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)
#> [1] "------------Nb of entities in each functional group--------------"
#> country
#> 83
#> [1] "------------Probability distributions on each network--------------"
#> [1] "bernoulli" "bernoulli"
#> [1] "-------------------------------------------------------------------"
#> [1] " ------ Searching the numbers of blocks starting from [ 1 ] blocks"
#> [1] "ICL : -2708.58 . Nb of blocks: [ 1 ]"
#> [1] "ICL : -1728.54 . Nb of blocks: [ 2 ]"
#> [1] "ICL : -1562.54 . Nb of blocks: [ 3 ]"
#> [1] "ICL : -1460.03 . Nb of blocks: [ 4 ]"
#> [1] "ICL : -1434.6 . Nb of blocks: [ 5 ]"
#> [1] "Best model------ ICL : -1434.6 . Nb of clusters: [ 5 ] for [ country ] respectively"
We can retrieve the clustering
clust_country_indep = MultiplexFitIndep$memberships[[1]]
sort(clust_country_indep)
#> United States of America Canada
#> 1 1
#> Belize El Salvador
#> 2 2
#> Colombia Ecuador
#> 2 2
#> Bolivia Uruguay
#> 2 2
#> Bahamas Cuba
#> 3 3
#> Haiti Dominican Republic
#> 3 3
#> Jamaica Trinidad and Tobago
#> 3 3
#> Barbados Dominica
#> 3 3
#> Grenada St. Lucia
#> 3 3
#> St. Vincent and the Grenadines Antigua & Barbuda
#> 3 3
#> St. Kitts and Nevis Mexico
#> 3 3
#> Guatemala Honduras
#> 3 3
#> Nicaragua Costa Rica
#> 3 3
#> Panama Venezuela
#> 3 3
#> Guyana Suriname
#> 3 3
#> Peru Brazil
#> 3 3
#> Paraguay Chile
#> 3 3
#> Argentina United Kingdom
#> 3 4
#> Netherlands Belgium
#> 4 4
#> Luxembourg France
#> 4 4
#> Spain Portugal
#> 4 4
#> Germany German Federal Republic
#> 4 4
#> Poland Hungary
#> 4 4
#> Czech Republic Italy
#> 4 4
#> Greece Norway
#> 4 4
#> Denmark Iceland
#> 4 4
#> Bavaria German Democratic Republic
#> 5 5
#> Baden Wuerttemburg
#> 5 5
#> Austria-Hungary Austria
#> 5 5
#> Czechoslovakia Slovakia
#> 5 5
#> Malta Albania
#> 5 5
#> Croatia Yugoslavia
#> 5 5
#> Bosnia and Herzegovina Cyprus
#> 5 5
#> Bulgaria Moldova
#> 5 5
#> Romania Russia
#> 5 5
#> Estonia Latvia
#> 5 5
#> Lithuania Ukraine
#> 5 5
#> Belarus Armenia
#> 5 5
#> Georgia Azerbaijan
#> 5 5
#> Finland Sweden
#> 5 5
#> Cape Verde Sao Tome and Principe
#> 5 5
#> Guinea-Bissau
#> 5
And we can plot the reorganized adjacency matrices or the corresponding expectations:
plot(MultiplexFitIndep,type="expected")
Fitting a multiplex SBM model where the two layers are assumed to be dependent
Now we assume that the two layers are dependent conditionally to the latent block variables. We then set dependent=TRUE
MultiplexFitdep = estimateMultiplexSBM(list(netA,netB),dependent = TRUE,estimOptions = list(verbosity=0))
We can retrieve the clustering and compare it to the one obtained in the independent case.
clust_country_dep = MultiplexFitdep$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,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.