We study a driven selection mechanism on a fixed heavy-tailed network. At each step, a power-normalization rule recomputes the direction of fresh injection from the current mass profile. A primitive mixing matrix then transports the existing stock and the new mass. The exponent $\theta$ sets the feedback. Positive values give more weight to larger coordinates, whereas negative values favor smaller ones. After removing the deterministic growth of total mass, we give an explicit mixing--forcing condition under which the injection profile converges to a nonlinear Perron--Frobenius-type fixed point on the simplex. Hilbert's projective metric makes the stability mechanism transparent. The discounted network response draws positive profiles closer together, while the escort map scales their projective distance by $|\theta|$. On heavy-tailed networks, the fixed point separates three effects that are often conflated: response or degree tilt, anomalous inverse-participation-ratio scaling, and genuine few-node localization. Positive feedback favors high-response nodes and, when response follows degree, tilts the selected profile toward the hubs. Negative feedback favors low-response nodes. It usually produces a broad peripheral cloud unless the lower tail of the response field is itself thin. Computations on finite networks illustrate convergence, forcing-rate dependence, and the sign law. Monte Carlo samples of truncated heavy-tailed response profiles display the predicted participation-ratio crossover. A uniform comparison bound gives conditions under which this scaling transfers to the full fixed point. The mechanism differs from conserved-mass condensation and graph growth because feedback selects a non-equilibrium profile on a fixed, heterogeneous network.
Delayed generalization, or grokking, remains poorly understood despite extensive empirical study. We identify an exactly solvable late-time relaxation mechanism for grokking in linear models trained with full-batch heavy-ball optimization and weight decay, together with a locally quadratic extension to nonlinear neural networks. Our analysis reveals a distinguished population-active component of the empirical null space, which we call the grokking subspace. Along this subspace, the training predictions remain unchanged, leaving weight decay as the sole restoring force and giving rise to a slow dissipative relaxation governed by an exact discrete-time and continuous-time law. We show that only this subspace contributes to the slow asymptotic decay of the population risk and derive explicit iteration-scale predictions for the grokking time, recovering the familiar $(1-\beta)/(\eta\lambda)$ scaling in the weak-regularization regime. The theory further predicts distinct effects of optimizer choice, distinguishing coupled $L_2$ regularization from decoupled weight decay, and yields causal predictions for interventions that modify the grokking component. We verify all theoretical identities without fitted parameters in a synthetic model where every subspace and relaxation rate is computable in closed form. We further observe genuine delayed generalization in modular addition, where the measured delay follows the predicted scaling and the late-time relaxation agrees closely with the theoretical clock.
This work shows that a dual-threshold bootstrap percolation model on random hypergraphs separates a connected active backbone from large-scale endogenous activation, providing a basis for predicting cascade risk and designing targeted node- and group-level interventions in complex systems.
We study approximate counting and sampling algorithms for the hard-core model on $\Delta$-regular bipartite graphs under a spectral expansion condition. Let $M_G$ be the biadjacency matrix of $G$. For every fixed $\xi\in(0,1)$, we give an FPRAS for the hard-core partition function and an efficient approximate sampler whenever \[ \lambda\leq \frac{1-\xi}{\sigma_2(M_G)}. \] The main idea is to introduce a family of quadratic tilts in the left-right occupation imbalance and show that each tilted measure can be sampled efficiently using Glauber dynamics. A discrete Gaussian identity expresses the original hard-core model as an exact positive mixture of these tilted measures; truncation and simulated annealing then yield efficient counting and sampling algorithms. For the complementary high-fugacity regime, we refine the polymer-model approach and show that the required phase-dominance and cluster expansion conditions follow from the singular-spectrum bound alone. Combining the two regimes, we obtain efficient approximate counting and sampling at every fugacity $\lambda>0$ whenever \[ \sigma_2(M_G)\leq c\left(\frac{\Delta^2}{\log(\mathrm e\Delta)}\right)^{1/3} \] for an absolute constant $c>0$. In particular, this recovers all-fugacity algorithms for random $\Delta$-regular bipartite graphs for all sufficiently large $\Delta$, while providing an efficiently verifiable certificate of their success on a given instance.
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.
Amit Tiwari, C. Hens, Prosenjit Kundu· 0 citations
We formulate feature learning as a geometric critical phenomenon of the lifted tensor-product learning metric. The central object is not a scalar overlap, but the target-active geometry of \[ \mathcal N_{0,L}=\frac1N\sum_{r=1}^{L}\Sigma_{r\to L}\otimes T_{0\to r-1}, \] which entangles forward pullback survival with backward push-forward visibility. The neutral phase is target-isotropic: after restriction to endpoint target-active states and trace normalization, the lifted metric is proportional to the identity. Learning corresponds to an instability of this target-isotropic fixed point and to the emergence of traceless target-aligned eigentensors. We derive discrete Dyson expansions for local anisotropic insertions and their continuous Callan--Symanzik flow. Crucially, before constructing the full temporal mean-field theory, we identify the local spatial source of the $\beta$-functions directly from microscopic kinematics: asynchronous gradient updates generate synchronous metric strains, whose target-active symmetric traceless components act as curvature-like defects. The Wilsonian depth RG flow is then governed by the transport, balance, and coarse-grained irrelevance of these defects. Heavy-tailed spectra arise, under a scale-free counting hypothesis, as the spectrum of the target-active lifted geometry, with exponent addition in the matched pullback--push-forward sector. Finally, we relate this depth RG picture to temporal stochastic training dynamics and to the kinematic imprint of the learned channel on empirical weight Gram matrices.
We formulate a non-Abelian theory of network compatibility in which dynamical transformations reside on the links. Gauge covariance follows from the freedom to choose local representation frames, while plaquette holonomies quantify the incompatibility of closed-loop transformations. For an SU(2) model on the complete simplicial 2-complex with quenched random plaquette couplings, parallel-tempering simulations reveal a continuous disorder-driven phase transition characterized by the network compatibility $M_P$. As the disorder strength approaches the critical value, the compatibility drops rapidly to a value that decreases with system size, while the frustration energy of the network sharply rises. Moreover, analysis of connected replica-overlap width provides no evidence for thermodynamic replica-symmetry breaking. Instead, the high-disorder regime sustains a nearly constant integrated adjacent correlation as the system size increases. Therefore, rather than a frozen gauge glass or a featureless disordered ``gas''phase, dense topological frustration produces a non-glassy correlated gauge liquid in which individual pair correlations are geometrically diluted while a finite integrated correlation survives.