# gst: directed extended-range percolation.

<< ️For standard percolation, directionality is known to increase the complexity of percolation, requiring the distinct statistical treatment of in messages and out messages for determining the percolation order parameters. >>

<< ️Here (AA) show that when connectivity is ensured by paths of length 𝑅≥2, network directionality, impeding backtracking, can significantly reduce the complexity of percolation. To illustrate this finding, (AA) introduce directed extended-range percolation (DERP), defined on directed networks with nonreciprocal edges and motivated by applications in quantum communication. >>

<< ️In this framework, message transmission is enabled between trusted nodes separated by a directed path of length at most 𝑅. Using a message-passing approach, (AA) show that, on locally treelike structures, directionality enables an exact determination of the percolation threshold and the anomalous critical indices with a number of distinct statistical messages smaller than the naive expectation, which would require twice as many messages as in the undirected version. This reduction arises because directionality suppresses backtracking paths, thereby considerably simplifying the resulting equations. >>

<< ️On random directed networks (AA) find that the critical behavior of DERP depends sensitively on degree correlations. These analytical predictions are corroborated by extensive Monte Carlo simulations, highlighting the profound impact of directionality and correlations on long-range connectivity in complex networks. >>

Wenbo Liu, Yiwen Zeng, Xueming Liu, et al. Directed extended-range percolation. Phys. Rev. E 114, L042301. Oct 26, 2026.

https://journals.aps.org/pre/abstract/10.1103/hs2n-6rsn

Also: network, transition, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, networks, transitions, percolation, directed extended-range percolation, percolation threshold, percolation phase transition, random directed networks, network phase transitions, long-range connectivity, critical phenomena.

# brain: minimal network of brain dynamics; hierarchy of approximations to quasicritical neural network dynamics.

<< ️(AA) present an interacting model of neural network dynamics that incorporates key biological features, including multiple forms of inhibitory interactions. (They) develop a hierarchy of analytical mean-field approximations to characterize nonequilibrium phase transitions between ordered, disordered, and chaotic regimes, complemented by a detailed stability analysis. >> 

<< ️(AA) show that inhibition generically enhances the stability of network dynamics. The model is consistent with the quasicriticality hypothesis, exhibiting regions of maximal dynamical susceptibility and mutual information, modulated by the strength of external stimuli. (They) further demonstrate that, at the mean-field level, the critical transition belongs to the mean-field directed percolation universality class, in agreement with prior experimental and theoretical studies. More broadly, (Their) framework may offer insights into neurological disorders, with the unstable regime exhibiting chaotic dynamics that may be associated with epileptic seizures. >>

Jeremy B. Goetz, Naruepon Weerawongphrom, Rashid V. Williams-García, et al. Minimal network of brain dynamics: Hierarchy of approximations to quasicritical neural network dynamics. Phys. Rev. E 114, 044321. Oct 22, 2026.

https://journals.aps.org/pre/abstract/10.1103/64b7-qb8p

arXiv: 2512.22093v2 [cond-mat.dis-nn]. Jul 15, 2026.

https://arxiv.org/abs/2512.22093

Also: brain, neuro, network, transition, in https://www.inkgmr.net/kwrds.html 

Keywords: brain, neurons, networks, transitions, neuronal network activity, percolation phase transition, cortical networks, criticality.

# gst: how mobile oscillators reshape dynamical robustness in complex networks.

<< ️While the robustness of collective dynamics in static networks is well established, how agent mobility and network topology jointly shape dynamical robustness remains largely unexplored. >>

<< ️In this letter, (AA) investigate the robustness of mobile Stuart-Landau oscillators performing random walks on networks and interacting diffusively when co-located. In the rapid-mobility regime, (They) reduce the dynamics to a mean-field framework linking network degree statistics to an effective global coupling strength. >> 

<< ️(AA) show analytically that increased degree heterogeneity in the network enhances the effective coupling strength, and strictly lowers the critical threshold of functional degradation for amplitude death. Numerical simulations confirm the predicted heterogeneity-dependent ordering of the transition thresholds, showing an earlier collective collapse in scale-free networks than in random or regular networks. >> 

