function [connectivity_vec, sub_minus_full] = connectivity(F)
% PURPOSE: Compute the degree of weak connectivity, unilateral connectivity, 
%          and strong connectivity of a fuzzy graph F. We also judge whether
%          each individual is a weekening point, neutral point, or 
%          strengthening point.
%-------------------------------------------------------------------------------
% USAGE: [connectivity_vec, sub_minus_full] = connectivity(F)
% where: F = (asymmetric) fuzzy graph (n x n)
%            F(i,j) is the dependence of variable i on variable j
%            all diagonal elements should be equal to 1.
%            [Example] F = [1.0, 0.8, 0.9;
%                           0.2, 1.0, 0.0;
%                           0.0, 0.5, 1.0];
%        n = # of variables
%-------------------------------------------------------------------------------
% RETURNS: connectivity_vec = [connectivity_w; connectivity_u; connectivity_s]
%                             (3 x 1)
%          connectivity_w = degree of weak connectivity of F
%          connectivity_u = degree of unilateral connectivity of F
%          connectivity_s = degree of strong connectivity of F
%          sub_minus_full = difference between connectivity of a subgraph F
%                           and fuzzy graph F  (n x 3)
%                           sub_minus_full(i,:) is concerned with individual i. 
%                           sub_minus_full(:,1) is concerned with weak connectivity.
%                           sub_minus_full(:,2) is concerned with unilateral connectivity.
%                           sub_minus_full(:,3) is concerned with strong connectivity.
%                           (e.g. If sub_minus_full(1,1) is NEGATIVE, then individual 1
%                                 is a STRENGTHENING point in terms of weak connectivity.
%                                 If sub_minus_full(1,1) is ZERO, then individual 1
%                                 is a NEUTRAL point in terms of weak connectivity. 
%                                 If sub_minus_full(1,1) is POSITIVE, then individual 1
%                                 is a WEAKENING point in terms of weak connectivity.)
% -------------------------------------------------------------------------------
% SEE ALSO: maxminTM.m
% -------------------------------------------------------------------------------
% REFERENCE: 
% H. Yamashita and T. Takizawa (2010). Fuzzy Theory. Tokyo: Kyoritsu
% Shuppan Co., Ltd. (in Japanese).
% ------------------------------------------------------------------------------
% Written by Kaiji Motegi, Faculty of Political Science and Economics,
% Waseda University.
% June 25, 2014.
% ------------------------------------------------------------------------------

% define "transmax":
% a function taking the element-wise maximum of a matrix A and its transpose
function outcome = transmax(A)
          outcome = max(A, A');
end

% max-min transition matrix of F
F_hat = maxminTM(F);
% max-min transition matrix of transmax of F
F_transmax_hat = maxminTM(transmax(F));

% the degree of weak connectivity is equal to the smallest element of 
% transition matrix of transmax of F
% (Note: direction does not matter when weak connectivity is concerned)
connectivity_w = min(min(F_transmax_hat));

% the degree of unilateral connectivity is equal to the smallest element of
% transmax of transition matrix of F
% (Note: direction does matter but only unilateral relationship is required
%        when unilateral connectivity is concerned)
connectivity_u = min(min(transmax(F_hat)));

% the degree of strong connectivity is equal to the smallest element of
% transition matrix of F
% (Note: direction does matter and bilateral relationship is required
%        when strong connectivity is concerned)
connectivity_s = min(min(F_hat));

% save results
connectivity_vec = [connectivity_w; connectivity_u; connectivity_s];

n = size(F,1);       % # of variables
sub_minus_full = zeros(n, 3);
for i = 1:n   % work on each variable
     % remove individual i in order to get a subfigure
     F_sub = F([1:(i-1) (i+1):end], [1:(i-1) (i+1):end]);
     % get transition matrix of the subfigure
     F_sub_hat = maxminTM(F_sub);
     % get transition matrix of transmax of subfigure
     F_sub_transmax_hat = maxminTM(transmax(F_sub));

     % weak connectivity of subfigure
     connectivity_sub_w = min(min(F_sub_transmax_hat));
     % unilateral connectivity of subfigure     
     connectivity_sub_u = min(min(transmax(F_sub_hat)));
     % strong connectivity of subfigure
     connectivity_sub_s = min(min(F_sub_hat));
      
     % collect results
     connectivity_sub_vec = [connectivity_sub_w; 
                             connectivity_sub_u; 
                             connectivity_sub_s];
     
     % difference between connectivity of subfigure and connectivity of
     % original figure.
     % If an element is NEGATIVE, then it means that connectivity has gone
     % down by removing individual i and hence individual i is a STRENGTHENING point.
     % If an element is ZERO, then it means that connectivity has not changed
     % by removing individual i and hence individual i is a NEUTRAL point.
     % If an element is POSITIVE, then it means that connectivity has gone up
     % by removing individual i and hence individual i is a WEAKENING point.     
     sub_minus_full(i,:) = (connectivity_sub_vec - connectivity_vec)';                      
end;    

end