our investigation on the statistical properties of these algorithms reveals several important and critical shortcomings that we make methodological improvements to address. Further, we carry out a performance evaluation of these algorithms using simulated networks and extend the methods from binary to count networks.

This paper proposes that common measures for network transitivity, based on the enumeration of transitive triples, do not reflect the theoretical statements about transitivity they aim to describe. These statements are often formulated as comparative conditional probabilities, but these are not directly reflected by simple functions of enumerations. We think that a better approach is obtained by considering the probability of a tie between two randomly drawn nodes, conditional on selected features of the network. Two measures of transitivity based on correlation coefficients between the existence of a tie and the existence, or the number, of two-paths between the nodes are developed, and called “Transitivity Phi” and “Transitivity Correlation.” Some desirable properties for these measures are studied and compared to existing clustering coefficients, in both random (Erdös–Renyi) and in stylized networks (windmills). Furthermore, it is shown that in a directed graph, under the condition of zero Transitivity Correlation, the total number of transitive triples is determined by four underlying features: size, density, reciprocity, and the covariance between in- and outdegrees. Also, it is demonstrated that plotting conditional probability of ties, given the number of two-paths, provides valuable insights into empirical regularities and irregularities of transitivity patterns.

Many real-world networks exhibit a multicores-periphery structure, with densely connected vertices in multiple cores surrounded by a general periphery of sparsely connected vertices. Identification of the multicores-periphery structure can provide a new lens to understand the structures and functions of various real-world networks. This paper defines the multicores-periphery structure and introduces an algorithm to identify the optimal partition of multiple cores and the periphery in general networks. We demonstrate the performance of our algorithm by applying it to a well-known social network and a patent technology network, which are best characterized by the multicores-periphery structure. The analyses also reveal the differences between our multicores-periphery detection algorithm and two state-of-the-art algorithms for detecting the single core-periphery structure and community structure.

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.

The study of complex brain networks, where structural or functional connections are evaluated to create an interconnected representation of the brain, has grown tremendously over the past decade. Many of the statistical network science tools for analyzing brain networks have been developed for cross-sectional studies and for the analysis of static networks. However, with both an increase in longitudinal study designs and an increased interest in the neurological network changes that occur during the progression of a disease, sophisticated methods for longitudinal brain network analysis are needed. We propose a paradigm for longitudinal brain network analysis over patient cohorts, with the key challenge being the adaptation of Stochastic Actor-Oriented Models to the neuroscience setting. Stochastic Actor-Oriented Models are designed to capture network dynamics representing a variety of influences on network change in a continuous-time Markov chain framework. Network dynamics are characterized through both endogenous (i.e. network related) and exogenous effects, where the latter include mechanisms conjectured in the literature. We outline an application to the resting-state functional magnetic resonance imaging setting with data from the Alzheimer’s Disease Neuroimaging Initiative study. We draw illustrative conclusions at the subject level and make a comparison between elderly controls and individuals with Alzheimer’s disease.

Though declining since the 1990s, adolescent pregnancy remains common in the United States. Social supports appear to improve outcomes for pregnant teens; however, teen pregnancy introduces social obstacles, such as stigma. This study investigates how currently or previously pregnant teens’ friendship networks differ from nonpregnant girls using data from the National Longitudinal Study of Adolescent to Adult Health (Add Health) and multilevel regression models. To mitigate concerns that background differences contribute to both pregnancy risk and social networks, girls who experience a pregnancy prior to one data collection time point are compared girls who experience their first pregnancy after this time point. This group who become pregnant after the time point is presumably more similar to already pregnant teens than those never experiencing teen pregnancy. When compared to these girls who become pregnant in the future, those who have already experienced a teen pregnancy report similar numbers of friends (out-nominations) and perceived social acceptance, but are predicted to have fewer peers reporting them as friends (in-nominations) and fewer reciprocated friendships. This suggests that pregnant teens may face stigmatization, of which they may be unaware. It further highlights a new comparison group to account for selection in studies of adolescent pregnancy.

Multilayer graphs consist of several graphs, called layers, where the vertex set of all layers is the same but each layer has an individual edge set. They are motivated by real-world problems where entities (vertices) are associated via multiple types of relationships (edges in different layers). We chart the border of computational (in)tractability for the class of subgraph detection problems on multilayer graphs, including fundamental problems such as maximum-cardinality matching, finding certain clique relaxations, or path problems. Mostly encountering hardness results, sometimes even for two or three layers, we can also spot some islands of computational tractability.

Identification of influential nodes is an important step in understanding and controlling the dynamics of information, traffic, and spreading processes in networks. As a result, a number of centrality measures have been proposed and used across different application domains. At the heart of many of these measures lies an assumption describing the manner in which traffic (of information, social actors, particles, etc.) flows through the network. For example, some measures only count shortest paths while others consider random walks. This paper considers a spreading process in which a resource necessary for transit is partially consumed along the way while being refilled at special nodes on the network. Examples include fuel consumption of vehicles together with refueling stations, information loss during dissemination with error-correcting nodes, and consumption of ammunition of military troops while moving. We propose generalizations of the well-known measures of betweenness, random-walk betweenness, and Katz centralities to take such a spreading process with consumable resources into account. In order to validate the results, experiments on real-world networks are carried out by developing simulations based on well-known models such as Susceptible-Infected-Recovered and congestion with respect to particle hopping from vehicular flow theory. The simulation-based models are shown to be highly correlated with the proposed centrality measures.

Reproducibility: Our code and experiments are available at https://github.com/hmwesigwa/soc_centrality

The recently developed bag-of-paths (BoP) framework consists in setting a Gibbs–Boltzmann distribution on all feasible paths of a graph. This probability distribution favors short paths over long ones, with a free parameter (the temperature T) controlling the entropic level of the distribution. This formalism enables the computation of new distances or dissimilarities, interpolating between the shortest-path and the resistance distance, which have been shown to perform well in clustering and classification tasks. In this work, the bag-of-paths formalism is extended by adding two independent equality constraints fixing starting and ending nodes distributions of paths (margins).When the temperature is low, this formalism is shown to be equivalent to a relaxation of the optimal transport problem on a network where paths carry a flow between two discrete distributions on nodes. The randomization is achieved by considering free energy minimization instead of traditional cost minimization. Algorithms computing the optimal free energy solution are developed for two types of paths: hitting (or absorbing) paths and non-hitting, regular, paths and require the inversion of an n × n matrix with n being the number of nodes. Interestingly, for regular paths on an undirected graph, the resulting optimal policy interpolates between the deterministic optimal transport policy (T → 0+) and the solution to the corresponding electrical circuit (T → ∞). Two distance measures between nodes and a dissimilarity between groups of nodes, both integrating weights on nodes, are derived from this framework.