statGraph (version 0.5.1)

fast.eigenvalue.probability: Degree-based eigenvalue probability

Description

fast.eigenvalue.probability returns the probability of an eigenvalue given the degree and excess degree probability.

Usage

fast.eigenvalue.probability(deg_prob, q_prob, all_k, z, n_iter = 5000)

Value

A complex number whose imaginary part absolute value corresponds to the probability of the given eigenvalue.

Arguments

deg_prob

The degree probability of the graph.

q_prob

The excess degree probability of the graph.

all_k

List of sorted unique degrees greater than 1 of the graph.

z

Complex number whose real part is the eigenvalue whose probability we want to obtain, the imaginary part is a small value (e.g., 1e-3).

n_iter

The maximum number of iterations to perform.

References

Newman, M. E. J., Zhang, X., & Nadakuditi, R. R. (2019). Spectra of random networks with arbitrary degrees. Physical Review E, 99(4), 042309.

Examples

set.seed(42)
G <- igraph::sample_smallworld(dim = 1, size = 10, nei = 2, p = 0.2)

# Obtain the degree distribution
deg_prob <- c(igraph::degree_distribution(graph = G, mode = "all"),0.0)
k_deg <- seq(1,length(deg_prob)) - 1

# Obtain the excess degree distribution
c <- sum(k_deg * deg_prob)
q_prob <- c()
for(k in 0:(length(deg_prob) - 1)){
  aux_q <- (k + 1) * deg_prob[k + 1]/c
  q_prob <- c(q_prob,aux_q)
}

# Obtain the sorted unique degrees greater than 1
all_k <- c(1:length(q_prob))
valid_idx <- q_prob != 0
q_prob <- q_prob[valid_idx]
all_k <- all_k[valid_idx]

# Obtain the probability of the eigenvalue 0
z <- 0 + 0.01*1i
eigenval_prob <- -Im(fast.eigenvalue.probability(deg_prob,q_prob,all_k,z))
eigenval_prob