Integrated trophic position as a proxy for food‐web complexity

There are two distinct approaches to describing the distributions of biomass and species in food webs: one to consider them as discrete trophic levels (TLs); and the other to consider them as continuous trophic positions (TPs). Bridging the gap between these two perspectives presents a nontrivial challenge in integrating biodiversity and food‐web structure. Food network unfolding (FNU) is a technique used to bridge this gap by partitioning the biomass of species into integer TLs to compute three complexity indices, namely vertical (DV), horizontal (DH) and range (DR) diversity (D indices), through decomposition of Shannon's index H′. Using FNU, the food web (a network of species with unique TPs) is converted to a linear food chain (a biomass distribution at discrete TLs). This enables us to expect that the unfolded biomass within species decreases exponentially as the TL increases. Under this condition, the mean TL value in unfolded food chains is hypothesized to have an exponential relationship with the vertical diversity, DV. To explore this, we implemented FNU and calculated D indices for food webs publicly available at EcoBase (n = 158) and calculated the integrated TP (iTP), defined as the biomass‐weighted average TP of a given food web. The iTP corresponds to the mean TL in unfolded food chains and can be empirically measured through compound‐specific isotope analysis of amino acids (CSIA‐AA). Although our analysis is biased towards marine ecosystems, we revealed an exponential relationship between iTP and DV, suggesting that iTP can serve as a measurable proxy for DV. Furthermore, we found a positive correlation between the iTP observed in the total communities (total iTP) and the iTPs of partial communities consisting only of species with 2.0 ≤ TP < 3.0 (partial iTP; r2 = 0.48), suggesting that DV can be predicted using partial iTP. Our findings suggest that the net effect of species diversity, excluding the effect of biomass (corresponding to H′ − DV), on food‐web complexity can be revealed by combining CSIA‐AA with biodiversity analysis (e.g. environmental DNA).


