%APCLUSTER Affinity Propagation Clustering (Frey/Dueck, Science 2007)% [idx,netsim,dpsim,expref]=APCLUSTER(s,p) clusters data, using a set % of real-valued pairwise data point similarities as input. Clusters % are each represented by a cluster center data point (the "exemplar"). % The method is iterative and searches for clusters so as to maximize % an objective function, called net similarity.% % For N data points, there are potentially N^2-N pairwise similarities; % this can be input as an N-by-N matrix ‘s‘, where s(i,k) is the % similarity of point i to point k (s(i,k) needn抰 equal s(k,i)).  In % fact, only a smaller number of relevant similarities are needed; if % only M similarity values are known (M < N^2-N) they can be input as % an M-by-3 matrix with each row being an (i,j,s(i,j)) triple.% % APCLUSTER automatically determines the number of clusters based on % the input preference ‘p‘, a real-valued N-vector. p(i) indicates the % preference that data point i be chosen as an exemplar. Often a good % choice is to set all preferences to median(s); the number of clusters % identified can be adjusted by changing this value accordingly. If ‘p‘ % is a scalar, APCLUSTER assumes all preferences are that shared value.% % The clustering solution is returned in idx. idx(j) is the index of % the exemplar for data point j; idx(j)==j indicates data point j % is itself an exemplar. The sum of the similarities of the data points to % their exemplars is returned as dpsim, the sum of the preferences of % the identified exemplars is returned in expref and the net similarity % objective function returned is their sum, i.e. netsim=dpsim+expref.% %     [ ... ]=apcluster(s,p,‘NAME‘,VALUE,...) allows you to specify %       optional parameter name/value pairs as follows:% %   ‘maxits‘     maximum number of iterations (default: 1000)%   ‘convits‘    if the estimated exemplars stay fixed for convits %          iterations, APCLUSTER terminates early (default: 100)%   ‘dampfact‘   update equation damping level in [0.5, 1).  Higher %        values correspond to heavy damping, which may be needed %        if oscillations occur. (default: 0.9)%   ‘plot‘       (no value needed) Plots netsim after each iteration%   ‘details‘    (no value needed) Outputs iteration-by-iteration %      details (greater memory requirements)%   ‘nonoise‘    (no value needed) APCLUSTER adds a small amount of %      noise to ‘s‘ to prevent degenerate cases; this disables that.% % Copyright (c) B.J. Frey & D. Dueck (2006). This software may be % freely used and distributed for non-commercial purposes.%          (RUN APCLUSTER WITHOUT ARGUMENTS FOR DEMO CODE)function [idx,netsim,dpsim,expref]=apcluster(s,p,varargin);if nargin==0, % display demo    fprintf(‘Affinity Propagation (APCLUSTER) sample/demo code\n\n‘);    fprintf(‘N=100; x=rand(N,2); % Create N, 2-D data points\n‘);    fprintf(‘M=N*N-N; s=zeros(M,3); % Make ALL N^2-N similarities\n‘);    fprintf(‘j=1;\n‘);    fprintf(‘for i=1:N\n‘);    fprintf(‘  for k=[1:i-1,i+1:N]\n‘);    fprintf(‘    s(j,1)=i; s(j,2)=k; s(j,3)=-sum((x(i,:)-x(k,:)).^2);\n‘);    fprintf(‘    j=j+1;\n‘);    fprintf(‘  end;\n‘);    fprintf(‘end;\n‘);    fprintf(‘p=median(s(:,3)); % Set preference to median similarity\n‘);    fprintf(‘[idx,netsim,dpsim,expref]=apcluster(s,p,‘‘plot‘‘);\n‘);    fprintf(‘fprintf(‘‘Number of clusters: %%d\\n‘‘,length(unique(idx)));\n‘);    fprintf(‘fprintf(‘‘Fitness (net similarity): %%g\\n‘‘,netsim);\n‘);    fprintf(‘figure; % Make a figures showing the data and the clusters\n‘);    fprintf(‘for i=unique(idx)‘‘\n‘);    fprintf(‘  ii=find(idx==i); h=plot(x(ii,1),x(ii,2),‘‘o‘‘); hold on;\n‘);    fprintf(‘  col=rand(1,3); set(h,‘‘Color‘‘,col,‘‘MarkerFaceColor‘‘,col);\n‘);    fprintf(‘  xi1=x(i,1)*ones(size(ii)); xi2=x(i,2)*ones(size(ii)); \n‘);    fprintf(‘  