<< ️(Their) results establish mobility and network heterogeneity as key determinants of robustness in mobile oscillator populations, with implications for resilient biological and engineered collectives. >>

Sayantan Nag Chowdhury, Srilena Kundu. How Mobile Oscillators Reshape Dynamical Robustness in Complex Networks. arXiv: 2610.16608v1 [nlin.AO]. Oct 12, 2026.

https://arxiv.org/abs/2610.16608

Also: network, random, walks, collapse, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, networks, randomness, walks, collapse, mobile oscillators, mobility, transition thresholds.

# life: timescale-induced synchronization of contrarians in adaptive dynamical networks.

<< ️From the firing of neurons to the formation of social consensus, the interplay between conformist and contrarian behaviors regulates collective dynamics. (AA) unravel how adaptation timescales induce distinct synchronization behaviors in mixed populations of conformists and contrarians. >>

<< While slow adaptation leaves contrarians in a persistent state of incoherence, fast adaptation traps them into a scaffold formed by conformists, stabilized by rapid feedback from the evolving network structure, leading to the emergence of synchronized clusters. Notably, contrarians are entrained only within an intermediate coupling window and require a conformist majority. >>

<< ️(AA) analytically reduce a low-dimensional description using a collective coordinate approach and demonstrate that a conformist majority and a threshold adaptation rate are necessary conditions for contrarians to synchronize in adaptive networks, which agrees well with numerical simulations. (Their) results highlight the critical role of adaptation timescales in reshaping collective behaviors in mixed populations, offering insights into consensus formation and polarization in adaptive social networks. >>

Akash Yadav, V.K. Chandrasekar, D.V. Senthilkumar. Timescale-induced synchronization of contrarians in adaptive dynamical networks. Phys. Rev. E 114, 044310. Oct 13, 2026.

https://journals.aps.org/pre/abstract/10.1103/v1vq-skr1

Also: behav, network, in https://www.inkgmr.net/kwrds.html 

Keywords: behav, networks, collective dynamics, synchronization, synchronization transition, coupled oscillators, threshold adaptation rate.

# gst: chaotic Griffiths phase in neuron map networks.

<< ️Griffiths phases provide a mechanism for sustaining critical dynamics over extended parameter regions without fine-tuning to an isolated transition point. This behavior has been mainly associated with structural heterogeneity in complex networks. >>

<< ️Here (AA) show that quenched heterogeneity in local parameters can generate a chaotic Griffiths phase in a network of map-based neurons. (They) study globally coupled Chialvo maps with quenched disorder in a local neuronal parameter, thereby isolating intrinsic parameter disorder from topological disorder. >>

<< ️As the coupling strength is varied, a chaotic Griffiths phase emerges between desynchronized and coherent chaotic dynamics, characterized by intermittent formation and breakup of neuronal clusters and large fluctuations in the fraction of clustered neurons. Within this regime, cluster sizes exhibit power-law distributions with coupling-dependent exponents, providing evidence of critical behavior over a finite parameter interval. >> 

<< ️The chaotic Griffiths phase is absent in the homogeneous limit and broadens progressively as the degree of parameter heterogeneity increases. (AA) further introduce topological disorder through small-world connectivity and find that its combined action with local parameter heterogeneity substantially extends the chaotic Griffiths phase. In both globally coupled and small-world networks, the number of positive Lyapunov exponents displays anomalous scaling with system size. >> 

<< ️These results demonstrate that intrinsic dynamical heterogeneity constitutes an independent mechanism for generating chaotic Griffiths phases and that its interplay with structural disorder enhances extended criticality in neuronal networks. >>

Juan Velez-Rojas, M.G. Cosenza. Chaotic Griffiths phase in neuron map networks. arXiv: 2610.03344v1 [nlin.CD]. Oct 2, 2026.

https://arxiv.org/abs/2610.03344

Also: network, brain, neuro, intermittency,  fluctuations, disorder, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, networks, brain, neurons, intermittency, disorder, fluctuations, quenched heterogeneity, quenched disorder, structural disorder, chaotic Griffiths phase, criticality.

