Pair approximation for the $q$-voter model with independence on multiplex networks

Pair approximation for the noisy threshold $q$-voter model

“…In this case, we replace Eqs. (9,10a) into the rate equations (13,14). It turns out that, due to some cancellation of the combinatorial numbers, the averages .…”

“…A difficulty of the numerical calculation is that the binomial coefficients appearing in the rate equations take huge values for large k, while ρ k takes very small ones, leading to a very delicate numerical evaluation. To circumvent this problem when solving numerically equations ( (13,14)) in order to find the fixed points [29], we made expansions using the moments of the binomial distribution and the negative moments of the degree distribution, instead of evaluating the bare expressions.…”

“…We have found numerically the fixed points of the rate equations (13,14) in order to obtain the stability diagrams for networks of increasing connectivity µ = m q (with m ∈ N and q = 12), predicted by the PA, for the rules (a) without and (b) with repetition. The results are displayed in Fig.…”

“…In order to perform the averages F (ℓ; k, q, q 0 , p) ρi and ℓF (ℓ; k, q, q 0 , p) ρi appearing in Eqs. (13,14) after replacing the probabilities Eq. ( 10) using the binomial distribution k ℓ ρ l i (1 − ρ i ) k−ℓ , we use the following results:…”

“…However, few works on q-voter models go beyond the mean-field dynamics. Among them, we find the standard q-voter in complex networks [10,11] and in multiplex networks [12,13], as well as the non-linear noisy voter [14,15], related to the qvoter, for which analytical results were obtained through the pair approximation (PA). In order to investigate the threshold q-voter on random networks we use the PA approximation complemented by Monte Carlo simulations.…”