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) . In this article they proposed an exponential family of probability distributions to model , where is a possible realisation of the random matrix .

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 () (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:

where are parameters, and (identifying constrains). can be interpreted as the mean tendency of **reciprocation**, can be interpreted as the **density** (size) of the network, can be interpreted as as the **productivity** (out-degree) of a node, can be interpreted as the **attractiveness** (in-degree) of a node.

The values and 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 , the model assumes that all with are independent.

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

Consider the pairs and describe the joint distribution of , assuming all are statistically independent. This can be done by parameterizing the probabilities

where .

Hence leading

where for , and for .

It can be seen that the parameters and can be interpreted as the reciprocity and differential attractiveness, respectively. With a bit of algebra we get:

and

Now, if we consider the following restrictions:

, and where .

With some algebra we get the proposed form of the model

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