# life: scaling laws for the spread of defection in evolutionary games on networks.

<< ️(AA) study the spread of defection in evolutionary games on networks using deterministic replicator dynamics posed on graphs. Each node hosts a mixed population of cooperators and defectors interacting through a Prisoner's Dilemma payoff matrix, with nearest-neighbor coupling determined by the network adjacency matrix. >>

<< ️(AA) develop two complementary analytical approximations that quantify how network architecture influences the timescale over which cooperation collapses. In a weak-coupling regime, internode interactions act as a small perturbation, yielding explicit approximate solutions on general graphs. For networks with strongly heterogeneous degree distributions, (They) derive a quasisteady approximation based on the natural separation of timescales between high- and low-degree nodes. >>

<< ️Using these approximations, (AA) introduce a defection spreading-time metric and show that the collapse of cooperation exhibits robust power-law scaling with system size across a broad range of network topologies. Analytical predictions are validated numerically on lattices, star graphs, incrementally connected graphs, small-world networks, and scale-free networks. >>

<< ️The (AA) analysis shows that, to leading order, defection dynamics are controlled by local nodal degree, while network coupling and degree heterogeneity provide secondary corrections that influence the global collapse dynamics. (They) further show that increasing degree heterogeneity increases the average defection time, providing a complementary spectral interpretation through the principal eigenvalue of the adjacency matrix and related measures of graph irregularity. >>

<< ️These results quantify how network architecture controls the transient dynamics of evolutionary collapse and establish new connections between evolutionary game dynamics and spreading processes on networks. >>

M. Giles, P.K. Newton. Scaling laws for the spread of defection in evolutionary games on networks. Phys. Rev. E 114, 044301. Oct 1, 2026.

https://journals.aps.org/pre/abstract/10.1103/25qf-sfsw

Also: game, network, collapse, transition, in https://www.inkgmr.net/kwrds.html 

Keywords: games, networks, collapse, transitions, Prisoner's Dilemma, defection, defection spreading-time, internode interactions, transient dynamics.

# gst: recovery random walks and extreme events on complex networks.

<< ️Extreme events are widely studied within simple random walk frameworks, where their probability is determined by the network structure and stationary walker distribution. >> 

<< ️Here, (AA) propose a recovery random walk (RRW) model in which extreme events temporally `freeze' the nodes where they occur for a fixed duration Δ (Δ=0 recovers the original model), trapping walkers and reducing the effective mobile population, thereby making the model more practical. >>

<< ️(AA) derive a first-principles description of this feedback and a delayed differential equation for the frozen-node fraction. This finite freezing produces an initial overshoot, followed by damped oscillatory relaxation to a steady state for the fraction of the frozen nodes. (They) find that freezing suppresses extreme event probability while preserving its degree dependence dynamics, and suppresses the EE frequency. >>

<< Further, the analytical prediction for the fraction of the frozen nodes agrees closely with simulations. This model closely reflects real-world scenarios, yielding more practical EE statistics. >>

Karan Singh, R.V.Narendran, V.K.Chandrasekar, et al. Recovery Random Walks and Extreme Events on Complex Networks. arXiv: 2609.39377v1 [nlin.AO]. Sep 30, 2026.

https://arxiv.org/abs/2609.39377

Also: walk, walking, random, network, transition, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, walk, walking, randomness, networks, transitions, random walk, extreme events (EE), stationary walker distribution, recovery random walk (RRW), frozen-node fractions, damped oscillatory relaxation. 

