[Paper Review] Realizing interdependent couplings as thermal or higher-order interactions

Ivan Bonamassa, Bnaya Gross+1 more|arXiv (Cornell University)|Oct 17, 2021

Complex Network Analysis Techniques66 references4 citations

TL;DR

This paper proposes that interdependent couplings in complex networks can be physically realized as thermal or higher-order spin interactions, mapping interdependent Ising spin networks to either adaptive thermal couplings or directed $K$-spin interactions. It analytically derives the phase diagram using the thermal portrait and replica-symmetric theory, revealing that interdependence amplifies thermal fluctuations, induces extreme vulnerability via supercooled states, and establishes a deep isomorphism between interdependent percolation and the ground state of random $K$-xor-sat problems.

ABSTRACT

Interdependence is a fundamental ingredient to analyze the stability of many real-world complex systems featuring functional liasons. Yet, physical realizations of this coupling are still unknown, due to the lack of a theoretical framework for their study. To address this gap, we develop an interdependent magnetization framework and show that dependency links between $K-1$ pairwise networks of Ising spins can be rigorously mapped to directed $K$-spin interactions or to adaptive thermal couplings. We adopt the thermal portrait to determine analytically the phase diagram of the model under different structural configurations and we corroborate our results by extensive simulations. We find that interdependence acts like an entropic force that amplifies site-to-site thermal fluctuations, yielding unusual forms of vulnerability and making the system's functioning often unrecoverable. Finally, we discover an isomorphism between the ground state of random multi-spin models and interdependent percolation on randomly coupled networks. This connection raises new perspectives of cross-fertilization, providing unfamiliar methods with relevant implications in the study of constraint satisfaction as well as to the functional robustness of interdependent systems.

Motivation & Objective

• To resolve the lack of a physical framework for interdependent couplings in complex systems.
• To map interdependent spin networks to thermal or higher-order interactions for physical realizability.
• To analyze the stability and phase transitions of interdependent networks using statistical mechanics.
• To uncover a theoretical isomorphism between interdependent percolation and constraint satisfaction problems like $K$-xor-sat.
• To provide a theoretical and simulation-based foundation for understanding systemic collapse and recovery in interdependent systems.

Proposed method

• Develops an interdependent magnetization framework for $K-1$ pairwise Ising spin networks.
• Maps dependency links to either directed $K$-spin interactions on hypergraphs or adaptive thermal couplings based on time-scale separation.
• Applies the thermal portrait with replica-symmetric solutions to solve the higher-order Hamiltonian via interdependent population dynamics.
• Uses the $K$-xor-sat freezing transition equation $\psi = \left(1 - e^{-K\alpha\psi}\right)^{K-1}$ to locate the coexistence threshold $\langle k\rangle_{cx}$ and structural spinodal $\langle k\rangle_{sp}$.
• Employs extensive Monte Carlo simulations on Erdős-Rényi networks to validate analytical results.
• Establishes a bijection between the ground state of diluted ferromagnetic multi-spin models and interdependent percolation on randomly coupled networks.

Experimental results

Research questions

• RQ1 How can interdependent couplings in complex networks be physically realized in terms of thermal or higher-order interactions?
• RQ2 What is the phase diagram of interdependent Ising spin networks under varying network connectivity and dependency fractions?
• RQ3 How does interdependence amplify thermal fluctuations and lead to extreme vulnerability and irreversible collapse?
• RQ4 What is the analytical connection between interdependent percolation and the ground state of random $K$-xor-sat problems?
• RQ5 Why do percolation cascades remain trapped in local minima, and what determines the critical threshold for systemic failure?

Key findings

• Interdependence acts as an entropic force that amplifies site-to-site thermal fluctuations, increasing systemic vulnerability.
• The system undergoes a first-order phase transition from a ferromagnetic (functioning) to a paramagnetic (malfunctioning) phase, with recovery often impossible due to supercooled states.
• For $K=3$, recovery is possible only if the fraction $q$ of interdependent spins is below a critical threshold.
• The structural spinodal $\langle k\rangle_{sp}$, determined by the $K$-xor-sat freezing transition equation, marks the onset of metastability and is the only critical singularity in the failure process.
• The coexistence threshold $\langle k\rangle_{cx}$, where the system can almost surely be dismantled, diverges with the number of layers.
• Percolation cascades remain trapped in local minima of the free-energy landscape, failing to reach global solutions due to hidden clusters of metastable states.