Inside the Voynich Network: graph analysis

[quimqu]

What happens if you change the window to only directly adjacent tokens? (Or pairs of tokens)? What if you make the edges directed? Do you see anything new?

When we limit the window to directly adjacent tokens and make the graph directed, we measure something closer to real word order instead of general co-occurrence:

Text	Reciprocity	Clustering C_obs	Small-world σ	Modularity	Communities	H mean (bits)	Frac H = 0
Voynich MS	0.102	0.0766	114.18	0.332	38	0.64	0.72
Tirant lo Blanch (Cat.)	0.107	0.0384	23.72	0.289	13	1.01	0.53
Culpeper (Eng.)	0.142	0.0429	26.77	0.281	18	1.19	0.46
Alchemical Herbal (Lat.)	0.056	0.0229	8.54	0.494	17	0.57	0.66
Timm generated	0.119	0.0762	25.32	0.303	14	0.97	0.54
Mauro pseudo-Voynich	0.111	0.0881	10.67	0.245	27	0.87	0.60

The main differences are:
- Reciprocity becomes clearer. Natural languages have slightly higher reciprocity (around 0.10–0.14) because word order is flexible: A → B sometimes also appears as B → A. In the Voynich, reciprocity stays low (0.10), meaning word pairs almost never reverse. But note that the lowest is the Alchemical Herbal in Latin. *
- Entropy drops a lot. More than 70% of Voynich tokens have only one possible follower (H = 0), while in normal texts it is about half. This suggests mechanical repetition rather than grammatical structure. Again, Alchemical Herbal is not so far away from the Voynich, and has an even lower entropy
- The small-world property remains but becomes extreme. Voynich has σ around 114, while natural languages range between 8 and 30. This means there are many short, repetitive loops, typical of a templated system.
- Community structure changes too. Natural texts form a bit more of a dozen broad clusters, but the Voynich splits into many small, tight ones. It still shows order, but it is too fragmented.

When using directed, adjacent links, the Voynich still looks organized, but its structure is too rigid and repetitive compared to real languages. This supports the idea that it is not a normal language but a constrained, procedural system that imitates word sequences without real syntax.

* To make it clear with RadioFM later You are not allowed to view links. Register or Login to view. and my You are not allowed to view links. Register or Login to view.: my measure of reciprocity is not the same as Fincher’s "word reversibility". Fincher looked at how often a word pair A → B also appears as B → A somewhere in the text. That is a frequency-based metric, tied to the linear sequence of the text. In my case, reciprocity in this post is calculated on a directed graph of bigrams. Each word is a node, and there is a link A → B if B ever follows A at least once. Reciprocity here then measures the topological symmetry of that network (the percentage of links that are bidirectional) not how frequently they occur. So Fincher’s measure reflects how often reversals happen, while mine reflects whether such connections exist at all. They are related but not equivalent, and they can move in opposite directions in repetitive or highly structured texts like the Voynich.