  Network data appear in a number of applications, such as online social
networks and biological networks, and there is growing interest in both
developing models for networks as well as studying the properties of such data.
Since individual network datasets continue to grow in size, it is necessary to
develop models that accurately represent the real-life scaling properties of
networks. One behavior of interest is having a power law in the degree
distribution. However, other types of power laws that have been observed
empirically and considered for applications such as clustering and feature
allocation models have not been studied as frequently in models for graph data.
In this paper, we enumerate desirable asymptotic behavior that may be of
interest for modeling graph data, including sparsity and several types of power
laws. We outline a general framework for graph generative models using
completely random measures; by contrast to the pioneering work of Caron and Fox
(2015), we consider instantiating more of the existing atoms of the random
measure as the dataset size increases rather than adding new atoms to the
measure. We see that these two models can be complementary; they respectively
yield interpretations as (1) time passing among existing members of a network
and (2) new individuals joining a network. We detail a particular instance of
this framework and show simulated results that suggest this model exhibits some
desirable asymptotic power-law behavior.
