Did unsupported citations about fungal networks double? It depends on the weighting

Karst, Jones and Hoeksema reported in 2023 that citations crediting forest fungal network studies with more than their results support had doubled in the past 25 years. That sentence is now part of a public dispute. We reran the claim on the table the authors deposited, under a plan hash-locked before the table was analysed. The table holds 268 study-year rates for 11 focal studies on network function and 7 on network structure, and no denominators, which we rebuilt from the paper's supplement. On the authors' negative binomial model, unsupported citations rose by a factor of 2.19 for network function (0.86 to 4.14) and 2.84 for network structure (1.18 to 6.62).c1 Weighted by citations, the rise was 1.77 for function (0.52 to 2.99) and 1.54 for structure (0.85 to 5.81), and neither interval excludes 1.c2 Inside individual focal studies the rate rose for both: 3.0 times for function (p 0.001) and 3.5 times for structure (p 0.009).c3 With this many focal studies, a true doubling gives an interval clear of 1 only 0.425 of the time, so an interval that includes 1 is no evidence against doubling.c4 "Doubled" fairly describes our refit of the authors' model; their own fitted curve was not available to us. Counted as a share of all citations the rise is smaller and less certain.

A common mycorrhizal networkDefinition: common mycorrhizal network forest ecology and mycology A set of fungal threads that colonise the roots of two or more plants at once and so may link them below ground. Excludes claims about what such a network does, such as moving carbon or signals between trees, which are the matters in dispute. full definition is a set of fungal threads that colonise the roots of two or more plants at once. Popular accounts have described such networks as channels through which trees share carbon and warnings. Karst, Jones and Hoeksema [1] reviewed the field evidence and the way it is cited. They read the later citations of 18 field studies, which we call focal studiesDefinition: focal study our usage, following Karst and colleagues One of the 18 field studies on these networks whose later citations the original authors read and coded, 11 on network function and 7 on network structure. Excludes the citing papers, which are the units counted inside each focal study. full definition, and coded each citation as supported or not by what the focal study found. In their abstract they report that unsupported citationsDefinition: unsupported citation citation analysis, as coded by Karst and colleagues A citation that credits a focal study with a claim that the original authors judged the study's results do not support. Excludes any judgement of ours: every coding in this paper is the original authors'. full definition "have doubled in the past 25 years". An Author Correction followed [2].

The paper drew replies. Klein and colleagues [4] accepted that some of its criticism is justified and argued that evidence for carbon transfer between trees through these networks is solid and growing. Simard, Ryan and Perry [5] replied to three of its critiques; among other points they called its rationale for counting citations by year vague. Neither reply, as far as we read them at abstract and summary level, reanalysed the deposited numbers.

The doubling figure is a statistic, and the authors deposited the table behind it under CC0. That makes it checkable at low cost. Our question was narrow: on that table, does the rate of unsupported citations double over the study period for both kinds of claim, and does the answer hold when each year is weighted by how many citations it holds? The decision the reread informs is how our own work may quote the figure. Searches run on 9 November 2026 for a numeric reanalysis of the deposit returned none. That is what the searches returned; it does not show that none exists.

The preregistrationDefinition: preregistration research method (metascience) A dated written plan for a study, fixed and hash-locked before any result is seen: the question, the data, the statistic, the controls, the forecasts and the kill conditions. Excludes a plan revised after looking at the outcome, and a protocol written only for the record after the run. full definition was hash-locked at 06:07:12 UTC on 31 October 2026, before the table was analysed. It was not blind, and it lists what had been seen: the deposit's metadata in full, the header row of the table, and one data row printed by accident while checking the header. From an earlier reading of ours we also knew the per-year numerators in the paper's supplementary Note 3 and the method in its Note 2: a negative binomial regression of the percentage on indexed year. The article body and its main figure were behind a paywall and unread, so the authors' fitted curve and their own definition of doubled were not available to us.

The plan fixed a definition. Doubled means: the ratio of the fitted rate at the end of each group's span to the fitted rate at its start has a 95 percent interval, from resampling focal studies, that excludes 1 and covers 2, and the point estimate beats year-shuffled rates. Function and structure are read separately.

The plan named five measures, set out in the methods, with a verdictDefinition: verdict research method (our usage) The pass or fail of one preregistered rule set against the run's numbers, read off by code. Excludes an interpretation written after reading the numbers. full definition for each and a calibration on planted data. Kill conditionsDefinition: kill condition research method (our usage, after Popper's falsification) A rule written in the preregistration that, when met, decides against the claim with no room for judgement after the run. Excludes reasons found after the run for setting a result aside. full definition, without discretion:

The plan also recorded our own forecasts, set against the outcomes in Table 2.

