Scale-Free Loopy Structure is Resistant to Noise in Consensus Dynamics in Complex Networks

Area of Science:

• Network Science
• Complex Systems
• Control Theory

Background:

• Real-world networks often exhibit scale-free properties, characterized by power-law degree distributions and sparsity.
• Consensus dynamics are fundamental in distributed systems, aiming for agreement among agents.
• Noise can significantly impact the performance and stability of consensus algorithms.

Purpose of the Study:

• To analyze the coherence of first-order consensus dynamics in binary scale-free networks under white noise.
• To quantify network coherence using the H2-norm, measuring agent tracking of the consensus value.
• To investigate the robustness of consensus dynamics in various scale-free network models.

Main Methods:

• Derivation of a lower bound for network coherence based on average degree, independent of network size.
• Numerical analysis of coherence in Barabási-Albert and random Apollonian networks.
• Analytical study of coherence in deterministically growing sparse networks using the Kirchhoff index.

Main Results:

• Network coherence in sparse, scale-free real-world networks approaches a constant value.
• Coherence in growing Barabási-Albert and Apollonian networks also converges to a constant.
• Analytical expressions for coherence in specific growing networks tend towards small constants, indicating noise has a negligible effect.

Conclusions:

• Scale-free topology and loopy structures confer strong robustness to noisy consensus dynamics.
• The inherent properties of power-law networks minimize the impact of external noise on achieving consensus.
• These findings suggest that scale-free networks are highly reliable for distributed systems operating in noisy environments.