Abstract
Saying that a network is a small world bundles together at least three separate claims. This paper separates them and shows in numbers which does not follow from which. NO NEW MATHEMATICAL THEOREM IS CLAIMED.
First, short paths come free. On $1000$ nodes of average degree $10$, a network with no structure whatever has a mean path length of $3.26$ — so a short path is no evidence of design (Section 2).
Second, and this is the core. Short paths and local clustering are carried by different edges. Rewiring $1\%$ of a ring's edges drops the mean path from $50.45$ to $8.75$, that is to $17.3\%$, while the clustering falls only from $0.6667$ to $0.6446$, that is to $96.7\%$ (Section 3).
Third, a heavy degree tail does not follow from either. The $10\%$-rewired ring has a largest degree only $1.40$ times the mean, while a network grown by preferential attachment reaches $14.50$ times, with a largest degree of $142$ (Section 4).
Fourth, the fence is the second moment. With every average degree at about $10$, the ratio of the second moment to the first came out as $10.00$, $11.00$, $10.10$ and $\mathbf{22.20}$ (Section 5).
Fifth,nothing is claimed here about any real network — only the four generated here were measured.
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM IS CLAIMED — the Watts–Strogatz construction, preferential attachment, and the appearance of the second moment in a spreading threshold are all standard. Nothing is claimed about real networks, social, biological or technical — what the degree distribution of any real system is is not a question this paper touches. What is treated is measurements of generated networks. On a word: Paper 176 wrote that “small” becomes two different things under measure and under category. The “small” here is neither of those but a third use — one word splitting again, somewhere else. What is added is separating three claims by what carries each, measuring that a $1\%$ rewiring moves the two independently, exhibiting a counterexample to the heavy tail following from short paths, and placing the fence at the second moment.
1. Notation and position
Four undirected networks on $1000$ nodes of average degree $10$: a ring joining each node to its ten nearest on a circle, a purely random network, a ring with a fraction of edges rewired, and one grown by preferential attachment. The mean path length is measured by breadth-first search from $200$ randomly chosen nodes, and the clustering is the average over nodes of the fraction of their neighbour pairs that are joined. All of it was computed for this paper.
Position: Paper 186 placed a fence at the tail of an error and Paper 190 placed the fence on rarity at a finite exponential moment. This paper adds how readily a network spreads to that row — and again the average is hiding it.
2. Short paths come free
Verification 2.1
| network | mean path length | clustering |
|---|---|---|
| ring | $50.4505$ | $\mathbf{0.6667}$ |
| purely random | $\mathbf{3.2648}$ | $0.0100$ |
(QED)
Observation 2.2 A network with no structure at all already has short paths. So an observation of the six-degrees kind is by itself no evidence of anything — it is what randomness supplies for nothing.
Observation 2.3 What is surprising is not the short path but that a short path and heavy clustering coexist. The ring is clustered and far; the random network is near and unclustered. Neither is a small world.
3. The core: different edges carry the two
Verification 3.1: Rewiring a fraction of the ring's edges.
| fraction | mean path | clustering | path vs ring | clustering vs ring |
|---|---|---|---|---|
| $0$ | $50.4505$ | $0.6667$ | $1.0000$ | $1.0000$ |
| $0.001$ | $34.0233$ | $0.6654$ | $0.6744$ | $0.9980$ |
| $0.01$ | $\mathbf{8.7452}$ | $\mathbf{0.6446}$ | $\mathbf{0.1733}$ | $\mathbf{0.9668}$ |
| $0.05$ | $5.3227$ | $0.5823$ | $0.1055$ | $0.8734$ |
| $0.2$ | $3.8093$ | $0.3356$ | $0.0755$ | $0.5034$ |
| $1.0$ | $3.2601$ | $0.0080$ | $0.0646$ | $0.0119$ |
(QED)
Observation 3.2 A one-percent rewiring cuts the path to $17\%$ while $97\%$ of the clustering survives. The two quantities are carried by different edges — a few shortcuts decide the distance, and many neighbourly ties decide the clustering.