| INTRODUC TI ON
Species interactions, such as prey-predator relationships, are a critical factor in determining ecosystem structure and function (Estes et al., 2011;Halpern et al., 2008;Pauly et al., 1998;Rooney & McCann, 2012).Trophic networks, that is a collection of all prey-predator relationships in a given ecosystem, have been extensively studied within this context (e.g.Christianen et al., 2017;Cucherousset & Villéger, 2015;Fulton et al., 2005;Layman et al., 2007;Shannon et al., 2014;Tam et al., 2017;Thompson et al., 2012;Ulanowicz, 2004;Wang & Brose, 2018).Representing trophic networks with a proxy is one of the most common measures to understand food-web complexities (e.g.Banker et al., 2022;Bascompte, 2009;Bridier et al., 2021;Ings et al., 2009;Thompson et al., 2007).The prey-predator network often focusses on a continuous number that gives each species a unique niche in a food web, namely trophic position (TP) (Thompson et al., 2012).In contrast, the energy-flow pyramid is based on a discrete integer number of times organic matter passes through the mouth of organisms in a linear food chain, namely trophic level (TL) (Lindeman, 1942).
The food network unfolding (FNU) technique (Higashi et al., 1989) provides one way to bring together the focal areas of the preypredator network and the energy-flow pyramid.FNU transforms a food web into a linear food chain when the energy-flow matrix, production and biomass of all species are available.After the food webs are unfolded into food chains, the total biomass in the system is distributed to discrete and integer TLs, with the original species still identifiable (Figure 1).Kato et al. (2018) used this approach to propose three complexity indices (D indices): horizontal diversity, D H (average species diversity within each TL); range diversity, D R (diversity of the TL to which a species belongs); and vertical diversity, D V (diversity of the number of TLs).However, it is practically difficult to estimate the D indices because what is empirically observed in nature are continuous TPs, rather than discrete TLs.The continuous TP is defined as the average length of the energy transfer pathways from the primary producers to the focal species.Although there are several methods for estimating the TP, it is often formulated as follows (Levine, 1980): where TP j is the TP (≥1) of the trophic species j, S is the number of trophic species in the focal food web, and F ij is the fraction of the energy flow from prey i to predator j relative to all the prey species for predator j (i.e. ∑ S i=1 F ij = 1).It is also formulated in unfolded food chains as follows: where k, B i (k) and B i are TL, the biomass of species i belonging to TL k and the biomass of species i, respectively.In traditional studies, ecologists have examined the gut contents of all trophic species belonging to a focal system to determine who eats whom, estimate the species' TP and represent the entire food web (e.g.Briand & Cohen, 1987;Winemiller, 1990).The TP, which is also estimated using stable isotope analysis (SIA) (Wada et al., 1987;Post, 2002), and more recently using compound-specific SIA of amino acids (CSIA-AA, Chikaraishi et al., 2009;Riekenberg et al., 2022), has contributed to addressing a variety of environmental issues, including the sustainable use of fisheries resources, management of endangered or invasive species and biomagnification of toxic substances (Kelly et al., 2007;Pauly et al., 1998;Vander Zanden et al., 1999).Of these, the biomass-weighted average TP of a community (hereafter referred to as integrated TP: iTP) has been a useful proxy for ecosystem status, especially in marine ecosystems that are affected by fishing (e.g.Fulton et al., 2005;Heymans et al., 2014;Pauly et al., 1998;Shannon et al., 2014;Tam et al., 2017).For example, Pauly et al. (1998) suggested that the continuous decline in the iTP of commercial fish worldwide over the late twentieth century was mainly due to selective overfishing of fish with a high TP.iTP is also measurable using CSIA-AA (Ishikawa et al., 2017), which is formulated and converted for unfolded food chain TLs using Equation (2) as follows: where B (k) and B T are the total biomass allocated to TL k, the total biomass of the focal community per unit area or space, respectively.
Here, we propose a methodology for estimating D indices using the empirically measurable iTP.Higashi et al. (1989) suggested that the production (unit: energy per space per time) of an unfolded food chain decreases exponentially with increasing TL, which allows pyramids of biomass (unit: energy per space) to be approximated as an exponential distribution if the difference in turnover time between TLs is negligible.It is hypothesized that there is an exponential relationship between vertical diversity, D V , and the expected value of TLs (i.e.iTP, Equation 3).To test this hypothesis, we analysed food-web data published worldwide and available from the EcoBase repository, which is open for various meta-analyses (e.g.Colléter et al., 2015).Using this dataset, the TP and iTP were calculated (except for TP = 1 species) and D indices were derived using the FNU technique.Additionally, we evaluated the extent to which a partial community consisting of species with low TPs, such as marine zooplankton (hereafter referred to as partial iTP), represents the total community, including all heterotrophic animals from herbivore to top predator (hereafter referred to as total iTP).This will provide essential information for future research in which iTP is estimated using another methodology (e.g.CSIA-AA).

| D indices
The three complexity D indices: horizontal diversity, D H (average species diversity within each TL); range diversity, D R (diversity of the (1) TL to which a species belongs); and vertical diversity, D V (diversity of the number of TLs) are defined as follows (Kato et al., 2018): where (k) and B i are the total biomass allocated to TL k, the total biomass of the focal community per unit area or space, the biomass of species i belonging to TL k and the biomass of species i, respectively.It should be noted that the basal species (TL = 1) are excluded from this calculation.In terms of food web ecology, D H represents functional redundancy, D R is related to the level of intraguild predation, and D V is relevant to the food chain length (FCL; Figure 1).Importantly, D indices provide a link between the trophic transfer of energy and biodiversity.This is because the D indices are strictly related to the Shannon H′ entropy index (Shannon & Weaver, 1963) with biomass as weight factor instead of abundance (Wilhm, 1968) as follows (Han, 1998;Kato et al., 2018):