The data are the authors' Dataverse deposit [3], released under CC0. Its table gives, for each focal study and each citing year, the study-year rateDefinition: study-year rate our usage for the deposited table The share of one focal study's citations in one calendar year that were coded unsupported, as stored in the deposit without its denominator. Excludes a count of citations, which the deposit does not hold. full definition of unsupported citations. A "Count" column holds 100 times the rate, a percentage. The file holds 272 rows; 4 had a blank rate and were skipped and counted, leaving 268 rates: 158 for 11 function studies over 1997 to 2022 and 110 for 7 structure studies over 1999 to 2022. The year 2022 is incomplete.

The table holds no denominators. We built each rebuilt denominatorDefinition: rebuilt denominator our usage in this study The number of citations a focal study received in a year, recovered by finding the whole number n for which a published count k gives the deposited rate k over n. Excludes a count read directly from any source; years with a rate of zero share what remains of the study's total and are the least certain. full definition from the per-year numerators printed in the paper's supplementary Note 3: for a nonzero rate and a known numerator $k$, the whole number $n$ with rate $k/n$. Where no whole number fits, the reduced fraction's denominator was used as a lower bound. Years with a rate of zero carry no numerator information and shared what remained of the study's reviewed total, the weakest step in the rebuild. The supplement is copyrighted, so this paper reports only derived values from it.

Two checks came first. The integrity check required the Count column to equal 100 times the rate, the expected numbers of studies, and group means of the yearly rate within 1.5 points of the means stated by the authors, 32.2 and 14.8 percent. The rebuild check required at least 0.7 of nonzero rates to give a whole-number denominator and a median gap of at most 0.2 between rebuilt numerators and the totals in the paper's tables. A failed rebuild would void only the citation-weighted measure.

Following Note 2, a negative binomial modelDefinition: negative binomial model statistics (count regression) A regression for counts whose variance can exceed their mean, fitted here with a log link so that the fitted rate changes by a fixed factor per year. Excludes models that weight each year by how many citations it holds. full definition with a log link was fitted to the percentage in each study-year on the indexed year $t$, using every row including the incomplete 2022:

The ratio over a group's span $T$ (25 years for function, 23 for structure) is

The same fit without 2022 is reported beside it. This model gives every study-year equal weight, a year with one citation as much as a year with thirty.

New focal studies enter the pool over time, so a pooled rise could come from which studies are cited when. The within-study modelDefinition: within-study model statistics (fixed effects) A regression with its own baseline level for each focal study, so the year slope is estimated only from change inside studies. Excludes change that comes from which focal studies happen to be cited in early and late years. full definition gives each focal study $s$ its own level:

fitted as a quasi-Poisson fitDefinition: quasi-Poisson fit statistics (count regression) A log-linear count regression whose standard errors are scaled by the observed spread, used here for the within-study model. Excludes a full probability model for the counts. full definition. It passes if the slope is above 0 and above the 95th percentile of its year-shuffled control.

The citation-weighted share modelDefinition: citation-weighted share model statistics (logistic regression) A regression of the number of unsupported citations out of all citations in each study-year on the year, so a year with thirty citations counts thirty times as much as a year with one. Excludes the equal weight per study-year that a model of the deposited percentages gives. full definition fits the rebuilt count $k$ of unsupported citations out of $n$ in each study-year by a binomial logistic regression on year:

and reports the ratio of the fitted share at the last year to the share at the first:

Intervals for the ratios come from study resamplingDefinition: study resampling statistics (cluster bootstrap) Drawing focal studies with replacement, refitting the model each time, and taking the middle 95 percent of the refitted ratios as the interval. Excludes resampling single study-years, which would ignore that citations of one study move together. full definition, because citations of one focal study move together. Reported without a verdict: the pooled share in each group's first ten years against 2012 to 2021, the rate in the first five indexed years against the last five up to 2021, and growth in total citations over the last five years against the five before.

The year-shuffled controlDefinition: control statistics and experimental design A version of the data with the structure under test removed (for example shuffled or pooled), scored the same way as the real data, so the real score has something to beat. Excludes a comparison with a different dataset or with a published number. full definition permutes the rates across years inside each focal study, 1,000 times, and refits the authors' model and the within-study model. A real ratio or slope must beat the control's 95th percentile. Its p value is a p value from drawsDefinition: p value (from draws) statistics The share of control draws scoring at or above the real score, with one added to the count and to the number of draws, so it is never zero. Excludes a p value from a textbook formula that assumes a distribution. full definition, so with 1,000 draws the smallest possible value is 1 in 1,001.

