Topological Symmetry Breaking in Antagonistic Dynamics

A dynamic concordia discors, a finely tuned equilibrium between opposing forces, is hypothesized to drive historical transformations. Similarly, a precise interplay of excitation and inhibition, often approximated by an 80:20 ratio, underlies the normal functionality of neural systems. In artificial neural networks, reinforcement learning enables the fine-tuning of internal signed connections, optimizing adaptive responses to complex stimuli and ensuring robust performance. Engineered systems with antagonistic interactions remain comparatively unexplored, particularly because their emergent phases are closely linked to frustration mechanisms in the underlying network. In this context, spin-glass theory has shown how apparently uncontrollable non-ergodic and chaotic behavior can arise from the complex interplay of competing interactions and frustration among units, leading to multiple metastable states that prevent the system from exploring all accessible configurations over time. Here, we show how topology constrains dynamics in systems with antagonistic interactions. We use the signed Laplacian operator to demonstrate how fundamental topological defects in lattices and networks percolate, shaping the geometrical structure and the complex energy landscape of the system. This reveals novel and highly robust multistable phases and establishes deep connections with spin glasses when thermal noise is considered, providing a natural topological and algebraic description of emergent multistability and non-ergodicity in frustrated systems.

💡 Research Summary

The paper investigates how antagonistic (positive‑negative) interactions embedded in lattices and networks give rise to topological symmetry breaking, multistability, and spin‑glass‑like non‑ergodic behavior. The authors introduce the signed Laplacian ¯L = ¯D − A, where A is the signed adjacency matrix and ¯D contains the absolute degree sums. Unlike the ordinary Laplacian, ¯L encodes “antipodal proximity”: neighbors linked by a negative edge contribute with opposite sign, pushing vertices toward opposite positions in a spectral embedding.

Using ¯L, they define a diffusion operator ρ̂(t)=e^{-t¯L}/Z(t) and the associated graph entropy S(t)=Tr