# gst: attraction-controlled torque organization and rotational states in frictional granular matter.

<< In this study, (AA) numerically investigate how interparticle attraction affects stress transmission, torque organization, and particle rotation in a two-dimensional frictional granular material. Although the attraction is a central force and exerts no direct torque on the particles, it strongly modifies the contact network and particle rotation. >>

<< Attraction produces a low-rate stress state and selects the high-stress state near the shear-thickening regime of the dry system, while leaving the high-rate thickened state almost unchanged. By decomposing the shear stress, (AA) find that the direct attractive contribution accounts for only a fraction of the total stress, while the repulsive contribution is strongly enhanced. >>

<< Attraction increases the coordination number and torque amplitude and can reverse the local torque-sign correlation. At low packing fractions and intermediate shear rates, a stress-collapse state emerges with strongly suppressed torque and rotation despite a finite contact network. >>

<< These (AA) results show that attraction controls macroscopic rheology through reorganization of the frictional contact network. >>

Kiwamu Yoshii. Attraction-controlled torque organization and rotational states in frictional granular matter. arXiv: 2609.35270v1 [cond-mat.dis-nn]. Sep 28, 2026.

https://arxiv.org/abs/2609.35270

Also: particle, network, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, particle, network, 2-D frictional granular particles, attraction, repulsion, shear stress, low- high- rate stress state, stress transmission, stress-collapse state, torque organization, torque amplitude, particle rotation, shear-thickening regime, frictional contact network.

# gst: avalanche assemblies in arbitrary connection topologies.

<< ️Complex systems are commonly modeled as networks of units that integrate external input and emit brief pulses of activity upon reaching a threshold. Such pulse-coupled interactions can give rise to cascades of activity propagating through the network in the form of avalanches. >>

<< While the statistics of these collective events are known to depend sensitively on network topology, rigorous analytical descriptions are usually restricted to homogeneous networks or mean-field approximations which are limited to particular aspects of generated avalanches like size and duration. >>

<< ️Here (AA) develop a mathematically rigorous theory of avalanche formation in recurrent networks of pulse-coupled units without assuming homogeneity. For arbitrary nonnegative coupling matrices, the framework provides a closed form for the avalanche assembly distribution, which quantifies the probability that a given subset of units is activated during an avalanche. >>

<< This (AA) approach represents a substantial conceptual advance over existing theories allowing analysis of the structure-function relation between network topology and detailed statistics of generated avalanches. This is particularly relevant for the irregular, highly structured, and asymmetric coupling topologies ubiquitous in natural systems, such as brain networks with pronounced columnar, modular, and hierarchical organization. >>

Maik Schünemann, Marc Kesseböhmer, Udo Ernst. Avalanche assemblies in arbitrary connection topologies. Phys. Rev. E 114, 034414. Sep 29, 2026.

https://journals.aps.org/pre/abstract/10.1103/nl7d-16n8

Also: network, neuro, brain, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, network, brain, complex systems, collective dynamics, pulse-coupled interactions, threshold cascades of activity, avalanches, avalanche assembly distribution, network topology, columnar- modular- hierarchical organization.

# life: emotions as intrinsic colored noise in biological systems.

<< ️The idea that emotions in biological systems are analogous to intrinsic colored noise (by AA) is advanced and justified. A model describing the dynamics of operation of biological networks under the influence of colored noise is suggested. >>

<< ️The agents of a biological network can be represented either by biological species, such as humans and animals, or by neurons of the brain, or by the nodes of a neural network. Operational actions, or decisions, in a biological noisy network are based not only on the evaluation of utility of alternatives, but also on the agents emotions. The model is probabilistic, with the choice of alternatives characterized by the related probabilities. >>

<< ️At the initial step, the agents make decisions individually and then start exchanging information with each other and imitating the actions of other agents, thus forming a biological network of interacting agents. Numerical simulations are accomplished for a heterogeneous society consisting of three groups of agents, one group possessing long-range memory, the other group, short-range memory, and the third group of super-rational agents acting strictly on the basis of utility, being deprived of emotions. >>

<< ️Dynamics of opinions in different groups can be smooth, oscillatory, or chaotic. Altogether, eight types of operation are found, depending on the dynamics of group decisions. Under strong imitation effect, there appears chaotic motion in the evolution of decision choice. (by AA) it is shown  how the Ellsberg paradox can be resolved and how the exchange of information influences the dynamics of this paradox. >>

V.I.Yukalov, E.P.Yukalova. Emotions as intrinsic colored noise in biological systems. arXiv: 2609.25970v1 [physics.soc-ph]. Sep 22, 2026.

https://arxiv.org/abs/2609.25970

Also: noise, brain, network, game, in https://www.inkgmr.net/kwrds.html 

Keywords: life, noise, brain, networks, biological noisy network, agents emotions, colored noise, affective operation, long-range memory, short-range memory, games.

