Worldwide bilateral geopolitical interactions network inferred from national disciplinary profiles

TL;DR

The paper tackles how national disciplinary structures relate to worldwide geopolitical interactions by defining country disciplinary profiles as unit vectors in an $n_d$-dimensional discipline space and analyzing annual cross-country correlations across two time windows separated by the Berlin Wall. It combines random matrix theory with a maximum-entropy framework to infer a bilateral interaction network $\{J_{ij}\}$ via a pseudo-likelihood optimization over $n_d$-dimensional spin variables, and then uses hierarchical clustering and PCA to extract geopolitical structure from the inferred couplings. The analysis reveals genuine information in the cross-correlation matrices, stable long-term patterns, and bilateral interaction clusters that correspond to historical geopolitical blocs, supporting a physics-inspired, quantitative view of international relations. These methods offer a principled way to study direct vs indirect couplings and historical shifts in international science and policy, with potential for deeper historical insights and policy-relevant analyses. In particular, the $n_d$-dimensional Boltzmann formulation and the pseudo-likelihood gradient provide analytic traction for Bayesian-type inference in high-dimensional spin-like models of geopolitical data.

Abstract

A disciplinary profile of a country is defined as the versor whose components are the numbers of articles produced in a given discipline divided the overall production of the country. Starting from the Essential Science Indicators (ESI) schema of classification of subject area, we obtained the yearly disciplinary profiles of a worldwide graph, where on each node sits a country, in the two time intervals [1980-1988] and [1992-2017], the fall of the Berlin Wall being the watershed. We analyse the empirical pairwise cross-correlation matrices of the time series of disciplinary profiles. The contrast with random matrix theory proves that, beyond measurement noise, the empirical cross-correlation matrices bring genuine information. Arising from the Shannon theorem as the least-structured model consistent with the measured pairwise correlations, the stationary probability distribution of disciplinary profiles can be described by a Boltzmann distribution related to a generalized $n_d$-dimensional Heisenberg model. The set of network interactions of the Heisenberg model have been inferred and to it they have been applied two clusterization methods, hierarchical clustering and principal component analysis. On a geopolitical plane this allow to obtain a characterization of the worldwide bilateral interactions based on physical modeling. A simple geopolitical analysis reveals the consistency of the results obtained and gives a boost to deeper historical analysis. In order to obtain the optimal set of pairwise interactions we used a Pseudo-Likelihood approach. We analytically computed the Pseudo-Likelihood and its gradient. The analytical computations deserve interest in whatever inference Bayesian problem involving a $n_d$-dimensional Heisenberg model.

Figures (13)

• Figure 1: Panel I. Histogram representation of the eigenvalues distribution of the empirical correlation matrix $\textbf{C}$ of disciplinary profiles in the time interval [1992-2017] (white bars) and of the isomorphic finite-dimensional RCM $\textbf{R}$ (red bars). Panel II Histogram representation of eigenvalues distribution of the matrix $\tilde{\textbf{C}}$ (white bars), obtained from standardized data, and $\tilde{\textbf{R}}$ (red bars). The solid curve shows theoretical predictions of RMT. Panel III Eigenvalues distribution of $C(k)$ corresponding to MAT. SCI. (light-grey bars) and of $\textbf{R}(k)$ (light-red bars). The solid curve shows the Marchenko-Pastur distribution. Panel III Eigenvalues distribution of $\tilde{\textbf{C}}(k)$ corresponding to MAT. SCI. (light-grey bars) and of $\tilde{\textbf{R}}$ (light-red bars). The solid curve shows the Marchenko-Pastur distribution. The insets show a zoom of the large-eigenvalues region of the eigenvalue distribution of empirical covariance matrices.
• Figure 2: Panel I. Distribution of eigenvector components (white bars) corresponding to the largest eigenvalue of C ($\lambda \gg \lambda_+$) contrasted with the eigenvector components distribution of test RCM (red bars) and RMT predictions (solid curve). Panel II. Distribution of eigenvector components corresponding to a bulk eigenvalue $\lambda$: $\lambda_-<\lambda<\lambda_+$. Panel III. Distribution of eigenvector components corresponding to an eigenvalue $\lambda<\lambda_-$. Panel IV. Inverse Partecipation Ratio (IPR) as a function of $\lambda$ of the empirical covariance matrix (open circles) and of RCM (stars).
• Figure 3: Number of eigenvalues different from zero in a matrix obtained by the sum of two empirical cross-correlation matrices related to two different disciplines, $\textbf{C}(k)+\textbf{C}(l)$. Each cross-correlation matrix is calculated for a single time t.
• Figure 4: Panels I./II.$\overline{O}(\tau)$ for two different delay time $\tau$. Panels III./IV.$O(t=1,\tau)$ for two different delay time $\tau$.
• Figure 5: $O(t=1,\tau)$ for only the eigenvectors corresponding to the five largest eigenvalues at different delay time $\tau$. The autocorrelation, $|\zeta_{\lambda}(t) \cdot \zeta_{\lambda}{t+\tau}|$ of the eigenvectors corresponding to the first three largest eigenvalue is shown for $t=1$ as a function of $\tau$.