line([x(ii,1),xi1]‘‘,[x(ii,2),xi2]‘‘,‘‘Color‘‘,col);\n‘);    fprintf(‘end;\n‘);    fprintf(‘axis equal tight;\n\n‘);    return;end;start = clock;% Handle arguments to functionif nargin<2 error(‘Too few input arguments‘);else    maxits=1000; convits=100; lam=0.9; plt=0; details=0; nonoise=0;    i=1;    while i<=length(varargin)        if strcmp(varargin{i},‘plot‘)            plt=1; i=i+1;        elseif strcmp(varargin{i},‘details‘)            details=1; i=i+1;        elseif strcmp(varargin{i},‘sparse‘)%             [idx,netsim,dpsim,expref]=apcluster_sparse(s,p,varargin{:});            fprintf(‘‘‘sparse‘‘ argument no longer supported; see website for additional software\n\n‘);            return;        elseif strcmp(varargin{i},‘nonoise‘)            nonoise=1; i=i+1;        elseif strcmp(varargin{i},‘maxits‘)            maxits=varargin{i+1};            i=i+2;            if maxits<=0 error(‘maxits must be a positive integer‘); end;        elseif strcmp(varargin{i},‘convits‘)            convits=varargin{i+1};            i=i+2;            if convits<=0 error(‘convits must be a positive integer‘); end;        elseif strcmp(varargin{i},‘dampfact‘)            lam=varargin{i+1};            i=i+2;            if (lam<0.5)||(lam>=1)                error(‘dampfact must be >= 0.5 and < 1‘);            end;        else i=i+1;        end;    end;end;if lam>0.9    fprintf(‘\n*** Warning: Large damping factor in use. Turn on plotting\n‘);    fprintf(‘    to monitor the net similarity. The algorithm will\n‘);    fprintf(‘    change decisions slowly, so consider using a larger value\n‘);    fprintf(‘    of convits.\n\n‘);end;% Check that standard arguments are consistent in sizeif length(size(s))~=2 error(‘s should be a 2D matrix‘);elseif length(size(p))>2 error(‘p should be a vector or a scalar‘);elseif size(s,2)==3    tmp=max(max(s(:,1)),max(s(:,2)));    if length(p)==1 N=tmp; else N=length(p); end;    if tmp>N        error(‘data point index exceeds number of data points‘);    elseif min(min(s(:,1)),min(s(:,2)))<=0        error(‘data point indices must be >= 1‘);    end;elseif size(s,1)==size(s,2)    N=size(s,1);    if (length(p)~=N)&&(length(p)~=1)        error(‘p should be scalar or a vector of size N‘);    end;else error(‘s must have 3 columns or be square‘); end;% Construct similarity matrixif N>3000    fprintf(‘\n*** Warning: Large memory request. Consider activating\n‘);    fprintf(‘    the sparse version of APCLUSTER.\n\n‘);end;if size(s,2)==3 && size(s,1)~=3,    S=-Inf*ones(N,N,class(s));     for j=1:size(s,1), S(s(j,1),s(j,2))=s(j,3); end;else S=s;end;if S==S‘, symmetric=true; else symmetric=false; end;realmin_=realmin(class(s)); realmax_=realmax(class(s));% In case user did not remove degeneracies from the input similarities,% avoid degenerate solutions by adding a small amount of noise to the% input similaritiesif ~nonoise    rns=randn(‘state‘); randn(‘state‘,0);    S=S+(eps*S+realmin_*100).*rand(N,N);    randn(‘state‘,rns);end;% Place preferences on the diagonal of Sif length(p)==1 for i=1:N S(i,i)=p; end;else for i=1:N S(i,i)=p(i); end;end;% Numerical stability -- replace -INF with -realmaxn=find(S<-realmax_); if ~isempty(n), warning(‘-INF similarities detected; changing to -REALMAX to ensure numerical stability‘); S(n)=-realmax_; end; clear(‘n‘);if ~isempty(find(S>realmax_,1)), error(‘+INF similarities detected; change to a large positive value (but smaller than +REALMAX)‘); end;% Allocate space for messages, etcdS=diag(S); A=zeros(N,N,class(s)); R=zeros(N,N,class(s)); t=1;if plt, netsim=zeros(1,maxits+1); end;if details    idx=zeros(N,maxits+1);    netsim=zeros(1,maxits+1);     dpsim=zeros(1,maxits+1);     expref=zeros(1,maxits+1); end;% Execute parallel affinity propagation updatese=zeros(N,convits); dn=0; i=0;if symmetric, ST=S; else ST=S‘; end; % saves