# Stochastic Block Models for Multiplex networks

### War and Alliance case study

#### team großBM

#### 2021-06-09

The `war` data set comes in the `sbm` package:

    library(sbm)
    data("war")

This data set contains a list of two networks (`belligerent` and `alliance`) where the nodes are countries; an edge in the network `belligerent` means that the two countries have been at war at least once between years 1816 to 2007; an edge in network `alliance` means that the two countries have had a formal alliance between years 1816 and 2012. The network `belligerent` have less nodes since countries which have not been at war at all are not considered.

These two networks were extracted from https://www.correlatesofwar.org/ (see Sarkees and Wayman (2010) for war data, and Gibler (2008) for formal alliance). Version 4.0 was used for war data and version 4.1 for formal alliance.

# Data manipulation

Since they don’t have the same size, we choose to only consider nodes (countries) which were at war with at least one other country. This corresponds to the first 83 nodes in the Alliance network.

    library(igraph)
    #>
    #> Attaching package: 'igraph'
    #> The following objects are masked from 'package:stats':
    #>
    #>     decompose, spectrum
    #> The following object is masked from 'package:base':
    #>
    #>     union
    A = as.matrix(get.adjacency(war$alliance))
    A = A[1:83,1:83]
    B = as.matrix(get.adjacency(war$belligerent))

We can start with a plot of this multiplex network:

    netA = defineSBM(A,model="bernoulli",dimLabels = "country")
    netB = defineSBM(B,model="bernoulli",dimLabels = "country")
    plotMyMultiplexMatrix(list(netA,netB))

# Fitting a multiplex SBM model where the two layers are assumed to be independent

We run the estimation of this multiplex model. By setting `dependent=FALSE`, we declare that we consider the two layers to be independent conditionally on the latent block variables.

    MultiplexFitIndep = estimateMultiplexSBM(list(netA,netB), dependent = FALSE, estimOptions = list(verbosity=0))

We can retrieve the clustering

    clust_country_indep = MultiplexFitIndep$memberships[[1]]
    sort(clust_country_indep)
    #>       United States of America                         Canada
    #>                              1                              1
    #>                         Belize                    El Salvador
    #>                              2                              2
    #>                       Colombia                        Ecuador
    #>                              2                              2
    #>                        Bolivia                        Uruguay
    #>                              2                              2
    #>                        Bahamas                           Cuba
    #>                              3                              3
    #>                          Haiti             Dominican Republic
    #>                              3                              3
    #>                        Jamaica            Trinidad and Tobago
    #>                              3                              3
    #>                       Barbados                       Dominica
    #>                              3                              3
    #>                        Grenada                      St. Lucia
    #>                              3                              3
    #> St. Vincent and the Grenadines              Antigua & Barbuda
    #>                              3                              3
    #>            St. Kitts and Nevis                         Mexico
    #>                              3                              3
    #>                      Guatemala                       Honduras
    #>                              3                              3
    #>                      Nicaragua                     Costa Rica
    #>                              3                              3
    #>                         Panama                      Venezuela
    #>                              3                              3
    #>                         Guyana                       Suriname
    #>                              3                              3
    #>                           Peru                         Brazil
    #>                              3                              3
    #>                       Paraguay                          Chile
    #>                              3                              3
    #>                      Argentina                 United Kingdom
    #>                              3                              4
    #>                    Netherlands                        Belgium
    #>                              4                              4
    #>                     Luxembourg                         France
    #>                              4                              4
    #>                          Spain                       Portugal
    #>                              4                              4
    #>                        Germany        German Federal Republic
    #>                              4                              4
    #>                         Poland                        Hungary
    #>                              4                              4
    #>                 Czech Republic                          Italy
    #>                              4                              4
    #>                         Greece                         Norway
    #>                              4                              4
    #>                        Denmark                        Iceland
    #>                              4                              4
    #>                        Bavaria     German Democratic Republic
    #>                              5                              5
    #>                          Baden                   Wuerttemburg
    #>                              5                              5
    #>                Austria-Hungary                        Austria
    #>                              5                              5
    #>                 Czechoslovakia                       Slovakia
    #>                              5                              5
    #>                          Malta                        Albania
    #>                              5                              5
    #>                        Croatia                     Yugoslavia
    #>                              5                              5
    #>         Bosnia and Herzegovina                         Cyprus
    #>                              5                              5
    #>                       Bulgaria                        Moldova
    #>                              5                              5
    #>                        Romania                         Russia
    #>                              5                              5
    #>                        Estonia                         Latvia
    #>                              5                              5
    #>                      Lithuania                        Ukraine
    #>                              5                              5
    #>                        Belarus                        Armenia
    #>                              5                              5
    #>                        Georgia                     Azerbaijan
    #>                              5                              5
    #>                        Finland                         Sweden
    #>                              5                              5
    #>                     Cape Verde          Sao Tome and Principe
    #>                              5                              5
    #>                  Guinea-Bissau
    #>                              5

And we can plot the reorganized adjacency matrices or the corresponding expectations:

    plot(MultiplexFitIndep)

    plot(MultiplexFitIndep,type="expected")

# Fitting a multiplex SBM model where the two layers are assumed to be dependent

Now we assume that the two layers are dependent conditionally to the latent block variables. We then set `dependent=TRUE`

