A recurring data mining task in complex networks is to determine how individual nodes contribute to system behavior. Existing approaches rely on either static-graph centralities or control-theoretic quantities such as controllability Gramians, which assume linear, time-invariant dynamics. Estimated systems, however, are typically nonlinear and time-varying. We define "emergent contribution (EC)," a finite-horizon measure of a node's dynamical leverage: the metric-weighted energy of its impulse response accumulated along the system trajectory. Computed from the Jacobians of any differentiable model, EC is estimator-agnostic and reduces exactly to average controllability in the linear, time-invariant limit. Our contribution is a characterization of when the two measures agree and diverge. Using a controlled synthetic family with known ground-truth contribution, we construct a phase diagram spanning nonlinearity, regime structure, persistence, and perturbation amplitude. EC and average controllability agree under static or smoothly drifting dynamics and both track ground truth. Divergence emerges under persistent regime switching, is strongest under persistent sign reversal, and disappears when the sign reversal is removed. At extreme perturbation amplitudes, both measures degrade, identifying the limits of local linearization. We place five estimated real systems from several domains within this phase space. Their placement serves as a diagnostic of when EC provides information beyond static controllability and therefore justifies its additional computational cost. On one panel examined in depth, a twenty-seed retraining ensemble reveals a robust variance--leverage dissociation: nodes whose perturbations propagate widely despite low within-system variance, which is not recovered by static centralities nor variance-based summaries.

Large language models can solve tasks that were not present in the training set.

We systematically analyze optimization dynamics in deep neural networks (DNNs) trained with stochastic gradient descent (SGD) and study the effect of learning rate $\eta$, depth $d$, and width $w$ of the neural network. By analyzing the maximum eigenvalue $\lambda^H_t$ of the Hessian of the loss, which is a measure of sharpness of the loss landscape, we find that the dynamics can show four distinct regimes: (i) an early time transient regime, (ii) an intermediate saturation regime, (iii) a progressive sharpening regime, and (iv) a late time edge of stability regime.

We aim to understand grokking, a phenomenon where models generalize long after overfitting their training set. We present both a microscopic analysis anchored by an effective theory and a macroscopic analysis of phase diagrams describing learning performance across hyperparameters. We find that generalization originates from structured representations, whose training dynamics and dependence on training set size can be predicted by our effective theory (in a toy setting). We observe empirically the presence of four learning phases: comprehension, grokking, memorization, and confusion. We find representation learning to occur only in a Goldilocks zone (including comprehension and grokking) between memorization and confusion. Compared to the comprehension phase, the grokking phase stays closer to the memorization phase, leading to delayed generalization. The Goldilocks phase is reminiscent of intelligence from starvation in Darwinian evolution, where resource limitations drive discovery of more efficient solutions. This study not only provides intuitive explanations of the origin of grokking, but also highlights the usefulness of physics-inspired tools, e.g., effective theories and phase diagrams, for understanding deep learning.

Despite the non-convex optimization landscape, over-parametrized shallow networks are able to achieve global convergence under gradient descent. The picture can be radically different for narrow networks, which tend to get stuck in badly-generalizing local minima. Here we investigate the cross-over between these two regimes in the high-dimensional setting, and in particular investigate the connection between the so-called mean-field/hydrodynamic regime and the seminal approach of Saad \& Solla. Focusing on the case of Gaussian data, we study the interplay between the learning rate, the time scale, and the number of hidden units in the high-dimensional dynamics of stochastic gradient descent (SGD). Our work builds on a deterministic description of SGD in high-dimensions from statistical physics, which we extend and for which we provide rigorous convergence rates.