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A Generalized Benford Framework for Threat Identification in Counter-Intelligence

Timothy Tarter

TL;DR

This work extends Benford's law beyond its traditional discrete leading-digit form by constructing a continuous Benford measure on bounded domains and pairing it with a frequency-based, matrix-analytic framework for counter-intelligence data. It defines a log-determinant based Benford Matrix A from pairwise site comparisons, introduces a Benford Test Statistic $\lambda = 2.0973 - \frac{\ln|\det(A)|}{n}$, and uses higher moments under the continuous Benford model to enable hypothesis testing for Benford-ness. The methodology provides a quantifiable way to detect hidden Benford patterns in suspects' multi-site activity, guiding investigators to prioritize sites whose inclusion most perturbs the Benford structure. Numerical Python simulations illustrate the approach and point to practical applications in threat identification and early warning in national-security contexts.

Abstract

In this paper, we develop a framework of 'Benford models' for counter-intelligence investigations which analyze frequency data of a suspect's visits to physical locations, online websites, and communication channels. We accomplish this by establishing the Benford measure for continuous & bounded domains, generalizing the accumulated percentage differences between sites in the frequency data with the log-determinant of 'Benford Matrices,' employing an estimator to determine a 'Benford Test Statistic,' and identifying maximal values of that test statistic across all permutations of included sites in our data. This framework is intended to complement outlier analysis models by finding where hidden Benford patterns 'break' in frequency data and telling investigators which sites they should investigate.

A Generalized Benford Framework for Threat Identification in Counter-Intelligence

TL;DR

This work extends Benford's law beyond its traditional discrete leading-digit form by constructing a continuous Benford measure on bounded domains and pairing it with a frequency-based, matrix-analytic framework for counter-intelligence data. It defines a log-determinant based Benford Matrix A from pairwise site comparisons, introduces a Benford Test Statistic , and uses higher moments under the continuous Benford model to enable hypothesis testing for Benford-ness. The methodology provides a quantifiable way to detect hidden Benford patterns in suspects' multi-site activity, guiding investigators to prioritize sites whose inclusion most perturbs the Benford structure. Numerical Python simulations illustrate the approach and point to practical applications in threat identification and early warning in national-security contexts.

Abstract

In this paper, we develop a framework of 'Benford models' for counter-intelligence investigations which analyze frequency data of a suspect's visits to physical locations, online websites, and communication channels. We accomplish this by establishing the Benford measure for continuous & bounded domains, generalizing the accumulated percentage differences between sites in the frequency data with the log-determinant of 'Benford Matrices,' employing an estimator to determine a 'Benford Test Statistic,' and identifying maximal values of that test statistic across all permutations of included sites in our data. This framework is intended to complement outlier analysis models by finding where hidden Benford patterns 'break' in frequency data and telling investigators which sites they should investigate.
Paper Structure (16 sections, 2 theorems, 15 equations, 5 figures)

This paper contains 16 sections, 2 theorems, 15 equations, 5 figures.

Table of Contents

  1. Overview
  2. Law of Anomalous Numbers
  3. Early Applications
  4. Defining the Benford Measure
  5. The Discrete Characterization of Benford Measure
  6. Generalization of Benford Measure to a Continuous Domain
  7. Counter-Intelligence Threat Identification
  8. Frequency Analysis
  9. Order Invariance of Sites & Generalizing $\mu$
  10. The Benford Test Statistic
  11. Higher Moments of the Benford Distribution
  12. Hypothesis Testing Formula
  13. Maximizing $\lambda$
  14. Numerical Simulation in Python
  15. Conclusion
  16. ...and 1 more sections

Key Result

Lemma 1

Let $f_i$ denote the number of visits to a site i, letting n be the total possible number of sites. Then, $g_{i,j} \simeq ln(f_i) - ln(f_{j})$ is the percent change between $f_i$ and $f_{j}$.

Figures (5)

  • Figure 1: Demonstration
  • Figure 2: Law of Anomalous Numbers Visualization (Frost, 2022)
  • Figure 3: Sample Benford Set
  • Figure 4: Sample 'Not' Benford Set
  • Figure :

Theorems & Definitions (2)

  • Lemma 1
  • Theorem 1