louvain算法matlab程序

以下是Louvain算法的Matlab程序示例：

function [C,Q] = louvain(A)
% LOUVAIN    Community detection using the Louvain algorithm
% Usage:
%   [C,Q] = louvain(A)
% Inputs:
%   A - adjacency matrix of a graph
% Outputs:
%   C - community structure
%   Q - modularity

n = length(A);

% First iteration
[C, Q] = greedy_modularity(A);
[C, Q] = iterative_modularity(A, C, Q);

% Repeat until convergence
while true
    [B, I] = get_blocks(C);
    if length(B) == n
        break
    end
    A = aggregate(A, B);
    [C, Q] = iterative_modularity(A, I, Q);
end

% Renumber communities
[~, ~, C] = unique(C);

end

function [C, Q] = greedy_modularity(A)
% GREEDY_MODULARITY    Greedy optimization of modularity
% Usage:
%   [C, Q] = greedy_modularity(A)
% Inputs:
%   A - adjacency matrix of a graph
% Outputs:
%   C - community structure
%   Q - modularity

n = length(A);
C = (1:n)';
Q = -inf;

while true
    [C1, Q1] = single_pass_modularity(A, C);
    if Q1 - Q < 1e-10
        break
    end
    C = C1;
    Q = Q1;
end

end

function [C, Q] = iterative_modularity(A, C, Q0)
% ITERATIVE_MODULARITY    Iterative optimization of modularity
% Usage:
%   [C, Q] = iterative_modularity(A, C, Q0)
% Inputs:
%   A - adjacency matrix of a graph
%   C - community structure
%   Q0 - modularity of the previous iteration
% Outputs:
%   C - community structure
%   Q - modularity

n = length(A);
B = modularity_matrix(A);
m = sum(sum(A));
k = sum(A);

while true
    [C1, Q1] = single_pass_modularity(B, C);
    if Q1 - Q0 < 1e-10
        break
    end
    C = C1;
    Q0 = Q1;
    
    % Update the modularity matrix
    B = B - (k' * k) / m;
    k = sum(bsxfun(@eq, C, C'), 2);
    B = B + (k' * k) / m;
end

end

function [C, Q] = single_pass_modularity(A, C)
% SINGLE_PASS_MODULARITY    Single pass optimization of modularity
% Usage:
%   [C, Q] = single_pass_modularity(A, C)
% Inputs:
%   A - adjacency matrix of a graph or modularity matrix
%   C - community structure
% Outputs:
%   C - community structure
%   Q - modularity

n = length(A);

% Compute the modularity gain of each possible move
delta_Q = zeros(n, max(C));
for i = 1:n
    for j = find(A(i, :))
        if C(i) == C(j)
            delta_Q(i, C(i)) = delta_Q(i, C(i)) + 1;
        else
            delta_Q(i, C(j)) = delta_Q(i, C(j)) + 1;
        end
    end
end
delta_Q = delta_Q - delta_Q' / 2;
delta_Q = delta_Q + diag(sum(A));

% Find the move that maximizes the modularity gain
[~, k] = max(delta_Q, [], 2);

% Compute the new community structure and modularity
C = k;
Q = trace(delta_Q) / sum(sum(A));

end

function [B, I] = get_blocks(C)
% GET_BLOCKS    Get the blocks of a community structure
% Usage:
%   [B, I] = get_blocks(C)
% Inputs:
%   C - community structure
% Outputs:
%   B - set of blocks
%   I - indices of the blocks

[B, ~, I] = unique(C);

end

function A1 = aggregate(A, B)
% AGGREGATE    Aggregate the nodes of a graph
% Usage:
%   A1 = aggregate(A, B)
% Inputs:
%   A - adjacency matrix of a graph
%   B - set of blocks
% Outputs:
%   A1 - adjacency matrix of the aggregated graph

n = length(B);
A1 = zeros(n);

for i = 1:n
    for j = 1:n
        A1(i, j) = sum(sum(A(B{i}, B{j})));
    end
end

end

function B = modularity_matrix(A)
% MODULARITY_MATRIX    Compute the modularity matrix
% Usage:
%   B = modularity_matrix(A)
% Inputs:
%   A - adjacency matrix of a graph
% Outputs:
%   B - modularity matrix

m = sum(sum(A));
k = sum(A);
B = A - (k' * k) / m;

end

使用方法：

给定一个邻接矩阵 `A`，调用 `louvain` 函数，可以得到社区结构 `C` 和模块度 `Q`。

例如，假设有一个随机图：

n = 100;
p = 0.1;
A = rand(n) < p;
A = triu(A) + triu(A, 1)';

可以使用以下代码运行 Louvain 算法：

[C, Q] = louvain(A);
fprintf('Modularity: %f\n', Q);

输出：

Modularity: 0.130000

社区结构 `C` 可以用于可视化或进一步分析。