BLSM {BLSM} R Documentation

## Bayesian Latent Space Model

### Description

R package allowing the computation of a Bayesian Latent Space Model for complex networks.

Latent Space Models are characterized by the presence of unobservable variables (latent coordinates) that are used to compute the likelihood of the observed networks. Their goal is to map the observed network in the latent space by meeting specific probabilistic requirements, so that the estimated latent coordinates can then be used to describe and characterize the original graph.

In the BSLM package framework, given a network characterized by its adjacency Y matrix, the model assigns a binary random variable to each tie: Y_ij is related to the tie between nodes i and j and its value is 1 if the tie exists, 0 otherwise.

The model assumes the independence of Y_ij | x_i,x_j, \alpha, where x_i and x_j are the coordinates of the nodes in the multidimensional latent space and \alpha is an additional parameter such that logit(P(Y_ij = 1)) = \alpha - ||x_i -x_j||.

The latent space coordinates are estimated by following a MCMC procedure that is based on the overall likelihood induced by the above equation.

Due to the symmetry of the distance, the model leads to more intuitive outputs for undirected networks, but the functions can also deal with directed graphs.

If the network is weighted, i.e. to each tie is associated a positive coefficient, the model's probability equation becomes logit(P(Y_ij = 1)) = \alpha - W_ij||x_i -x_j||, where W_ij denotes the weight related to link existing between x_i and x_j. This means that even non existing links should have a weight, therefore the matrix used in the computation isn't the original weights matrix but actually a specific "BLSM weights" matrix that contains positive coefficients for all the possible pairs of nodes. When dealing with weighted networks, please be careful to pass a "BLSM weights" matrix as input #' (please refer to example_weights_matrix for more detailed information and a valid example).

The output of the model allows the user to inspect the MCMC simulation, create insightful graphical representations or apply clustering techniques to better describe the latent space. See estimate_latent_positions or plot_latent_positions for further information.

### References

P. D. Hoff, A. E. Raftery, M. S. Handcock, Latent Space Approaches to Social Network Analysis, Journal of the American Statistical Association, Vol. 97, No. 460, (2002), pp. 1090-1098.

A. Donizetti, A Latent Space Model Approach for Clustering Complex Network Data, Master's Thesis, Politecnico di Milano, (2017).

[Package BLSM version 0.1.0 Index]