| Dataset preparation
We accessed EcoBase (http:// ecoba se.ecopa th.org) on 10 September 2021 and downloaded 158 published food webs that provide quantitative diet matrices (i.e.DC ij ) and vectors for production and biomass (i.e.B i ) of all trophic species therein, which are necessary in our analysis.Data from the worldwide food webs were obtained (Figure 2), with most of the data coming from marine ecosystems, except for three cases from freshwater ecosystems (Table S1).Before analysing these data, we made the following assumptions.First, the unassimilated diet fractions were assumed to be identical for all species.Therefore, the diet matrix DC ij provided (4) F I G U R E 1 Graphical abstract conceptualizing the present study that aims to bridge the prey-predator network and the energy-flow pyramid.We analysed 158 published food webs available at EcoBase.Letters denote trophic species.After implementing a food network unfolding (FNU) approach, food webs composed of species with various trophic positions (TPs) were transformed to food chains in which all the biomass fractions were allocated to discrete trophic levels (TLs).Three complexity indices, horizontal diversity (D H ), range diversity (D R ) and vertical diversity (D V ) were computed.We hypothesized that the integrated trophic position (iTP) is a proxy for D V , which can characterize the configuration of biomass pyramids.
in each food-web model was used as a surrogate of the energy-flow matrix F ij as follows: where G ij is the transposed F ij matrix, which is used in the FNU approach (Higashi et al., 1989) described in the next section.Second, the self-loop interactions (i.e.cannibalism) were removed from the original diet matrix and eventually ignored in the TP calculation (i.e. DC ii = F ii = 0).However, in some food-web models in the dataset, single taxonomic species were divided into multiple trophic species with different size classes or growth stages, where cannibalism was sometimes found as a prey-predator interaction within the single taxonomic species.The taxonomic resolution in the EcoBase models tends to be low in low TP organisms such as zooplankton or benthos.Third, the contribution of the microbial loop to energy flow was considered but likely underestimated because (1) few diet matrices registered microbes as trophic species and (2) the taxonomic resolution of microorganisms was not sufficiently high to quantify the detrital trophic pathways in which they play important roles.
It should also be mentioned that the spatiotemporal scale of each food-web model was not standardized in this study.
The TP vectors (y) for S elements (i.e.trophic species) were calculated using the S × S identity matrix (I), the S × S Q matrix (i.e. F ij ), and the 1 vector with S elements using the following absorbing Markov chain, which is derived mathematically from Equation (1) (Levine, 1980): iTP in each food web was calculated using TP, the biomass vectors B i , and Equation (3), excluding primary producers (TP = 1).No species exhibited 1 < TP < 2 (e.g.mixotrophic organisms) in our food-web dataset.
The number of trophic species, S, in our food-web models was 5-91.The Shannon's H′ index was calculated using Equation ( 7) and B i for each food web.The top predator (i.e. a proxy for FCL) was defined as the highest TP found in each food web.

| FNU approach
To implement FNU (Higashi et al., 1989)  , represents the organic matter that has experienced trophic transfer steps for k − 1 times before reaching species i. B i (k) was calculated for 2 ≤ k ≤ n as follows (Kato et al., 2018): Geographical distribution of the food webs analysed in the present study (n = 158).Each point represents the centre of the study area where the food web was modelled.Colour and size gradients correspond to latitude and spatial scale, respectively.
where G ij k−1 denotes G ij to the power of k − 1, and T i and F j0 are throughflow, which corresponds to the production (unit: energy per space per time) per biomass (unit: energy per space) times biomass, P/B × B, in the EcoBase dataset, and the production, P, of the basal species j, respectively.B i and the total biomass allocated to TL k, B (k) , were determined using B i (k) as follows: Thus, the total biomass of the focal community B T is the sum of either biomass of all trophic species in the community or fractional biomass of all unfolded integer TLs, which was formulated as follows: Although most of the unfolded food chains extend infinitely, unless the original food web does not include cyclic pathways of trophic flows, fractional biomass becomes negligible at some point.The necessary and sufficient condition of Equation ( 13) was given using the maximum TL n as follows: In practice, TL was extended to n = 20 in the present study.It should be noted that the unfolded TL can be extended much longer (almost infinitely; Figure S1) than the original folded TP, which is known to be at most 5-6 in nature, because the food web often contains cycles (e.g. species A is consumed by species B, some of which are consumed by species A).

