pubfile.m

无标度网络的生成模型

%% B-A Scale-Free Network Generation and Visualization
% *By Mathew Neil George*

%% Description and Cautions
% The *SFNG* m-file is used to simulate the B-A algorithm and returns scale-free
% networks of given node sizes. Understanding the B-A algorithm is key 
% to using this code to its fullest. Due to Matlab resource limitations, it may not be
% possible to generate networks much larger than 15000 nodes, and increasing the
% *mlinks* variable increases processing time severely. This code was
% developed so that one could generate a network of small size, and then
% use that network as a seed to build a greater sized network, continuing
% this process until the actual desired network size is reached. This is for
% processor and time management purposes. However, realize that the initial
% seed does not have to have scale-free properties, while the later seeds
% may happen to have these properties. Therefore, it is prudent not to make the
% initial seed size much larger than a few nodes (most commonly 5
% interconnected nodes). In addition, the *mlinks* should be kept constant
% throughout the creation of the scale-free network.
%
% The *PLplot* m-file takes a scale-free network in adjacency matrix format
% and draws a best fit line to the frequency of degrees distribution of the
% nodes. Degree is the number of links that connect to and from a single node
% For scale-free networks, the frequency of degrees distribution forms a 
% power-law curve, with an exponent usually between -2 and -3. This code is
% designed to allow only non-zero frequencies to be graphed in log-log format.
% The function returns the equation of the power-law fit in a cfit variable.
%
% The *CNet* m-file function creats a network graph using the *gplot*
% function with circular coordinates. It allows for a simple, yet
% intuitive, visualization of a given network.

%% Parameters
% *SFNG*
%
% * *Nodes* is the desired network size, including the seed network size
%   (i.e. Nodes minus seed network size equals the number of nodes to be
%   added).
%
% * *mlinks* controls the number of links a new node can make to the existing
%   network nodes.
%
% * *seed* is the original network to which the B-A algorithm links
%   additional nodes with a specific preferential attachment procedure.
%   This undirected adjacency matrix can be created manually, or one could
%   use the *Adjacency Matrix GUI*. Each node must have at least one link.
%   The *seed* variable can be replaced with a developed scale-free network
%   to generate a larger one. Make sure the new *Nodes* variable is greater
%   than the size of the *seed* network.
% 
% *PLplot*
%
% * *Net* is the input network which is to be graphed.
%
% *CNet*
%
% * *Net* is the input network which is to be graphed.
%
% Note that variables *Nodes*, *mlinks*, and *size* must be whole numbers and 
% variables *seed* and *Net* must be undirected adjacency matrices. The 
% diagonol elements of any adjacency matrix used with these functions must
% all be zero.

%% Sample Output
% Here is a small example to demonstrate how to use the code. This code
% creates a seed network of 5 nodes, generates a scale-free network of 300 nodes from
% the seed network, and then performs the two graphing procedures.
seed =[0 1 0 0 1;1 0 0 1 0;0 0 0 1 0;0 1 1 0 0;1 0 0 0 0]
Net = SFNG(100, 1, seed);
PL_Equation = PLplot(Net)
CNet(Net)

%%用pajek画图
Num = max(size(Net));
fid = fopen('Node.net', 'w');
fprintf(fid, '*Vertices   %d \r\n', Num);

for i = 1 : Num 
    fprintf(fid, '  %s   "v%s"           %6.2f  %6.2f  %6.2f  ', num2str(i), num2str(i), rand(1), rand(1), 0.5);
    fprintf(fid,'\r\n');
end 

fprintf(fid, '*Edges\r\n');

for i = 1 : Num 
    for j = i + 1 : Num
        if Net(i,j) == 1
            fprintf(fid, '  %d  %d  %d', i, j, 1);            
            fprintf(fid,'\r\n');
        end 
    end 
end

fclose(fid);
            

% % we will calculate the average clustering coefficient ONLY for those
% % vertices where the number of neighbors is >1. We can calculate it
% % for all vertices as well, by defining that if the vertex has zero 
% % neighbors, its clustering coeff is zero, and if it has one neighbor,
% % its clustering coeff is one.
% A_ud=Net+Net';             % sum A and its transpose
% 
% triangles=sum(Net,2);
% k=sum(A_ud,2); % the number of edges for each vertex
% 
% c_avg=0;
% N_k_morethanone=0;
% 
% for i=1:length(Net);
%     if k(i)>1
%         c_avg=c_avg+2*triangles(i)/(k(i)*(k(i)-1));
%         N_k_morethanone=N_k_morethanone+1; 
%     end 
% end
% 
% c_avg=c_avg/N_k_morethanone;
% 
% fprintf('The average clustering coefficient for the %d vertices where k>1 is %1.2f\n',N_k_morethanone,c_avg);
% 
% %%computing potential
% 
% for i=1:length(Net)
%     for j=1:length(Net)
%         if (i~=j) & (Net(i,j) == 0)
%            Net(i,j)=inf;
%         end        
%     end
% end    
% 
% 
% %%计算最短路径
% Mat_Distance = Fun_Distance(Net);
% %初始化
% Int_EffectRange = Inf;%%影响范围
% Flt_Sigma = 1;
% Lin_Mass = ones(1, length(Net));%%节点质量
% 
%  % %%(优化因子，该优化因子对应的势熵)
% [Flt_H] = Fun_Sigma(Net, Lin_Mass); 
% Min=Fun_List(Flt_H(:,1)');
% Flt_Sigma = Flt_H(Min(size(Min,1), 1), 2);
% Lin_Potential = zeros(1, length(Net));
% 
% 
% 
% %画sigma图
% figure
% plot(Flt_H(:,2),Flt_H(:,1))
% xlabel('影响因子(sigma)')
% ylabel('势熵(Potential Entropy H)')
% 
% 
% %计算各个节点的势值
% for i = 1: length(Net)
%     Lin_Potential(i) = Fun_ComputePotential (Mat_Distance(i,:),i, Flt_Sigma, Lin_Mass,Int_EffectRange);
% end
% %各个节点的势值排序
% Mat_ListPotential = Fun_List ( Lin_Potential)
% 
% %计算平均距离
% d=  Fun_D(Mat_Distance)
% 
% 
% %计算各个节点的度&排序
% [Lin_Degree, Mat_RelationD] = Fun_Degree ( Net);
% Mat_ListDegree = Fun_List (Lin_Degree);
% 
% 
% %% References
% % One explanation of the B-A Algorithm can be found on this PDF website
% % http://arxiv.org/PS_cache/cond-mat/pdf/0107/0107420.pdf
% %
% % Undirected Adjecency Matrices are defined on Wikipedia.org
% % http://en.wikipedia.org/wiki/Adjacency_matrix
% %
% % The *Adjacency Matrix GUI* file by Steve Chuang can be found on the Matlab File Exchange
% % http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=6937&objectType=file
% 
% %% Acknowledgements
% % Special thanks to Mark Ballerini with the Massapequa High School Science 
% % Research Program and Zoltan Dezso at the University of Notre Dame for 
% % their invaluable help in researching network theory as well as to my 
% % family for providing motivation and encouragement in pursuing science.