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版本更新 · v2arXiv:2504.15779

Shannon observables: A scalable approach to information decomposition

Aaron J. Gutknecht, Fernando E. Rosas, David A. Ehrlich, Abdullah Makkeh, Pedro A. M. Mediano, Michael Wibral

原文摘要

Distributed systems, such as biological and artificial neural networks, process information via complex interactions engaging multiple subsystems, resulting in high-order structure of the information within the system with distinct properties across scales. Investigating how these systems process information remains challenging due to difficulties in defining appropriate multivariate metrics and ensuring their scalability to large systems. To address these challenges, we introduce a novel framework based on what we call "Shannon observables": quantities that capture essential properties of high-order information processing, yet can be computed directly from Shannon entropies and scale efficiently to large systems. Our theoretical results demonstrate how Shannon observables can be used to resolve long-standing ambiguities regarding the interpretation of widely used multivariate information-theoretic measures. On the practical side, we demonstrate the framework's versatility across several domains. An analytical study of error-correcting codes shows how Shannon observables trace out distinct encoding regimes as channel noise varies; analyses of deep classifiers and convolutional autoencoders reveal characteristic information-processing signatures across layers and over training; and applied to magnetoencephalography recordings from the human brain, the framework yields a cortical atlas exposing pronounced regional structure in directed information transfer. Overall, our framework resolves fundamental limitations in analysing high-order information phenomena and offers broad opportunities for theoretical developments and empirical analyses.

cs.IT · cs.AI · cs.LG · nlin.AO · physics.data-an

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.13146

Everything in its right place: attention is a radio head

Lluís Arola-Fernández, Aissam Khadraoui, Alex Arenas

原文摘要

Attention, the core of language models, is a digital calculation over tokens: an exponential selects them, a division normalizes and a positional code sets the order. We show that attention can instead emerge from the synchronization of a receiver with its context tokens. Each past token is an oscillator whose state carries its content and whose frequency drifts with its age. The present token selects them by content, aligning an antenna with its query, and by position, setting the frequency of a receiver with its dial. Solving Kuramoto's problem for this receiver yields an attention layer in closed form with two phases: above a critical coupling set by the context, the receiver locks and its weights normalize; below it, it stays silent and its weights sum to less than one. Trained networks use both phases to select, abstain and count, fail where the theory predicts and keep reading far beyond the training length. Each part of attention finds its place in one device, a radio head: it tunes in to what to hear and when, and stays silent when nothing relevant is on air.

physics.soc-ph · cond-mat.dis-nn · cs.AI · cs.LG · nlin.AO

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.11734

A Bifurcation-Theoretic Framework for the Controllability of Emergent Intelligent Systems

Hao Li, Shaowen Lu

原文摘要

Real-world intelligent systems must reconcile two demands: the adaptability conferred by emergence and the steerability required for accountable deployment. We show that these demands are governed by two distinct bifurcations on the same object and that, consequently, increased capability does not imply increased controllability. Emergence is a bifurcation of the capability variable $S$, with criterion $Γ:=\mathcal F'(0)>1$ equivalent to the existence of a unique positive stable fixed point of a self-consistency equation, following solely from the concavity of $\mathcal F$. Controllability is a bifurcation of the permission variable $c$, whose dynamics include an erosion term and exhibit a saddle-node fold with hysteresis as the gate criterion $ρ^\ast$ crosses unity. Three quantitative results follow: the critical exponent distinguishes families rather than classifying the transition; the gating-cost bound $K(M)\propto\log(1+M)$ grows with scope, not capability; and the fold threshold is $γ_c\approx4μκ$. Calibrated in both directions, the criteria reproduce the two-dimensional Ising transition, admit two true-feedback order parameters, and reject nine candidate quantities, supporting the conclusion that the order parameter is geometric rather than capability-related. Since permission cannot be endogenized and no deployed system has been measured to satisfy $Γ>1$ and $ρ^\ast>1$ simultaneously, increased capability does not imply controllability.

cs.AI · eess.SY · nlin.AO

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.10691

A Geometric Theory for Nonequilibrium Transformer Dynamics

Aythami Bethencourt de León, Darryl D. Holm

原文摘要

We develop a variational and geometric theory based on statistical physics for the joint evolution of Transformer attention and token representations at fixed model parameters. To this end, we introduce a free energy for the attention and model the evolution of the attention weights by its Fisher--Rao gradient flow. At fixed token representations, we prove that the softmax matrix is the unique free-energy minimiser and that every positive attention trajectory converges to it. For prescribed time-dependent representations, we show that the attention dynamics becomes linear after a change to logit variables. This result yields tracking bounds relative to the instantaneous softmax and sufficient conditions for attention to converge to a fixed matrix. Hamiltonian dynamics for the representations and their momenta are then coupled to the attention relaxation driven by the same energy. This coupling results in an energy-dissipation law. We also develop two alternative Hamiltonians incorporating the attention value map. We prove global well-posedness of the attention flow and all three coupled systems.

