Papers

Strict thermodynamic scaling relations, such as the Rushbrooke inequality, are fundamentally established for equilibrium critical phenomena in the thermodynamic limit. In finite-size dynamical systems exhibiting synchronization, the direct application of such identities is hindered both by the finiteness of the network and the nonequilibrium nature of the spontaneous synchronization transition. To bypass this difficulty, we rigorously study the dynamical counterparts of the order parameter, susceptibility, and specific heat in finite systems of dynamical oscillators with nonlinear coupling. By measuring these quantities as a function of system size, we extract the associated critical exponents governing the transition. The validity of our thermodynamic mapping is tested by directly confirming the Rushbrooke inequality. Our results establish that standard equilibrium thermodynamic scaling architectures can be systematically applied to the finite-size scaling of nonequilibrium synchronization dynamics.

Granovetter's threshold model provides a classical framework for social mobilization, where collective action spreads through cascades as individuals join once movement size reaches their personal threshold. Here, we characterize social tipping points-the minimum seed required for global mobilization-using the initial fraction of instigators, $\rho_0$, as a control parameter. For a finite population of size $N$ with Beta-distributed thresholds $(\alpha, \beta)$, we present an exact analytical study of the cascade dynamics. By evaluating the asymptotic active fraction $\rho_\infty$, we map the thermodynamic phase diagram separating partial cascades ($\rho_0 \le \rho_\infty < 1$) from complete mobilization ($\rho_\infty = 1$), revealing continuous and discontinuous transition lines that meet seamlessly at a critical endpoint. For interior-peaked distributions ($\alpha > 1, \beta > 1$), the regimes are separated by a hybrid phase transition combining a first-order discontinuity with second-order bottleneck singularities. Combining exact finite-$N$ combinatorial formulations with large-deviation theory, we establish how finite-size fluctuations smooth these singularities. For power-law thresholds ($\alpha > 1, \beta = 1$), the critical scaling window shrinks as $N^{-1/3}$, while the expected inactive fraction vanishes as $N^{-1/3}$ at criticality. For interior-peaked distributions ($\beta > 1$), the order parameter is bimodally distributed: realizations either achieve full mobilization or stall near a bottleneck $\rho^*$. Excluding fully mobilized trajectories, $\rho^* - \langle \rho_\infty \rangle$ vanishes as $N^{-1/4}$ and the scaling window compresses to $N^{-1/2}$. Together, these results establish an exact finite-size scaling theory for threshold-driven tipping phenomena.

We propose a minimal adaptive network model with product-form coupling that reduces dimensionality while capturing key features of synaptic plasticity. The system spontaneously self-organizes into adaptive chimera states, where identical oscillators separate into synchronized and desynchronized groups through adaptive weight dynamics. These adaptive chimeras organize into branches with fixed coherent cluster fraction and exhibit transitions between stationary, breathing, and chaotic collective dynamics, revealing a collective bifurcation structure. Crucially, the resulting attractor landscape is highly multistable: repeated cluster reorganizations generate distinct dynamical pathways that coexist within the same parameter regime and depend sensitively on initial conditions.

We explore quantum Turing pattern zoology, where the same Lindblad equation supports a morphology atlas of stripes, spots, holes, labyrinths, and defects. The stable stripe species provides a quantitatively controlled case in which morphology and Gaussian witness loss separate parametrically. In particular, visible Turing stripes can remain after two Gaussian witness margins associated with the same $k_*$ mode cross zero in a completely positive Lindblad lattice. The witness thresholds on the exact shell fall as $\mathcal{N}^{-1}$. The stripe nematic threshold tends to a nonzero value at fixed lattice size, time window, and morphology criterion. The ratio of the morphology threshold to either witness threshold therefore grows with $\mathcal{N}$. Imaging and momentum-resolved covariance measurements probe these sectors separately.

Reputation is widely recognized as a key mechanism for sustaining cooperation. However, most existing game-theoretic models treat reputation primarily as an external factor that modulates payoffs, interaction structures, or strategy update rules. In many social contexts, though, reputation operates primarily as information – it shapes how individuals interpret their own experiences and assess the behavior of others. To bridge this gap, we propose a spatial prisoner's dilemma game grounded in the reinforcement learning paradigm, in which agents equipped with Q-learning integrate both individual and social information via a locally defined reputation metric to guide their decisions. Our results reveal that reputation-modulated learning significantly promotes the emergence of cooperative behavior, and we observe a discontinuous phase transition from full cooperation to full defection as the temptation increases. Cooperation spreads through the nucleation of cooperative clusters, whereas the disintegration of these clusters drives the system into an absorbing state of complete defection. Overall, this study demonstrates that reputation facilitates cooperation not only by providing direct incentives but also by reshaping the social information landscape that agents rely on for learning and adaptation.

