Abstract
A citation network can only point backwards in time, so by construction it should contain no cycles. What this paper shows is that this corpus does not obey that, carrying $\mathbf{304}$ edges that run the other way, and that these are explained as traces of revision. No new theorem or law is claimed.
First, $\mathbf{304}$ of $\mathbf{3219}$ directed edges point forward in number——$\mathbf{9.44}$ per cent, with $\mathbf{39}$ mutually citing pairs (Section 2).
Second, the largest strongly connected component holds $\mathbf{131}$ papers——spanning numbers $3$ to $151$, with $\mathbf{4}$ components of size two or more totalling $\mathbf{138}$ papers (Section 3).
Third, most of the corpus can still be ordered——$\mathbf{389}$ papers, or $\mathbf{73.81}$ per cent, lie in no cycle (Section 3).
Fourth, and this is the core. Papers with a backward citation carry revision logs——$\mathbf{67.31}$ per cent of the $\mathbf{52}$ that have one, against $\mathbf{1.26}$ per cent of the $\mathbf{475}$ that do not, a ratio of $\mathbf{53.29}$ (Section 4).
Fifth, the backward citations do not reach next door——the median distance is $\mathbf{62}$ and the maximum $\mathbf{146}$, while distance one accounts for only $\mathbf{11}$ edges, or $\mathbf{3.62}$ per cent (Section 4).
Sixth, the separator is whether an edge is read by writing order or by reference order——one edge points opposite ways on two clocks (Section 5).
Keywords: citation network, strongly connected component, trace of revision, cumulative advantage, corpus of papers.
Scope (what is not claimed)
No new theorem or law is claimed——citation-network analysis, strongly connected components and Price’s cumulative advantage are all standard. The quality of the corpus is not measured——neither many nor few backward citations is treated as an indicator of quality; only structure is measured. No comparison with other corpora is made——the general rate of backward citation in the literature was not measured, so whether $\mathbf{9.44}$ per cent is high or low cannot be said. The content of revisions is not examined——only whether a revision log exists, not what was changed. No causal claim is made——co-occurrence of revision logs and backward citations is measured, but which came first is not decidable with these tools. No power law is claimed——a fit to the citation distribution is reported without any claim about mechanism. Relation to earlier papers: Paper 118 showed that coordinate time, proper time and entropy are three different logical types——the two clocks here are a fourth: writing order and reference order generate different orderings. Paper 81 treated the Poincare recurrence theorem and the arrow of time——that arrow is physical and this one bibliographic, and cycles are at issue in both. Paper 200 measured the corpus as its own data——this paper belongs to that line, taking the direction of citation as its object. Paper 400 showed that the instrument broke before the thing it was counting——here too, an outdated ceiling in the scanner would have silently discarded $168$ citations, so the tool’s self-test was run first. Paper 526 showed that the scope note was a habit adopted along the way——the traces of revision here likewise record when a habit changed, in the form of paper numbers. Paper 300 showed that sharing a root is decidable——the two orderings are separate roots under the criterion of what a quantity carries. What is added is counting $\mathbf{304}$ backward citations and $\mathbf{39}$ mutual pairs, computing the strongly connected components to obtain a largest of $\mathbf{131}$, finding $\mathbf{389}$ papers outside every cycle, measuring the co-occurrence with revision logs at $\mathbf{53.29}$, and placing the separator on the two clocks.
1. Notation and the Question
Setting: in a citation network a later paper cites an earlier one, so edges always run back in time and cycles cannot form. When Price treated citation networks in $1965$, this acyclicity was assumed.
Observation 1.1 that holds only if papers do not change after publication. Revise one and a citation to a later paper can be added——at that instant the edge stops running back in time.
Observation 1.2 there are two orderings. The order in which papers were first written, and the order the references now point in——just as Paper 118 separated coordinate time, proper time and entropy into three logical types, there are two clocks here.
Question: how far from acyclic is this corpus?
2. Measuring
Computation 2.1: extracting internal citation edges from every bibliography as of $10$ September $2026$, using the corpus’s own scanner and passing its self-test first, namely agreement with a known case and zero unresolvable internal references.
| quantity | value |
|---|---|
| nodes | $\mathbf{527}$ papers |
| directed edges | $\mathbf{3219}$ |
| backward in number (citing earlier) | $\mathbf{2913}$ |
| forward in number (citing later) | $\mathbf{304}$ ($\mathbf{9.44}$ per cent) |
| self-citations | $\mathbf{2}$ |
| mutual pairs | $\mathbf{39}$ |
Observation 2.2 the three counts sum to the total. $2913+304+2=3219$——a check that nothing was dropped, not a claim.
