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Evolving scale-free networks and node-based random edge deletion

Everton R. Constantino, Alberto Saa

TL;DR

It is shown that the node-based edge deletion mechanism is less likely to disrupt the scale-free (power-law) regime than the uniform edge deletion, and the precise balance between preferential attachment and node-based deletion can transform a scale-free network into a critical one with stretched exponential decay.

Abstract

We investigate a growing network model that combines preferential and uniform attachment with two distinct mechanisms of edge deletion. In addition to the usual uniform probability edge deletion, we introduce a novel node-based rule in which uniformly chosen non-isolated nodes lose one of their incident edges. This mechanism differs fundamentally from uniform edge deletion and leads to a nonlinear evolution for the stationary degree distribution due to the nonlinear dependence on the fraction of isolated nodes. We solve the general problem in the stationary regime and obtain closed-form expressions for the degree distribution in terms of hypergeometric and confluent hypergeometric functions. Depending on the balance between attachment and deletion rates, three asymptotic regimes for the degree distribution arise: power-law, exponential, and a critical regime characterized by a stretched exponential decay. We show that the node-based edge deletion mechanism is less likely to disrupt the scale-free (power-law) regime than the uniform edge deletion. Moreover, we also demonstrate that the precise balance between preferential attachment and node-based deletion can transform a scale-free network into a critical one with stretched exponential decay. Extensive numerical simulations exhibit excellent agreement with the theoretical predictions.

Evolving scale-free networks and node-based random edge deletion

TL;DR

It is shown that the node-based edge deletion mechanism is less likely to disrupt the scale-free (power-law) regime than the uniform edge deletion, and the precise balance between preferential attachment and node-based deletion can transform a scale-free network into a critical one with stretched exponential decay.

Abstract

We investigate a growing network model that combines preferential and uniform attachment with two distinct mechanisms of edge deletion. In addition to the usual uniform probability edge deletion, we introduce a novel node-based rule in which uniformly chosen non-isolated nodes lose one of their incident edges. This mechanism differs fundamentally from uniform edge deletion and leads to a nonlinear evolution for the stationary degree distribution due to the nonlinear dependence on the fraction of isolated nodes. We solve the general problem in the stationary regime and obtain closed-form expressions for the degree distribution in terms of hypergeometric and confluent hypergeometric functions. Depending on the balance between attachment and deletion rates, three asymptotic regimes for the degree distribution arise: power-law, exponential, and a critical regime characterized by a stretched exponential decay. We show that the node-based edge deletion mechanism is less likely to disrupt the scale-free (power-law) regime than the uniform edge deletion. Moreover, we also demonstrate that the precise balance between preferential attachment and node-based deletion can transform a scale-free network into a critical one with stretched exponential decay. Extensive numerical simulations exhibit excellent agreement with the theoretical predictions.
Paper Structure (8 sections, 41 equations, 1 figure)

This paper contains 8 sections, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The model
  3. The solutions
  4. The power-law case: $\alpha_0>\alpha_2$
  5. The exponential case: $\alpha_0<\alpha_2$
  6. The critical case
  7. Discussion
  8. Mathematics details

Figures (1)

  • Figure 1: Degree distributions for random networks obtained by running $10^7$ steps of our algorithm (blue dots) and the theoretical predictions in terms of hypergeometric functions (solid red line). Top left: Power-law case with all rules of our algorithm, corresponding to $p_1=0.5$, $p_2=0.3$, $p_3 = p_4 = 0.1$, $m_1=3$, $m_2=2$, and $m_3=m_4=1$. From the theoretical predictions, we have $P_0\approx 2.63\times 10^{-3}$, while from the simulation we obtained $P_0\approx 2.51\times 10^{-3}$. Top right: Another power-law case with PA and the two deletion rules, corresponding to $p_1=0.8$, $p_2=0$, $p_3 = p_4 = 0.1$, and $m_1 =3$, and $m_3=m_4=1$. The theoretically predicted and the observed values for $P_0$ in this case are, respectively $1.40\times 10^{-3}$ and $1.38\times 10^{-3}$. Bottom left: Critical case with PA and uniform edge deletion, corresponding to $p_1=2/3$, $p_3=1/3$, $p_2 = p_4 = 0$, and $m_1=m_3=1$. Since we do not have the node-based edge deletion in this case ($\beta_2=0$), we do not need to solve the nonlinear equation for determine $P_0$. Bottom right: Exponential case with PA and the two deletion rules, corresponding to $p_1=0.6$, $p_2=0$, $p_3 = 0.3$, $p_4 = 0.1$, and $m_1=m_3=m_4=1$. The theoretically predicted and observed values for $P_0$ given by, respectively, $5.80\times 10^{-1}$ and $5.89\times 10^{-1}$. For all figures, the red lines correspond to the respective theoretical prediction (\ref{['hyperPL']}), (\ref{['hyperPLexp']}), or (\ref{['kummer']}). We obtained very good overall agreement for all considered cases. It is worth to notice that for the last cases, namely the sub-exponential and the exponential decaying cases, it is intrinsically more expensive computationally to attain good statistical convergence. For all cases we have considered, the nonlinear equations for $P_0$ based on (\ref{['plC']}), (\ref{['expC']}), and (\ref{['ccC']}) had a unique solution.