# grorbitDist: New Graphlet Degree Distribution term for the 'ergm' Package... In ergm.graphlets: ERG Modeling Based on Graphlet Properties

## Description

This is a new term for the `ergm` package that evaluates the graphlet degree distribution of a network for a given graphlet orbit.

## Details

The function `ergm` is used to fit linear exponential random graph models, in which the probability of a given network, y, on a set of nodes is \exp{θ * g(y)}/c(θ), where g(y) is a vector of network statistics for y, θ is a parameter vector of the same length and c(θ) is the normalizing constant for the distribution.

The network statistics g(y) are entered as terms in the function call to `ergm`. The `ergm` package contains a wide range of terms. For the details on the possible `<model terms>`, see `ergm-terms`.

The `ergm.userterms` package provides a template for adding new terms. The terms can be used throughout the `ergm` package and behave identically to the supplied terms.

## The Graphlet Degree Distribution Term

`grorbitDist(grorbit, d)`

The graphlet degree distributions of various graphlet orbits can be included into the ERGM by using the `grorbitDist` term. The question that the change score function of this term answers is: how do the number of nodes having graphlet degree n for orbit i change when an edge is flipped? This term has two arguments: `grorbit` and `d`. The `grorbit` argument is a vector of distinct integers representing the list of graphlet orbits to include into the model. The `d` argument is a vector of distinct integers. This terms adds one network statistic to the model for each pairwise combination of the arguments in `grorbit` and `d` vectors. The statistic for the combination of (i, j) is equals to the number of nodes in the network that have graphlet degree j for orbit i. This term is defined for the 15 graphlet orbits corresponding to graphlets with up to 4 nodes. Therefore, `grorbit` accepts values between 0 and 14. Graphlets of size 5 are omitted for this term because of the high computational complexity of the change score computation of the term. This term is defined for undirected networks.

## References

Yaveroglu ON, Fitzhugh SM, Kurant M, Markopoulou A, Butts CT, Przulj N (2015). ergm.graphlets: A Package for ERG Modeling Based on Graphlet Properties, Journal of Statistical Software 65(12), 1-29, URL http://www.jstatsoft.org/v65/i12/.