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<html xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd"> <head> <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> <!--This HTML is auto-generated from an M-file.To make changes, update the M-file and republish this document. --> <title>B-A Scale-Free Network Generation and Visualization</title> <meta name="generator" content="MATLAB 7.1"> <meta name="date" content="2006-08-12"> <meta name="m-file" content="pubfile"><style>body { background-color: white; margin:10px;}h1 { color: #990000; font-size: x-large;}h2 { color: #990000; font-size: medium;}/* Make the text shrink to fit narrow windows, but not stretch too far in wide windows. On Gecko-based browsers, the shrink-to-fit doesn't work. */ p,h1,h2,div.content div { /* for MATLAB's browser */ width: 600px; /* for Mozilla, but the "width" tag overrides it anyway */ max-width: 600px; /* for IE */ width:expression(document.body.clientWidth > 620 ? "600px": "auto" );}pre.codeinput { background: #EEEEEE; padding: 10px;}span.keyword {color: #0000FF}span.comment {color: #228B22}span.string {color: #A020F0}span.untermstring {color: #B20000}span.syscmd {color: #B28C00}pre.codeoutput { color: #666666; padding: 10px;}pre.error { color: red;}p.footer { text-align: right; font-size: xx-small; font-weight: lighter; font-style: italic; color: gray;} </style></head> <body> <div class="content"> <h1>B-A Scale-Free Network Generation and Visualization</h1> <introduction> <p><b>By Mathew Neil George</b></p> </introduction> <h2>Contents</h2> <div> <ul> <li><a href="#1">Description and Cautions</a></li> <li><a href="#2">Parameters</a></li> <li><a href="#3">Sample Output</a></li> <li><a href="#4">References</a></li> <li><a href="#5">Acknowledgements</a></li> </ul> </div> <h2>Description and Cautions<a name="1"></a></h2> <p>The <b>SFNG</b> 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 <b>mlinks</b> 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 <b>mlinks</b> should be kept constant throughout the creation of the scale-free network. </p> <p>The <b>PLplot</b> 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. </p> <p>The <b>CNet</b> m-file function creats a network graph using the <b>gplot</b> function with circular coordinates. It allows for a simple, yet intuitive, visualization of a given network. </p> <h2>Parameters<a name="2"></a></h2> <p><b>SFNG</b></p> <div> <ul> <li><b>Nodes</b> 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). </li> </ul> </div> <div> <ul> <li><b>mlinks</b> controls the number of links a new node can make to the existing network nodes. </li> </ul> </div> <div> <ul> <li><b>seed</b> 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 <b>Adjacency Matrix GUI</b>. Each node must have at least one link. The <b>seed</b> variable can be replaced with a developed scale-free network to generate a larger one. Make sure the new <b>Nodes</b> variable is greater than the size of the <b>seed</b> network. </li> </ul> </div> <p><b>PLplot</b></p> <div> <ul> <li><b>Net</b> is the input network which is to be graphed. </li> </ul> </div> <p><b>CNet</b></p> <div> <ul> <li><b>Net</b> is the input network which is to be graphed. </li> </ul> </div> <p>Note that variables <b>Nodes</b>, <b>mlinks</b>, and <b>size</b> must be whole numbers and variables <b>seed</b> and <b>Net</b> must be undirected adjacency matrices. The diagonol elements of any adjacency matrix used with these functions must all be zero. </p> <h2>Sample Output<a name="3"></a></h2> <p>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. </p><pre class="codeinput">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(300, 1, seed);PL_Equation = PLplot(Net)CNet(Net)</pre><pre class="codeoutput">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 0PL_Equation = General model Power1: PL_Equation(x) = a*x^b Coefficients (with 95% confidence bounds): a = 204.1 (200.4, 207.7) b = -2.177 (-2.264, -2.09)</pre><img vspace="5" hspace="5" src="pubfile_01.png"> <img vspace="5" hspace="5" src="pubfile_02.png"> <h2>References<a name="4"></a></h2> <p>One explanation of the B-A Algorithm can be found on this PDF website <a href="http://arxiv.org/PS_cache/cond-mat/pdf/0107/0107420.pdf">http://arxiv.org/PS_cache/cond-mat/pdf/0107/0107420.pdf</a></p> <p>Undirected Adjecency Matrices are defined on Wikipedia.org <a href="http://en.wikipedia.org/wiki/Adjacency_matrix">http://en.wikipedia.org/wiki/Adjacency_matrix</a></p> <p>The <b>Adjacency Matrix GUI</b> file by Steve Chuang can be found on the Matlab File Exchange <a href="http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=6937&objectType=file">http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=6937&objectType=file</a></p> <h2>Acknowledgements<a name="5"></a></h2> <p>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. </p> <p class="footer"><br> Published with MATLAB® 7.1<br></p> </div> <!--##### SOURCE BEGIN #####%% 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(300, 1, seed);
PL_Equation = PLplot(Net)
CNet(Net)
%% 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.##### SOURCE END #####--> </body></html>?? 快捷鍵說明
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