[Paper Review] Self-organized and driven phase synchronization in coupled map scale free networks

Sarika Jalan, R. E. Amritkar|ArXiv.org|Jan 27, 2002

Neural Networks Stability and Synchronization1 references3 citations

TL;DR

This paper investigates phase synchronization in coupled logistic maps on scale-free networks, revealing two distinct synchronization mechanisms: self-organized (driven by intra-cluster couplings at low coupling strength) and driven (driven by inter-cluster couplings at high coupling strength). The transition between these regimes involves cluster reorganization, with the scale-free topology enabling both synchronization types, highlighting the role of network structure in dynamical coherence.

ABSTRACT

As a model of evolving networks, we study coupled logistic maps on scale free networks. For small coupling strengths nodes show turbulent behavior but form phase synchronized clusters as coupling increases. We identify two different ways of cluster formation. For small coupling strengths we get {\it self-organized clusters} which have mostly intra-cluster couplings and for large coupling strengths there is a crossover and reorganization to {\it driven clusters} which have mostly inter-cluster couplings. In the novel driven synchronization the nodes of one cluster are driven by those of the others.

Motivation & Objective

• To understand how phase synchronization emerges in complex dynamical networks with scale-free topology.
• To investigate the role of coupling strength in determining synchronization mechanisms—self-organized vs. driven—within such networks.
• To analyze the structural and dynamical reorganization of clusters as coupling strength increases.
• To determine whether the scale-free nature of the network is essential for observing phase synchronization.
• To draw parallels between the model's behavior and real-world systems like social groups or sports fans.

Proposed method

• Modeling a network of N nodes using coupled logistic maps with local dynamics f(x) = μx(1−x) and identity coupling g(x) = x.
• Generating scale-free networks via preferential attachment, ensuring a power-law degree distribution P(k) ∼ k⁻³.
• Defining phase synchronization via a phase distance d_ij = 1 − 2n_ij/(n_i + n_j), where matching minima indicate d_ij = 0.
• Using a time-averaged coupling rule: x_i^{t+1} = (1−ε)f(x_i^t) + ε/(∑_j C_ij) ∑_j C_ij g(x_j^t), with ε as coupling strength.
• Analyzing cluster formation and synchronization through numerical simulations across varying μ and ε values.
• Classifying clusters as self-organized (intra-cluster dominance) or driven (inter-cluster dominance) based on coupling patterns.

Experimental results

Research questions

• RQ1 How does coupling strength ε influence the emergence of phase-synchronized clusters in scale-free networks?
• RQ2 What distinguishes self-organized synchronization (intra-cluster couplings) from driven synchronization (inter-cluster couplings) in this system?
• RQ3 How does the network topology—specifically scale-free structure—affect the stability and formation of synchronized clusters?
• RQ4 Does the transition between self-organized and driven synchronization involve reorganization of node membership in clusters?
• RQ5 Are there real-world analogs of this dual synchronization mechanism in social or biological systems?

Key findings

• For small coupling strengths (ε), phase synchronization arises via self-organized clusters dominated by intra-cluster couplings, with minimal inter-cluster connections.
• As coupling strength increases, a crossover occurs to driven synchronization, where clusters are synchronized primarily through inter-cluster couplings.
• The transition involves reorganization of nodes into new cluster configurations, indicating a dynamic shift in network structure.
• The driven synchronization regime is characterized by nodes in one cluster being driven by nodes in other clusters, not by internal cohesion.
• The scale-free topology is essential: non-scale-free networks (exponential degree distribution) do not exhibit phase synchronization.
• The phenomenon persists across large network sizes (N = 1000), with self-organized regions stable and driven regions slightly shrinking as m increases.