Question

The degree distribution pk expresses the probability that a randomly selected node has k neighbors. However, if we randomly select a link, the probability that a node at one of its ends has degree k is qk = akpk, where a is a normalization factor. a. Find the normalization factor a, assuming that the network has a power law degree distribution with 2 < γ < 3, with minimum degree kmin and maximum degree kmax. b. In the configuration model qk is also the probability that a randomly chosen node has a neighbor with degree k. What is the average degree of the neighbors of a randomly chosen node? c. Calculate the average degree of the neighbors of a randomly chosen node in a network with n = 104, γ = 2.3, kmin = 1 and kmax = 1000. Compare the result with the average degree of the network 〈k⟩?

Answer 1

The question requires finding a normalization constant and the average degree of neighbors in a network with power-law distribution. Exact calculations depend on the full formula for the distribution. Generally, the average degree is compared to the network's mean degree.

Step-by-step explanation:

The question involves finding normalization factors and average degrees within a network characterized by a power-law degree distribution. The normalization constant a ensures that the probability distribution sums to 1. However, calculating a specifically requires the full formula for the power-law distribution, which isn't provided, but generally, it involves summing the probabilities multiplied by degree k over all degrees. As for the average degree of neighbors (qk distribution), it is calculated by multiplying each degree by its probability and summing these products, normalized by the constant a.

The provided information for n = 104 nodes, γ = 2.3, kmin = 1, and kmax = 1000 can be used to calculate specific values, but without the specific formulas for the power-law distribution, the computation can't be completed in this answer. The result can be compared to the network's mean degree ⟨k⟩ to understand the structure of the network and the typical connectivity of a node.