Reciprocity—the mutual provisioning of support/goods—is a pervasive feature of social life. Directed networks provide a way to examine the structure of reciprocity in a community. However, measuring social networks involves assumptions about what relationships matter and how to elicit them, which may impact observed reciprocity. In particular, the practice of aggregating multiple sources of data on the same relationship (e.g., “double-sampled” data, where both the “giver” and “receiver” are asked to report on their relationship) may have pronounced impacts on network structure. To investigate these issues, we examine concordance (ties reported by both parties) and reciprocity in a set of directed, double-sampled social support networks. We find low concordance in people’s responses. Taking either the union (including any reported ties) or the intersection (including only concordant ties) of double-sampled relationships results in dramatically higher levels of reciprocity. Using multilevel exponential random graph models of social support networks from 75 villages in India, we show that these changes cannot be fully explained by the increase in the number of ties produced by layer aggregation. Respondents’ tendency to name the same people as both givers and receivers of support plays an important role, but this tendency varies across contexts and relationships type. We argue that no single method should necessarily be seen as the “correct” choice for aggregation of multiple sources of data on a single relationship type. Methods of aggregation should depend on the research question, the context, and the relationship in question.

State preferences play an important role in international politics. Unfortunately, actually observing and measuring these preferences are impossible. In general, scholars have tried to infer preferences using either UN voting or alliance behavior. The two most notable measures of state preferences that have flowed from this research area are ideal points (Bailey et al., 2017) and S-scores (Signorino & Ritter, 1999). The basis of both these models is a spatial weighting scheme that has proven useful but discounts higher-order effects that might be present in relational data structures such as UN voting and alliances. We begin by arguing that both alliances and UN voting are simply examples of the multiple layers upon which states interact with one another. To estimate a measure of state preferences, we utilize a tensor decomposition model that provides a reduced-rank approximation of the main patterns across the layers. Our new measure of preferences plausibly describes important state relations and yields important insights on the relationship between preferences, democracy, and international conflict. Additionally, we show that a model of conflict using this measure of state preferences decisively outperforms models using extant measures when it comes to predicting conflict in an out-of-sample context.

We develop a model of strategic network formation of collaborations to analyze the consequences of an understudied but consequential form of heterogeneity: differences between actors in the form of their production functions. We also address how this interacts with resource heterogeneity, as a way to measure the impact actors have as potential partners on a collaborative project. Some actors (e.g., start-up firms) may exhibit increasing returns to their investment into collaboration projects, while others (e.g., established firms) may face decreasing returns. Our model provides insights into how actor heterogeneity can help explain well-observed collaboration patterns. We show that if there is a direct relation between increasing returns and resources, start-ups exclude mature firms and networks become segregated by types of production function, portraying dominant group architectures. On the other hand, if there is an inverse relation between increasing returns and resources, networks portray core-periphery architectures, where the mature firms form a core and start-ups with low-resources link to them.

Decentralized coordination is one of the fundamental challenges for societies and organizations. While extensively explored from a variety of perspectives, one issue that has received limited attention is human coordination in the presence of adversarial agents. We study this problem by situating human subjects as nodes on a network, and endowing each with a role, either regular (with the goal of achieving consensus among all regular players), or adversarial (aiming to prevent consensus among regular players). We show that adversarial nodes are, indeed, quite successful in preventing consensus. However, we demonstrate that having the ability to communicate among network neighbors can considerably improve coordination success, as well as resilience to adversarial nodes. Our analysis of communication suggests that adversarial nodes attempt to exploit this capability for their ends, but do so in a somewhat limited way, perhaps to prevent regular nodes from recognizing their intent. In addition, we show that the presence of trusted nodes generally has limited value, but does help when many adversarial nodes are present, and players can communicate. Finally, we use experimental data to develop computational models of human behavior and explore additional parametric variations: features of network topologies and densities, and placement, all using the resulting data-driven agent-based (DDAB) model.

We study the detection and the reconstruction of a large very dense subgraph in a social graph with n nodes and m edges given as a stream of edges, when the graph follows a power law degree distribution, in the regime when $m=O(n. \log n)$ . A subgraph S is very dense if it has $\Omega(|S|^2)$ edges. We uniformly sample the edges with a Reservoir of size $k=O(\sqrt{n}.\log n)$ . Our detection algorithm checks whether the Reservoir has a giant component. We show that if the graph contains a very dense subgraph of size $\Omega(\sqrt{n})$ , then the detection algorithm is almost surely correct. On the other hand, a random graph that follows a power law degree distribution almost surely has no large very dense subgraph, and the detection algorithm is almost surely correct. We define a new model of random graphs which follow a power law degree distribution and have large very dense subgraphs. We then show that on this class of random graphs we can reconstruct a good approximation of the very dense subgraph with high probability. We generalize these results to dynamic graphs defined by sliding windows in a stream of edges.

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.

Personalized PageRank has found many uses in not only the ranking of webpages, but also algorithmic design, due to its ability to capture certain geometric properties of networks. In this paper, we study the diffusion of PageRank: how varying the jumping (or teleportation) constant affects PageRank values. To this end, we prove a gradient estimate for PageRank, akin to the Li–Yau inequality for positive solutions to the heat equation (for manifolds, with later versions adapted to graphs).

