transitivity_wd.m

function T=transitivity_wd(W)
%TRANSITIVITY_WD    Transitivity
%
%   T = transitivity_wd(W);
%
%   Transitivity is the ratio of 'triangles to triplets' in the network.
%   (A classical version of the clustering coefficient).
%
%   Input:      W       weighted directed connection matrix
%
%   Output:     T       transitivity scalar
%
%   Reference:  Rubinov M, Sporns O (2010) NeuroImage 52:1059-69
%               based on Fagiolo (2007) Phys Rev E 76:026107.
%
%
%   Mika Rubinov, UNSW/U-Cambridge
%   Christoph Schmidt, Friedrich Schiller University Jena
%   2007-2013

% Modification history:
% 2007: original (MR)
% 2013: removed test for absence of nodewise 3-cycles (CS)


%   Methodological note (also see note for clustering_coef_bd)
%   The weighted modification is as follows:
%   - The numerator: adjacency matrix is replaced with weights matrix ^ 1/3
%   - The denominator: no changes from the binary version
%
%   The above reduces to symmetric and/or binary versions of the clustering
%   coefficient for respective graphs.
T=0;
A=W~=0;                             %adjacency matrix
S=W.^(1/3)+(W.').^(1/3);            %symmetrized weights matrix ^1/3
K=sum(A+A.',2);                     %total degree (in + out)
cyc3=diag(S^3)/2;                   %number of 3-cycles (ie. directed triangles)
CYC3=K.*(K-1)-2*diag(A^2);          %number of all possible 3-cycles
    T=sum(cyc3)./sum(CYC3);         %transitivity