# gst: disorder-promoted stability.

<< Previous studies of network dynamics have suggested that heterogeneity among nodes inhibits stability, at odds with the ubiquity of inherently heterogeneous natural and engineered systems. >>

<< ️Here, (AA) show that this conclusion arises from model reductions introduced for mathematical tractability and breaks down when nodal dynamics are higher-dimensional, yielding non-Hermitian Jacobians. In such systems, including neural, power-grid, and material networks, nodal heterogeneity can instead enhance stability, even when parameters are randomly disordered. >>

<< ️Non-Hermiticity also underlies the stabilizing effects of network heterogeneity, which can arise even in one-dimensional nodal dynamics through nonreciprocal interactions, as shown for ecological networks. >>

<< ️(AA) framework reveals disorder not as a liability but as a general resource for stabilizing complex systems. >>

Arthur N.Montanari, Pietro Zanin, Adilson E.Motter. Disorder-promoted stability. arXiv: 2609.25226v1 [cond-mat.dis-nn]. Sep 21, 2026.

https://arxiv.org/abs/2609.25226

Science. 393: 6817, 1241-1249. Sep 17, 2026. DOI: 10.1126/science.aeg3946.

https://www.science.org/doi/10.1126/science.aeg3946

Also: noise, disorder, network, game, poetry, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, noise, disorder, networks.

# gst: collective hysteresis and multistability in threshold networks.

<< ️In a mechanistic model of the dawn chorus, Kaye showed that heterogeneous activation thresholds and a shared feedback signal determined by the population's active fraction can produce abrupt collective activation and hysteresis. >>

<< ️(AA) extend this mechanism to a network of interacting agents. Each node has a continuous activation level, and a nonnegative row-stochastic matrix determines how node activities contribute to individual feedback. (They) prove that sufficiently weak feedback yields a unique globally attracting equilibrium. For all feedback strengths, the homogeneous dynamics exactly reproduce Kaye's scalar equation; consequently, network topology does not alter the folds or cusp of the homogeneous branch, and no heterogeneous mode becomes unstable before the homogeneous mode. >>

<< ️For equitable partitions, the network admits an exact quotient system in which nodes within a block receive the same aggregate input from every block. When blocks are uncoupled, the quotient reduces to independent copies of Kaye's scalar equation. (AA) show that every assignment of stable scalar equilibria to blocks persists under sufficiently weak interblock coupling, producing a combinatorial family of stable quotient equilibria that lift to stable full-network equilibria and remain under small perturbations that break exact equitability. >>

<< ️For two symmetrically coupled blocks, branches with unequal block activities terminate at a pair of symmetry-related cusp bifurcations. Near the onset of bistability, (AA) derive scaling laws for the interblock coupling at which these bifurcations occur, the activity difference between the blocks at bifurcation, and the corresponding shift of the external stimulus from the scalar cusp. Numerical continuation confirms the scaling laws for gamma, logistic, and normal threshold distributions. >>

Moses Boudourides. Collective Hysteresis and Multistability in Threshold Networks. arXiv: 2609.10580v1 [cs.SI]. Sep 5, 2026.

https://arxiv.org/abs/2609.10580

Also: network, instability, random, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, hysteresis, collective hysteresis, networks, threshold networks, stability, bistability, multistability, randomness, small perturbations, bifurcations, cusp bifurcations, Kaye's dawn chorus, stochasticity, heterogeneous activation thresholds, weak feedbacks, shared feedback signals, attracting equilibrium.

