Multivariate Power Laws and Preferential Attachment Modeling

In one-dimension, heavy tails or power-laws are easily understood to represent Pareto like behavior where data plotted on a log-log scale looks roughly linear. The generalization to higher dimensions is not always obvious and the infinite variety of dependence possibilities can be daunting. Multivariate regular variation of measures is a clean, flexible and clear way forward. A variety of mathematical and statistical techniques guide a user.

Network modeling of social networks using preferential attachment presents other challenges. Models can be difficult to analyze and only occasionally do simulations from these models leave a comfortable impression that simulation matches reality. One glaring discrepancy is “reciprocity”, meaning the percentage of directed edges that link to nodes (network users) in both directions. (You like me and I like you. You reference my paper and I reference yours.) Real data exhibits higher reciprocity compared to what is given by simulations from traditional preferential attachment.

In- and out-degree sequence data for many social networks marginally exhibit the expected straight line power law behavior and preferential attachment models theoretically predict this both marginally and in the two-dimensional sense. We can add reciprocity to the model by assuming something like “when I connect to you, you flip a coin to decide if you want to connect with me.” This and its generalizations correct the empirical under-prediction of reciprocity and introduces the feature that asymptotically the limit measure of regular variation concentrates on a line. This means large values of in- and out-degree tend to always be present simultaneously, a property called “asymptotic full dependence”. Without reciprocity, preferential attachment leads to in- and out-degree having a limit measure of regular variation that concentrates on the full positive quadrant meaning that a large value of either in- or out-degree can be associated with a variety of values in the other.

This is part of the lecture series funded by the NSF RTG grant DMS 2134107.

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