Introducing Graph Cumulants: What is the Variance of Your Social Network?

Understand

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.