# gst: noise-shaped synchrony in neuronal oscillator networks.

<< ️Stochastic fluctuations often act as promoters of activity in excitable and oscillatory systems, giving rise to phenomena such as coherence resonance. Here, (AA) show that in the low-intensity regime, noise plays a dual role in weakly coupled neuronal units: It can both inhibit collective spiking and, with increasing intensity, restore coordinated activity. For a ring of diffusively coupled, oscillatory FitzHugh-Nagumo neurons, (They) demonstrate how the interplay of noise and coupling strength organizes a rich set of collective dynamical states. >> 

<< ️(AA) systematically classify emergent dynamical scenarios by an in-depth time-series analysis that integrates multiple complementary measures. (They) are able to automatically identify five distinct dynamical clusters in parameter space: pure noise, quiescent state, noisy synchronization, complete synchronization, and intermittent switching. (Their) workflow provides a general, automated framework for characterizing collective dynamics in coupled oscillator networks. >>

Max Contreras, Philipp Hövel. Noise-shaped synchrony in neuronal oscillator networks. Phys. Rev. E 114, L022302. Aug 26, 2026. 

https://journals.aps.org/pre/abstract/10.1103/sxl4-6mdm

Also: noise, brain, neuro, disorder & fluctuations, random, intermittency, network, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, noise, brain, disorder, fluctuations, randomness, networks, coupled oscillator networks, stochasticity, stochastic fluctuations, coherence resonance, collective spiking inhibition, restored coordinated activity, pure noise, quiescent state, noisy synchronization, complete synchronization, intermittent switching.

# gst: self-organizing chimera states in adaptive networks.

<< ️(AA) 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. >>

Felix Augustsson, Rok Cestnik, Matthias Wolfrum, et al. Self-organizing Chimera States in Adaptive Networks. arXiv: 2608.20517v1 [nlin.AO]. Aug 20, 2026.

https://arxiv.org/abs/2608.20517

Also: chimera, self-assembly, network, chaos, transition, brain, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, chimera, self-assembly, networks, transitions, brain, oscillators, adaptive chimera, synaptic plasticity, synchronized- desynchronized stationary states, breathing, chaotic collective dynamics, collective bifurcations, multistable attractors.

# gst: twisted state stability under local order parameter adaptation.

<< ️Twisted states commonly arise in nonlocally coupled ring networks. (AA) study the stability of twisted states in networks with higher-order interactions under the adaptive mechanism based on the local order parameter. >>

<< ️The adaptive feedback dynamically regulates the coupling strength and thus reshapes the collective dynamics of the system. Analytical and numerical results show that higher-order adaptivity reduces the linear stability of twisted states, whereas pairwise adaptivity enhances it. Numerical simulations further demonstrate that increasing the exponent of higher-order adaptation drives the system toward more ordered states and correspondingly enlarges the basin of twisted states. >>

<< Moreover, a larger coupling range reduces twisted-state stability, while in random hypergraphs, higher-order adaptivity enlarges the synchronization basin without significantly changing the linear stability. These findings reveal a nonequilibrium relationship between linear stability and basin stability and highlight the important role of adaptive regulation in higher-order network dynamics. >>

Xueqin Wang, Hong He, Wende Tang, et al. Twisted state stability under local order parameter adaptation. Phys. Rev. E 114, 014209. Jul 13, 2026. 

https://journals.aps.org/pre/abstract/10.1103/p2wd-nfbr

Also: network, stability, chimera, order, disorder, fluctuations, random, twist, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, networks, stability, chimera, order, disorder, fluctuations, randomness, synchronization, coupled oscillators, adaptive feedbacks, twisted states, basin stability. 

