WS以及NW小世界網絡的生成（MATLAB）

WS小世界網絡生成算法，一般小世界網絡生成算法速度慢，節點度分布與數學推導不符，在網絡仿真中造成不便，這里針對實際網絡動力學仿真過程撰寫了WS小世界網絡的MATLAB生成算法，並考慮了矩陣化，具有較高的速度。

以下是対應的代碼：

% The simulation of WS-smallworld network
% the algorithm of WS-smallworld's generation has been improved in speed,
% and tend to be easily understood
% writen by [e-mail]
% Example:
% N = 100; %network size (number of nodes)
% m = 6; %2*m is the average edges of each nodes
% p = 0.1; %rewiring probability
% matrix = small_world_WS_new(N,m,p);
function matrix = small_world_WS_new(N,m,p)
rng('default')
rng('shuffle')
matrix=zeros(N,N);
% generate regular network
for i=m+1:N-m
    matrix(i,i-m:i+m)=1;
end
for i=1:m
    matrix(i,1:i+m)=1;
end
for i=N-m+1:N
    matrix(i,i-m:N)=1;
end
for i=1:m
    matrix(i,N-m+i:N)=1;
    matrix(N-m+i:N,i)=1;
end
% rewiring the network
for i = 1:N
    % then rewiring the edges with the probability of p
    [series1,series2] = range_sort(N,m,i);
    index0 = series1(rand(2*m,1)>1-p);
    if(~isempty(index0))
        matrix(i,index0) = 0;
        matrix(i,series2(randperm(length(series2),length(index0))))=1;
    end
end
matrix = matrix -diag(diag(matrix));
end

function [series1,series2] = range_sort(N,m,i)
% select the index of nodes in row i for rewiring
if(i-m>0 && i+m<=N)
    series1 = i-m:i+m;
    series2 = setdiff(1:N,series1);
elseif(i-m<=0)
    series1 = [1:i+m,N-m+i:N];
    series2 = setdiff(1:N,series1);
else
    series1 = [1:m-N+i,i-m:N];
    series2 = setdiff(1:N,series1);
end
% Without considering the connection of diagonal elements
series1(series1==i) = [];
end

　　

參考文獻：

Watts D J, Strogatz S H. Collective dynamics of ‘small-world’networks[J]. nature, 1998, 393(6684): 440-442.

NW小世界網絡的生成方法相對簡單，我這里附加對應代碼：

% 基於Matlab 的小世界網絡仿真
% 經過矩陣化修改后，生成速度已經大大加快
function matrix = small_world_NW(N,m,p)
% N=50;m=3;p=0.1;
% matrix=sparse([]);
matrix = zeros(N,N);
for i=m+1:N- m
    matrix(i,i- m:i+m)=1;
end
for i=1:m
    matrix(i,1:i+m)=1;
end
for i=N- m+1:N
    matrix(i,i- m:N)=1;
end
for i=1:m
    matrix(i,N- m+i:N)=1;
    matrix(N- m+i:N,i)=1;
end
% Random add edge
kk=(rand(N,N)<p);
matrix = logical(matrix + kk);
matrix = matrix -diag(diag(matrix));

　　

對應生成網絡的測試圖的代碼：

clear,clc,close all
% load A.txt
N=10;
m=2;
p=0.1;
% A= small_world_WS_new(N,m,p);
A = small_world_NW(N, m, p);
t=linspace(0,2*pi,N+1);
x=sin(t);
y=cos(t);
figure
set(gcf,'color','w')
plot(x,y,'o','markerfacecolor','k'),hold on
for i=1:N
    for j=1:N
        if (A(i,j)==1)
        fp1=plot([x(i),x(j)],[y(i),y(j)],'r-'); hold on
        set(fp1,'linesmoothing','on')
        end
    end
end
axis([-1.05,1.05,-1.05,1.05])
axis square
axis off
sum(sum(A))