| Biomass distribution, D V and iTP
In this section, we demonstrate the exponential relationship between iTP and D V , assuming that biomass and TL follow an exponential distribution.This is because iTP and D V represent the mean values of the biomass distributions across TL (Equation 3) and the Shannon entropy of the biomass distributions (Equation 6), respectively.
Given that the biomass transfer efficiency between any pair of two neighbouring TLs is <1, the biomass at TL k (B (k) /B T ) in each food web was expected to follow an exponential distribution, which was formulated as follows: where f(k) ex and C are the expected B (k) /B T value and the constant, respectively.Given that the unfolded TL is extended infinitely (i.e. n → ∞ ), the expected iTP and D V values were then determined as follows: Equations ( 17) and ( 18) yielded the following equation that predicts an exponential relationship between the iTP and D V : Equation ( 16) also indicates that the biomass distribution can be estimated only using the iTP because the exponential distribution is characterized by a single parameter, A ex , and C. Thus, Equations ( 15) and ( 16) yielded the following equation characterizing the biomass distribution B (k) /B T using the iTP: The difference between the observed and expected iTPs was assessed using the standardized residual χ as follows:

| Partial iTP
To explore the relationship between total and partial iTPs, we constructed four partial food webs consisting of species with TP from 2.0 to 2.5, 3.0, 3.5 and 4.0, respectively.The partial iTP was calculated using the following equation: where S P and B P represent the number of species and biomass in the partial community, respectively.The partial iTP was then plotted against the total iTP.The slopes and determination coefficients of regression lines (r 2 ) were compared between partial and total iTPs to determine to what extent the partial iTP represented the total food web. (10)

| Statistical analyses and graphing
The effects of latitude and spatial scale (log-transformed) in the analysed food webs on iTP were examined using a general linear model (GLM) with the Gaussian distribution for the residuals.Datasets were prepared using R (R Core Team, 2022), and all analyses were performed using MATLAB 2023a with the Statistics and Machine Learning Toolbox and the Curve Fitting Toolbox (Mathworks, Natick, MA, USA).The R and MATLAB codes are available at the Dryad Digital Repository (https:// doi.org/ 10. 5061/ dryad.4xgxd 25dp).

