% Fast multi-scale detection of communities using Arenas et al.'s parameter
% described in (Arenas, Fernandez, Gomez; Analysis of the structure of
% complex networks at different resolution levels, New Journal of Physics,
% 2008)
%
% Author: Erwan Le Martelot
% Date: 09/11/11
%
% Input
% - adj: adjacency matrix
% - ps : list of r values to consider (from high to low)
%
% Output
% - coms: community (crip) membership per node
% - Qs : criterion values of the corresponding partitions
%
% Comments:
% - This implementation uses Arenas et al. criterion. The code specific to
% it is placed between comment signs lines and can be replaced with code
% optimising any other criterion. It was placed within the algorithm and
% not in external functions for speed optimisation purposes only.
function [coms,Qs] = mscd_afg(adj, ps)
% Check there is at least one scale parameter
if (nargin < 2) || isempty(ps)
error('One scale parameter value at least is required: ms_afg(adj,ps)');
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Neighbours list for each node
for i=1:length(adj)
Nbs{i} = find(adj(i,:));
Nbs{i}(Nbs{i}==i) = [];
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Initial community partition: each node in one separate community
com = 1:length(adj);
% Compute community partition for the current parameter
for p_idx=1:length(ps)
% Current parameter value
p = ps(p_idx);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%fprintf('r = %g\n',p);
% Compute current adj
adj_p = adj + p*eye(length(adj));
% Total weight m and its double m2
m2 = sum(sum(adj_p));
m = m2/2;
% Degree vector
d = sum(adj_p,2);
% Total weight of a community
ucom = unique(com);
for i=1:length(ucom)
wcom(i) = sum(d(com==ucom(i)));
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Initial Q value
Q = compute_Q(adj_p, com, m2, d);
% While changes can be made
check_nodes = true;
check_communities = true;
while check_nodes
% disp('AFG big loop');
% Nodes moving
moved = true;
while moved
%fprintf('Node loop\n');
moved = false;
% Create list of nodes to inspect
l = 1:length(adj_p);
% While the list of candidates is not finished
while ~isempty(l)
% Pick at random a node n from l and remove it from l
idx = randi(length(l));
n = l(idx);
l(idx) = [];
% Find neighbour communities of n
ncom = unique(com(Nbs{n}));
ncom(ncom == com(n)) = [];
% For each neighbour community of n
best_dQ = 0;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
nb = Nbs{n};
nb1 = nb(com(nb) == com(n));
sum_nb1 = -sum(adj_p(n,nb1));
w1 = wcom(com(n)) - d(n);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for i=1:length(ncom)
% Compute dQ for moving n to current community
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
c = ncom(i);
nb2 = nb(com(nb) == c);
dQ = sum_nb1+sum(adj_p(n,nb2));
dQ = (dQ + (d(n)*(w1-wcom(c)))/m2)/m;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% If best so far, keep track of the best
if dQ > best_dQ
best_dQ = dQ;
new_c = ncom(i);
end
end
% If a move is worth it, do it
if best_dQ > 0
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Update total weight of communities
wcom(com(n)) = wcom(com(n)) - d(n);
wcom(new_c) = wcom(new_c) + d(n);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Update community of n
com(n) = new_c;
% Update Q
Q = Q + best_dQ;
%%%%%%%%%% DEBUG %%%%%%%%%
% % %fprintf('Move node %g: Q=%g\n',n,Q);
% eqQ = compute_Q(adj_p, com, m2, d);
% if abs(Q - eqQ) >= 0.00001
% fprintf('Node Warning: found Q=%f, should be Q=%f. Diff = %f\n',Q, eqQ, abs(Q-eqQ));
% fprintf('Correcting\n'); Q = eqQ;
% end
%%%%%%%%%%%%%%%%%%%%%%%%%%
% A move occured
moved = true;
check_communities = true;
end
end
end % Nodes
check_nodes = false;
if ~check_communities
break;
end
% Community merging
moved = true;
while moved
%fprintf('Community loop\n');
moved = false;
% Create community list cl
cl = unique(com);
% While the list of candidates is not finished
while ~isempty(cl)
% Pick at random a community cn from cl and remove it from cl
idx = randi(length(cl));
cn = cl(idx);
cl(idx) = [];
% Find neighbour communities of cn
ncn = find(com==cn);
nbn = unique([Nbs{ncn}]);
ncom = unique(com(nbn));
ncom(ncom == cn) = [];
% For each neighbour community of cn
best_dQ = 0;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
sum_dn1 = sum(d(ncn));
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for ncom_idx=1:length(ncom)
% Compute dQ for merging cn with current community
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
n2 = com==ncom(ncom_idx);
dQ = (sum(sum(adj_p(ncn,n2))) - sum_dn1*sum(d(n2))/m2)/m;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% If positive, keep track of the best
if dQ > best_dQ
best_dQ = dQ;
new_cn = ncom(ncom_idx);
end
end
% If a move is worth it, do it
if best_dQ > 0
% disp('Move com');
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Update total weight of communities
wcom(new_cn) = wcom(new_cn) + wcom(cn);
wcom(cn) = 0;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Merge communities
com(ncn) = new_cn;
% Update Q
Q = Q + best_dQ;
%%%%%%%%%% DEBUG %%%%%%%%%
% % %fprintf('Merge communities %g and %g: Q=%g\n',cn,new_cn,Q);
% eqQ = compute_Q(adj_p, com, m2, d);
% if abs(Q - eqQ) >= 0.00001
% fprintf('Com Warning: found Q=%f, should be Q=%f. Diff = %f\n',Q, eqQ, abs(Q-eqQ));
% fprintf('Correcting\n'); Q = eqQ;
% end
%%%%%%%%%%%%%%%%%%%%%%%%%%
% A move occured
moved = true;
check_nodes = true;
end
end
end % Communities
check_communities = false;
end % while changes can be made
% Reindexing communities
ucom = unique(com);
for i=1:length(com)
com(i) = find(ucom==com(i));
end
% Storing best community found for current parameter value
coms{p_idx} = com';
Qs(p_idx) = Q;
end % for p
end
% Compute the value of Q for the given partition
function [Q] = compute_Q(adj, com, m2, d)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Q = 0;
for i=1:length(adj)
Q = Q + adj(i,i);
for j=i+1:length(adj)
if com(i) == com(j)
Q = Q + 2*(adj(i,j) - (d(i)*d(j))/m2);
end
end
Q = Q - (d(i)*d(i))/m2;
end
Q = Q / m2;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
end