memory if it‘s symmetricwhile ~dn    i=i+1;     % Compute responsibilities    A=A‘; R=R‘;    for ii=1:N,        old = R(:,ii);        AS = A(:,ii) + ST(:,ii); [Y,I]=max(AS); AS(I)=-Inf;        [Y2,I2]=max(AS);        R(:,ii)=ST(:,ii)-Y;        R(I,ii)=ST(I,ii)-Y2;        R(:,ii)=(1-lam)*R(:,ii)+lam*old; % Damping        R(R(:,ii)>realmax_,ii)=realmax_;    end;    A=A‘; R=R‘;    % Compute availabilities    for jj=1:N,        old = A(:,jj);        Rp = max(R(:,jj),0); Rp(jj)=R(jj,jj);        A(:,jj) = sum(Rp)-Rp;        dA = A(jj,jj); A(:,jj) = min(A(:,jj),0); A(jj,jj) = dA;        A(:,jj) = (1-lam)*A(:,jj) + lam*old; % Damping    end;        % Check for convergence    E=((diag(A)+diag(R))>0); e(:,mod(i-1,convits)+1)=E; K=sum(E);    if i>=convits || i>=maxits,        se=sum(e,2);        unconverged=(sum((se==convits)+(se==0))~=N);        if (~unconverged&&(K>0))||(i==maxits) dn=1; end;    end;    % Handle plotting and storage of details, if requested    if plt||details        if K==0            tmpnetsim=nan; tmpdpsim=nan; tmpexpref=nan; tmpidx=nan;        else            I=find(E); notI=find(~E); [tmp c]=max(S(:,I),[],2); c(I)=1:K; tmpidx=I(c);            tmpdpsim=sum(S(sub2ind([N N],notI,tmpidx(notI))));            tmpexpref=sum(dS(I));            tmpnetsim=tmpdpsim+tmpexpref;        end;    end;    if details        netsim(i)=tmpnetsim; dpsim(i)=tmpdpsim; expref(i)=tmpexpref;        idx(:,i)=tmpidx;    end;    if plt,        netsim(i)=tmpnetsim;        figure(234);        plot(((netsim(1:i)/10)*100)/10,‘r-‘); xlim([0 i]); % plot barely-finite stuff as infinite        xlabel(‘# Iterations‘);        ylabel(‘Fitness (net similarity) of quantized intermediate solution‘);%         drawnow;     end;end; % iterationsI=find((diag(A)+diag(R))>0); K=length(I); % Identify exemplarsif K>0    [tmp c]=max(S(:,I),[],2); c(I)=1:K; % Identify clusters    % Refine the final set of exemplars and clusters and return results    for k=1:K ii=find(c==k); [y j]=max(sum(S(ii,ii),1)); I(k)=ii(j(1)); end; notI=reshape(setdiff(1:N,I),[],1);    [tmp c]=max(S(:,I),[],2); c(I)=1:K; tmpidx=I(c);    tmpdpsim=sum(S(sub2ind([N N],notI,tmpidx(notI))));    tmpexpref=sum(dS(I));    tmpnetsim=tmpdpsim+tmpexpref;else    tmpidx=nan*ones(N,1); tmpnetsim=nan; tmpexpref=nan;end;if details    netsim(i+1)=tmpnetsim; netsim=netsim(1:i+1);    dpsim(i+1)=tmpdpsim; dpsim=dpsim(1:i+1);    expref(i+1)=tmpexpref; expref=expref(1:i+1);    idx(:,i+1)=tmpidx; idx=idx(:,1:i+1);else    netsim=tmpnetsim; dpsim=tmpdpsim; expref=tmpexpref; idx=tmpidx;end;if plt||details    fprintf(‘\nNumber of exemplars identified: %d  (for %d data points)\n‘,K,N);    fprintf(‘Net similarity: %g\n‘,tmpnetsim);    fprintf(‘  Similarities of data points to exemplars: %g\n‘,dpsim(end));    fprintf(‘  Preferences of selected exemplars: %g\n‘,tmpexpref);    fprintf(‘Number of iterations: %d\n\n‘,i);    fprintf(‘Elapsed time: %g sec\n‘,etime(clock,start));end;if unconverged    fprintf(‘\n*** Warning: Algorithm did not converge. Activate plotting\n‘);    fprintf(‘    so that you can monitor the net similarity. Consider\n‘);    fprintf(‘    increasing maxits and convits, and, if oscillations occur\n‘);    fprintf(‘    also increasing dampfact.\n\n‘);end;

s=[1 0.85 0.9 0.5 0.45 0.5 0.4 0.4 0.5 0.45;   0.85 1 0.85 0.6 0.65 0.7 0.6 0.55 0.8 0.7;   0.9 0.85 1 0.75 0.7 0.65 0.55 0.5 0.6 0.5;   0.5 0.6 0.75 1 0.9 0.7 0.7 0.85 0.5 0.45;   0.45 0.65 0.7 0.9 1 0.9 0.9 0.85 0.6 0.65;   0.5 0.7 0.65 0.7 0.9 1 0.85 0.75 0.75 0.75;   0.4 0.6 0.55 0.7 0.9 0.85 1 0.85 0.5 0.55;   0.4 0.55 0.5 0.85 0.85 0.75 0.85 1 0.3 0.25;   0.5 0.8 0.6 0.5 0.6 0.75 0.5 0.3 1 0.9;   0.45 0.7 0.5 0.45 0.65 0.75 0.55 0.25 0.9 1;    ];p=median(median(s));

idx =     1     1     1     5     5     5     5     5     9     9netsim =    8.1875dpsim =    6.2000expref =    1.9875