Observation 2.3 there are $\mathbf{2}$ self-citations. Paper 1 and Paper 2 list themselves in their own bibliographies——the only self-loops in the corpus, and these are also the two that cite no other paper. Saying they “cite nothing” is inaccurate without adding “other than themselves”.
Observation 2.4 the instrument was doubted first. If the scanner’s ceiling on paper numbers is left outdated, citations above it are silently discarded——that happened once and cost $168$ citations; the ceiling is now read from the folders. Exactly the pattern of Paper 400.
3. The Shape of the Cycles
Computation 3.1: computing the strongly connected components.
| quantity | value |
|---|---|
| strongly connected components | $\mathbf{393}$ |
| largest component | $\mathbf{131}$ papers (numbers $3$ to $151$) |
| components of size two or more | $\mathbf{4}$ (totalling $\mathbf{138}$ papers) |
| papers in no cycle | $\mathbf{389}$ ($\mathbf{73.81}$ per cent) |
Observation 3.2 the cycles are concentrated in one place. The largest holds $\mathbf{131}$ papers and the other three together only $\mathbf{7}$——the corpus is not tangled throughout; one early-to-middle band is.
Observation 3.3 that band coincides with the period of heavy revision. Numbers $3$ to $151$——many second and third editions live there, and adding citations after the fact makes such a paper point at later ones that point back.
Observation 3.4 most of the corpus can still be ordered. $\mathbf{389}$ papers, $\mathbf{73.81}$ per cent, lie in no cycle——“not acyclic” and “without order” are different: three quarters can be sorted.
Observation 3.5 connectivity is a separate matter. Ignoring direction, there is $\mathbf{1}$ component covering $\mathbf{100}$ per cent, with diameter $\mathbf{5}$ and mean shortest path $\mathbf{2.7059}$——no paper is isolated. Whether cycles exist is independent of whether the graph is connected.
4. Backward Citations Come with Revisions
Computation 4.1: comparing the share carrying a revision log between papers with and without a backward citation.
| group | papers | share with a revision log |
|---|---|---|
| with a backward citation | $\mathbf{52}$ | $\mathbf{0.6731}$ |
| without | $\mathbf{475}$ | $\mathbf{0.0126}$ |
| ratio | — | $\mathbf{53.29}$ |
Observation 4.2 this is the core. A factor of $\mathbf{53.29}$——backward citations are not scattered at random but gather in papers that carry revision logs.
Observation 4.3 causation cannot be stated. Did revising add the backward citation, or did adding one call for a revision log——co-occurrence is measurable; order is not, with these tools. That is a missing separator, not a weak conclusion.
Computation 4.4: measuring the distance of each backward citation, the cited number minus one’s own.
| quantity | value |
|---|---|
| minimum | $\mathbf{1}$ |
| median | $\mathbf{62}$ |
| maximum | $\mathbf{146}$ |
| distance one (next door) | $\mathbf{11}$ edges ($\mathbf{3.62}$ per cent) |
Observation 4.5 these are not citations to the paper next door. Distance one is only $\mathbf{3.62}$ per cent and the median is $\mathbf{62}$——too large to be explained by renumbering. Reaching $\mathbf{62}$ papers ahead requires touching the text after that paper existed.
Observation 4.6 the maximum of $\mathbf{146}$ spans a quarter of the corpus. Out of $527$ papers——an early paper citing a middle one, an edge that only revision can create.
5. The Separator
Observation 5.1 the separator is whether the edge is read by writing order or reference order. The same edge points to the future on one clock and to the past on the other——neither is the right one; there are two.
Observation 5.2 a fourth logical type for Paper 118. That paper separated coordinate time, proper time and entropy——this adds bibliographic order. A paper number is a scale for writing order, not for reference order.
Observation 5.3 a separate root from the arrow of Paper 81. There the cycle is Poincare recurrence, certain to return but not before the universe ends——here $\mathbf{39}$ pairs are closed in fact. One is about possibility, the other about measurement.
Observation 5.4 they part under the criterion of Paper 300. Writing order needs a date as a belonging——reference order needs only the text, which is why backward citations can be counted from numbers while recovering writing order needs a separate record.
Observation 5.5 what is not said. Many backward citations are called neither good nor bad——and with no comparison against other corpora, whether $\mathbf{9.44}$ per cent is high or low cannot be said.
6. Roll Call and Closing
- $\mathbf{3219}$ directed edges (Computation 2.1): over $\mathbf{527}$ nodes.
- $\mathbf{304}$ backward citations (Computation 2.1): $\mathbf{9.44}$ per cent.
- $\mathbf{39}$ mutual pairs (Computation 2.1): cycles exist in fact.
- $\mathbf{2}$ self-citations (Observation 2.3): Paper 1 and Paper 2 list themselves.
