```% S. Boyd, et. al., "Convex Optimization of Graph Laplacian Eigenvalues"
% ICM'06 talk examples (www.stanford.edu/~boyd/cvx_opt_graph_lapl_eigs.html)
% Written for CVX by Almir Mutapcic 08/29/06
% (figures are generated)
%
% In this example we consider a graph described by the incidence matrix A.
% Each edge has a weight W_i, and we optimize various functions of the
% edge weights as described in the referenced paper; in particular,
%
% - the fastest distributed linear averaging (FDLA) problem (fdla.m)
% - the fastest mixing Markov chain (FMMC) problem (fmmc.m)
%
% Then we compare these solutions to the heuristics listed below:
%
% - maximum-degree heuristic (max_deg.m)
% - constant weights that yield fastest averaging (best_const.m)
% - Metropolis-Hastings heuristic (mh.m)

% small example (incidence matrix A)
A = [ 1  0  0  1  0  0  0  0  0  0  0  0  0;
-1  1  0  0  1  1  0  0  0  0  0  0  1;
0 -1  1  0  0  0  0  0 -1  0  0  0  0;
0  0 -1  0  0 -1  0  0  0 -1  0  0  0;
0  0  0 -1  0  0 -1  1  0  0  0  0  0;
0  0  0  0  0  0  1  0  0  0  1  0  0;
0  0  0  0  0  0  0 -1  1  0 -1  1 -1;
0  0  0  0 -1  0  0  0  0  1  0 -1  0];

% x and y locations of the graph nodes
xy = [ 1 2   3 3 1 1 2   3 ; ...
3 2.5 3 2 2 1 1.5 1 ]';

% Compute edge weights: some optimal, some based on heuristics
[n,m] = size(A);

[ w_fdla, rho_fdla ] = fdla(A);
[ w_fmmc, rho_fmmc ] = fmmc(A);
[ w_md,   rho_md   ] = max_deg(A);
[ w_bc,   rho_bc   ] = best_const(A);
[ w_mh,   rho_mh   ] = mh(A);

tau_fdla = 1/log(1/rho_fdla);
tau_fmmc = 1/log(1/rho_fmmc);
tau_md   = 1/log(1/rho_md);
tau_bc   = 1/log(1/rho_bc);
tau_mh   = 1/log(1/rho_mh);

fprintf(1,'\nResults:\n');
fprintf(1,'FDLA weights:\t\t rho = %5.4f \t tau = %5.4f\n',rho_fdla,tau_fdla);
fprintf(1,'FMMC weights:\t\t rho = %5.4f \t tau = %5.4f\n',rho_fmmc,tau_fmmc);
fprintf(1,'M-H weights:\t\t rho = %5.4f \t tau = %5.4f\n',rho_mh,tau_mh);
fprintf(1,'MAX_DEG weights:\t rho = %5.4f \t tau = %5.4f\n',rho_md,tau_md);
fprintf(1,'BEST_CONST weights:\t rho = %5.4f \t tau = %5.4f\n',rho_bc,tau_bc);

% Plot results
figure(1), clf
plotgraph(A,xy,w_fdla);
text(0.55,1.05,'FDLA optimal weights')

figure(2), clf
plotgraph(A,xy,w_fmmc);
text(0.55,1.05,'FMMC optimal weights')

figure(3), clf
plotgraph(A,xy,w_md);
text(0.5,1.05,'Max degree optimal weights')

figure(4), clf
plotgraph(A,xy,w_bc);
text(0.5,1.05,'Best constant optimal weights')

figure(5), clf
plotgraph(A,xy,w_mh);
text(0.46,1.05,'Metropolis-Hastings optimal weights')
```
```
Calling sedumi: 73 variables, 59 equality constraints
------------------------------------------------------------
SeDuMi 1.21 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
Split 1 free variables
eqs m = 59, order n = 19, dim = 131, blocks = 3
nnz(A) = 145 + 0, nnz(ADA) = 2791, nnz(L) = 1425
it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
0 :            1.70E+00 0.000
1 :   6.61E-01 4.22E-01 0.000 0.2482 0.9000 0.9000   0.59  1  1  2.1E+00
