A 2D lattice

% A 2D lattice example for the noise-corrupted leader-selection problem.
%
% form the incidence matrix and graph Laplacian matrix for 2D lattice
%
% number of nodes for the N x N 2D lattice
n = 81;
N = sqrt(n);

% compute the number of edges
ne = 0.5*( 2*4 + (N-2)*3*4 + (N-2)^2*4 );

% form the index set of edges
idx = zeros(2,ne);

% evaluate the indices for those edges on the rows of the grid
for i = 1:N
    idx(1,(N-1)*(i-1)+1:(N-1)*i) = N*(i-1)+1:N*(i-1)+(N-1);
    idx(2,(N-1)*(i-1)+1:(N-1)*i) = N*(i-1)+2:N*(i-1)+N;
end

% evaluate the indices for those edges on the columns of the grid
for i = 1:N
    idx(1,ne/2+ ((N-1)*(i-1)+1:(N-1)*i) ) = i:N:(i+(N-2)*N);
    idx(2,ne/2+ ((N-1)*(i-1)+1:(N-1)*i) ) = (i:N:(i+(N-2)*N))+N;
end

% form the incidence matrix Eg of the 2D lattice
Eg = incmat(idx);

% form the graph Laplacian
L = Eg*Eg';

% kappa is taken as the diagonal of L
kappa = diag(L);

% Start solving the leader selection problem
%
% choose the number of leaders
kval = 1:1:40;

% pre-allocate memory for data collection
Jlow = zeros(size(kval));
Jup = zeros(size(kval));
LSgreed = zeros(n,length(kval));

for i = 1:length(kval)

    Nl = kval(i);
    flag = 1; % for the noise-corrupted leader selection formulation
    [Jlow(i),Jup(i),LSgreed(:,i)] = leaders(L,Nl,kappa,flag);

end