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ISFCClustering/2019_03_03_BCT/resource_efficiency_bin.m
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resource_efficiency_bin.m
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function [Eres,prob_SPL] = resource_efficiency_bin(adj,lambda,SPL,M)
% RESOURCE_EFFICIENCY_BIN Resource efficiency and shortest-path probability
%
% [Eres,prob_SPL] = resource_efficiency_bin(adj,lambda,SPL,M)
%
% The resource efficiency between nodes i and j is inversly proportional
% to the amount of resources (i.e. number of particles or messages)
% required to ensure with probability 0 < lambda < 1 that at least one of
% them will arrive at node j in exactly SPL steps, where SPL is the
% length of the shortest-path between i and j.
%
% The shortest-path probability between nodes i and j is the probability
% that a single random walker starting at node i will arrive at node j by
% following (one of) the shortest path(s).
%
% Inputs:
%
% adj,
% Unweighted, undirected adjacency matrix
%
% lambda,
% Probability (0 < lambda < 1)
% set to NAN if computation of Eres is not desired
%
% SPL,
% Shortest-path length matrix (optional)
%
% M,
% Transition probability matrix (optional)
%
%
% Outputs:
%
% Eres,
% Resource efficiency matrix.
%
% prob_SPL,
% Shortest-path probability matrix
%
% Notes:
%
% Global measures for both Eres and prob_SPL are well defined and can
% be computed as the average across the off-diagonal elements:
% GEres = mean(Eres(~eye(N)>0));
% Gprob_SPL = mean(rob_SPL(~eye(N)>0));
%
%
% Reference: Goñi J, et al (2013) PLoS ONE
%
%
% Joaquin Goñi, IU Bloomington, 2012
N = size(adj,1);
EYE = logical(eye(N,N));
flagResources = ~isnan(lambda);
if flagResources
if lambda<=0 || lambda>=1
error('p_req_values must be non-zero probabilities')
end
z = zeros(N,N);
end
if nargin<4
SPL = distance_wei_floyd(adj);
end
if nargin<5
M = diag(sum(adj,2))\adj;
end
Lvalues = unique(SPL(:));
Lvalues = Lvalues(~(Lvalues==0));
prob_SPL = zeros(N,N); % a priori zero probability of going through SPL among nodes
for indexSPL=1:length(Lvalues) %for each possible value of SPL for current component
SPLvalue = Lvalues(indexSPL);
[~, hcols] = find(SPL==SPLvalue);
hvector = unique(hcols); clear hrows hcols
entries = SPL==SPLvalue;
if flagResources % compute Eres
[prob_aux,z_aux] = prob_first_particle_arrival(M,SPLvalue,hvector,lambda);
else % not compute Eres
[prob_aux] = prob_first_particle_arrival(M,SPLvalue,hvector,[]);
end
prob_aux(~entries) = 0;
prob_SPL = prob_SPL + prob_aux;
if flagResources
z_aux(~entries) = 0;
z = z + z_aux;
end
end
prob_SPL(EYE) = 0;
if flagResources
z(prob_SPL==1) = 1;
Eres = 1./z;
Eres(EYE) = 0;
else
Eres = nan;
end
function [prob,resources] = prob_first_particle_arrival(M,L,hvector,lambda)
N = size(M,1);
prob = zeros(N,N);
if nargin<4
hvector=1:N;
end
flagResources = ~isnan(lambda);
if flagResources
if lambda<=0 || lambda>=1
error('p_req_values must be non-zero probabilities')
end
resources = zeros(N,N);
end
for hindex=1:length(hvector) %for each destination node h
h = hvector(hindex);
B_h = M;
B_h(h,:) = 0; B_h(h,h) = 1; % h becomes absorbant state.
B_h_L = B_h^L;
term = 1-B_h_L(:,h);
prob(:,h)= 1-term;
if flagResources
resources(:,h) = repmat(log(1-lambda),N,1)./repmat(log(term),1);
end
end
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