The planted checkDefinition: planted check research method; called a positive control in experimental science Data built so that the right answer is known, run through the same analysis before the real data, to show that the analysis can see the effect when it is there. Excludes any check made on the real data, which cannot show what the analysis misses. full definition kept the real studies, years and rebuilt denominators and drew new counts from a known truth: a study effect (lognormal, standard deviation 0.4) and a true ratio of 1 or 2 over the span. It ran 60 simulations per group per truth, 120 per truth in all, with 150 study resamples each. The calibration passed if a flat truth was called a rise at most 0.15 of the time, the doubled truth gave a median ratio between 1.6 and 2.4, and its interval covered the truth at least 0.8 of the time.

The authors' stated means were recovered. Row means came out at 33.2 and 15.3 percent against the stated 32.2 and 14.8, within the 1.5 point allowance. The mean of study means gave 32.3 and 14.7, which identifies the definition the authors used.

The rebuild passed: 143 of 169 nonzero rates, 0.846, gave a whole-number denominator, and the rebuilt numerators matched the paper's table totals with a median gap of 0. The per-study totals are in Appendix B.

The calibration met its preregistered bar. With a flat truth the pipeline called a rise 0.092 of the time. With a doubled truth the median ratio was 2.11 and the interval covered the truth 0.89 of the time, against a bar of 0.8. The intervals are narrower than their stated level: they are 95 percent intervals, and they covered a planted doubling about 0.89 of the time where 0.95 was claimed. Read every interval on this page as somewhat too narrow. The calibration also measured statistical powerDefinition: statistical power statistics The chance that a test detects an effect of a given size when that effect is real, measured here on planted data with a known doubling. Excludes the chance that an effect is real given that the test detected it. full definition: with a true doubling, the interval excluded 1 only 0.425 of the time.

Table 1 gives the ratios from all three models and Figure 1 draws them.

On the authors' negative binomial model, unsupported citations rose by a factor of 2.19 for network function (0.86 to 4.14) and 2.84 for network structure (1.18 to 6.62).c1 Both beat their year-shuffled controls. By the letter of the plan, function fails because its interval reaches below 1, and structure passes. Dropping the incomplete 2022 gave 2.26 and 2.43.

Inside individual focal studies the rate rose for both: 3.0 times for function (p 0.001) and 3.5 times for structure (p 0.009).c3 The slopes, 0.044 and 0.054 per year, beat their shuffled cuts of 0.022 and 0.036. The rise does not come from which focal studies happen to be cited early and late.

Weighted by citations, the rise was 1.77 for function (0.52 to 2.99) and 1.54 for structure (0.85 to 5.81), and neither interval excludes 1.c2 The fitted share went from 0.205 to 0.363 for function and from 0.113 to 0.175 for structure. In pooled shares, function went from 21.5 percent in its first ten years to 34.9 percent in 2012 to 2021, about 1.6 times; structure went from 13.6 to 14.7 percent, about 1.1 times. Figure 2 shows why the weighting matters: the early structure years hold few citations each.

The preregistered kill text for structure applies. Its pass on the authors' model leans on thin early years: the first five structure years, 1999 to 2003, each hold about 3.1 rebuilt citations and a rate of zero. On five-year windows, function's rate went from 0.253 to 0.377, a ratio of 1.49; structure's first five years are all zero, so its ratio has no finite value.

Total citations also grew. Over the last five years against the five before, function's citations grew 1.86 times and structure's fell to 0.85 times.

With this many focal studies, a true doubling gives an interval clear of 1 only 0.425 of the time, so an interval that includes 1 is no evidence against doubling.c4 The failures of the citation-weighted verdicts show that the data cannot confirm a doubling on that weighting. They do not show that the rise was smaller than double.

The reread supports the direction of the original claim and narrows its size. On the authors' own model, "doubled" is a fair description of the fitted curve, and the within-study fits show the rise inside the cited studies, the strongest part of their case. The size depends on a choice of unit. The deposited Count column is a percentage fitted as a count, so each study-year carries equal weight. A year with one citation coded unsupported counts as much as a busy year. Weighted by citations, function rose by about 1.6 to 1.8 times and structure by 1.1 to 1.5 times.

The power figure governs how to read the failures. With 11 and 7 focal studies, even a true doubling usually fails to give an interval clear of 1. The reread therefore does not refute a doubling. It shows that the deposit cannot pin the size down closely on either weighting.

For citing practice we take one rule from this: when a rate is modelled as a count, report the citation-weighted ratio beside it. The sentence we use for the claim is "rose, about doubled on the authors' model", with the interval.