| RE SULTS
The observed TP (mean ± SD = 3.24 ± 0.78) and iTP (mean ± SD = 2.43 ± 0.21) fell within the ranges 2.00-5.40(n = 4785) and 2.01-3.27(n = 158), respectively (Figure 3).Observed TP was dominated by trophic species with 3.0 ≤ TP < 3.5 (26%), followed by those with 2.0 ≤ TP < 2.5 (22%) and 3.5 ≤ TP < 4.0 (21%).The iTP was most frequently observed as 2.2 ≤ iTP < 2.4 (35%), followed by 2.4 ≤ iTP < 2.6 (28%) and 2.6 ≤ iTP < 2.8 (18%).The GLM revealed that iTP was significantly affected by spatial scale (t = 2.68, p = 0.008) but not by latitude (t = 0.65, p = 0.52) nor by the interaction between spatial scale and latitude (t = −0.91,p = 0.36).A large variation in the biomass distribution of the unfolded food chains was found: TL and fractional biomass (B (k) /B T ) showed exponentially negative relationships (Figure 4).B (k) /B T in most food chains did not converge to zero, even when the TL k was extended to 20, because the original diet matrix, DC ij , contained cycles.However, for all food chains, B (k) /B T became smaller than 10 −10 at k = 18 (Figure 4), suggesting that the cyclic fraction is diminished according to the power law and that the maximum TL set in the present study, n = 20, is sufficient to hold Equation ( 14).Although some of the biomass trajectories towards TL appeared to be nonlinear until TL k = 5, most became linear at k > 5 (Figure 4).data exhibited distributions that were close to exponential distributions (Figure S5).Therefore, our prediction that biomass will exponentially decrease across unfolded TLs was supported.
The correlation between iTP and range diversity, D R , was also strong (adjusted r 2 = 0.5474).Therefore, the iTP detects the food chain length (FCL), corresponding to D V , and the average omnivory in the system, corresponding to D R .Other correlations among horizontal diversity, D H (adjusted r 2 = 0.1612), the conventional Shannon diversity index, H′ (adjusted r 2 = 0.2739), and iTP were also positive, but these were not as strong as those mentioned above.
According to the results of exponential fitting (Equation 22), we set the constant C at 1.5 (Equation 18).Regarding Equation ( 16), the observed iTP − 1.5 values fairly explained the exponential fitting parameter (i.e.A ex ) for TL and biomass, although they tended to deviate from the 1:1 line when the iTP was low or high (Figure 6a).Most χ values of the exponential fitting (χ ex ) were <0.2 when the observed iTP was <3.0 (Figure 6b).The χ ex values <0.2, <0.1, and <0.05 were 96%, 67% and 28%, respectively.If a distribution completely follows the exponential distribution, all of its information should be recoverable from its partial information.Therefore, we attempted to reconstruct the iTP using the biomass distribution at low TLs and the central difference equation and compared them with partial iTP (see Supporting Information and Figures S7 and S8).
We compared the partial iTP with the iTP of the original food web, that is the total iTP (Figure 7).When trophic species with TPs ≥2.5 were excluded from the partial food web, the regression line between partial iTP (y) and total iTP (x) was given as follows: y = 0.126 x + 1.80 (r 2 = 0.10; p < 0.001).The slope and r 2 value of the regression line increased as more species were included in the partial food web: y = 0.386 x + 1.29 (r 2 = 0.48; p < 0.001) for 2.0 ≤ TP < 3.0; y = 0.720 x + 0.603 (r 2 = 0.89; p < 0.001) for 2.0 ≤ TP < 3.5; and y = 0.899 x + 0.223 (r 2 = 0.98; p < 0.001) for 2 ≤ TP <4.0 (Figure 7).When more species with a higher TP were included in the partial food web, the slope became larger and closer to 1. Thus, partial iTP might be a proxy for total iTP, and the precision of the total iTP estimation increases with the increasing r 2 value in a partial food web that accommodates more species with higher TPs.

| Overview
In our analysis using 155 marine and 3 freshwater food webs, the FNU revealed that the D V value can be estimated using iTP, as the biomass fraction decreases exponentially as the TL increases.
Although the observed TPs were 2-5, the highest iTP was 3.27, suggesting that the biomass of primary and secondary consumers is responsible for the observed iTP.In fact, the iTP of the partial community, which includes trophic species with TP ≤ 3.0, explains the iTP for 48% of the total community.This shows that collecting an operationally definable partial community is useful for estimating iTP in natural ecosystems, as most biomass is concentrated in low TPs.
The variation in iTPs found in the food web (2.43 ± 0.21; n = 158) was similar to that determined using the compound-specific isotope analysis of amino acids (CSIA-AA) method for stream macroinvertebrate communities (2.60 ± 0.13; n = 15; Ishikawa et al., 2017) and marine mesozooplankton communities (2.33 ± 0.34; n = 14; Ishikawa, Tadokoro, et al., 2023).The results suggest that the partial iTP should be used instead of the total iTP, since holistic and quantitative sampling of organisms from bacteria to whales in marine ecosystems is unrealistic.Furthermore, the iTP index is compa-  16) and ( 20).Black line indicates 1:1.Colour and size are same as Figure 2.
estimations based on the CSIA-AA method.This allows for generalization of our findings, despite the fact that the food web based on stomach content analysis ('who eats whom' networks) may not necessarily be identical to those based on SIA ('who assimilates whom' networks) (Nielsen et al., 2018;Ishikawa, 2018).
However, few studies have linked iTP-like indices to both biodiversity and food-web complexity, which will be key to filling the gap between the prey-predator network and the energy-flow pyramid.
Moreover, indices equivalent to iTP have been largely neglected in stable isotope ecology in favour of other indices, such as FCL, which have played an important role for more than two decades (Post et al., 2000).This neglect was in part due to the unavoidable calibration of isotopic baselines when using the bulk tissues of organisms in the conventional method.However, such calibration can now be avoided in iTP through the use of advanced methods such as CSIA-AA (Ishikawa, 2018).