The emergence of organized spatiotemporal patterns is ubiquitous in oscillatory systems, from neural populations to engineered networks. Identifying these patterns and tracking how they evolve over time remains challenging, particularly when systems exhibit transient dynamics. Here, we introduce a framework based on spatial ordinal patterns to characterize the spatiotemporal dynamics of oscillatory systems. Our approach acts directly on the phase rather than the amplitude, with additional patterns introduced to account for near-equal phases. This symbolic representation encodes local spatial ordering relations, capturing both phase gradients and synchronized clusters within a single framework. From this construction, we define a spatial permutation entropy that quantifies the diversity of spatiotemporal patterns at each point in time, enabling the detection of transient dynamics and regime transitions as they occur. We show that this approach distinguishes phase-locked states with identical levels of global synchronization but distinct spatial organization and also characterizes partially synchronized states. We demonstrate the method on synthetic oscillator networks across multiple spatiotemporal regimes, and on resting-state EEG recordings from human volunteers, where it distinguishes different conditions within individual volunteers.

Adjoint methods provide a powerful route for gradient-based optimization, but their physical implementation is obstructed in generic nonlinear systems because the adjoint dynamics requires backward-time evolution, Jacobian transposition, and terminal-value constraints. Here we show that nonlinear parity-time ($\mathcal{PT}$)-symmetric systems overcome this obstruction. Using a class of nonlinear non-Hermitian resonator networks, we establish symmetry relations that map the formal adjoint dynamics onto experimentally accessible forward-time evolutions supplemented by controlled injections. This construction enables exact in-situ evaluation of adjoint gradients without requiring explicit backward propagation or matrix transposition. We demonstrate the approach in nonlinear $\mathcal{PT}$-symmetric resonator chains, where the resulting optimization protocol autonomously discovers parameter configurations that realize prescribed spatio-temporal functionalities, including uniform energy redistribution and targeted wave transport at predefined time windows. Our results identify $\mathcal{PT}$ symmetry as a resource for implementing computational sensitivities within physical systems and establish a route toward self-optimizing nonlinear machines.

Spatial early warning signals (EWSs) seek evidence of an approaching tipping point from a single snapshot of many interacting elements. Existing theory largely assumes spatial homogeneity, whereas networks introduce systematic differences among nodes that may obscure fluctuation-based warning signals. We develop a mathematical framework for spatial EWSs in stochastic dynamical systems on networks. We find that the expected spatial variance, a popular spatial EWS, decomposes exactly into a structural contribution from heterogeneity in the equilibrium state and a fluctuation contribution determined by the stationary covariance. Near a simple steady-state bifurcation, the potentially divergent covariance concentrates along the critical eigendirection: the left eigenvector determines how strongly noise excites the critical fluctuation, while the right eigenvector determines its spatial pattern. Consequently, the spatial variance has a divergent fluctuation contribution when the limiting critical eigendirection is noise-excited and spatially nonuniform after centering. In contrast, the spatial coefficient of variation generally saturates, while skewness, kurtosis, and Moran's $I$ approach network-dependent limits without a universal warning direction. We also derive results for homogeneous networks, node-wise baseline subtraction as preprocessing, and Hopf bifurcations, for which the limiting distributions are qualitatively different. These results clarify when spatial EWSs provide reliable warnings and why their performance depends on network structure, noise, and preprocessing.

