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.

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.

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.

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.

The presence of unobserved node-specific heterogeneity in exponential random graph models (ERGM) is a general concern, both with respect to model validity as well as estimation instability. We, therefore, include node-specific random effects in the ERGM that account for unobserved heterogeneity in the network. This leads to a mixed model with parametric as well as random coefficients, labelled as mixed ERGM. Estimation is carried out by iterating between approximate pseudolikelihood estimation for the random effects and maximum likelihood estimation for the remaining parameters in the model. This approach provides a stable algorithm, which allows to fit nodal heterogeneity effects even for large scale networks. We also propose model selection based on the Akaike Information Criterion to check for node-specific heterogeneity.