| FNU and D indices
To our knowledge, the present study is the first in which FNU was applied to transform a complex food web into a simple and linear food chain.Kato et al. (2018) partially applied FNU to stream macroinvertebrate TPs and biomass data and thereby computed D indices under the assumption that every trophic species belongs to at most two neighbouring TLs.Using the EcoBase dataset, the application of FNU was generalized by relaxing this assumption.It is found that many species actually register their biomass on more than two TLs (Thompson et al., 2007) because they consume prey with different TPs (i.e.omnivory, as indicated by D R ); therefore, their biomass is partitioned into multiple TLs.
The correlations among D indices and H′ were similar to those found by Kato et al. (2018), who studied stream macroinvertebrates, that is (1) a strong positive correlation between D H and H′ and (2) a positive correlation between D V and D R (Figure S6).

| iTP, D indices and biomass distribution
The iTP index can serve as an excellent proxy for D V .D V can be interpreted as the entropy of TLs, which increases as the trophic transfer of biomass increases.This notion is also supported by the significant correlation between iTP and top predator TP (another proxy for FCL) (Figure S9).The variation in iTP and D V is a result of the variation in the parameters of the exponential fitting function for biomass and TL.Equation ( 16) indicates that the biomass distribution can be estimated solely with the iTP-derived fitting parameter A ex .Reconstructions of iTP from A ex values are highly precise (χ ex < 0.2) when 2.2 < iTP < 2.8, but precision decreases when iTPs fall outside this range.This is because an increase in iTP is associated with (1) a biomass distribution biased towards higher TLs and (2) an increase in deviation from the exponential fitting.
Other distributions, such as the Weibull distribution, may be more suitable for reconstructing the biomass distribution across TLs (see Supporting Information).In such cases, however, another metric, for example the biomass or trophic throughflow of each species, should also be considered.
The taxonomic resolution in the EcoBase food-web data clearly differed among TPs, which is known to have potential effects on food-web analysis (Cohen et al., 1993;Martinez, 1993).For example, some food webs provided remarkably low taxonomic resolution of trophic species (i.e.organisms grouped into an approximate biological category such as order or class).Because most of the foodweb models in EcoBase were designed for fisheries management (Colléter et al., 2015), fish, mammals and seabirds tended to be more taxonomically resolved than zooplankton or benthos.Thus, trophic links may be missing within species with a low TP that are combined into a single trophic species.Furthermore, this difference in taxonomic resolution affects the computation of D H and D R but not D V , depending on whether the species identity is considered.D V is independent of species identity (i.e.i) because neither B i nor B i (k) is included in its definition.Therefore, the strong positive correlation between iTP and D V indicates that iTP is robust against variation in taxonomic resolution.This is consistent with the findings of previous studies in which taxonomic resolution of food webs had limited effects on FCL (Thompson & Townsend, 2000).
In our previous studies, the correlation between iTP and H′ was negative (Ishikawa et al., 2017;Ishikawa, Tadokoro, et al., 2023), which is inconsistent with the present study.However, the positive correlation between iTP and H′ was more variable than that between iTP and D V or iTP and D R .This may have arisen because (1) the effects of D V and D R on H′ were offset and (2) H′ was mainly controlled by D H (Kato et al., 2018), which also had a weak positive correlation with iTP.These results suggest that species diversity affects mainly the food-web structure through biomass distribution within a single TL.In other words, an increase in species diversity mainly concentrated at the same TL does not make a substantial contribution to the increase of FCL (Figure S10).Furthermore, the dataset analysed in the present study was much larger (n = 158) than in our previous studies (n = 15: stream macroinvertebrates; n = 14: marine mesozooplankton).Thus, the negative correlation between iTP and H′ reported in the previous studies may be hidden in the more general pattern reported in the present study.The positive correlation between the iTP and the omnivory index, D R , is also inconsistent with the trophic omnivory hypothesis (Ishikawa et al., 2017), according to which predator omnivory is expected to decrease their own

