MATLAB实例：构造网络连接图(Network Connection)及计算图的代数连通度(Algebraic Connectivity)

1. 图的代数连通度(Algebraic Connectivity)

图的代数连通度：Laplace图谱的次小特征值。

2. 网络连接图(Network Connection)的构造

随机生成一个具有50个节点的传感器网络。节点随机放置在3.5 x 3.5方形区域内，通信距离为0.8。如下图所示，共有159条边，其代数连通度为：0.3007。

3. MATLAB程序

demo_Create_Network_Connection.m

%创建无向图 网络连接图 Network Connection.

clc;

close all;

clear;

Conf.Square = 3.5; %方形区域的边长

Conf.NodeNumber = 50; %节点个数

Conf.CommDist = 0.8; %最大通信距离

is_create_network = 1;

if is_create_network == 1

[ Network, Dists ] = CreateNetworksFunc(Conf);

save Network_1.mat Network

else

load Network_1.mat

end

nodenum = size(Network.Nodes.loc,1); %节点个数

lap_matrix = zeros(nodenum); %节点数*节点数 图的Laplace矩阵：diag(d1,d2,...dn)-邻接矩阵，di为节点i的度

for i=1:nodenum

idx = Network.Nodes.neighbors{i}; %邻接节点的id

lap_matrix(i,idx) = -1; %负的邻接矩阵

lap_matrix(i,i) = length(idx); %对角线元素为节点的度

end

eig_val = eig(lap_matrix); %lap_matrix的特征值

eig_val = sort(eig_val,'ascend'); %从小到大排序，最小特征值为0

algeb_conn = eig_val(2) % algebraic connectivity 代数连通度：lap_matrix的第二小特征值>0，连通图

avg_deg = sum(diag(lap_matrix))/nodenum % average values 节点度的均值

DrawNetworks(Network);

% DrawNetworks(Network, Dists); %把所有的边的长度(通信距离)都标出来了

print(gcf,'-dpng','Network_1.png'); %保存图片

CreateNetworksFunc.m

function [ Network, Dists ] = CreateNetworksFunc(Conf)

% 创建无向图 网络连接图 Network Connection.

num = Conf.NodeNumber; %节点个数

square = Conf.Square; %方形区域的边长

maxDist = Conf.CommDist; %最大通信距离

loc = square*rand(num,2) - square/2; %num*2的随机数 节点坐标

Dists = Euclid_Dist(loc(:,1),loc(:,2)); %节点数*节点数，对角线元素为0

% without self-loop 不存在节点自己到自己的路径，对角线上的元素为无穷大

Dists = Dists + 10*maxDist*eye(num);

Neighbors = cell(num,1);

maxDegree = 0; %节点的最大度，与节点相邻的最大边数

edges = 0; %图的总边的个数，无向图的度/2

for i=1:num

Neighbors{i} = find(Dists(i,:)<=maxDist); %找邻接节点的id

if length(Neighbors{i}) > maxDegree

maxDegree = length(Neighbors{i}); %节点的最大度

end

edges = edges + length(Neighbors{i});

end

Nodes.loc = loc;

Nodes.neighbors = Neighbors;

Network.maxDegree = maxDegree;

Network.edges = edges/2; %% undirected graph

Network.Conf = Conf;

Network.Nodes = Nodes;

end

function dist = Euclid_Dist(X,Y)

% 求两两节点之间的距离，输出[节点*节点]的矩阵，距离矩阵

len = length(X);

xx = repmat(X,1,len); %节点数*节点数

yy = repmat(Y,1,len);

dist = sqrt((xx-xx').^2+(yy-yy').^2); %节点数*节点数

end

DrawNetworks.m

function fig = DrawNetworks( Network )

%画无向图 网络连接图 Network Connection.

% function fig = DrawNetworks( Network, Dists ) %把所有的边的长度(通信距离)都标出来了

num = Network.Conf.NodeNumber; %节点个数

loc = Network.Nodes.loc; %节点坐标

square = Network.Conf.Square; %方形区域的边长

Neighbors = Network.Nodes.neighbors; %邻接节点的id

fig = figure;

plot(loc(:,1),loc(:,2),'ro','MarkerSize',8,'LineWidth',2); %节点是红色圆圈

side=ceil(square/2);

axis([-side,side,-side,side]);

for i=1:num

for k = 1:length(Neighbors{i})

j = Neighbors{i}(k);

% c = num2str(Dists(i,j),'%.2f');

% text((loc(i,1) + loc(j,1))/2,(loc(i,2) + loc(j,2))/2,c,'Fontsize',10); %把所有的边的长度(通信距离)都标出来了

% hold on;

line([loc(i,1),loc(j,1)],[loc(i,2),loc(j,2)],'LineWidth',0.8,'Color','b'); %线是蓝色

end

end

set(gcf, 'Color', 'w'); %白色

end

4. 参考文献

[1] Hua J, Li C. Distributed variational Bayesian algorithms over sensor networks[J]. IEEE Transactions on Signal Processing, 2015, 64(3): 783-798.

[2] 肖恩利, 束金龙, 闻人凯. 图的代数连通度及其点连通度[J]. 华东师范大学学报(自然科学版), 2003, 2003(4):1-4.