The article *An Exponential Family of Probability Distributions for Directed Graphs*, published by Holland and Leinhardt (1981), set the foundation for the now known exponential random graph models (ERGM) or *p** models, which model jointly the whole adjacency matrix (or graph) $latex X$. In this article they proposed an exponential family of probability distributions to model $latex P(X=x)$, where $latex x$ is a possible realisation of the random matrix $latex X$.

The article is mainly focused on directed graphs (although the theory can be extended to undirected graphs). Two main *effects* or *patterns* are considered in the article:* Reciprocity*, which relates to appearance of symmetric interactions ($latex X_{ij}=1 \iff X_{ji}=1$) (see nodes 3-5 of the Figure below).

and, the ** Differential attractiveness** of each node in the graph, which relates to the amount of interactions each node “receives” (in-degree) and the amount of interactions that each node “produces” (out-degree) (the Figure below illustrates the differential attractiveness of two groups of nodes).

The model of Holland and Leinhardt (1981), called *p*1 model, that considers jointly the reciprocity of the graph and the differential attractiveness of each node is:

$latex p_1(x)=P(X=x) \propto e^{\rho m + \theta x_{**} + \sum_i \alpha_i x_{i*} + \sum_j \beta_j x_{*j}}, $

where $latex \rho,\theta,\alpha_i,\beta_j $ are parameters, and $latex \alpha_*=\beta_*=0 $ (identifying constrains). $latex \rho $ can be interpreted as the mean tendency of **reciprocation**, $latex \theta$ can be interpreted as the **density** (size) of the network, $latex \alpha_i $ can be interpreted as as the **productivity** (out-degree) of a node, $latex \beta_j $ can be interpreted as the **attractiveness** (in-degree) of a node.

The values $latex m, x_{**}, x_{i*}$ and $latex x_{*j}$ are: the number of reciprocated edges in the observed graph, the number of edges, the out-degree of node i and the in-degree of node j; respectively.

Taking $latex D_{ij}=(X_{ij},X_{ji})$, the model assumes that all $latex D_{ij}$ with $latex i<j $ are independent.

To better understand the model, let’s review its derivation:

Consider the pairs $latex D_{ij}=(X_{i,j},X_{j,i}),\,i<j $ and describe the joint distribution of $latex \{D_{ij}\}_{ij}$, assuming all $latex D_{ij}$ are statistically independent. This can be done by parameterizing the probabilities

$latex P(D_{ij}=(1,1))=m_{ij} \text{ if } i<j,$

$latex P(D_{ij}=(1,0))=a_{ij} \text{ if } i\neq j,$

$latex P(D_{ij}=(0,0))=n_{ij} \text{ if } i<j,$

where $latex m_{ij}+a_{ij}+a_{ji}+n_{ij}=1,\, \forall \, i<j $.

Hence leading

$latex P(X=x)=\prod_{i<j} m_{ij}^{x_{ij}x_{ji}} \prod_{i\neq j}a_{ij}^{x_{ij}(1-x_{ji})} \prod_{i<j}n_{ij}^{(1-x_{ij})(1-x_{ji})}

=e^{\sum_{i<j} {x_{ij}x_{ji}} \rho_{ij} + \sum_{i\neq j}{x_{ij}} \theta_{ij}} \prod_{i<j}n_{ij}, $

where $latex \rho_{ij}=log(m_{ij}n_{ij} / a_{ij}a_{ji})$ for $latex i<j$, and $latex \theta_{ij}=log(a_{ij}/n_{ij})$ for $latex i\neq j$.

It can be seen that the parameters $latex \rho_{ij}$ and $latex \theta_{ij}$ can be interpreted as the reciprocity and differential attractiveness, respectively. With a bit of algebra we get:

$latex exp(\rho_{ij})=[ P(X_{ij}=1|X_{ji}=1)/P(X_{ij}=1|X_{ji}=0) ]/[ P(X_{ij}=1|X_{ji}=0) / P(X_{ij}=0|X_{ji}=0) ], $

and

$latex exp(\theta_{ij})=P(X_{ij}=1|X_{ji}=0)/P(X_{ij}=0|X_{ji}=0). $

Now, if we consider the following restrictions:

$latex \rho_{ij}=\rho,\, \forall i<j$, and $latex \theta_{ij}=\theta+\alpha_i + \beta_j,\, \forall i\neq j $ where $latex \alpha_*=\beta_*=0 $.

With some algebra we get the proposed form of the model

$latex p_1(x)=P(X=x) \propto e^{\rho m + \theta x_{**} + \sum_i \alpha_i x_{i*} + \sum_j \beta_j x_{*j}}.$