| Limitations
We acknowledge that the present study had the following limitations.First, all the original data used in this study were downloaded from EcoBase, which may include some issues, such as auto-balancing of biomass data and mass-balance assumptions that rely on constant parameters such as production, respiration and mortality (Christensen & Pauly, 1992) in Ecopath, and citation of diet data from the literature.These factors could potentially introduce errors in our subsequent analysis.Second, although the detritus was operationally considered TP = 1, this assumption may be inappropriate, given that the detritus is a mixture of living and nonliving organic matter in an ecosystem.Both the present study and previous studies were strongly biased towards aquatic realms that are affected by human disturbances, such as fishing; therefore, future studies should address other realms, including deep-sea and terrestrial ecosystems, in which detritus plays a more important role in trophic transfer (Hagen et al., 2012;Vetter, 1995).Since the trophic structure is often different between ecosystem types (Riede et al., 2011;Shurin et al., 2002), the findings of this study may not be simply applied to other ecosystems.Third, it was assumed that  Hernández et al., 2017), and other species interactions that drive trophic transfer of energy, such as host-parasite relationships (Lafferty et al., 2006), were partially but not fully considered in our study.These unexplored research areas could be addressed using cutting-edge toolboxes such as CSIA-AA (Riekenberg et al., 2021).

| Bias that may have affected the results
We are unable to rule out the possibility that the aforementioned bias using EcoBase led to our finding of the exponential relationship between iTP and D V .The finding is dependent on the exponential distribution of biomass along TLs, which is not necessarily modelled in some previous studies (e.g.Barbier & Loreau, 2019;Fath & Killian, 2007;Wang et al., 2009).If the biomass distribution significantly deviated from the exponential distribution (e.g.due to resource subsidy, trophic cascade or variation in turnover time), the exponential relationship between iTP and D V would be invalid.In this case, it is not conclusive that iTP is a proxy for food-web complexity.Although we analysed aquatic and marine ecosystems where the top-heavy and nonexponential biomass distributions are sometimes observed (e.g.Gasol et al., 1997;McCauley et al., 2018;Mourier et al., 2016;Trebilco et al., 2016), the results show that the biomass distribution is neither top-heavy nor nonexponential in the EcoBase dataset.We are still unsure whether this is because of the bias using EcoBase or because of the FNU conversion from TPs to TLs.However, it should be mentioned that biomass distributions across unconverted TPs do not follow exponential distributions in the EcoBase dataset (Figure S4).Since the present study is among the first to use the FNU approach to convert TPs into TLs, more studies are needed to test whether nonexponential biomass distributions across continuous TPs are restructured to exponential biomass distributions across discrete TLs in a variety of ecosystems.
We speculate that our finding might be rather reinforced by involving terrestrial ecosystems where the bottom-heavy and exponential biomass distributions are often observed.

| CON CLUS IONS
The partial iTP can be empirically determined using CSIA-AA.In practice, the partial community would involve targeting species with low TP, such as invertebrates.Although such species are abundant in terms of biomass in most ecosystems, the present study suggests that the effects of species with a higher TP on total iTP are not negligible.
One solution would involve using a regression line for the partial and total iTPs (Fig. 7).For example, if the stream macroinvertebrate and marine mesozooplankton communities assessed in our previous studies were assumed to accommodate trophic species with TP of 2.0-3.0, the mean values of the observed partial iTP, 2.60 and 2.33, respectively, would have been calibrated to the total iTP of 2.78 and 2.40, respectively, using the regression line.Furthermore, these calibrated values fall within the 2σ error of the mean iTP (2.43 ± 0.42, n = 158) observed in the present study, although the precision (i.e.r 2 value) of the regression line should be carefully considered.Another approach to estimate the total iTP is using the central difference equation that is applicable to exponential distributions (see Supporting Information).
However, the deviation from expectation increases with increasing iTP because the more biomass distributed at higher TLs, the less precise the exponential fitting becomes.If these problems can be overcome by successfully scaling the partial community up to the total community, the iTP index has the potential to bridge stable isotope ecology, biodiversity informatics and food-web science.Given that iTP is a proxy for D V , both D V and Shannon's index H′ can be measured simultaneously by developing a new research network that monitors both iTP and environmental DNA.In particular, the value H′-D V , which corresponds to D H -D R , is expected to characterize the net effect of species diversity, excluding the effect of biomass, on food-web complexity.In other words, H′-D V indicates the excess of biodiversity that does not contribute to the trophic transfer efficiency.For example, H′-D V might be useful in examining the effect of invasive species, which contribute to community biomass but do not contribute to trophic flow, as they are not consumed by others, on ecosystem stability.This information will fill the gap between the prey-predator network and the energy-flow pyramid.

