Depth-Scalability and Path-Plasticity

• Depth-scalability and path-plasticity are intertwined phenomena that describe the interplay between hierarchical depth and adaptability in algebraic decompositions and neural models.
• They capture how repeated operations stabilize homological invariants and enable flexible adaptations in system structures through iterative adjustments.
• Applications span algebraic combinatorics, neural network design, and statistical physics, offering actionable insights for optimizing model performance and resilience.

Depth-Scalability and Path-Plasticity are intertwined mathematical and algorithmic phenomena emerging across algebraic combinatorics, neural architectures, and statistical physics models. Both encapsulate the interaction between hierarchical depth and the adaptability (“plasticity”) of functional pathways or algebraic decompositions. In algebraic contexts, they refer to the stabilization and flexibility of homological invariants (e.g., depth, Stanley depth) under iterative operations on ideals; in computational models, they describe the capacity of layered or networked systems to maintain expressivity as they increase in size, while selectively adapting or reusing substructures.

1. Algebraic Foundations: Path Ideals and Their Invariants

Let $S = K[x_1,\ldots,x_n]$ be a polynomial ring over a field $K$. For a graph $G$, the $m$-path ideal encodes all length-$m$ paths: for the path graph, this is

$$I_{n,m} = (x_1x_2\cdots x_m,\ x_2x_3\cdots x_{m+1}, \ldots, x_{n-m+1}\cdots x_n),$$

while for the cycle graph,

$$J_{n,m} = I_{n,m} + \text{wraparound monomials of length } m,$$

covering the entire cycle. Homological invariants of their powers—specifically, the depth and Stanley depth—capture the interplay between combinatorial covering and algebraic regularity.

Depth-scalability in this setting refers to the behavior of $\operatorname{depth}(S/I_{n,m}^t)$ or $\operatorname{depth}(S/J_{n,m}^t)$ as the exponent $t$ increases. Path-plasticity manifests in the flexibility with which these invariants descend and stabilize, including the exact or near-coincidence of depth and Stanley depth for certain powers, and the fine combinatorial control offered by the ideal's structure (Balanescu et al., 2023, Balanescu et al., 2023).

2. Depth-Scalability: Stabilization Phenomena

2.1 For Path and Cycle Ideals

In powers of path and cycle ideals, depth-scalability refers to the systematic descent and eventual stabilization of depth. For the $K$0-path ideal of a path graph,

$K$1

exhibiting a “staircase” decay followed by stabilization at $K$2 for all higher powers (Balanescu et al., 2023). For the cycle ideal, the upper bound sharpens further: $K$3 where $K$4 (Balanescu et al., 2023). If $K$5, depth collapses to zero; for $K$6, the bound reflects the minimal intersection complexity among subrings associated with the cycle's symmetry.

2.2 Mechanistic Interpretation

This depth stabilization occurs because, combinatorially, repeated powering of the ideal enforces redundant covering, which reduces independent generators in minimal free resolutions. Homologically, colon ideals with respect to carefully constructed monomials (e.g., products over all variables) precisely capture associated prime intersections dictating the stable depth.

3. Path-Plasticity: Flexibility and Decompositional Control

Path-plasticity denotes the system's—or algebraic object's—capacity to flexibly realize, adapt, or preserve certain structural responses (e.g., decompositions, homological invariants, or computational pathways) under changing conditions or operations.

3.1 In Commutative Algebra

For path and cycle ideals, this is encapsulated by:

• Exact coincidence of the depth and Stanley depth in specific families, such as $K$7 for all $K$8 (Balanescu et al., 2023).
• The capacity for flexible adjustment: successive powers of the ideal allow sdepth to “bend” downward, but owing to combinatorial symmetries, can sometimes preserve equality or strict bounds with depth (Balanescu et al., 2023).
• The crucial monotonicity and bounding properties:

$K$9

and $G$0, guaranteeing structural plasticity without violating established inequalities.

3.2 Stanley Decomposition Perspective

Path-plasticity is further mirrored in the construction of Stanley decompositions for module quotients, particularly when inductive “pruning” procedures or cyclic symmetry are present. These decompositions can be tuned to match the drop in depth, explicitly reflecting the combinatorics of the generating paths of the ideal (Balanescu et al., 2023).

