MATLAB实例:构造网络连接图(Network Connection)及计算图的代数连通度(Algebraic Connectivity)
作者:凯鲁嘎吉 - 博客园 http://www.cnblogs.com/kailugaji/
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. 连通度与代数连通度
图的连通度侧重的是图的结构性质,而代数连通度侧重的是矩阵的代数性质。
- 图的代数连通度:
图的Laplace矩阵的次小特征值。
- 点连通度:
一个具有N个点的图G中,在去掉任意K−1个顶点后(1<=K<=N)所得的子图仍然连通,去掉K个顶点后不连通,则称G是K连通图,K称作图G的点连通度,记作K(G)。
- 边连通度:
一个具有N条边的图G中,在去掉任意K−1条边后(1<=K<=N)所得的子图仍然连通,去掉K条边后不连通,则称G是K连通图,K称作图G的边连通度,记作K(G)。
5. 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.
[3] Junhao Hua. Distributed Variational Bayesian Algorithms. Github, 2017.