Adjacency matrix with periodic boundaries

I have a 2-dimensional square lattice of nodes .

I need to find the distance between the nodes (adjacency matrix), but I should account for periodic boundary conditions. This means that the 2d sheet of nodes must first be wrapped into a torus, and then I should assess the distances between the nodes. I can find the adjacency matrix for a few number of nodes. But I wonder if there is a more general method that I could use to code a function that given the number of nodes in the 2D lattice would return the adjacency matrix.

I read that the adjacency matrix could be computed by using a sum of kronecker products. But I did not understand it.

Any help?

Answer 1

[Ayush Anand]

Hi,

I understand you are trying to compute the adjacency matrix for a 2D lattice with periodic boundary conditions. You can use the concept of a Kronecker product for the same. The Kronecker product allows you to construct a larger matrix by taking all possible combinations of elements from two smaller matrices.

You can follow the steps below to construct the adjacency matrix for a 2D lattice with periodic boundary conditions:

1. Define the adjacency matrix for a single row or column of nodes: You can start by creating a matrix that represents the adjacency relationships between neighboring nodes in a single row or column.

2. Compute the adjacency matrix for a 2D lattice: To obtain the adjacency matrix for the entire 2D lattice, you can use the Kronecker product to combine the adjacency matrices for the rows and columns.

3. Apply periodic boundary conditions: To account for periodic boundary conditions, you need to consider the wrap-around connections between the first and last rows/columns of the lattice. This can be achieved by taking the Kronecker product of the adjacency matrix with a circulant matrix that represents the wrap-around connections.

You can use the "kron" function in MATLAB to compute the Kronecker product and the "circulant" function to generate the circulant matrix. A sample code for the above steps could be as follows:

% Define the dimensions of the lattice
m = 3; % number of rows
n = 4; % number of columns

% Adjacency matrix for a single row
row_adjacency = eye(n) + circshift(eye(n), [0, 1]) + circshift(eye(n), [0, -1]);

% Adjacency matrix for a single column
column_adjacency = eye(m) + circshift(eye(m), [1, 0]) + circshift(eye(m), [-1, 0]);

% Compute adjacency matrix for the 2D lattice
adjacency = kron(row_adjacency, eye(m)) + kron(eye(n), column_adjacency);

% Apply periodic boundary conditions
wrap_around_adjacency = kron(circulant(ones(m, 1)), eye(m*n)) + kron(eye(m*n), circulant(ones(n, 1)));
adjacency = adjacency + wrap_around_adjacency;

You can refer to the following page for more details on the "kron" function and the "circulant" function:

https://www.mathworks.com/help/matlab/ref/kron.html

https://www.mathworks.com/matlabcentral/fileexchange/22876-circulant

I hope this helps!

[JOHN NIKI]

wrap_around_adjacency = kron(circulant(ones(m, 1)), eye(m*n)) + kron(eye(m*n), circulant(ones(n, 1)));

Arrays have incompatible sizes for this operation.