Stratified Interaction Analysis

• Stratified Interaction Analysis is a framework that partitions systems into distinct layers or classes to study how interactions are modulated by these structures.
• It integrates mathematical models, network metrics, and simulation techniques—such as stratification assortativity and permutation-based tests—to quantify intra- and inter-class dynamics.
• Its applications span diverse fields, from social and economic networks to layered physical systems, offering deeper insights than aggregated models.

Stratified interaction analysis encompasses mathematical and computational methodologies for investigating how interactions are modulated, constrained, or structured by explicit stratification—whether by socioeconomic status, network position, spatial layer, attribute class, or physical layering—across a wide range of disciplines. Stratified analysis systematically incorporates tiered, layered, or class-based partition structure into the modeling of interaction, transmission, or coupling processes, yielding insights not accessible by aggregated or homogeneous models. Theoretical frameworks include agent-based network models, graph-theoretic stratification metrics, spatiotemporal dynamical systems, stratified sampling in statistics and explainable AI, and physical models of stratified wave and material layer interactions.

1. Formalism and Definitions

Central to stratified interaction analysis is the explicit partitioning of system entities, nodes, or variables into distinct strata (classes, layers, or tiers), with interactions conditioned upon, or modulated by, this stratification. The mathematical structure can be:

• Stratified Networks: Nodes are assigned to classes or "strata," and the adjacency/interaction structure is analyzed with respect to these assignments. The Stratification Assortativity (StA) metric quantifies the segregation of connections across classes, controlling for degree distribution and network density. For a network $G=(V,E)$ with status scores $s(u)$, StA compares observed intra-class similarity with a degree-preserving null model, returning a normalized score in $[-1,1]$ (Jalali et al., 2022).
• Stratified Agent Interaction: In economic models, stratification is implemented via a threshold on agent attributes (e.g., wealth). Only agents whose attribute difference falls within a prescribed class-width $u$ may interact—enforcing emergent class boundaries and affecting global exchange patterns (Herrera et al., 2010).
• Multi-Stratum Graphs/Networks: The Multi-Stratum Model (MSM) represents multiple networks (strata) linked by identity mappings, allowing the synthesis of metrics (centrality, reach, betweenness) across diverse platforms while capturing cross-stratum propagation and redundancy (Magnani et al., 2012).
• Stratified Material and Field Models: In physics and engineering, layered (stratified) structures are modeled as stacks with distinct wave or transport behavior at each interface—using transfer matrices, path summations, or action-angle dynamical systems to resolve the interplay between layers (Mechelen et al., 2014, Heifetz et al., 2017, Guha et al., 2012).
• Stratified Sampling and Statistical Testing: In data science and XAI, stratified sampling and permutation underpins robust estimation of interaction indices (e.g., Shapley interaction, SII) and significance tests for attribute-class specific dependency structures (Kolpaczki et al., 24 Jan 2024, Henelius et al., 2016).

2. Quantitative Metrics and Methodologies

Stratified interaction analysis employs a suite of metrics adapted to diverse phenomenological domains:

• Network Stratification Assortativity (StA): Quantifies the extent of intra-class vs. inter-class connection, normalized against a degree-sequence preserving randomization. Computed as

$$\mathrm{StA}(G) = \frac{S_\mathrm{strat}(G) - \mathbb{E}[S_\mathrm{strat}(G')]}{k - \mathbb{E}[S_\mathrm{strat}(G')]}$$

with $S_\mathrm{strat}$ an average of class-conditional edge similarities (Jalali et al., 2022).

• l-Adjacency Clustering Coefficient: Defines $C_i^{(l)}$ for a node $i$ as the weighted mean of the clustering coefficients of nodes at distance $l$—thereby profiling stratified community cohesion at multiple scales (Cerqueti et al., 2019).
• Stratified Shapley-based Sampling: SVARM-IQ introduces a stratified mean estimator for any-order Shapley interaction indices, partitioning coalitions both by size and subset identity, leading to unbiased, low-variance approximation of feature interactions (Kolpaczki et al., 24 Jan 2024).
• Permutation-based Stratified Significance Tests: ASTRID and related frameworks use within-class permutation of features to test for class-conditional independence, automatically discovering the finest attribute partition that preserves class-conditional predictive structure (Henelius et al., 2016).
• Gini Coefficient and Activity in Economic Models: Quantifies concentration (inequality) and long-run exchange rate (activity) as a function of class-width $u$, risk-favor parameter $f$, and network topology ($k$, $p$) in stratified economic systems (Herrera et al., 2010).

