I want to implement the following optimization problem from the following paper Randomized Gossip Algorithms, Page 10 Eq 53

The screenshot of the optimization problem.

1- In this problem, W, P, and P_{ij} are n-by-n matrices. I would appreciate if you help me with implementing the following constraint in CVX.

W=\frac{1}{n}\sum_{i,j=1}^{n}P_{i,j}W_{i,j}

2- Also, in this problem, E is a set of neighbors of a nod i. Constraint P_{ij}=0 if {i,j}\not\in{E} means that P_{ij} is zero if node i and j are not neighbors. Does anyone can help with how to implement this neighborhood relationship?

For $n=3$, neighbors.xlsx can look like:

screenshot of neighbors.xlsx

This means node 1 is neighbor with node 2, node 2 is neighbor with node 1 and 3, and node 3 is neighbor with node2.

I have the written the following piece of code for that in Matlab.

cvx_begin sdp
    agt = struct([]);
    neighbors = readcell('neighbors.xlsx');
    N = 2;
    for i = 1:N
      agt(i).neighbors = neighbors{i};
    end
    variable s
    variable P(N,N) symmetric
    variable W_ij(N,N) symmetric
    expression W
    
    minimize (s)
    
subject to     

P(:) >= 0;

    j = 1;
    for i = 1:N
        D =[i,j];
        if ~ismember(D,agt(i).neighbors)
            P(i,j)== 0;
        end
        j = j+1;
    end


    for i = 1:N
        for j = 1:N
            W = P(i,j).*W_ij;
        end    
    end
    W = (1/N).*W;
    W-(1/N)*ones(N,1)*ones(1,N) - s*eye(N) == semidefinite(N);

cvx_end

It does not work, and I get the following error. Any help is greatly appreciated.

Error using  .*  (line 262)
Disciplined convex programming error:
    Invalid quadratic form(s): not a square.


Error in lambda (line 35)
            W = P(i,j).*W_ij;