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Abstract Reciprocity in social networks is a measure of information exchange between two individuals, and indicates interaction patterns between pairs of users. A recent study finds that the reciprocity coefficient of a classical directed preferential attachment (PA) model does not match empirical evidence. Towards remedying this deficiency, we extend the classical three-scenario directed PA model by adding a parameter that controls the probability of creating a reciprocal edge. This proposed model also allows edge creation between two existing nodes, making it a realistic candidate for fitting to datasets. We provide and compare two estimation procedures for fitting the new reciprocity model and demonstrate the methods on simulated and real datasets. One estimation method requires careful analysis of the heavy tail properties of the model. The fitted models provide a good match with the empirical tail distributions of both in- and out-degrees but other mismatched diagnostics suggest that further generalization of the model is warranted. 

Reciprocity, or the tendency of individuals to mirror behavior, is a key measure that de- scribes information exchange in a social network. Users in social networks tend to engage in different levels of reciprocal behavior. Differences in such behavior may indicate the existence of communities that reciprocate links at varying rates. In this paper, we de- velop methodology to model the diverse reciprocal behavior in growing social networks. In particular, we present a preferential attachment model with heterogeneous reciprocity that imitates the attraction users have for popular users, plus the heterogeneous nature by which they reciprocate links. We compare Bayesian and frequentist model fitting techniques for large networks, as well as computationally efficient variational alternatives. Cases where the number of communities is known and unknown are both considered. We apply the presented methods to the analysis of Facebook and Reddit networks where users have non- uniform reciprocal behavior patterns. The fitted model captures the heavy-tailed nature of the empirical degree distributions in the datasets and identifies multiple groups of users that differ in their tendency to reply to and receive responses to wallposts and comments. 

Preferential attachment (PA) network models have a wide range of applications in various scientific disciplines. Efficient generation of large-scale PA networks helps uncover their structural properties and facilitate the development of associated analytical methodologies. Existing software packages only provide limited functions for this purpose with restricted configurations and efficiency. We present a generic, user-friendly implementation of weighted, directed PA network generation with R package wdnet. The core algorithm is based on an efficient binary tree approach. The package further allows adding multiple edges at a time, heterogeneous reciprocal edges, and user-specified preference functions. The engine under the hood is implemented in C++. Usages of the package are illustrated with detailed explanation. A benchmark study shows that wdnet is efficient for generating general PA networks not available in other packages. In restricted settings that can be handled by existing packages, wdnet provides comparable efficiency.