  In an increasingly interconnected world, understanding and summarizing the
structure of these networks becomes increasingly relevant. However, this task
is nontrivial; proposed summary statistics are as diverse as the networks they
describe, and a standardized hierarchy has not yet been established. In
contrast, vector-valued random variables admit such a description in terms of
their cumulants (e.g., mean, (co)variance, skew, kurtosis). Here, we introduce
the natural analogue of cumulants for networks, building a hierarchical
description based on correlations between an increasing number of connections,
seamlessly incorporating additional information, such as directed edges, node
attributes, and edge weights. These graph cumulants provide a principled and
unifying framework for quantifying the propensity of a network to display any
substructure of interest (such as cliques to measure clustering). Moreover,
they give rise to a natural hierarchical family of maximum entropy models for
networks (i.e., ERGMs) that do not suffer from the "degeneracy problem", a
common practical pitfall of other ERGMs.