We investigate stochastic resonance in a hierarchy of delayed excitable systems, from a single FitzHugh–Nagumo neuron to single-layer and duplex networks with autaptic, pairwise, triadic higher-order, and interlayer interactions. We show that the order and organization of interactions generate distinct resonance regimes. Among the autapse-free single-layer architectures, pairwise and triadic coupling attain comparable maximal responses, but over different ranges of nominal coupling strength: the pairwise enhancement persists across the investigated coupling range, whereas the triadic enhancement is confined to weak coupling. Under the matched participation degrees used here, this difference is largely attributable to the doubled effective coupling generated by the hyperedge construction. Delayed autaptic feedback primarily shifts the optimal noise level and, within the configurations investigated, yields a genuine enhancement only when pairwise and triadic interactions coexist, revealing a non-additive coupling effect. In duplex networks, interlayer coupling acts predominantly as a response-equalization mechanism: it enhances the weaker layer, generally at the expense of the stronger one, with the redistribution controlled mainly by the uncoupled resonance-capacity gap. Across the sampled duplex configurations, increasing the interlayer delay largely suppresses the response within the investigated delay range. These results identify interaction order, resonance mismatch, and delay as key control parameters for noise-assisted signal processing in multilayer excitable systems.

Spectrally stable cascades can undergo enormous transient growth before eventually dying out. We show that this amplification is controlled by the directed architecture of the interactions, independently of the eigenvalues that determine asymptotic stability. More remarkably, when several critical subsystems are connected sequentially, their depth becomes a new control parameter for critical fluctuations: each additional critical stage generates a new cascade-size exponent, producing an infinite hierarchy of universality classes. This result follows because the fluctuating population produced at one stage becomes the random input to the next. The mechanism persists for both finite-variance and heavy-tailed reproduction and strongly enhances the probability of rare terminal events. Applied to multitype earthquake triggering, it provides a stationary mechanism for the anomalously large abundance of foreshock-mainshock sequences. Directed architecture can therefore control both transient amplification and critical statistics in cascade processes even when it leaves spectral stability unchanged.

Environmental variability is a defining feature of natural ecosystems, yet many theories of ecological stability assume static environments. Here, we develop an analytical framework for local stability in complex ecological communities subjected to periodic environmental forcing. We show that, in slowly varying environments and under a large-system approximation, the maximal Lyapunov exponent can be approximated by the time-averaged rightmost spectral edge of the instantaneous interaction matrix, yielding explicit stability criteria for large ecological communities. Within the random-matrix interaction ensembles considered here, our minimal theoretical framework predicts a topology-dependent ordering of local perturbation stability under periodic temporal modulation in high-dimensional ecological networks, which is supported by numerical simulations. Beyond the adiabatic regime, rapid environmental oscillations dynamically stabilize otherwise unstable ecosystems, revealing a high-frequency rescue effect that is absent from static theories. These results extend ecological stability theory beyond autonomous systems and provide a general framework for understanding resilience in fluctuating environments.

We derive a one-dimensional reduction for nonlinear dynamics on simplicial complexes containing both pairwise and triangular (higher-order) interactions. The effective state is defined using a mixed weight determined by the pairwise and triangular degrees of each node. The resulting reduced equation retains two structural coefficients, associated separately with the pairwise and higher-order coupling channels. A fluctuation expansion identifies the closure assumptions underlying the reduction and shows how deviations of individual node states from the effective state contribute to the approximation error. We numerically validate the proposed framework on Gene-regulatory dynamics, the double-well system, and SIS spreading. The states of the reduced model are compared with full-network simulations through coupling-parameter sweeps, steady-state branch calculations, and progressive node-removal experiments on synthetic and real-world networks. The reduced model successfully reproduces the principal transitions and steady-state branches in all three dynamical systems considered. Agreement is strongest for relatively homogeneous networks and deteriorates when structural heterogeneity produces a broader distribution of node states. The closure diagnostics account for this loss of accuracy and indicate when a single effective state is no longer sufficient. The reduction therefore provides a tractable description of resilience in systems with coexisting pairwise and higher-order interactions.

Many traditional models of electoral dynamics emphasize opinion change, social influence and persuasion mechanisms. In contrast, contemporary polarized societies often exhibit relatively stable partisan blocs coexisting with an electorally relevant mobile population. We propose a minimal dynamical model in which electoral competition is governed not by direct persuasion between opposing blocs, but by the dynamics and allocation of a mobile electoral interface. The electorate is partitioned into two stable partisan blocs and a mobile fraction, while direct transitions between the polarized blocs are strongly suppressed. Mobile voters are allocated between the competing blocs through a Fermi-like probabilistic rule governed by rejection asymmetries. Analytical calculations show that the electoral susceptibility to political shocks is proportional to the stationary size of the mobile electoral interface, identifying electoral mobility as the key quantity controlling the macroscopic response of polarized systems to external perturbations. The model naturally predicts a continuum of mobility regimes, ranging from frozen polarization to highly responsive electoral states characterized by a broad mobile interface. These results suggest that, in strongly polarized elections, aggregate electoral changes may be governed primarily by fluctuations at the mobile electoral interface rather than by large-scale ideological conversion.