4. Manifestations in Neural and Statistical Models

Depth-scalability and path-plasticity generalize beyond commutative algebra, appearing as organizing themes in diverse domains:

4.1 Hierarchical Neural Models

In unsupervised spatiotemporal Hebbian hierarchies, increasing the depth of the network correlates with monotonic increases in both accuracy and mutual information, up to certain architectural limits. Path-plasticity is realized through staged synaptic updates: shallow layers rapidly adapt to new input transformations, while deep layers “lock in” invariants, reflecting diminished synaptic plasticity as representations become more abstract (Kouh, 2014).

4.2 Random Graph SNNs

In CogniSNN, depth-scalability is achieved by random graph architectures combined with spiking residual nodes employing OR-gate skip connections, which maintain identity mappings and avert gradient-collapse in deep paths. Path-plasticity is enabled via critical-path-based learning-without-forgetting (LwF), where only the parameters on high- or low-centrality paths are fine-tuned according to task similarity, thus reusing or adapting relevant sub-networks with minimal interference (Huang et al., 9 May 2025).

4.3 Statistical Physics and Phenotypic Switching

In evolutionary spin-glass models, spectral analysis reveals that evolved genotypes admitting robust and plastic switching between phenotypes rely on a 1D switching path in the space spanned by the first two eigenmodes of the interaction matrix. The free-energy landscape forms a deep, narrow valley (robustness) with nearly flat curvature along the path (plasticity), and the valley's depth scales linearly with system size, i.e., displays depth-scalability (Sakata et al., 2023).

5. Mathematical Structures Underlying Scalability and Plasticity

5.1 Stabilization, Plateau, and “Drop” Patterns

Both in algebraic and computational instances, depth or accuracy frequently follows a staircase pattern: plateaus interrupted by discrete drops, then stabilization. For example, in algebraic path ideals, this directly results from the combinatorics of overlapping generators and the thresholds at which new minimal primes dominate (Balanescu et al., 2023, Balanescu et al., 2023).

5.2 Gradient-Controlled and Centrality-Based Adaptation

In dynamic network models, neuron addition/pruning by gradient potential (as in Neuroplastic Expansion) (Liu et al., 2024) or critical-path centrality (CogniSNN) (Huang et al., 9 May 2025) provides algorithmic instantiations of path-plasticity: sub-networks most relevant for ongoing or new tasks are targeted for modification, thereby balancing stability and adaptability.

6. Applications, Implications, and Limitations

Depth-scalability and path-plasticity enable systems:

• To maintain or increase capacity and expressivity (algebraic or neural) as depth increases, while avoiding collapse of relevant invariants or loss of functional pathways.
• To flexibly adapt to new regimes, tasks, or boundary conditions, often through selective fine-tuning or reconfiguration of a small subset of paths or components.
• To reconcile stability (robust maintenance of learned or evolved features) with plasticity (capacity for reversible or task-driven adaptation).

Limitations noted in the literature include computational overhead (due to dynamic masking or centrality computations), stabilization bottlenecks, and parameter sensitivity. Future work is directed toward fully automated depth-scalability (e.g., automated layer insertion) and explicit path-regularization for broader generalization (Liu et al., 2024).

7. Representative Results and Comparative Summary

Domain	Depth-Scalability Mechanism	Path-Plasticity Mechanism	Key Reference
Path/Cycle Ideals	Stabilization of depth as $G$1 increases	Flexible Stanley decompositions, monotonicity	(Balanescu et al., 2023, Balanescu et al., 2023)
Spiking Neural Networks	OR-gate residual node in RGA	Critical-path-based LwF	(Huang et al., 9 May 2025)
Spin-Glass Evo Models	Deep/geometric valley in free-energy	1D switching path, Hessian low mode	(Sakata et al., 2023)
Deep Hebbian Hierarchies	Monotonic info/accuracy vs. depth	Layer-wise decay of synaptic plasticity	(Kouh, 2014)
Dynamic RL Topologies	Gradient-stimulated layer/neuron growth	Experience-consolidation & path diversity	(Liu et al., 2024)

Across these domains, both phenomena structurally underwrite the balance of efficiency and adaptability, whether through algebraic, statistical, or algorithmic mechanisms.