Create a `graphlet.null_model` module for heterogeneous networks

[rjurney]

In order to determine whether a graphlet is a network motif, we need to compare its frequency versus a null model to determine if it is statistically significant. This means we need heterogeneous null models... and I can't find any libraries such as networkx that contain null models for heterogeneous networks.

Create a graphlet.null_model Module

We should create a module graphlet.null_model with networkx style generators that accept properties such as:

• Total number of nodes of each node type
• Summary of degrees of each edge type

Or whatever we find in the literature for various heterogeneous null models.

NetworkX Generators

While networkx Generators cover a range of randomly generated networks, they do not handle multiple types of nodes found in a property graph, also known as a heterogeneous network.

Literature Review

• Structure of Heterogeneous Networks - creates a type-specific null model for modularity
• Null Model and Community Structure in Heterogeneous Networks - lots about heterogeneous null models in Section 3.
• Analytical Models for Motifs in Temporal Networks: Discovering Trends and Anomalies

The null model of the heterogeneous networks refers to those network models that has the same set of types of nodes T, number of homogeneous nodes N, number of heterogeneous U, distribution of homogeneous node degree P(k) and distribution of heterogeneous node degree P(u) with the original network, while otherwise is taken to be an instance of the random network.

For each two types of the nodes, there is a distribution of heterogeneous node degree. Therefore, there are |T|2 − |T| distribution of heterogeneous node degree in a heterogeneous network, where T refers to the set of types of nodes.

Uses random walks - a Markov process - that considers the odds of walking across different edge types (source_type, dest_type) - reaches a steady state that incorporates homogeneous and heterogeneous degrees. Also includes modularity function using this null model.