Low-dimensional reductions provide a useful framework for studying high-dimensional dynamics on complex networks, but most existing approaches are restricted to pairwise interactions. Here, we develop a one-dimensional reduction for dynamical systems on networks with purely higher-order interactions. The reduction is formulated through an effective higher-order interaction strength ($\beta_{\Delta}$), associated with the triangular interactions of the underlying network and the dynamical system's effective state. We present a theoretical framework for the dimension-reduction approach and validate it across three dynamical models with exclusively higher-order interactions. We find that the reduction accuracy is mainly determined by the homogeneity of node states, i.e., the deviations in state values become very small. Numerical results on synthetic and real networks show that the reduced model captures the effective steady states and transitions of the full system with good accuracy.

In our previous paper, we described the proposal and validation of a new physical implementation-type simulator (spontaneous symmetry breaking machine (SSBM)) for a combinatorial optimization problem that utilizes a phenomenon dual to the model in which a pseudo-spin many-body system is created by spontaneous symmetry breaking. Furthermore, we performed numerical simulations with different initial fluctuations on a large-scale benchmark problem (K_{2000}) and reported that we identified a condition under which pseudo-spin patterns gradually converge during the solution search process to ultimately form a single pattern and that the cut value of this converged solution reached 99.7% of the known best. In this paper, we discuss the conditions under which the SSBM can exhibit behavior analogous to that of a continuous-variable-type Ising machine, and report that by adding an effect similar to physical annealing to these conditions, the SSBM can be made to search for the best known cuts in K_{2000}.

Microbial populations generally evolve in fluctuating environments under time-varying conditions. These are often described by binary switching models, sometimes seen as coarse-grained feast-famine cycles, in which resource availability switches abruptly between abundant and scarce conditions. However, experimental studies suggest that feast-famine environments actually exhibit more complex temporal dynamics. Here, we study how two strains, one growing slightly slower than the other, compete for the same resources in fluctuating environments comprising a finite number of intermediate states, each having its own carrying capacity. Environmental switching between these states and their carrying capacities represents gradual changes in nutrient availability. This class of multi-state stochastic switching models can be interpreted as a coarse-grained description of feast–famine cycles and allows us to investigate strain competition under the gradual recovery and depletion of resources. By computational and analytical means, we characterise the population dynamics in these multi-state fluctuating environments. In particular, we study how the switching rates and distribution of carrying capacities affect the population-size statistics, fixation probability, and mean fixation time. By comparing these results with their counterparts in binary environments, we clarify how the frequency and amplitude of environmental fluctuations influence population dynamics in coarse-grained feast-famine cycles.

The minimum-time feedback law for driving a population of coupled oscillators into synchrony is unknown. Here we settle it for identical Kuramoto oscillators under an instantaneous power constraint. Greedy control, maximizing $\dot r$ at each instant, is exactly optimal at $N=2$ and suboptimal above, as dynamic programming confirms at $N=3,4$. The obstruction is geometric: the greedy closed loop is a reparametrized gradient flow of $r$, fixing its path independently of the power budget, and as a first-harmonic forcing it cannot leave a single Möbius orbit. A three-harmonic policy trained on the DP fields, a trajectory expert, and a smooth first-hitting-time objective beats greedy by $10$–$14\%$ at $N=10$–$100$ and matches DP to within $0.3\%$ where ground truth exists. Reading the policy rather than deploying it collapses it to a two-constant law, $u_i\propto-\sin\phi_i+a_2\sin2\phi_i+a_3\sin3\phi_i$, which recovers $84$–$99\%$ of the network's advantage, with the same functional form holding from $N=3$ to $N=1000$. Machine learning discovered a closed-form law beyond the reach of direct analytical methods.

