Inside the Voynich Network: graph analysis

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Let's continue with the analysis of the Voynich MS though graphs.

This first part of the analysis is based on directed word-to-word graphs, where each edge connects a token A → B if word B follows A in the text. This approach keeps the natural direction of information flow, unlike undirected co-occurrence graphs that only record proximity. Each corpus was converted into a directed graph and measured using a set of topological and information-theoretic metrics. These metrics describe structure, hierarchy, and randomness in how words connect.

The comparison includes three groups of texts:

• Natural languages: medieval, classical, and modern works in Latin, English, French, Spanish, and others.
  
• Voynich texts: the full Voynich corpus and its internal subsets (A, B).
  
• Artificial ciphers: the Naibe cipher and Torsten Timm’s generated text.
  

The table below summarizes the main indicators for all groups, showing the range of values for each metric and how Voynich differs from natural and artificial systems.

KPI	Natural texts (min–max)	Voynich (min–max)	Artificial (Naibe, Timm)	Description	Voynich vs Natural / Artificial
H0_mean_lifetime	0.79–0.97	0.81–0.85	0.79–0.89	Mean persistence of connected components. Measures graph cohesion at 0-dim topology.	Voynich is slightly lower than natural texts, close to artificial, suggesting weaker component stability.
H1_frac_inf	≈0.00 (none)	≈0.00 (none)	≈0.00 (none)	Fraction of infinite 1-dim holes. Reflects whether loops persist indefinitely.	No difference; all show finite loops only.
avg_clustering	0.38–0.53	0.50–0.57	0.45–0.82	Local density of triangles. Measures how often neighbors are connected.	Voynich has moderate clustering, slightly higher than natural texts but lower than some artificial graphs.
spectral_gap	1.6–5.9	2.9–4.5	5.9	Second Laplacian eigenvalue gap. Indicates overall graph connectivity and mixing speed.	Voynich falls mid-range; less connected than natural, less random than artificial.
kcore_max	4–17	5–12	8–17	Maximal k-core index. Captures how strongly nodes are mutually connected in dense cores.	Voynich has mid-core density, lower than typical artificial graphs.
flow_hierarchy	0.76–0.91	0.58–0.65	0.51–0.64	Fraction of edges in acyclic subgraphs. Quantifies directionality and information flow.	Voynich shows weaker directionality, closer to artificial graphs → more isotropic structure.
edge_reciprocity	0.08–0.24	0.34–0.42	0.36–0.48	Proportion of mutual (bidirectional) edges. Measures how often relations are symmetric.	Voynich has high reciprocity, like artificial ciphers, unlike natural directional flow.
entropy_rate	0.53–1.59	0.89–1.87	1.74–2.01	Average information rate of random walk transitions. Higher = less predictable.	Voynich entropy sits between natural and artificial → partially random transition dynamics.
degree_assortativity	-0.10–0.15	≈0.00	≈0.00	Correlation between node degrees. Positive = hubs connect to hubs.	Voynich similar to artificial: nearly neutral assortativity (no social-like structure).
avg_shortest_path	2.0–2.9	2.1–2.2	2.0–2.1	Mean minimal steps between nodes. Reflects overall navigability.	Voynich is very close to artificial; both slightly shorter paths than natural languages.

To make it visual, I have reduced all the KPI's to a two dimension vector with PCA method and plotted. Ypou can see that Torsten Timm's generated text is fairly away from any other text, while Voynich (full, A or B) are even closer to natural languages than naibe cipher. Here are the results:

   

I thought that reducing the KPI's to two dimensions with PCA method helps positioning each text. So I did the same experiment on the co-ocurrence graphs generated at the first part of this study.

This plot shows the position of each corpus based on ten structural metrics of its word co-occurrence graph. Each point represents a text, and the axes (PCA1 and PCA2) combine multiple graph properties into two main dimensions that explain most of the variation. The analysis includes natural languages, the Voynich text (EVA and CUVA) and its internal subsets (A and B), and thre artificial or generated corpora (Naibe Cipher, Timm and Mauro's Markov). By reducing all metrics to a two-dimensional map, we can see how similar or different the internal network organization of each text is:

   

The Voynich samples form a compact cluster close to natural languages, but slightly below them along the first axis, suggesting similar small-world and modular properties but lower diversity and reciprocity. The artificial texts are more scattered and show larger deviations, with the Markov and Naibe models standing apart in the upper area (with non european text or the Rohon), indicating higher uniformity and weaker hierarchical patterns. Overall, the Voynich graphs behave more like "European" natural languages than synthetic ones or asiatic ones, though they remain slightly detached, showing a distinct but structured internal topology.