2 :   6.11E-01 1.41E-01 0.000 0.3331 0.9000 0.9000   2.38  1  1  3.7E-01
3 :   6.60E-01 2.95E-02 0.000 0.2094 0.9000 0.9000   1.20  1  1  7.1E-02
4 :   6.46E-01 5.39E-03 0.000 0.1829 0.9000 0.9000   1.10  1  1  1.3E-02
5 :   6.44E-01 2.24E-04 0.000 0.0416 0.9900 0.9900   1.02  1  1  5.2E-04
6 :   6.43E-01 3.51E-06 0.000 0.0157 0.9900 0.9317   1.00  1  1  2.0E-05
7 :   6.43E-01 2.99E-07 0.201 0.0853 0.9458 0.9450   1.00  1  1  1.8E-06
8 :   6.43E-01 5.40E-08 0.248 0.1805 0.9156 0.9000   1.00  1  1  3.7E-07
9 :   6.43E-01 9.46E-09 0.000 0.1754 0.9099 0.9000   1.00  1  1  6.5E-08
10 :   6.43E-01 2.88E-10 0.156 0.0304 0.9900 0.9901   1.00  2  2  2.0E-09

iter seconds digits       c*x               b*y
10      0.1   Inf  6.4333140650e-01  6.4333140760e-01
|Ax-b| =   3.5e-10, [Ay-c]_+ =   1.2E-09, |x|=  3.5e+00, |y|=  9.5e-01

Detailed timing (sec)
Pre          IPM          Post
1.000E-02    7.000E-02    1.000E-02
Max-norms: ||b||=6.250000e-01, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 99.2911.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.643331

Calling sedumi: 94 variables, 80 equality constraints
------------------------------------------------------------
SeDuMi 1.21 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
Split 1 free variables
eqs m = 80, order n = 40, dim = 152, blocks = 3
nnz(A) = 180 + 0, nnz(ADA) = 4438, nnz(L) = 2296
it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
0 :            7.51E-01 0.000
1 :   2.31E-01 2.60E-01 0.000 0.3466 0.9000 0.9000   3.03  1  1  1.2E+00
2 :   6.39E-01 8.43E-02 0.000 0.3238 0.9000 0.9000   1.54  1  1  2.9E-01
3 :   6.99E-01 2.04E-02 0.000 0.2420 0.9000 0.9000   1.32  1  1  6.0E-02
4 :   6.82E-01 4.67E-03 0.000 0.2291 0.9000 0.9000   1.14  1  1  1.3E-02
5 :   6.81E-01 1.18E-03 0.000 0.2530 0.9000 0.9000   1.02  1  1  3.3E-03
6 :   6.81E-01 2.49E-04 0.000 0.2102 0.9000 0.8437   1.00  1  1  7.6E-04
7 :   6.81E-01 7.95E-06 0.000 0.0320 0.9900 0.9900   1.00  1  1  2.4E-05
8 :   6.81E-01 8.85E-07 0.193 0.1114 0.9450 0.9118   1.00  1  1  2.9E-06
9 :   6.81E-01 1.35E-07 0.041 0.1520 0.9021 0.9000   1.00  1  1  4.6E-07
10 :   6.81E-01 1.75E-08 0.052 0.1304 0.9070 0.9000   1.00  1  1  7.1E-08
11 :   6.81E-01 1.34E-09 0.472 0.0762 0.9900 0.9900   1.00  1  1  5.4E-09

iter seconds digits       c*x               b*y
11      0.1   8.9  6.8096067815e-01  6.8096067720e-01
|Ax-b| =   6.2e-09, [Ay-c]_+ =   2.1E-09, |x|=  3.7e+00, |y|=  1.2e+00

Detailed timing (sec)
Pre          IPM          Post
0.000E+00    8.000E-02    1.000E-02
Max-norms: ||b||=8.750000e-01, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 40.1861.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.680961

Results:
FDLA weights:		 rho = 0.6433 	 tau = 2.2671
FMMC weights:		 rho = 0.6810 	 tau = 2.6025
M-H weights:		 rho = 0.7743 	 tau = 3.9094
MAX_DEG weights:	 rho = 0.7793 	 tau = 4.0093
BEST_CONST weights:	 rho = 0.7119 	 tau = 2.9422
```