We propose a novel tie-oriented model for longitudinal event network data. The generating mechanism is assumed to be a multivariate Poisson process that governs the onset and repetition of yearly observed events with two separate intensity functions. We apply the model to a network obtained from the yearly dyadic number of international deliveries of combat aircraft trades between 1950 and 2017. Based on the trade gravity approach, we identify economic and political factors impeding or promoting the number of transfers. Extensive dynamics as well as country heterogeneities require the specification of semiparametric time-varying effects as well as random effects. Our findings reveal strong heterogeneous as well as time-varying effects of endogenous and exogenous covariates on the onset and repetition of aircraft trade events.

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.

Antisocial behavior can be contagious, spreading from individual to individual and rippling through social networks. Moreover, it can spread not only through third-party influence from observation, just like innovations or individual behavior do, but also through direct experience, via “pay-it-forward” retaliation. Here, we distinguish between the effects of observation and victimization for the contagion of antisocial behavior by analyzing large-scale digital trace data. We study the spread of cheating in more than a million matches of an online multiplayer first-person shooter game, in which up to 100 players compete individually or in teams against strangers. We identify event sequences in which a player who observes or is killed by a certain number of cheaters starts cheating and evaluate the extent to which these sequences would appear if we preserve the team and interaction structure but assume alternative gameplay scenarios. The results reveal that social contagion is only likely to exist for those who both observe and experience cheating, suggesting that third-party influence and “pay-it-forward” reciprocity interact positively. In addition, the effect is present only for those who both observe and experience more than once, suggesting that cheating is more likely to spread after repeated or multi-source exposure. Approaching online games as models of social systems, we use the findings to discuss strategies for targeted interventions to stem the spread of cheating and antisocial behavior more generally in online communities, schools, organizations, and sports.

Graphlet counting is a widely explored problem in network analysis and has been successfully applied to a variety of applications in many domains, most notatbly bioinformatics, social science, and infrastructure network studies. Efficiently computing graphlet counts remains challenging due to the combinatorial explosion, where a naive enumeration algorithm needs O(Nk) time for k-node graphlets in a network of size N. Recently, many works introduced carefully designed combinatorial and sampling methods with encouraging results. However, the existing methods ignore the fact that graphlet counts and the graph structural information are correlated. They always consider a graph as a new input and repeat the tedious counting procedure on a regular basis even if it is similar or exactly isomorphic to previously studied graphs. This provides an opportunity to speed up the graphlet count estimation procedure by exploiting this correlation via learning methods. In this paper, we raise a novel graphlet count learning (GCL) problem: given a set of historical graphs with known graphlet counts, how to learn to estimate/predict graphlet count for unseen graphs coming from the same (or similar) underlying distribution. We develop a deep learning framework which contains two convolutional neural network models and a series of data preprocessing techniques to solve the GCL problem. Extensive experiments are conducted on three types of synthetic random graphs and three types of real-world graphs for all 3-, 4-, and 5-node graphlets to demonstrate the accuracy, efficiency, and generalizability of our framework. Compared with state-of-the-art exact/sampling methods, our framework shows great potential, which can offer up to two orders of magnitude speedup on synthetic graphs and achieve on par speed on real-world graphs with competitive accuracy.

Community-based physical activity programs, such as the Recreovía, are effective in promoting healthy behaviors in Latin America. To understand Recreovías’ challenges and scalability, we characterized its social network longitudinally while studying its participants’ social cohesion and interactions. First, we constructed the Main network of the program’s Facebook profile in 2013 to determine the main stakeholders and communities of participants. Second, we studied the Temporal network growth of the Facebook profiles of three Recreovía locations from 2008 to 2016. We implemented a Time Windows in Networks algorithm to determine observation periods and a scaling model of cities’ growth to measure social cohesion over time. Our results show physical activity instructors as the main stakeholders (20.84% nodes of the network). As emerging cohesion, we found: (1) incremental growth of Facebook users (43–272 nodes), friendships (55–2565 edges), clustering coefficient (0.19–0.21), and density (0.04–0.07); (2) no preferential attachment behavior; and (3) a social cohesion super-linear growth with 1.73 new friendships per joined user. Our results underscore the physical activity instructors’ influence and the emergent cohesion in innovation periods as a co-benefit of the program. This analysis associates the social and healthy behavior dimensions of a program occurring in natural environments under a systemic approach.

The network Laplacian spectral density calculation is critical in many fields, including physics, chemistry, statistics, and mathematics. It is highly computationally intensive, limiting the analysis to small networks. Therefore, we present two efficient alternatives: one based on the network’s edges and another on the degrees. The former gives the exact spectral density of locally tree-like networks but requires iterative edge-based message-passing equations. In contrast, the latter obtains an approximation of the spectral density using only the degree distribution. The computational complexities are 𝒪(|E|log(n)) and 𝒪(n), respectively, in contrast to 𝒪(n3) of the diagonalization method, where n is the number of vertices and |E| is the number of edges.