    MultiplexFitdep = estimateMultiplexSBM(list(netA,netB),dependent = TRUE,estimOptions = list(verbosity=0))

We can retrieve the clustering and compare it to the one obtained in the independent case.

    clust_country_dep = MultiplexFitdep$memberships[[1]]
    sort(clust_country_indep)
    #>       United States of America                         Canada
    #>                              1                              1
    #>                         Belize                    El Salvador
    #>                              2                              2
    #>                       Colombia                        Ecuador
    #>                              2                              2
    #>                        Bolivia                        Uruguay
    #>                              2                              2
    #>                        Bahamas                           Cuba
    #>                              3                              3
    #>                          Haiti             Dominican Republic
    #>                              3                              3
    #>                        Jamaica            Trinidad and Tobago
    #>                              3                              3
    #>                       Barbados                       Dominica
    #>                              3                              3
    #>                        Grenada                      St. Lucia
    #>                              3                              3
    #> St. Vincent and the Grenadines              Antigua & Barbuda
    #>                              3                              3
    #>            St. Kitts and Nevis                         Mexico
    #>                              3                              3
    #>                      Guatemala                       Honduras
    #>                              3                              3
    #>                      Nicaragua                     Costa Rica
    #>                              3                              3
    #>                         Panama                      Venezuela
    #>                              3                              3
    #>                         Guyana                       Suriname
    #>                              3                              3
    #>                           Peru                         Brazil
    #>                              3                              3
    #>                       Paraguay                          Chile
    #>                              3                              3
    #>                      Argentina                 United Kingdom
    #>                              3                              4
    #>                    Netherlands                        Belgium
    #>                              4                              4
    #>                     Luxembourg                         France
    #>                              4                              4
    #>                          Spain                       Portugal
    #>                              4                              4
    #>                        Germany        German Federal Republic
    #>                              4                              4
    #>                         Poland                        Hungary
    #>                              4                              4
    #>                 Czech Republic                          Italy
    #>                              4                              4
    #>                         Greece                         Norway
    #>                              4                              4
    #>                        Denmark                        Iceland
    #>                              4                              4
    #>                        Bavaria     German Democratic Republic
    #>                              5                              5
    #>                          Baden                   Wuerttemburg
    #>                              5                              5
    #>                Austria-Hungary                        Austria
    #>                              5                              5
    #>                 Czechoslovakia                       Slovakia
    #>                              5                              5
    #>                          Malta                        Albania
    #>                              5                              5
    #>                        Croatia                     Yugoslavia
    #>                              5                              5
    #>         Bosnia and Herzegovina                         Cyprus
    #>                              5                              5
    #>                       Bulgaria                        Moldova
    #>                              5                              5
    #>                        Romania                         Russia
    #>                              5                              5
    #>                        Estonia                         Latvia
    #>                              5                              5
    #>                      Lithuania                        Ukraine
    #>                              5                              5
    #>                        Belarus                        Armenia
    #>                              5                              5
    #>                        Georgia                     Azerbaijan
    #>                              5                              5
    #>                        Finland                         Sweden
    #>                              5                              5
    #>                     Cape Verde          Sao Tome and Principe
    #>                              5                              5
    #>                  Guinea-Bissau
    #>                              5
    aricode::ARI(clust_country_indep,clust_country_dep)
    #> [1] 0.8886699

On top of the clustering comparison, we can compare the ICL criteria to see which of the dependent or independent models is a best fit:

    MultiplexFitdep$ICL
    #> [1] -1470.276
    MultiplexFitIndep$ICL
    #> [1] -1434.602

We can do the same plots for the reorganized matrices and the corresponding expectations. Note that the expectations correspond to the marginal expectation of each layer and it may be relevant to have a look to the conditional expectations.

    plot(MultiplexFitdep)

    plot(MultiplexFitdep,type="expected")
    #> Warning in self$predict(): provided expectations are the marginal expectations
    #> in the dependent case

For instance, we may want to compare the marginal distribution for two countries in their given blocks to have being at war while they are allied at some point (before or after):

    p11 = MultiplexFitdep$connectParam$prob11
    p01 = MultiplexFitdep$connectParam$prob01
    p10 = MultiplexFitdep$connectParam$prob10
    # conditional probabilities of being at war while having been or will be allied
    round(p11/(p11+p10),2)
    #>      [,1] [,2] [,3] [,4]
    #> [1,] 0.09 0.10 0.03 0.43
    #> [2,] 0.10 0.11 0.09 0.14
    #> [3,] 0.03 0.09 0.05 0.03
    #> [4,] 0.43 0.14 0.03 0.01
    # marginal probabilities of being at war
    round(p11+p01,2)
    #>      [,1] [,2] [,3] [,4]
    #> [1,] 0.07 0.04 0.03 0.14
    #> [2,] 0.04 0.10 0.04 0.13
    #> [3,] 0.03 0.04 0.05 0.03
    #> [4,] 0.14 0.13 0.03 0.02

## References

Gibler, Douglas M. 2008. International Military Alliances, 1648-2008. CQ Press.
Sarkees, Meredith Reid, and Frank Whelon Wayman. 2010. Resort to War: A Data Guide to Inter-State, Extra-State, Intra-State, and Non-State Wars, 1816-2007. Cq Pr.