%%%%%%% ESTIMATING LAMBDA %%%%%%%

doc errorbar
% this function is really helpful, so it's worth reading the documentation
% on how it works!
% use 'doc XYZ' to open the documentation file for any function XYZ

X = [1/1024,1/512,1/256,1/128];
% our vector for 1/L
Y = [1.65,1.65,1.66,1.67];
% our vector of measured values of \lambda_c
E = [0.01,0.01,0.01,0.01];
% our errors (here assumed to be symmetric)
errorbar(X,Y,E)
% plot the results!!

% From here, we can fit a linear (or polynomial...) function from
% the plot window by clicking 'tools > basic fitting'

% Alternatively, we can click the tab labelled 'APPS' at the top of the MATLAB window
% and open up cftool, the curve fitting tool. From here, we specify the X
% data, the Y data and the type of function we want to fit, choose the
% parameters of that function and the cftool should perform the fitting
% automatically. 
% From there, we click the fitted model in the 'Table of fits' (at the bottom
% of the cftool window, and right click to bring up the option for 'save
% untitled fit to workspace'. This will create a few variables in the
% workspace with silly names.
% To plot the fitted model, you type 'plot(fittedmodel)' and there you have
% it! The fit is stored in a special kind of data structure that will
% automatically plot in a red line if used as an argument in the 'plot'
% function.
% N.B. make sure to type 'hold on' if you would like to plot this fit on 
% top of an existing plot containing data.

%%%%%%% WIGNER SEMICIRCLE LAW %%%%%%%

% Firstly, let's generate an Erdos-Renyi random graph matrix:
% Let's have N=1000 and z=1.5, therefore p = z/N = 1.5/1000
adj = random_graph(1000,0.015);

% Now we want the eigenvectors and eigenvalues of this matrix
[V,E] = eig(adj);

% N.B. E is a diagonal vector with the eigenvalues along the diagonal, so
% if we ask for 'sum(E)' we will get a 1D array of the row-sums, which will
% thus just be a 1D array of the eigenvalues

% So now, let's plot the graph spectrum using ksdensity, firstly it might
% be nice to remind ourselves what ksdensity can and will do:
doc ksdensity

% So, it will fit a gaussian around each point, and even around points
% where we have no samples, so in the second argument it ksdensity, we need
% to tell it only to generate the density plot at the points where we have
% actually collected data i.e. at the actual eigenvalues themselves! So:

[f,x] = ksdensity(sum(E),sum(E),'bandwidth',0.1);

% We've asked for a gaussian of bandwidth 0.1 fitted around the points
% sum(E) and only sampled at points sum(E), f and x now contain the desnity
% function and sample points respectively, let's plot them in some subtle
% blue circles

plot(x,f,'bo')

% Now, to compare this to the Wigner SemiCircle Law
% The spectrum should be approximately semicircular, with radius \Sigma


s = 0.015*(1-0.015);
% variance of bernoulli rv
X = [-2*(sqrt(1000*s)):0.01:2*(sqrt(1000*s))];
% points to plot the semicircle at (from -radius to radius)
Y = (1/(2*pi*1000*s))*sqrt(4*1000*s-X.*X);
% semicircle function
hold on
% hold our previous plot!!
plot(X,Y,'r-')
% plot the semicircle!

% Now we just do the same thing for a few other values of z and see how the
% spectrum compares to the semicircle in each case
% N.B. it is totally expected that the actual graph spectrum will look a
% bit 'wobbly' for certain parameters values (even the spectrum can end up
% looking quite rude...) and you may want to comment on why this is

% Afterwards, we can use the preferential attachment code to generate the
% adjacency matrix for a preferential attachment graph and see how the
% spectrum compares