cs.LG · math-ph · math.PR · nlin.AO · stat.ML

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arXiv ↗PDF ↗
版本更新 · v4arXiv:2610.25113

A learner-free swarm approximates the explore-exploit equilibrium allocation: A self-organized attractor in place of reinforcement-learning machinery

Galen J. Wilkerson

原文摘要

Reinforcement learning (RL) solves the explore-exploit problem by training a learner on reward; its multi-agent, resource-sharing form has an equilibrium allocation as its optimum -- the ideal-free distribution (IFD), the Nash equilibrium of a congestion game. We show a minimal swarm approximates it with no learner at all. We present a decentralized, learner-free method for this problem: agents explore by heavy-tailed (Levy) flights, recruit neighbours to a discovery through a degree-normalized threshold cascade, and disperse as a patch crowds. The method drives the collective to the shape of the equilibrium allocation (pooled spatial matching slope 1.14, R^2=0.61) with no training, no stored parameters, and no reward in its dynamics; it reaches this as a self-organized attractor and holds it, re-finding it when the resources move -- though it reaches the shape, not the exact equilibrium, and stays measurably exploitable (Nash-gap 1.46). Coupling recruitment to movement causes the allocation: removing it collapses the matching (slope 0.006). Against Proximal Policy Optimization (PPO) learners (feedforward, centralized-critic, graph message-passing, and recurrent) trained in the same sensing-limited environment, the method at zero training is the only arm left unbiased w.r.t. the IFD gradient (slope ~1, CI containing 1, vs learners 0.69-0.97, all under-concentrating) and ties the best learner's harvest where resource is clustered, while the learner collects more where resource is scarce (0.33 vs 0.22) and holds a tighter per-capita intake. Learning buys precision and transfer across reward structures, not a better allocation shape. A companion paper develops the mechanism.

cs.LG · cs.AI · cs.SI · math-ph · nlin.AO · physics.bio-ph

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2605.17510

Scale-Dependent Collective Adaptation in Self-Amending LLM Societies: A Cross-Family Study of Emergent Governance

Kazuya Horibe, Masaomi Hatakeyama, Gen Masumoto, Takashi Hashimoto, Peter Romero

原文摘要

We study group decision-making in artificial societies where the rules of play are themselves subject to collective amendment. Using the self-amending game Nomic, we compare multiple scales across two LLM families and find that collective adaptation does not improve monotonically with model size. Instead, both families exhibit a narrow mid-scale regime that supports sustained rule adoption, diverse amendments, and balanced consensus. Smaller models tend to remain rule-inert, whereas larger models often converge on restrictive voting patterns, and heterogeneous mixed-size groups collapse into veto-driven gridlock. These cross-scale contrasts persist under temperature perturbations and under a shift from unanimity to majority voting, although latent-state structure varies by family and scale. Hidden-state divergence alone does not explain collective performance: high representational divergence can coincide with poor behavioural outcomes. Linear probes reveal regime-selective coupling between latent vote-predictive signals and collective behaviour, but decodability is necessary rather than sufficient for adaptive play. Overall, the recurring regularity is non-monotonicity, not the particular scale at which the optimum appears. Self-amending games therefore provide a controlled testbed for studying collective adaptation in artificial societies beyond raw model scale.

nlin.AO · cs.MA

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2609.38037

Emergence in Network Systems is Bounded by Boundary-Crossing Paths

Johnny Jingze Li, Gabriel A. Silva

原文摘要

Emergence, features of a whole that no part shows alone, is central to complex systems but rarely measured so as to locate its source. Viewing a system structurally, as a network of interacting parts, we evaluate emergence as what observing the coupled whole reveals beyond, or erases from, the combined observations of the parts. For observations that build what they see route by route, we show that every emergent feature is produced by its own route across the interface between parts, a route that traces it to the components and interactions responsible. We show the number of all such boundary-crossing paths bounds the total emergence discrepancy, and equals the emergence of measures that count routes. The bound is computed locally around the interface, validated through network examples, and in a nervous system the routes show that interneurons relay most of the emergence. Our work associates emergence with path structure in the network system that can be anticipated, attributed, and engineered.

