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Evolving Networks Created by Preferential Attachment and Decay

Justin Downes

TL;DR

The paper tackles preserving a scale-free, power-law degree distribution in evolving networks where edges can be added and removed over time. It proposes a unified Evolution Model that couples standard preferential attachment with edge turnover, employing inverted deletion probabilities to counteract excessive growth of high-degree nodes, and applies it across three PA-based base models: Barabási-Albert, Bianconi-Barabási, and the Relevance model. In simulations with 10,000 nodes and 300 evolution steps (adding/removing ~1,000 edges per step), the BB model generally best maintains the power-law, while BA tends toward a normal-like distribution and Relevance can strengthen the heavy tail under certain decay conditions, though limits arise from decay. The work provides a practical, simulation-based framework for generating and evaluating dynamic, scale-free networks and offers directions for analytic treatment and agent-based extensions.

Abstract

Growing synthetic networks that follow power law distributions of a node's degree often involves adding one node at a time. Each node is added to the network with a fixed amount of edges and those edges are frozen for all future time steps. Yet real world networks often continuously evolve with edges being added and removed while new nodes are added to the network. Many existing growth models based on preferential attachment do not account for this evolutionary capability and when you extend their growth methods to add and remove edges to existing nodes the node degree distribution quickly loses its scale-free structure. This paper will go over a method to extend well known preferential attachment growth models to allow for the evolution of edges within a network while still maintaining a power law node degree distribution.

Evolving Networks Created by Preferential Attachment and Decay

TL;DR

The paper tackles preserving a scale-free, power-law degree distribution in evolving networks where edges can be added and removed over time. It proposes a unified Evolution Model that couples standard preferential attachment with edge turnover, employing inverted deletion probabilities to counteract excessive growth of high-degree nodes, and applies it across three PA-based base models: Barabási-Albert, Bianconi-Barabási, and the Relevance model. In simulations with 10,000 nodes and 300 evolution steps (adding/removing ~1,000 edges per step), the BB model generally best maintains the power-law, while BA tends toward a normal-like distribution and Relevance can strengthen the heavy tail under certain decay conditions, though limits arise from decay. The work provides a practical, simulation-based framework for generating and evaluating dynamic, scale-free networks and offers directions for analytic treatment and agent-based extensions.

Abstract

Growing synthetic networks that follow power law distributions of a node's degree often involves adding one node at a time. Each node is added to the network with a fixed amount of edges and those edges are frozen for all future time steps. Yet real world networks often continuously evolve with edges being added and removed while new nodes are added to the network. Many existing growth models based on preferential attachment do not account for this evolutionary capability and when you extend their growth methods to add and remove edges to existing nodes the node degree distribution quickly loses its scale-free structure. This paper will go over a method to extend well known preferential attachment growth models to allow for the evolution of edges within a network while still maintaining a power law node degree distribution.
Paper Structure (13 sections, 6 equations, 4 figures)

This paper contains 13 sections, 6 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Barabási-Albert Model
  4. Bianconi-Barabási Model
  5. Relevance Model
  6. Methodology
  7. Evolution Model
  8. Results
  9. Discussion
  10. Negative Attempts
  11. Non Power Law Models
  12. Agent Based Models
  13. Conclusion

Figures (4)

  • Figure 1: Evolution of a network where the edges selected to delete were based off of the Barabási-Albert model, degree distributions are approaching a normal distribution. Note the plot is in log-log format.
  • Figure 2: Evolution of a network where the edges selected to delete were based off of the inverse of the Barabási-Albert model, a closer approximation of the power law distribution
  • Figure 3: Starting and end plots of evolved networks after 300 steps of evolution with approx. 1000 edges added and deleted at each step.
  • Figure 4: A network that has evolved to have a normal distribution of nodes with $k$ degrees.