Observation 3.3 So what the phrase “small world” names is neither property but the pair. Measuring one of them does not license the phrase.
4. A heavy degree tail does not follow
Verification 4.1
| network | mean degree | largest | standard deviation | largest / mean |
|---|---|---|---|---|
| ring | $10.00$ | $10$ | $0.00$ | $1.00$ |
| random | $10.00$ | $21$ | $3.17$ | $2.10$ |
| rewired $10\%$ | $10.00$ | $14$ | $0.98$ | $\mathbf{1.40}$ |
| preferential | $9.79$ | $\mathbf{142}$ | $10.50$ | $\mathbf{14.50}$ |
(QED)
Observation 4.2 The $10\%$-rewired ring is a perfectly good small world and its degrees are all much alike. The grown network has a node fourteen and a half times the average. A heavy tail does not follow from short paths — they are separate claims about separate things.
Observation 4.3 And what carries the heavy tail is how the network was grown — only a rule under which the already-well-joined get joined again produces it.
5. The fence is the second moment
Observation 5.1 For something spreading on a network, what governs the threshold is not the mean degree but the ratio of the second moment to the first.
Verification 5.2
| network | mean degree | second moment over first | ratio to the mean |
|---|---|---|---|
| ring | $10.000$ | $10.000$ | $1.000$ |
| random | $10.000$ | $11.004$ | $1.100$ |
| rewired $10\%$ | $10.000$ | $10.099$ | $1.010$ |
| preferential | $9.786$ | $\mathbf{22.197}$ | $\mathbf{2.268}$ |
(QED)
Observation 5.3 Every average degree sits at about $10$, and the quantity that enters the threshold differs by more than a factor of two. The average is hiding it.
Observation 5.4 This is the third time the corpus meets the same shape — Paper 186 put the fence at the tail of an error and Paper 190 put the fence on rarity at a finite exponential moment. In each, the average does not settle it and the tail does. Here too, equal average degrees do not give equal behaviour.
6. The census
Observation 6.1
| what is called small | what carries it | caution |
|---|---|---|
| short paths | a few shortcuts | randomness gives it free |
| local clustering | many neighbourly edges | different edges from the paths |
| a heavy degree tail | how it was grown | does not follow from short paths |
| spreads readily | the second moment | not settled by the mean degree |
Observation 6.2 These four are not a staircase. The first two are statements about distance, the third about a distribution, and the fourth about what happens on top. One phrase over all of them makes it seem that seeing one settles the rest.
Observation 6.3 A word on the word. Paper 176 wrote that “small” comes apart into measure zero and first category. The “small” here is neither — one word splitting again, somewhere else.
7. Conclusion
- Short paths come free (Ver. 2.1): $3.26$ in a purely random network.
- Different edges carry the two (Ver. 3.1): a $1\%$ rewiring leaves path at $17\%$ and clustering at $97\%$.
- A heavy tail does not follow (Ver. 4.1): largest over mean of $1.40$ against $14.50$.
- The fence is the second moment (Ver. 5.2): $10.10$ against $22.20$ at the same average degree.
- The four are not a staircase (Obs. 6.2).
References
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- Yamagishi, Y. (2026). “The Curse of Dimensionality” Is Not One Curse [Paper 182]. Zenodo. 10.5281/zenodo.22104170
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- Yamagishi, Y. (2026). The Gaussian Is Special for More Than One Reason [Paper 187]. Zenodo. 10.5281/zenodo.22105639
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- Yamagishi, Y. (2026). The Thing That Smooths Is What Makes the Pattern [Paper 191]. Zenodo. 10.5281/zenodo.22106439
Reference
Yamagishi, Y. (2026). “A Small World” Is Three Separate Claims [Paper 192]. YOI: https://www.yamagishiyuuki.com/yoi/192
DOI: 10.5281/zenodo.22106544
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 192. This paper is a preprint and has not been peer reviewed.