Matlab wrong results with array calculations

Question

I'm trying to reproduce the results from a paper for which I give a link to avoid writing down all the math needed:

On Modeling and Simulation of Game Theory-based Defense Mechanisms against DoS and DDoS Attacks

More specifically what I'm having a problem with is the Figure 3 plot. The plot gives in the z axis the results of equation 3 given the two variables m and M. The other equations that will be needed are 5,6,7 and there are also two small ones in the paragraph before equation 6. Also in order to see what Xi is check the 4.2 part. All the variable values needed are given before the plot.

Now to get to the point, I'm trying to create the exact same plot in matlab but I've failed and I need help because my matlab skills are not so good.

I have a script file in which I have the following:

w1  = 1000;
w2  = 1000;
w3  = 10;  
B   = 2000;
n   = 20;  
r_l = 60;  
s_l = 20;  
g   = 10;  
a_f = 5000;
b   = 20;  

vx = 0 : 1 : 500;
vy = 0 : 1 : 90;

[x,y] = meshgrid(vx,vy);

z = payoff(w1, w2, w3, y, r_l, n, g, B, b, x, s_l, a_f);

h = surfc(x,y,z);
set(h, 'edgecolor','none')
xlabel('Firewall Midpoint (M)')
ylabel('Number of zombies')
zlabel('Attackers payoff')
view(-41,11);

Payoff is a function that is as follows:

function out = payoff(w1, w2, w3, m, r_l, n, g, B, b, M, s_l, a_f)

r_a = a_f./ m;
r_a_dash = r_a.*(1-Fx(r_a, b, M, B));
r_l_dash = r_l.*(1-Fx(r_l, b, M, B));

v_b = ( m .* r_a_dash ) ./ ( n .* r_l_dash + m .* r_a_dash );
v_n = normcdf(( g .* ( n .* r_l_dash + m .* r_a_dash ) ./ B ), r_l, s_l);

out = w1 * v_b + w2 * v_n - w3 * m;

Fx again is a function that does the following:

function out = Fx(x,b,M,B)

out=1./(1+exp(-b.*(x-M)./B));

I don't know where exactly is the mistake but the plot I get is the following which is not the same as the one in the paper.

The figure in the paper has a U shaped curve along the Firewall Midpointaxis whereas mine is monotonically increasing. My results

Can anyone spot any mistake(s) that I have? Thanks in advance.

I didn't read all your code but I know that the arrays generated by meshgrid are very unintuitive. It might help if you describe the error as well as possible. — denahiro
– denahiro, Commented Jul 30, 2012 at 13:56
Well I don't know where exactly the error occurs but the plot I get is different to the one in the paper. I added an image with the plot I generate for comparison. H hope it helps. — George Karanikas
– George Karanikas, Commented Jul 30, 2012 at 14:04
I just saw that for figure 3 they never explicitly state their value for beta. You assume that it is the same as in figure 2, which might not be true. — denahiro
– denahiro, Commented Jul 30, 2012 at 14:39
I tried configuring the number of beta. I gave it various values to see how the plot changes but no matter the value it never gets a shape similar to the one in the paper. If needed I can also include some pictures with various beta values. — George Karanikas
– George Karanikas, Commented Jul 30, 2012 at 14:43
I think that normcdf is giving me wrong results compared to the paper. Do I miss something in its usage? — George Karanikas
– George Karanikas, Commented Jul 31, 2012 at 6:50

Eitan T · Accepted Answer · 2012-07-31 17:33:18Z

3

The big thing I noticed was in your code you used:

v_n = normcdf(( g .* ( n .* r_l_dash + m .* r_a_dash ) ./ B ), r_l, s_l);

When you should have used (I think):

v_n = normcdf(( g .* ( n .* r_l_dash + m .* r_a_dash ) ./ B ), r_l_dash, s_l);

In the paper, they state:

Recall that r_l represents the expected rate of a legitimate flow. Let the average rate of legitimate flows passing through the firewall be r_l′.

In the normcdf function, the second argument should be the average, mu. This gives me a U-shaped curve along the Firewall Midpoint, however I can see it's not exact to the picture and I believe it's due to the value of b, as someone had already stated was not given.

Hope this helps. There may still be a calculation error as I've played around with various values of b and still can't match the image in the paper.

edited Jul 31, 2012 at 17:33

Eitan T

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answered Jul 31, 2012 at 13:34

TZPike05

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1 Comment

George Karanikas Over a year ago

Thank you. Indeed it needs r_l_dash.

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