Time delays are ubiquitous in physical and biological networked systems, playing a fundamental role in the emergence and stability of collective behavior. Yet, existing theoretical methods for synchronization analysis of delay-coupled systems are largely limited to networks with homogeneous degree distributions. Here, we extend the master stability function framework to a broad class of degree-heterogeneous, weighted, and directed delay-coupled networks. The analysis reveals that synchronization can be enhanced in heterogeneous networks compared to homogeneous ones, including all-to-all networks, which are known to be optimal in non-delayed systems. To identify optimal heterogeneous structures, we develop a network optimization method that finds directional, edge-weighted configurations maximizing synchronization stability. Our results show that, in delay-coupled networks, heterogeneity and nonreciprocity are key resources for synchronization.

Previous works have shown that strong coupling among heterogeneous agents enforces practical synchronization, leading to collective behavior governed by emergent dynamics. However, it remains an open question whether emergent behavior persists in the presence of communication delays, as high-gain methods are typically sensitive to delays. To address this problem, we instead slow down the agent dynamics, reproducing the synchronization mechanism of strong coupling on a slow time scale. As a result, the emergence of collective behavior is guaranteed despite arbitrary constant communication delays. This, however, comes at a cost: delays induce a derivative-like term, thereby scaling down the vector field of the emergent dynamics. The amount of the scaling is explicitly characterized as a function of the coupling weights and the delay lengths. We further suggest that this delay-induced slowdown, which is of independent interest, may be a general feature of diffusive coupling under suitable assumptions.

The physics of spreading in static networks is well understood through mappings to percolation. We show that spreading dynamics on temporal networks can analogously be mapped to reachability in temporal event graphs. This provides a theoretical and computational framework for a class of processes, such as variants of the susceptible-infected-susceptible model. Without explicit simulations, through the component analysis of event graphs, we obtain epidemic prevalence and derive epidemic thresholds for temporal networks with arbitrary degree and inter-event time distributions, with significant computational advantages as compared to explicit simulations.

Excitable and oscillatory responses are widespread in cell signaling, but how these arise in the same system is unclear. We propose a minimal model of a saddle node bifurcation on an invariant circle, where slow fluctuations modulate transitions between excitable and oscillatory states, and noise drives excitability. Fitting the model to cell signaling data from embryonic stem cells reveals that these cells operate close to the bifurcation. Proximity to the bifurcation allows the cell to tune different dynamic states, suggesting a general principle for versatile cell signaling.

We propose a mathematical model that represents the influence of an external actor (influencer) on a network of actors (influenced). The model is an adaptation of control systems within the family of formulations inspired by the Kuramoto Model of synchronisation. Capturing influence as a capacity to affect the character or behaviour of another, we study an external node's ability to influence the synchronisation of the Kuramoto system while its global behaviour is pulled to the external node's frequency and away from its natural mean frequency. In our work, stochastically generated dynamical weights assigned to the links between the network and the external influencer whereby links from one system to the other are assigned via one-sided heavy tail noise, generated by the Gamma distribution. We perform numerical experiments to examine transition points in the ability of the external node to alter the behaviour of the influenced network; either to disrupt synchronisation or to drive it to collective frequencies determined by the external node. We examine the dependence of transition points on the external node's frequency, where too ambitious a driving frequency fails to influence the system while retaining a synchronised state, and reducing achieves a state of synchronisation at a frequency shifted from the mean natural frequency. We also look at the analytic approximation for the system close to synchronisation and fragmentation to understand its behaviour around this limit.

Phase reduction provides a principled relationship between high-dimensional oscillator dynamics and low-dimensional phase models such as the Kuramoto model. However, connecting analytical results derived from the Kuramoto framework to experimental data remains an open challenge, as many physical platforms violate the weak-coupling assumptions underlying standard phase reduction. Here, we present an experimental platform based on electronic quadrature oscillators whose phase dynamics are, under certain assumptions, equivalent to the Kuramoto model. Crucially, this equivalence is achieved for both weak and strong coupling through a non-diffusive coupling circuit, and the design approach supports arbitrary (weighted and directed) network topologies while preserving scalability and oscillation regularity. Across different coupling schemes, we demonstrate that the platform reliably reproduces phase transitions predicted by the Kuramoto model for both global and cluster synchronization. This cost-effective platform enables systematic experimental validation of analytical and numerical results for synchronization in complex networks.