# gst: dynamical origin of extreme events in mutually coupled and networked Brusselator.

<< This (AA) study investigates the dynamical origins and statistical properties of extreme events (EEs) in a diffusively coupled theoretical Brusselator system, extending from pairwise interactions to globally coupled networks. >>

<< Statistically, the emergence of EEs is characterized by heavy-tailed probability density functions and exponential interevent interval distributions, alongside an analysis of the complementary cumulative distribution function and return periods. To elucidate the underlying generative mechanism, (They) employ invariant manifold computations and vector field divergence analysis. >>

<< (Their) results demonstrate that EEs are triggered by a critical region formed by the geometric alignment of stable and unstable manifolds near a saddle-type invariant set governed by the Shilnikov mechanism. Furthermore, extending the analysis to a globally coupled network reveals that network dimensionality does not affect the prevalence of EEs. >>

<< Finally, these numerical findings are corroborated by experimental observations from an analog electronic circuit, confirming the robustness of these dynamical phenomena in physical systems. >>

S.V. Manivelan, S.Sabarathinam, K.Thamilmaran, et al. Dynamical origin of extreme events in mutually coupled and networked Brusselator. Phys. Rev. E 114, 014217. Jul 15, 2026.

https://journals.aps.org/pre/abstract/10.1103/byp5-gxrz  

Also: network, chaos, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, networks, chaos, Brusselator, coupled oscillators, extreme event statistics, criticality, saddle-type invariant set.

# life: apropos of bifurcations from stable consensus to periodic multiconsensus, consensus-breaking global Hopf bifurcation in memory-based Multi-Agent Systems.

<< ️This (AA) dissertation provides the first systematic study of symmetric consensus-breaking bifurcation to periodic multiconsensus in multi-agent systems. It analyzes this for three classes of multi-agent systems based on three different types of memory, whose closed-loop dynamics equations form delay differential equations of retarded type, neutral type, and pseudoneutral type - a subclassification of retarded type equations introduced in this dissertation which bridges retarded and neutral type delay equations. >>

<< ️Equivariant twisted degree is used to analyze the symmetric global Hopf bifurcation problem in these systems, i.e. bifurcation from a stable consensus to periodic multiconsensus. This shows how the effects of memory allow self-organizing agents to move beyond mere stationary consensus. Theoretical results for the global Hopf bifurcation and symmetric classification of periodic multiconsensus solutions across all three systems are provided, and numerical results are conducted to both validate and enhance the theoretical predictions by providing stability information on the branches which is not obtainable by the degree alone. >>

<< These principles are demonstrated in three real-world applications: one involving the control of formations of UAVs (Unmanned Aerial Vehicles), allowing them to maintain their overall spatial relationships while dancing in complex selectable oscillations; and two more in networked asset markets featuring different traders with different memory-based strategies, showing how similar mechanisms can be responsible for economic cycles of bubbles and crashes. >>

<< Finally, (AA) also numerically investigate resonant double Hopf bifurcations in the neutral delay system, showing strong evidence of a breakdown to chaos via the Ruelle-Takens-Newhouse scenario and the existence of riddled basins. >>

Casey Maikalani Crane. Consensus-Breaking Global Hopf Bifurcation in Memory-Based Multi-Agent Systems. Dissertation. The University of Texas at Dallas. Aug 2026. Supervising Professor: Wieslaw Krawcewicz, Chair.  arXiv: 2607.02388v1 [math.DS]. Jul 2, 2026.

https://arxiv.org/abs/2607.02388

Also: network, swarm, bubble, crack, chaos, transition, dance, in https://www.inkgmr.net/kwrds.html 

Keywords: life, gst, networks, swarm, bubbles, crack, chaos, transitions, dance, multi-agent systems, symmetric consensus-breaking bifurcation, stable consensus, periodic multiconsensus, symmetric global Hopf bifurcation, self-organizing agents, economic cycles of bubbles and crashes, riddled basins. 

