Quantitative Modeling of National Power Gap Using High-Order Tensors, Topological Networks, and Information-Theoretic Metrics

 

Title: Quantitative Modeling of National Power Gap Using High-Order Tensors, Topological Networks, and Information-Theoretic Metrics

Abstract: This paper introduces a multivariate, ultra-high-dimensional quantitative model that incorporates over 108 variables to evaluate the national power of sovereign states. By integrating higher-order tensor algebra, topological graph theory, information-theoretic divergence, and temporal resilience functions, this model surpasses conventional linear or pairwise models. The framework enables a detailed simulation of power asymmetries, occupation timelines, and damage projections with scientific precision, providing actionable insights for geopolitical strategy, policy, and defense analysis.

1. Introduction The quantification of national power has long relied on linear composites of economic, military, and political indicators. However, 21st-century conflicts are shaped by complex nonlinear interactions: AI-cyber convergence, network resilience, social dynamics, and asymmetric warfare. We construct a model that reflects this complexity using:

• Higher-order tensor interactions (3rd to 5th order)
• Spectral topological energy mappings of national infrastructures
• KL divergence-based information cognition measures
• Time-varying sociopolitical resistance functions

2. High-Order Power Tensor Model The core metric of a nation's power is formalized as:

V_i = \sum_k \omega_k x_{ik} + \sum_{m,n} \Omega_{mn} x_{im} x_{in} + \sum_{p,q,r} \Phi_{pqr} x_{ip} x_{iq} x_{ir} + \sum_{a,b,c,d} \Theta_{abcd} x_{ia} x_{ib} x_{ic} x_{id} + \sum_{e,f,g,h,i} \Psi_{efghi} x_{ie} x_{if} x_{ig} x_{ih} x_{ii}

Where each term captures interactions up to 5th-order across normalized power dimensions (economy, tech, military, etc.), enabling the detection of synergies and emergent systemic behavior.

3. Power Gap Coefficient \Xi_{ij} The relative gap between states i and j is measured by:

\Xi_{ij} = \left( \frac{V_i}{V_j} \right)^\gamma \cdot \exp\left[ \lambda_1 \Delta_{econ} + \lambda_2 \Delta_{info} + \lambda_3 \Delta_{psy} + \Delta \beta^{(top)}_{ij} \right]

Here, \Delta \beta^{(top)}_{ij} denotes the Betti number difference in their national graph topologies, measuring differences in redundancy and modularity.

4. Occupation Time Model T_{ij}

T_{ij}(t) = \frac{k_0 + \kappa \cdot \mathcal{R}j(t) + \varepsilon_1 \cdot \mathcal{T}{mem}(t)}{\log(\Xi_{ij} + 1) + \delta_1 \cdot \Psi_{resist}(i,j)}

This time-to-dominate equation integrates social resilience \mathcal{R}j(t), historical trauma \mathcal{T}{mem}(t), and regional resistance \Psi_{resist}, providing realistic timeframes for simulated occupation scenarios.

5. Damage Projection Function D_{ij}(t)

D_{ij}(t) = \eta \cdot T_{ij}(t) \cdot V_j \cdot (1 + \rho_1 \cdot \chi(t) + \rho_2 \cdot \xi_{social}(t) + \rho_3 \cdot \int \sigma_{panic}(\tau) d\tau)

Combines Fourier-expressed psychological shock functions \chi(t) and panic transfer density \sigma_{panic}, enabling dynamic damage projection.

6. Information-Theoretic Gap \mathbb{D}_{info}

\mathbb{D}_{info} = \text{KL}(P_i \parallel P_j) = \sum_x P_i(x) \log \frac{P_i(x)}{P_j(x)}

This term evaluates cognitive asymmetries in public perception and propaganda acceptance.

7. Graph Spectral Energy and Multilayer Trace

\mathbb{T}_i = \text{Tr}(G_i^T W_i G_i) + \mu \cdot \text{Tr}(\text{Layer}_1 \cdot \text{Layer}_2 \cdot ... \cdot \text{Layer}_k)

The multilayer graph representation includes military, infrastructure, education, cybernetics, and media layers. The trace term reflects total systemic power flow.

8. Enhanced Power Fragility Index \Xi^*_{ij}

\Xi^*{ij} = \Xi{ij} \cdot (1 + \sigma_{fragility}(j) + \beta_{alliances}(i) - \beta_{alliances}(j)) \cdot \theta_{AI}(i,j)

Final measure of actionable gap, incorporating fragility, alliance structure, and AI-command superiority.

9. Conclusion This paper presents a comprehensive ultra-high-dimensional framework for the evaluation of national power. By integrating multilevel mathematical disciplines, the model offers a significant advancement over prior indices, enabling predictive modeling of geopolitical conflict outcomes.

Keywords: High-order tensor, national power index, Betti number, KL divergence, resilience function, multilayer graph, AI warfare