%% Generating an Erdos-Renyi graph  
% The Erdos-Renyi model is connect _n_ vertices with probability _p_. 
% Let's begin by setting these parameters for an example.

n = 150;
p = 0.01;

%%
% With these two parameters, we can instantiate the graph. The 
% variable |G| is the adjacency matrix for the graph. However,
% the first step doesn't treat edges symmetrically. The last
% two operations fix this and yield a symmetric adjacency matrix.

rand('seed',100); % reseed so you get a similar picture
G = rand(n,n) < p;
G = triu(G,1);
G = G + G'; 

%%
% See, that was pretty easy right?

%% Seeing giant component
% From the theory of Erdos-Renyi graphs, we know that a 
% _giant connected component_ should emerge whenever
% _p_ > _1/(n-1)_

thres_giant = 1/(n-1)

%%
% Because _p_ = 0.01 > 0.0067, we expect our graph to have such a giant 
% component. 

%%
% Using the Matlab interface to the AT&T Graphvis program neato, we 
% can generate a layout for this graph and visualize it using the 
% matlab gplot function. That way, we can see if it really does have a 
% giant component.

[x y] = draw_dot(G);
gplot(G, [x' y'], 'o-'); 
axis off;

%%
% Notice how there is only one large connected set of vertices.
% Everything else is a small disconnected component.