newmanGirvan.m

%##################################################################
% Newman-Girvan community finding algorithm
% Source: Newman, Girvan, "Finding and evaluating 
%                          community structure in networks"
% Algorithm idea:
% 1. Calculate betweenness scores for all edges in the network.
% 2. Find the edge with the highest score and remove it from the network.
% 3. Recalculate betweenness for all remaining edges.
% 4. Repeat from step 2.
%
% INPUTs: adjacency matrix (nxn), number of modules (k)
% OUTPUTs: modules (components) and modules history - 
%          each "current" module, Q - modularity metric
%
% Other routines used: edgeBetweenness.m, isConnected.m,
%                findConnComp.m, subgraph.m, numEdges.m
% GB: last updated, Oct 11 2012
%##################################################################


function [modules,module_hist,Q] = newmanGirvan(adj,k)

n=size(adj,1);
module_hist{1} = [1:n]; % current component
modules{1}=[1:n];

curr_mod=1;

adj_temp=adj;


while length(modules)<k
    
    w=edgeBetweenness(adj_temp);
    [wmax,indmax]=max(w(:,3));   % need to remove el(indmax,:)
    adj_temp(w(indmax,1),w(indmax,2))=0;
    adj_temp(w(indmax,2),w(indmax,1))=0;   % symmetrize
    
    if isConnected(adj_temp); continue; end % keep on removing edges
    
    comp_mat = findConnComp(adj_temp);
    for c=1:length(comp_mat); modules{length(modules)+1}=modules{curr_mod}(comp_mat{c}); end

    % remove "now" disconnected component (curr_mod) from modules
    modules{curr_mod}=modules{1};
    modules=modules(2:length(modules));
    
    modL=[];
    for j=1:length(modules); modL(j)=length(modules{j}); end
    [maxL,indL]=max(modL);
    curr_mod=indL;
    
    module_hist{length(module_hist)+1}=modules{indL};
    adj_temp=subgraph(adj,modules{indL});
    
end % end of while loop

% computing the modularity for the final module break-down
% Defined as: Q=sum_over_modules_i (eii-ai^2) (eq 5) in Newman and Girvan.
% eij = fraction of edges that connect community i to community j
% ai=sum_j (eij)

nedges=numEdges(adj); % compute the total number of edges

Q = 0;
for m=1:length(modules)
  module=modules{m};
  adj_m=subgraph(adj,module);
  
  e_mm=numEdges(adj_m)/nedges;
  a_m=sum(sum(adj(module,:)))/nedges-e_mm;
  
  Q = Q + (e_mm - a_m^2);
end