Topological Symmetry Breaking in Antagonistic Dynamics

TL;DR

The paper introduces topological symmetry breaking as a unifying lens to study antagonistic dynamics on signed networks. By leveraging the discrete signed Laplacian, it shows how topological defects percolate to fragment systems into mesoscopic states, producing robust multistable phases and non-ergodic behavior related to spin-glass physics. The work links spectral properties, percolation thresholds, and field-theoretic formalisms to neural-inspired and heterogeneous architectures, predicting Griffiths-like relaxation and phase structure as a function of defect density and temperature. These insights provide a topological-algebraic backbone for analyzing and potentially guiding learning algorithms and brain-inspired computation in systems with strong excitation-inhibition antagonism.

Abstract

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.

Figures (5)

• Figure 1: Signed Laplacian. (a) The antipodal proximity concept in the 2D spectral embedding with the kunegis2010spectral. The position $\bm{x}_i$ of the $i^{\mathrm{th}}$ node is computed as the average position of the neighbors $\bm{x}_j$ weighted by the adjacency index $A_{ij}$. When $\mathop{\mathrm{sign}}\nolimits(A_{ij}) = -1$, the position $\bm{x}_i$ is updated in the direction opposite to $\bm{x}_j$. Black arrows correspond to coordinate axes. (b) Sketch of a 2D square lattice with random positive (blue links) and negative (red links) interactions. (c) Heat capacity, $C$, versus the temporal resolution parameter of the system, $\tau$, for different values of $p$ averaged over multiple disorder realizations (see legend). (d) Probability distribution, $-\log[P(\tau^{(0)}_{\max})]$, for the position, $\tau^{(0)}_{\max}$, of the first detected maximum of $C$ at varying $p$. Its shape changes from a single minimum for low $p$ to a bistable configuration at $p_{\rm c}$.
• Figure 2: Percolation of topological defects in regular structures. (a) Partitioning of a square 2D lattice as indicated by the positive and negative signs of $|\lambda_0\rangle.$ A set of fractal clusters emerges beyond the critical value, $p_{\rm c}$. $\textbf{(b)}$ The lower part of each figure shows the partitioning of a 3D lattice as indicated by the positive and negative signs of $|\lambda_0\rangle$ (blue and red nodes, respectively). The upper part of the figure enhances only the positive ones. (c) Order parameter, $P_{\infty}$ versus the fraction of negative links, $p$ for a squared 2D lattice of variable size (see legend). Inset: Variance of the order parameter, $\chi$, as a function of the fraction of negative links, $p$. Note that $\chi$ diverges as the size of the system increases. Black dashed lines represent the expected value for the spin-glass transition at $T=0$. (d) Cluster size distribution for a 2D square lattice of different system sizes (see legend). The black dashed line represents the estimated Fisher exponent, $\tau\approx1.8$. (e) Order parameter, $P_{\infty}$ versus the fraction of negative links, $p$, for a 3D cubic lattice of variable size (see legend). Inset: Variance of the order parameter, $\chi$, as a function of the fraction of negative links, $p$. Note that $\chi$ diverges as the size of the system increases. Black dashed lines represent the expected value for the spin-glass transition at $T=0$. (f) Cluster size distribution for a cubic lattice of different system sizes (see legend). The black dashed line represents the estimated Fisher exponent, $\tau=2.18$. (g) Sketch of topological defects in a squared lattice: single links (S), Z defects representing the elementary cell, and X defects representing "anti-nodes". (h)$P_{\infty}$ versus the fraction of Z defects, $p$ for a squared 2D lattice of variable size (see legend). Inset: Variance of the order parameter, $\chi$, as a function of the fraction of Z defects, $p$. The black dashed vertical line corresponds to $p_{\rm c}=0.1$. (i)$P_{\infty}$ versus the fraction of X defects, $p$ for a squared 2D lattice of variable size (see legend). Inset: Variance of the order parameter, $\chi$, as a function of the fraction of X defects, $p$. The black dashed vertical line corresponds to $p_{\rm c}=0.1$.
