[Paper Review] Fractal and Small-World Networks Formed by Self-Organized Critical Dynamics

Akitomo Watanabe, Shogo Mizutaka+1 more|arXiv (Cornell University)|Jul 21, 2015

Complex Network Analysis Techniques3 citations

TL;DR

This paper proposes a self-organized critical (SOC) network model where dynamic growth and cascading overload failures drive the emergence of fractal and small-world network structures. The model demonstrates that critical cascades induce fractal networks, while non-critical states yield small-world properties, with a clear crossover from fractal to small-world topology during evolution, confirming SOC behavior through power-law distributions of avalanche sizes, cluster sizes, and inter-avalanche intervals.

ABSTRACT

We propose a dynamical model in which a network structure evolves in a self-organized critical (SOC) manner and explain a possible origin of the emergence of fractal and small-world networks. Our model combines a network growth and its decay by failures of nodes. The decay mechanism reflects the instability of large functional networks against cascading overload failures. It is demonstrated that the dynamical system surely exhibits SOC characteristics, such as power-law forms of the avalanche size distribution, the cluster size distribution, and the distribution of the time interval between intermittent avalanches. During the network evolution, fractal networks are spontaneously generated when networks experience critical cascades of failures that lead to a percolation transition. In contrast, networks far from criticality have small-world structures. We also observe the crossover behavior from fractal to small-world structure in the network evolution.

Motivation & Objective

• To explain the origin of fractal and small-world network structures in real-world complex systems.
• To investigate how self-organized criticality (SOC) dynamics can generate both fractal and small-world topologies.
• To resolve the apparent conflict between fractal scaling and small-world properties in real networks.
• To develop a dynamic network model where SOC behavior arises from the interplay of growth and failure-induced decay.
• To identify the conditions under which fractal networks emerge and crossover to small-world structures during network evolution.

Proposed method

• The model combines network growth via new node addition with decay due to cascading overload failures in large networks.
• Node load is calculated as the sum of edge weights, and failure occurs when load exceeds a tolerance threshold, triggering cascades.
• A load reduction parameter r(N) is introduced, which decreases slowly with network size N to maintain criticality.
• The system evolves through cycles of growth and failure, with critical cascades occurring when r(N) reaches a critical value rc.
• The network's topological properties are analyzed using fractal dimension dB via box-covering method and average path length l to assess small-world behavior.
• Power-law distributions of avalanche size, cluster size, and inter-avalanche time intervals are used to confirm SOC characteristics.

Experimental results

Research questions

• RQ1 How can self-organized criticality give rise to both fractal and small-world network structures?
• RQ2 What dynamical mechanism enables the spontaneous emergence of fractal networks during critical cascades?
• RQ3 How does the network topology evolve from fractal to small-world as it grows away from criticality?
• RQ4 What parameter conditions are necessary for sustained SOC behavior in a growing network with failure dynamics?
• RQ5 Can the crossover from fractal to small-world structure be explained by the addition of new nodes introducing short-cut edges?

Key findings

• The model exhibits clear self-organized criticality, with power-law distributions for avalanche size, cluster size, and inter-avalanche time intervals.
• After critical cascades, the giant component displays fractal structure with a fractal dimension dB ≈ 3.0, consistent with Erdős-Rényi random graphs at criticality.
• Networks far from criticality exhibit small-world properties, with average path length ⟨l⟩ ∝ log N.
• A crossover from fractal to small-world structure is observed during network evolution, driven by short-cut edges introduced by new nodes.
• The system remains in a universal SOC class regardless of parameter choices, provided the condition N_st ≫ N_pre is satisfied.
• The critical cascade size distribution follows a power law with exponent τ ≈ 1.5, confirming SOC behavior.