# gst: delay coordinates synchronization and induces abrupt transition in excitable networks.

<< ️Neuronal communication is inherently time-delayed, due to the finite speed of signal propagation. Although often considered challenging or disruptive, such time delays can also endow neural circuits with useful capabilities. Here, (AA) show that delays in excitatory connections between excitable neurons coordinate their synchronization patterns by creating self-sustained oscillations that may be out-of-phase or in-phase. The emergence of these oscillations leads to an abrupt, explosive, transition to in-phase synchronized regimes due to small changes in connection strength or time-delay. >>

<< ️(They) describe the mechanism underlying these phenomena as an interaction between the neuron's excitable dynamics and the delay in signal transmission, explaining many aspects of how the oscillations emerge. (They) show this phenomenon in different network connectivities, neuronal models, with and without excitation, with and without noise, highlighting the generality of the mechanism. >>

Bruno R.R. Boaretto, Kalel L. Rossi, Lyle E. Muller, et al. Delay coordinates synchronization and induces abrupt transition in excitable networks. arXiv: 2606.21703v1 [q-bio.NC]. Jun 19, 2026.

https://arxiv.org/abs/2606.21703

Also: network, brain, pause, noise, transition, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, networks, brain, pause, noise, transitions, neuronal communication, time delay, synchronization patterns, self-sustained oscillations.

# aibot: critical percolation as a synthetic data model for interpretability.

<< ️Neural networks learn features that reflect the hierarchical, multi-scale structure of natural data. Synthetic datasets used to evaluate interpretability methods typically lack this structure, limiting their value as realistic toy models. >>

<< ️To close this gap, (AA) introduce a family of synthetic datasets consisting of hierarchical functions defined on critical mean-field percolation clusters embedded in a high-dimensional data space. The percolation data consists of sparse, low-dimensional fractal clusters with a power-law size distribution. Latent variables modeling a taxonomic hierarchy generate each data point's target value. >>

<< The data model is analytically tractable with known critical exponents that fix its properties without requiring hyperparameter tuning. (They) leverage a mapping between percolation clusters, random trees, and additive coalescence to propose an almost linear-time algorithm to jointly sample a random tree and its hierarchical latent decomposition, enabling data generation at arbitrary scale. >>

<< ️Using probing experiments, (AA) find that the model's ground-truth latent variables can be linearly decoded from neural network activations. Together, sparsity, self-similarity, power-law statistics, and analytical tractability make critical percolation a principled testbed for interpretability research. >>

Aryeh Brill, Tom Ingebretsen Carlson. Critical Percolation as a Synthetic Data Model for Interpretability. arXiv: 2606.20347v1 [cs.LG]. Jun 18, 2026.

https://arxiv.org/abs/2606.20347

Also: ai, artificial intell, bot, network, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, ai, artificial intell, bot, networks, neural networks, percolation, criticality, critical percolation, critical mean-field percolation clusters, interpretability research, random trees, fractals, hierarchical latent decomposition, sparsity, self-similarity, power-law statistics.

# gst: collective drift and pinning in active rotator networks with Kuramoto coupling and mixed-sign feedback disorder.

<< ️Active rotator models provide a minimal phase description of excitable and oscillatory systems, and have long been used to study mutual entrainment, synchronization, and collective transitions. >>

<< ️Here, (AA) investigate fully connected active rotator networks with Kuramoto coupling, where a common intrinsic drive competes with local feedback amplitudes drawn from a zero-mean Gaussian distribution. This produces a competition between local pinning and collective phase alignment. >>

<< ️Using mean absolute late-time drift and the fractions of positive and negative drifting oscillators, (They) construct numerical regime maps in the feedback-disorder-coupling plane. At weak coupling, increasing the feedback disorder strength suppresses drift, while stronger coupling can restore positive late-time drift when feedback disorder is not too strong. (They) interpret these regimes using analytical limits for the uncoupled and coherent strong-coupling cases. >>

<< ️(They) also examine finite-size effects and zero-mean distributed intrinsic frequencies. Together, these results show that mixed-sign local feedback alone can reshape the balance between pinning and drifting in coupled active rotator networks, even when the intrinsic drive is homogeneous. >>

Arpan Dey. Collective drift and pinning in active rotator networks with Kuramoto coupling and mixed-sign feedback 

disorder. arXiv: 2606.10032v1 [nlin.AO]. Jun 8, 2026.

https://arxiv.org/abs/2606.10032

Also: networks, disorder, transition, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, networks, disorder, transitions, active rotator networks, excitable- oscillatory- systems, Kuramoto coupling, local feedbacks, feedback disorder.