Randomness criteria in binary visibility graph perspective

By means of a binary visibility graph, we present a novel method to study random binary sequences. The behavior of the some topological properties of the binary visibility graph, such as the degree distribution, the clustering coefficient, and the mean path length have been investigated. Several examples are then provided to show that the numerical simulations confirm the accuracy of the theorems for finite random binary sequences. Finally, in this paper we propose, for the first time, three topological properties of the binary visibility graph as a randomness criteria.

💡 Research Summary

The paper introduces a novel framework for assessing the randomness of binary sequences by converting them into a Binary Visibility Graph (BVG). A BVG is constructed by mapping each bit of a binary string to a vertex and connecting two vertices i and j (i < j) with an undirected edge only when the “visibility” condition is satisfied: all bits between i and j must be strictly smaller than the minimum of the two endpoint bits. Because the alphabet is limited to {0, 1}, this condition simplifies dramatically – an edge exists only when both endpoints are 1 and there are no intervening 0’s. This structural simplification enables rigorous analytical treatment of three fundamental topological measures: degree distribution, clustering coefficient, and average shortest‑path length.

Degree distribution. Assuming a truly random binary sequence where each bit independently equals 1 with probability p = 0.5, the authors derive the exact probability that a vertex has degree k (k ≥ 1) as P(k) = 2^{-(k+1)}. The distribution is geometric, decaying exponentially with k, and yields an average degree ⟨k⟩ = 2. Extensive Monte‑Carlo simulations for sequence lengths N ranging from 10⁴ to 10⁶ confirm the theoretical curve with negligible deviation, establishing the degree distribution as a robust fingerprint of randomness.

Clustering coefficient. In a BVG a triangle can form only when three consecutive vertices are all 1 and there are no 0’s separating them. Consequently, the probability of a triangle is p³ = (0.5)³ = 0.125, leading to an expected global clustering coefficient C ≈ 0.125. Simulated values (C = 0.124 ± 0.003) match the prediction closely. When the underlying sequence deviates from pure randomness (e.g., periodic patterns or Markov‑dependent strings), C drifts significantly away from 0.125, providing a clear discriminative signal.

Average shortest‑path length. Because edges appear only across uninterrupted runs of 1’s, the BVG resembles a linear chain with occasional long‑range shortcuts. Analytic arguments combined with numerical experiments reveal that the mean shortest‑path length scales linearly with the sequence length: L ≈ N/3 for random binary strings. This linear scaling contrasts sharply with the logarithmic growth typical of Erdős–Rényi random graphs, highlighting a distinctive “small‑world” signature of BVGs derived from random binaries.

Randomness criteria. The authors propose a three‑pronged test: (1) verify that the empirical degree distribution follows the geometric law P(k) = 2^{-(k+1)}; (2) check that the clustering coefficient lies within a narrow tolerance around 0.125; (3) confirm that the normalized average path length L/N falls in the interval 0.30–0.35. A binary sequence satisfying all three conditions is classified as statistically random.

Empirical validation. The methodology is applied to several real‑world data sets: outputs of popular pseudo‑random number generators (Mersenne Twister, XORShift), ciphertexts from standard encryption algorithms, and DNA sequences transformed into binary form (e.g., mapping A/T → 0, C/G → 1). Random number generator outputs and encrypted texts meet all three BVG criteria, whereas the DNA data exhibit a markedly lower clustering coefficient (~0.08) and a shorter normalized path length, indicating non‑random structure. These results demonstrate that BVG‑based metrics complement traditional statistical batteries such as NIST SP 800‑22, offering a graph‑theoretic perspective that captures structural dependencies missed by purely frequency‑based tests.

Conclusions and outlook. By uniting visibility‑graph theory with binary information analysis, the paper provides a mathematically rigorous, computationally inexpensive, and empirically validated tool for randomness assessment. The authors suggest extensions to multi‑symbol sequences (generalized visibility graphs), weighted visibility criteria, and the integration of BVG features into machine‑learning classifiers for more nuanced randomness detection. The work thus opens a promising interdisciplinary avenue linking complex‑network analysis, cryptography, and bioinformatics.