- largest component $\mathbf{131}$ (Computation 3.1): numbers $3$ to $151$.
- $\mathbf{389}$ papers in no cycle (Computation 3.1): $\mathbf{73.81}$ per cent.
- one component undirected (Observation 3.5): nothing is isolated.
- revision ratio $\mathbf{53.29}$ (Computation 4.1): the core.
- median distance $\mathbf{62}$ (Computation 4.4): not next door.
- maximum $\mathbf{146}$ (Computation 4.4): only revision creates it.
- the separator is two clocks (Observation 5.1): writing or reference order.
Closing: A citation network can only point backwards in time, so by construction it should contain no cycles. This corpus does not obey that——across $\mathbf{527}$ papers and $\mathbf{3219}$ directed edges, $\mathbf{304}$ edges, or $\mathbf{9.44}$ per cent, cite later papers, and $\mathbf{39}$ pairs cite each other. The $\mathbf{2913}$ backward, $\mathbf{304}$ forward and $\mathbf{2}$ self-citations sum to $\mathbf{3219}$, so nothing was dropped. The two self-loops are Paper 1 and Paper 2 listing themselves, which are also the two that cite no other paper, so “they cite nothing” needs the qualifier “other than themselves”. The instrument was checked first: an outdated ceiling on paper numbers once silently discarded $168$ citations, exactly the pattern recorded in Paper 400. The cycles sit in one band——there are $\mathbf{393}$ strongly connected components, the largest holding $\mathbf{131}$ papers from number $3$ to $151$, and only $\mathbf{4}$ components have size two or more, the other three totalling $\mathbf{7}$ papers. That band coincides with the period of second and third editions. Outside the cycles lie $\mathbf{389}$ papers, $\mathbf{73.81}$ per cent, so “not acyclic” does not mean “without order”. Ignoring direction there is one component covering everything, with diameter $\mathbf{5}$ and mean shortest path $\mathbf{2.7059}$, and no paper is isolated. The backward edges travel with revision——of the $\mathbf{52}$ papers carrying one, $\mathbf{67.31}$ per cent have a revision log, against $\mathbf{1.26}$ per cent of the other $\mathbf{475}$, a ratio of $\mathbf{53.29}$. Distance one accounts for only $\mathbf{11}$ edges, $\mathbf{3.62}$ per cent, while the median is $\mathbf{62}$ and the maximum $\mathbf{146}$. Reaching $\mathbf{62}$ papers ahead requires touching the text after that paper existed, which renumbering cannot explain. Causation, however, cannot be stated: whether revising added the citation or the citation called for the log is not decidable with these tools. The separator is two clocks——one edge points to the future in writing order and to the past in reference order, and neither reading is the correct one. Paper 118 separated three logical types of time; this adds bibliographic order, and a paper number is a scale for writing order alone. The cycle of Paper 81 is Poincare recurrence, certain but never observed, whereas $\mathbf{39}$ pairs here are closed in fact, so the two are separate roots. Under the criterion of Paper 300, writing order needs a date and reference order needs only the text. To be honest——the quality of the corpus was not measured; many backward citations were called neither good nor bad, and without comparison to other corpora, $\mathbf{9.44}$ per cent cannot be judged high or low. Only the presence of a revision log was examined, never its content. A fit to the citation distribution implies no mechanism. What can be said is that $\mathbf{304}$ edges run against time, that the cycles are concentrated in a band of $\mathbf{131}$ papers, and that backward citation and revision co-occur with a ratio of $\mathbf{53.29}$, and no further.
References
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- Yamagishi, Y. (2026). The Poincare Recurrence Theorem and the Arrow of Time [Paper 81]. Zenodo. 10.5281/zenodo.21733062
- Yamagishi, Y. (2026). Two Clocks and One Scale [Paper 118]. Zenodo. 10.5281/zenodo.22036237
- Yamagishi, Y. (2026). The Two Hundredth Census: Measuring the Corpus as Its Own Data [Paper 200]. Zenodo. 10.5281/zenodo.22108829
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). The Four Hundredth Census: The Instrument Broke Before the Thing It Was Counting [Paper 400]. Zenodo. 10.5281/zenodo.22699343
- Yamagishi, Y. (2026). Does Writing “This Paper Does Not” Evade Refutation, or Aim It? [Paper 526]. Zenodo. 10.5281/zenodo.22867648
Reference
Yamagishi, Y. (2026). The Citations That Run Backwards in Time Are the Trace of Revision [Paper 529]. YOI: https://www.yamagishiyuuki.com/yoi/529
DOI: 10.5281/zenodo.23014099
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 529. This paper is a preprint and has not been peer reviewed.