physics.soc-ph · cs.SI · nlin.AO

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2506.11796

Solving Stochastic Inverse Problems in Embedding Space

Elias Najarro, Nicolas Bessone, Sebastian Risi

原文摘要

Solving an inverse problem consists in finding the unknown parameters of a system from observations. This task is particularly challenging when observations have a strong stochastic component, producing diverse yet equivalent patterns. Self-organizing systems are a notable case: simple local rules can generate complex stochastic patterns. Many natural systems rely on such processes, making self-organization central to understanding natural phenomena. Inverse methods based on direct comparison of observed values struggle in this setting, as such comparisons cannot capture feature similarities between variable outcomes; as a result, the loss fails to meaningfully inform the optimizer. In this work, we show that the difficulty posed by stochasticity in observable patterns can be overcome by reformulating the inverse problem in an embedding space. By mapping stochastic patterns onto an invariant embedding space, where equivalent patterns have similar representations, we can effectively recover unknown causal parameters without the need for handcrafted objective functions or heuristics. We show that embedding models trained on large image datasets produce robust representations that capture these invariances. We evaluate the method on three self-organizing systems: a physical, a biological, and a social one; namely, a reaction-diffusion system, a model of embryonic development, and an agent-based model of social segregation. We show that the method reliably recovers parameters despite stochasticity in the pattern outcomes. We further apply the method to real biological patterns, highlighting its potential as a tool for both theorists and experimentalists to investigate the dynamics underlying complex stochastic pattern formation.

nlin.AO · cond-mat.dis-nn · cs.CV · cs.LG

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.02047

Beliefs in feedback: Factored representations of interacting agents in transformers

David A. Ehrlich, Manuel Baltieri, Fernando E. Rosas, Alexander B. Boyd

原文摘要

Complex systems, from ecosystems to brains, are usually modeled as modular components coupled through feedback rather than as indivisible wholes. Can a transformer trained on the joint behavior of a group of interacting agents be understood in the same way? In this work, we develop a computational-mechanics framework to study agents interacting in feedback loops. We formally prove that any joint stochastic process can be presented as resulting from the behavior of agents interacting in feedback, and the resulting process is generated by a hidden Markov model whose Bayesian beliefs factorize into separate components per agent. Furthermore, our experiments show that transformers trained on such feedback sequences make use of this separability, disentangling the beliefs about the underlying factors into separate subspaces of their residual stream. This representation is found to scale linearly (instead of exponentially) with the number of agents. Moreover, interventions show that these representations are causally efficacious, allowing for targeted steering of individual agents' beliefs by selectively inserting counterfactuals. However, coarse-graining the behavior is found to disrupt the factorization, and can force a higher-dimensional representation. Together, the results establish both the dimensional advantage of a factorized representation and the ability of transformers to exploit it, laying the groundwork for investigating traces of compositional predictive architecture in pretrained models.

cs.LG · cs.AI · cs.MA · nlin.AO

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2512.10047

Detailed balance in large language model-driven agents

Zhuo-Yang Song, Qing-Hong Cao, Ming-xing Luo, Hua Xing Zhu

原文摘要

Large language model (LLM)-driven agents are emerging as a powerful new paradigm for solving complex problems. Despite the empirical success of these practices, a theoretical framework to understand and unify their macroscopic dynamics remains lacking. This paper proposes a method based on the least action principle to estimate the underlying generative directionality of LLMs embedded within agents. By experimentally measuring the transition probabilities between LLM-generated states, we report evidence consistent with a tendency toward detailed balance in LLM-generated transitions, indicating that LLM generation may not be achieved by generally learning rule sets and strategies, but rather by implicitly learning a class of underlying potential functions that may transcend different LLM architectures and prompt templates. To our knowledge, this provides an effective macroscopic description of LLM generative dynamics that does not depend on specific model details. This work is an attempt to establish a macroscopic dynamics theory of complex AI systems, aiming to elevate the study of AI agents from a collection of engineering practices to a science built on effective measurements that are predictable and quantifiable.

cs.LG · cs.AI · nlin.AO

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.34428

Matter that learns what matters

Natalia G. Berloff

原文摘要

Can a material learn what deserves its attention? We show that it can in a model optical network, in which action outcomes train a persistent record in each link. The record biases whether a source is consulted and sets how strongly it changes a stored answer, so useful sources gain attention and influence without being identified in advance. The network's experience thus shapes the information available for its future learning. Resonant readouts make this learning fragile: reinforcement can carry a record past resonance, where it persists but its influence weakens. We derive bounds within which relevant signals guide the answer, irrelevant signals are rejected and stored answers remain open to revision. These bounds guide a learning rule that limits reinforcement as records form; in simulations, it acquires useful records and keeps them functional. When relevance changes, an obsolete record can block its own replacement. A local state that strengthens negative writing after unsuccessful actions releases obsolete records and completes the learning sequence more often. How a memory is read should decide how it is written.

cs.ET · nlin.AO

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arXiv ↗PDF ↗