Implementing Barabasi-Albert Method for Creating Scale-Free Networks

Question

I'm trying to implement a very simple preferential attachment algorithm for creating scale-free networks. These have degree distributions that follow a power-law, i.e. P(k) ~ k^-g, where g is the exponent. The algorithm below should produce degree distributions with the exponent equal 3 +/- 0.1, my implementation does not the exponents are closer to 2.5 +/- 0.1. I'm clearly not understanding something somewhere and continue to get it wrong.

I'm sorry if this is in the wrong place, I couldn't decide whether it should be in stackoverflow or maths.stackexchange.com.

The Algorithm:
Input: Number of Nodes N; Minimum degree d >= 1.
Output: scale-free multigraph
G = ({0,....,N-1}, E)
M: array of length 2Nd
for (v=0,...,n-1)
   for (i=0,...,d-1)
      M[2(vd+i)] = v;
      r = random number selected uniformly at random from {0,.....,2(vd+i)};
      M[2(vd+i)+1] = M[r];
   end
end

E = {};
for (i=0,...,nd-1)
   E[i] = {M[2i], M[2i+1]}
end

My Implementation in C/C++:

void SF_LCD(std::vector< std::vector<int> >& graph, int N, int d) {
    if(d < 1 || d > N - 1) {
        std::cerr << "Error: SF_LCD: k_min is out of bounds: " << d;
    }

    std::vector<int> M;
    M.resize(2 * N * d);

    int r = -1;
    //Use Batagelj's implementation of the LCD model
    for(int v = 0; v < N; v++) {
        for(int i = 0; i < d; i++) {
            M[2 * (v * d + i)] = v;
             r = mtr.randInt(2 * (v * d + i));
            M[2 * (v * d + i) + 1] = M[r];
        }
    }

    //create the adjacency list
    graph.resize(N);
    bool exists = false;
    for(int v = 0; v < M.size(); v += 2) {
        int m = M[v];
        int n = M[v + 1];

        graph[m].push_back(n);
        graph[n].push_back(m);
    }
}

Here's an example of a degree distribution I get for N = 10,000 and d = 1:

1   6674
2   1657
3   623
4   350
5   199
6   131
7   79
8   53
9   57
10  27
11  17
12  20
13  15
14  12
15  5
16  8
17  5
18  10
19  7
20  6
21  5
22  6
23  4
25  4
26  2
27  1
28  6
30  2
31  1
33  1
36  2
37  2
43  1
47  1
56  1
60  1
63  1
64  1
67  1
70  1
273 1

Answer 1

Okay, so I couldn't figure out how to make this particular algorithm work correctly, instead I used another one.

The Algorithm:
Input: Number of Nodes N; 
       Initial number of nodes m0; 
       Offset Exponent a; 
       Minimum degree 1 <= d <= m0.
Output: scale-free multigraph G = ({0,....,N-1}, E).

1) Add m0 nodes to G.
2) Connect every node in G to every other node in G, i.e. create a complete graph.
3) Create a new node i.
4) Pick a node j uniformly at random from the graph G. Set P = (k(j)/k_tot)^a.
5) Pick a real number R uniformly at random between 0 and 1.
6) If P > R then add j to i's adjacency list.
7) Repeat steps 4 - 6 until i has m nodes in its adjacency list.
8) Add i to the adjacency list of each node in its adjacency list.
9) Add i to to the graph.
10) Repeat steps 3 - 9 until there are N nodes in the graph.

Where k(j) is the degree of node j in the graph G and k_tot is twice the number of edges (the total number of degrees) in the graph G.

By altering the parameter a one can control the exponent of the degree distribution. a = 1.22 gives me an exponent g (in P(k) ~ k^-g) of 3 +/- 0.1.