Consider the following simple and rather unrealistic mathematical model of a network. Each of $n$ nodes belongs to one of several groups. The $m$-th group has $n_m$ nodes and each node in that group is connected to others in the group with independent probability $p_m = A(n_m−1)^{−\beta}$ , where $A$ and $\beta$ are constant, but not to any nodes in other groups. Thus this network takes the form of a set of disjoint clusters or communities.
- Calculate the average degree $\langle k\rangle$ of a node in group $m$.
- Calculate the average value $\bar{C}_m$ of the local clustering coefficient for nodes in group $m$.
- Hence show that $\bar{C}_m \sim \langle k\rangle^{-\frac{\beta}{1−\beta}}$.
- What value would $\beta$ have to have for the average value of the local clustering to fall off with increasing degree as $\langle k\rangle^{−\frac34}$?
My attempt:
Knowledge: The degree distribution of a network $p(k)$ is the frequency distribution of its nodesʼ degrees. $p(k)$ is the probability that a node selected uniformly at random will have the degree $k$. The mean degree is given by $\langle k\rangle=\sum_kkp(k)$.
The local clustering coefficient is the node-level measure:$$C_i=\frac{(\text{number of triangles including the i})}{(\text{number of pairs of neighbors of i})}=\frac{(\text{number of triangles including the i})}{\frac12k_i(k_i-1)}$$
For 1:Since each node has the same indepenent probability to be connected to another node in the group $\langle k \rangle=\sum_kkp(k)=kA(n_m-1)^{-\beta}$
For 2: $\bar{C}_m=\frac{2\cdot(\text{number of triangle including the i})}{kA(n_m-1)^{-\beta}(kA(n_m-1)^{-\beta}-1)}$
For 3: I still don't know how to do it, I guess my previous results are wrong/incomplete, since there is no obvious proportionality (I guess the "$\sim$" notation stands for that).
For 4: Still no ideas
Can anyone tell me where do I miss things?