With the increasing availability of behavioral data from diverse digital sources, such as social media sites and cell phones, it is now possible to obtain detailed information about the structure, strength, and directionality of social interactions in varied settings. While most metrics of network structure have traditionally been defined for unweighted and undirected networks only, the richness of current network data calls for extending these metrics to weighted and directed networks. One fundamental metric in social networks is edge overlap, the proportion of friends shared by two connected individuals. Here, we extend definitions of edge overlap to weighted and directed networks and present closed-form expressions for the mean and variance of each version for the Erdős–Rényi random graph and its weighted and directed counterparts. We apply these results to social network data collected in rural villages in southern Karnataka, India. We use our analytical results to quantify the extent to which the average overlap of the empirical social network deviates from that of corresponding random graphs and compare the values of overlap across networks. Our novel definitions allow the calculation of edge overlap for more complex networks, and our derivations provide a statistically rigorous way for comparing edge overlap across networks.

Many real-world networks, including social networks and computer networks for example, are temporal networks. This means that the vertices and edges change over time. However, most approaches for modeling and analyzing temporal networks do not explicitly discuss the underlying notion of time. In this paper, we therefore introduce a generalized notion of discrete time for modeling temporal networks. Our approach also allows for considering nondeterministic time and incomplete data, two issues that are often found when analyzing datasets extracted from online social networks, for example. In order to demonstrate the consequences of our generalized notion of time, we also discuss the implications for the computation of (shortest) temporal paths in temporal networks. In addition, we implemented an R-package that provides programming support for all concepts discussed in this paper. The R-package is publicly available for download.

This study considered the role of adult children in the core networks of U.S. older adults with varying levels of functional health. Taking a multidimensional perspective of the ego network system, we considered (a) presence of child(ren) in the network, (b) contact with children network members, and (c) embeddedness of children within the network. We observed older parents from three waves of the National Social Life, Health, and Aging Project (NSHAP). The common ‘important matters’ name generator was used to construct egocentric network variables, while self-reported difficulty with activities of daily life was used to measure disablement transitions. Parameters were estimated with Generalized Estimating Equations (GEE). Though child turnover was common in parents’ core networks, there was no evidence linking disablement transitions to systematic forms of child reshuffling. Children that remained in parents’ networks, however, showed increased contact with parents and with other members of the network when the parent underwent disability progression. Disability onset was not significantly linked to either outcome. There was limited evidence of gender variation in these patterns. Overall, results strengthen the view that children are distinctive members of older adults’ core networks. Further, the role of adult children shifts most noticeably at advanced stages of the disablement process.

We describe here a notion of diffusion similarity, a method for defining similarity between vertices in a given graph using the properties of random walks on the graph to model the relationships between vertices. Using the approach of graph vertex embedding, we characterize a vertex vi by considering two types of diffusion patterns: the ways in which random walks emanate from the vertex vi to the remaining graph and how they converge to the vertex vi from the graph. We define the similarity of two vertices vi and vj as the average of the cosine similarity of the vectors characterizing vi and vj. We obtain these vectors by modifying the solution to a differential equation describing a type of continuous time random walk.

This method can be applied to any dataset that can be assigned a graph structure that is weighted or unweighted, directed or undirected. It can be used to represent similarity of vertices within community structures of a network while at the same time representing similarity of vertices within layered substructures (e.g., bipartite subgraphs) of the network. To validate the performance of our method, we apply it to synthetic data as well as the neural connectome of the C. elegans worm and a connectome of neurons in the mouse retina. A tool developed to characterize the accuracy of the similarity values in detecting community structures, the uncertainty index, is introduced in this paper as a measure of the quality of similarity methods.

Superbubbles are acyclic induced subgraphs of a digraph with single entrance and exit that naturally arise in the context of genome assembly and the analysis of genome alignments in computational biology. These structures can be computed in linear time and are confined to non-symmetric digraphs. We demonstrate empirically that graph parameters derived from superbubbles provide a convenient means of distinguishing different classes of real-world graphical models, while being largely unrelated to simple, commonly used parameters.

Measurement errors are omnipresent in network data. Most studies observe an erroneous network instead of the desired error-free network. It is well known that such errors can have a severe impact on network metrics, especially on centrality measures: a central node in the observed network might be less central in the underlying, error-free network. The robustness is a common concept to measure these effects. Studies have shown that the robustness primarily depends on the centrality measure, the type of error (e.g., missing edges or missing nodes), and the network topology (e.g., tree-like, core-periphery). Previous findings regarding the influence of network size on the robustness are, however, inconclusive. We present empirical evidence and analytical arguments indicating that there exist arbitrary large robust and non-robust networks and that the average degree is well suited to explain the robustness. We demonstrate that networks with a higher average degree are often more robust. For the degree centrality and Erdős–Rényi (ER) graphs, we present explicit formulas for the computation of the robustness, mainly based on the joint distribution of node degrees and degree changes which allow us to analyze the robustness for ER graphs with a constant average degree or increasing average degree.