3. Model Architectures and Dynamical Regimes

Stratified interactions generate rich dynamical and structural regimes:

• Emergent Phases in Stratified Exchange Models: With varying class-width $u$, agent-based systems display laminar (frozen), intermittent (mixed), and turbulent (active) regimes. Activity $A$ and Gini $G$ provide sharp phase diagnostics, with transitions governed by percolation of class boundaries and favor-parameter tuning (Herrera et al., 2010).
• Stratified Community Structure: l-adjacency clustering reveals compound, multi-scale community architecture, enabling detection of nested, hierarchical, or peripheral/apical organization in business or infrastructural networks (e.g., airports) (Cerqueti et al., 2019).
• Wave-Interaction and Instability in Stratified Shear Flows: Stratified flows with interfacial or vorticity jumps are treated as coupled-wave systems; action-angle Hamiltonian or kinematic oscillator frameworks yield phase-locking and resonance instability criteria, linking exponential modal growth to non-modal transient amplification and generalized Fjørtoft/Howard-Miles-type criteria (Guha et al., 2012, Heifetz et al., 2017).
• Layered Material Characterization: Multilayered media are modeled using characteristic matrix products for optical/THz response; each stratum's properties are recovered via global fit of transfer function to time/frequency domain observations, with Drude-Lorentz parametrizations for dispersive permittivity (Mechelen et al., 2014).

4. Empirical Insights and Applications

Empirical studies employing stratified interaction analysis have yielded several domain-specific findings:

• Social Mobility and Stratification: Urban mobility analyses show that socioeconomic stratification tightly couples with spatial constraints—lockdown periods intensify within-class mobility while rendering cross-class flows less predictable. Pull (destination) socioeconomic factors more strongly determine mobility than push (origin) factors (Hilman, 2022).
• Cross-layer Social Role Discovery: MSM exposes user behavior, centrality, and reachability beyond single-platform analysis; cross-stratum shortest paths reveal bridging roles and hidden connectivity that elude monoplex approaches. Degree-uncertainty quantifies the error in single-stratum approximations (Magnani et al., 2012).
• Longitudinal Social Stratification: In co-authorship networks, the StA index evidences an increase in stratification over decades, with career trajectories increasingly determined by one's initial network position. This is in contrast with scalar assortativity metrics, which may register increased intra-class diversity even as stratification sharpens (Jalali et al., 2022).
• XAI and Feature Interaction Estimation: SVARM-IQ's stratified sampling approach dramatically reduces mean square error and improves precision of discovered interactions, especially in high-order, high-dimensional settings. Empirical precision@10 for identifying strongest feature couplings substantially improves at fixed sampling budgets (Kolpaczki et al., 24 Jan 2024).

5. Computational Techniques and Complexity

Efficient stratified interaction analysis requires methodological and algorithmic innovations:

• Efficient Sampling in High-dimensional Spaces: SVARM-IQ achieves non-asymptotic variance bounds and unbiasedness by stratifying over coalition-size and subset-identity—space complexity per-order is $O\big({n\choose k} (n-k) 2^k\big)$, with border enumeration for extreme coalition sizes and stratified averaging for each $K$ (Kolpaczki et al., 24 Jan 2024).
• Block-permutation for Attribute Partition Recovery: ASTRID implements a two-phase heuristic—attribute sorting followed by block segmentation—that scales as $O(m^2 R' \times \text{training time})$, enabling stratified class-conditional discovery in datasets with dozens of attributes (Henelius et al., 2016).
• Transfer Matrix Propagation: In N-layer stratified optical models, characteristic matrices provide compositional reflection/transmission coefficients; nonlinear least-squares fitting (e.g., Levenberg-Marquardt) extracts geometric and material parameters from spectral or time-domain data (Mechelen et al., 2014).
• Graph Traversal and Clustering Aggregation: Computation of l-adjacency clustering for diameter $L$ networks requires $O(nm + n^2 \log n)$ time, as Dijkstra or BFS is repeated from each node, followed by weighted aggregation over neighbor strata (Cerqueti et al., 2019).

6. Significance, Limitations, and Implications

Stratified interaction analysis systematically reveals how the imposition of hierarchical or layered structure alters the emergent properties of complex systems, often reversing or muting effects predicted by homogeneous models. Key cross-cutting findings include:

• Criticality and Transition Thresholds: Emergent transitions—between inactivity and turbulent exchange, or between low and high stratification—typically occur at threshold values of class width, stratification index, or coupling, not predictable from average node or agent properties alone (Herrera et al., 2010, Jalali et al., 2022).
• Role of Local Topology and Clustering: Stratified inequality measures (Gini, StA) are primarily driven by local clustering and neighborhood size, rather than by global rewiring or added long-range links, supporting an emphasis on local policy interventions for equity (Herrera et al., 2010).
• Limited Predictive Power of Single-layer or Non-stratified Models: Analysis ignoring stratification may severely misestimate influence, centrality, or resilience, and miss higher-order or cross-layer interactions essential for system function or vulnerability assessment (Magnani et al., 2012).
• Expanded Application Domains: Stratified frameworks extend naturally to complex business, financial, biological, and infrastructural networks, as well as to layered physical and material systems, providing a unified analytical structure across fields (Cerqueti et al., 2019, Mechelen et al., 2014).

A plausible implication is that future research and applications in network science, explainable AI, urban studies, economic modeling, and material characterization will increasingly adopt stratified interaction analysis frameworks to better resolve the nuanced, emergent behaviors induced by explicit layer, class, or community boundaries.