This paper develops an exact low-dimensional reduction theory for populations of Stuart-Landau oscillators. The theory covers two cases: in one case, coupling enters only through the coefficients of the Stuart-Landau equations, and in the other, coupling also enters through terms outside those coefficients. Under suitable assumptions, the two cases can be reduced exactly to three and seven-dimensional systems, respectively. The class of additional coupling terms for which the reduction applies is sufficiently general to allow a broad range of collective dynamics, such as clustering. In particular, the present reduction is shown to capture dynamics in which amplitude plays an essential role and which cannot be described by populations of phase oscillators or their low-dimensional reductions. The proposed framework is also shown to be a powerful tool for capturing complex nonequilibrium dynamics such as chaos.

We study Braess-type paradoxes in coupled oscillator networks where the addition of an edge to the network leads to an increase in the critical coupling required for the existence of a stable phase-locked synchronized solution. By introducing a continuous edge-weight parameter and constructing an augmented system that allows application of the Implicit Function Theorem, we derive an explicit first-order formula for the sensitivity of the critical coupling, $K'_c(0)$, to small changes in the network structure. This formula reveals that an edge is locally Braess if the ordering of the oscillators being coupled is opposite the ordering of their components in the critical eigenvector at the saddle-node bifurcation, providing a simple criterion for Braess paradox to occur. We then employ this local approximation as a predictor for the occurrence of Braess paradox when the edge is fully incorporated in the network. This prediction is tested numerically in several classes of random networks, demonstrating both high predictive fidelity and the nonlinear limitations of the local approximation. We also examine several explicit examples, highlighting possible motifs by which Braess-type paradoxes occur. Together, these results establish a predictive framework for understanding how incremental changes in network topology can produce counterintuitive synchronization outcomes.

We analyze inverse stochastic resonance (ISR) in a bistable FitzHugh–Nagumo neuron driven by additive noise in the voltage variable, focusing on how neural excitability and timescale separation regulate the noise-induced modulation of spiking activity. A codimension-two bifurcation analysis identifies a narrow bistable region in which a stable fixed point and a stable limit cycle coexist, separated by an unstable periodic orbit. Finite-time Monte Carlo simulations show that the occupation of the limit-cycle basin may appear to depend on the initial basin when rare transitions are not fully resolved. We prove that this dependence is not asymptotic: the stochastic system admits a unique invariant probability measure, so long-time firing statistics are independent of the initial basin of attraction. The parameter dependence of ISR is characterized by quasi-potential barriers computed with a geometric minimum action method for the degenerate noise. The difference between the limit-cycle and fixed-point quasi-potentials partitions the bistable wedge into two escape-dominated regimes. A reduced metastable two-state Markov approximation yields a weak-noise formula for the limit-cycle basin occupation probability, a sign criterion for genuine ISR, and a semiquantitative prediction of the ISR-minimizing noise amplitude. In the fitted regime with a negative effective exponent, a genuine ISR minimum occurs only when the limit-cycle quasi-potential exceeds that of the fixed point. These results provide an escape-balance mechanism linking intrinsic neuronal parameters to asymptotic noise-induced spike suppression.

Decentralized populations of Large Language Model (LLM) agents can spontaneously reach consensus on shared conventions, yet the microscopic mechanisms by which their internal stochasticity shapes macroscopic ordering remain unexplored. We study a minimal LLM Naming Game in which the listener's decision is a single-token LLM call at decoding temperature $T$, replacing the inventory check of the deterministic Naming Game. Each interaction decomposes into an in-inventory and an out-inventory channel with conditional rates $\pi(T)\!\equiv\!P(\text{YES}\mid w\in P_j)$ and $\phi(T)\!\equiv\!P(\text{YES}\mid w\notin P_j)$, whose balance controls an ordering-disordering drift. A mean-field theory of the two-rate dynamics yields an analytical ordering condition that generalizes the consensus threshold of the stochastic Naming Game to a critical line in the $(\pi,\phi)$ plane. Across three open-weight architectures, consensus is always reached, but through three distinct listener regimes: permissive (repaint-noise dominated), near-deterministic, and conservative (missed-collapse dominated). The effective finite-size exponent $\beta(T)$ in $t_{\rm conv}\!\sim\!N^{\beta}$ shifts with temperature, and the temperature-sensitivity $\alpha$ in $t_c\!\sim\!e^{\alpha T}$ ranges from ${\approx}\,0.67$ to ${\approx}\,0$ across architectures. Decoding temperature thus emerges as an architecture-dependent control parameter for decentralized LLM populations, quantitatively characterized by the statistical-physics toolkit.