# alphaRef: Compute Intercept Value in the Reference Network In spaceNet: Latent Space Models for Multidimensional Networks

## Description

Computes the value for the intercept in the reference network.

## Usage

 `1` ```alphaRef(Y, D = 2, sender = NULL, receiver = NULL) ```

## Arguments

 `Y` A three-dimensional array or list of n x n adjacency matrices composing the multidimensional network. A list will be converted to an array. If an array, the dimension of `Y` must be `(n,n,K)`, where `n` is the number of nodes and `K` the number of networks. Structural missing values (`NA`) are allowed, but no imputation is performed; see details. `D` The dimension of the latent space, with `D > 0`. The default value is `D = 2`. `sender, receiver` The type of node-specific sender and receiver effects to be included in the model. If specified, these effects can be set to constant (`"const"`) or/and variable (`"var"`). By default, node-specific effects are not included in the model (`NULL`).

## Details

The function computes the value for the intercept in the reference network (first network) of the multiplex. It is calculated taking into account the average effect of the latent space on edge probabilities (approximated with the constant 2) and the observed mean probability of an edge in the first network (Y_1):

p_1 = sum( Y_1) \ ( n * ( n - 1 ) ),

with n the number of nodes in the network. Then, the reference intercept is computed as:

log ( p_1 \ ( 1 - p_1) ) + 2.

When sender and/or receiver effects are included in the model, the intercept for the reference network is forced to be positive, see references.

## Value

The function returns the intercept value in the reference network.

## References

D'Angelo, S. and Murphy, T. B. and Alf<c3><b2>, M. (2018). Latent space modeling of multidimensional networks with application to the exchange of votes in the Eurovision Song Contest. arXiv.

D'Angelo, S. and Alf<c3><b2>, M. and Murphy, T. B. (2018). Node-specific effects in latent space modelling of multidimensional networks. arXiv.

`multiNet`
 ```1 2``` ```data(vickers) alphaR <- alphaRef(vickers, D = 2) # model without node-specific effect ```