77 lines
2.6 KiB
Matlab
77 lines
2.6 KiB
Matlab
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% Find the "optimal" number of communities given a network using an eigenvector method
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% Source: MEJ Newman: Finding community structure using the eigenvectors of matrices,
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% arXiv:physics/0605087
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% Newman, "Modularity and community structure in networks",
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% arxiv.org/pdf/physics/0602124v1
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%
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% Q=(s^T)Bs, Bij=Aij-kikj/2m
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% Bij^g=Bij - delta_ij * (sum k over g)B_ik
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% Bij^g=(Aij-kikj/2m)-delta_ij(sum k over g)(A(g)_ik-deg(g)_i deg(k)_j/2(m_g))
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% Bij^g=(Aij-kikj/2m)-delta_ij(k_i^(g)-k_i*sum(deg^(g)/2m)
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%
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% STEPS:
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% 1 define current modularity matrix
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% 2 compute eigenvector corresp. to largest eigenvalue
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% 3 separate into 2 modules based on signs in eigenvector
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% terminate when max eigenvalue is 0 for all subgraphs
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%
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% Other functions used: numEdges.m, degrees.m, subgraph.m, isConnected.m
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% GB: last modified, Oct 12, 2012
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function modules=newmanEigenvectorMethod(adj)
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modules={};
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n=length(adj);
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m=numEdges(adj);
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[deg,~,~]=degrees(adj);
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queue{1}=[1:n]; % append all nodes to the queue
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while length(queue)>0 % while there is always a divisible subgraph
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% compute modularity matrix- Bg
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G=queue{1}; % nodes in current (sub)graph to partition
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adjG=subgraph(adj,G); % first adjG same as original adj
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[degG,~,~]=degrees(adjG);
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Bg=zeros(length(G));
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for i=1:length(G) % nodes are G(i)
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for j=i:length(G) % nodes are G(j)
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% Bij = adj(G(i),G(j))-deg(G(i))*deg(G(j))/(2*m)
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% delta_ij = (i==j)
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% k_i^g - k_i (d_g)/2m = degG(i)-deg(G(i))*sum(degG)/(2*m)
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Bg(i,j)=( adj(G(i),G(j))-deg(G(i))*deg(G(j))/(2*m) )-(i==j)*(degG(i)-deg(G(i))*sum(degG)/(2*m));
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Bg(j,i)=Bg(i,j);
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end
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end
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[V,E]=eig(Bg); % eigenvalues
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if abs(max(diag(E)))<=10^(-5) % terminate - indivisible
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queue=queue(2:length(queue));
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modules={modules{1:length(modules)} G};
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continue
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end
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[~,indmax]=max(diag(E));
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u1=V(:,indmax);
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comm1=find(u1>0); comm2=find(u1<0); % indices in G
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if not(isConnected(adj(G(comm1),G(comm1)))) || not(isConnected(adj(G(comm2),G(comm2))))
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queue=queue(2:length(queue)); % remove(G)
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modules={modules{1:length(modules)} G}; % keep original G as indivisible
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% found disconnected subgraphs - do not explore this option
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continue
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end
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queue={queue{1:length(queue)}, G(comm1), G(comm2)}; % add comm1 and comm2 to the queue
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queue=queue(2:length(queue));
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end
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