• Figure 3: Ising dynamics. (a) Qualitative $T-p$ phase diagram for the 2D squared lattice with S defects. The lower part shows the energy levels corresponding to the first 100 eigenstates of $\bar{L}$ (see colorbar), for two values of $p$, respectively, in the ferromagnetic phase and the spin-glass one. The lowest eigenstate of $\bar{L}$ has been marked in red. Note that, inside the spin-glass phase, all eigenstates come closer in energy, and their ordering is lost at the 'onset of degeneracy' at $p_{\rm c}$. (b) Qualitative $T-p$ phase diagram for X defects computed with the eigenstates through numerical simulations in a 2D square lattice of size $N=64^2$. The breakup of the ferroelectric phase translates into the emergence of a metastable one where the eigenvectors now define multiple degenerate stable states characterized by local order up to the paraelectric/ferroelectric phase transition. At a certain $p$, stable minima becomes unstable, leading to a spin-glass phase. The lower part shows the energy levels corresponding to the first 100 eigenstates. In this case, the energy degeneracy is not observed before $p=0.4$, where the spin-glass phase emerges. (c) Annealing and sudden quenching initializing the system in a low eigenmode $|\lambda_2\rangle$ spin configuration on a 2D squared lattice with $p=0.25$ fraction of X defects. We heat the system up to $T\approx1.5$, i.e., smaller but closer to the critical Ising temperature. After relaxation, the system is suddenly quenched, observing that, up to border effects, it does not escape from the local minima associated to $\mathop{\mathrm{sign}}\nolimits(|\lambda_2\rangle)$. (d) Same as the previous case but using S defects with the lowest eigenmode at the $p_{\rm c}$. In this case, the system is locally stable but starts the attractor surfing at much lower temperatures. (e) Local magnetization order parameter, $m^{\rm GC}$, as a function of the temperature and the fraction of X defects for $N=64\times64$. (f)- (g) Spin overlap, $\braket{\lambda_i}{\sigma_\infty}$, as a function of the temperature for different lattice sizes for (f) X defects, using $|\lambda_2\rangle$, and (g) S defects, using $|\lambda_2\rangle$. The critical temperature is indicated in the insets as a vertical dashed line. Note how $T_{\rm c}$ is significantly greater for the case of X defects, while, in contrast, the initial configuration loses its stability at much lower temperatures for S defects. All curves have been averaged over $n_{\rm avg} = 1000$ independent realizations.
• Figure 4: Heterogeneous architectures. Order parameter, $P_{\infty}$ versus the fraction of negative links, $p$ for a: (a) network with $\langle \kappa \rangle=10$, (b) Small World network from a 2D lattice (rewiring probability $p_{\rm rew}=0.05$) and (c) a diluted 3D lattice ($p_{\rm dil}=0.25$) with normally weighted edges $p\sim\mathcal{N}(1, \sigma)$. Inset: Variance of the order parameter, $\chi$, as a function of the fraction of negative links, $p$. (d)$T-p$ Phase diagram of the network with a fraction $p$ of X defects measured by the local Ising order parameter $m^{\rm GC}_\infty$, and using$\mathop{\mathrm{sign}}\nolimits(|\lambda_2\rangle)$.
• Figure 5: (a) Spectral gap ($\Delta$) and Energy Gap ($E_0-E_1$) versus $J_0/J$ for a fully connected network with symmetric Gaussian interactions, highlighting the closure of the gap at the transition point. (b) Activity decay for the antagonistic contact process with a fraction of inhibitory links, $p=0.15$. Note the slow decay region indicative of Griffiths phases that emerge beyond $p>0.1$. This confirms that the previously identified topological phase transition is consistent with the interpretation that it governs the divergence of dynamical timescales. This dynamical slowing down is analytically linked to the proliferation of vanishingly small eigenvalues in the Signed Laplacian spectrum, which emerge as topological defects disrupt the spectral gap. Thus, the topological fragmentation of the network provides a structural encoding of the broad distribution of relaxation timescales observed in the dynamics. Inset: Temporal decay for a 2D lattice with no inhibitory links ($p=0$), showing standard bistable behavior. Parameters: L=96.