S U PP O RTI N G I N FO R M ATI O N
Additional supporting information can be found online in the Supporting Information section at the end of this article.
Table S1: Metadata of food-web models downloaded from EcoBase on 10 September 2021.
), three complexity indices (D H , D R and D V ) were derived (Equations 4-6, Kato et al., 2018).It should be noted that (1) the obtained D H , D R and D V values

Figure
Figure S4 and Figure 4 contrast the tangled biomass distributions of species over the TP gradient (Figure S4, before FNU) with linear trajectories of fractional biomass across TLs (Figure 4, after FNU).
iTP was positively correlated with vertical diversity, D V .We performed an exponential fitting of the data in Figure5c, obtaining the following equation (n = 158; adjusted r 2 = 0.9538):where mean a and b (with 95% prediction bounds) were −0.8998 (−0.9345 and −0.8651) and 1.5 (1.467 and 1.533), respectively, which are fairly consistent with Equation (18).As a slight deviation was observed in a (i.e.expected to be −1), the observed data were not entirely controlled by an exponential function.When fitted with the Weibull distribution, a generalized form of the exponential distribution, most (22) iTP = exp a + D V + b, F I G U R E 3 Frequency of (a) the TP (n = 4785) and (b) the iTP (n = 158) of the food webs analysed.Distribution of biomass fractions (biomass at trophic level k, B (k) , divided by the total biomass, B T , on a log scale) across unfolded trophic levels (2 ≤ k ≤ 20).Colour is same as Figure 2. versus three complexity indices, that is (a) horizontal diversity (D H ), (b) range diversity (D R ), (c) vertical diversity (D V ) and (d) Shannon's diversity index H′.Colour and size are same as Figure 2.
rable among different methodologies, such as direct observations or mass-balanced models based on gut content analysis, and indirect F I G U R E 6 Plots for (a) the exponential fitting parameter (A ex ) versus the integrated trophic position (iTP) − 1.5 and (b) the standardized residual χ of the exponential fitting parameter (χ ex ) versus the iTP.See Equations (

F
Regression lines with 95% confidence bounds for the partial integrated trophic position (iTP) (red: 2.0 ≤ TP < 2.5; purple: 2.0 ≤ TP < 3.0; orange: 2.0 ≤ TP < 3.5; green: 2.0 ≤ TP < 4.0) versus the total iTP.See Equation (21).These results suggest that the H′ is mainly determined by the D H , whereas the D V and D R represent the complexity of food webs that is not provided by H′.AlthoughKato et al. (2018) suggested that D indices are sensitive to environmental changes (e.g.anthropogenic disturbance), structural change in food webs had previously not been characterized using D indices.Our results suggest that D V is a proxy for the configuration of biomass distributions across TL because a sharper decrease in biomass along the TL leads to a decrease in D V value and eventually shortens the FCL.However, estimating the D V , D H and D R indices remains empirically difficult because it requires both the species diet matrix and their production and biomass, which cannot be obtained without conducting an extensive examination of all members of the focal food web.On the contrary, the iTP index, which is estimated using partial iTP via CSIA-AA, might offer a more practical solution, as discussed in the following sections.
TPs.As shown by the strong positive correlation between D V and D R , an increase in omnivory is associated with an increase in the top predator TP (i.e.FCL).However, unlike the FCL index D V , D R holds the identity of the species in its definition.Therefore, D R is sensitive to the taxonomic resolution of the original EcoBase dataset, which could affect the observed correlation between iTP and D R .
the energy flow in each food web is in a steady state.The modelled areas ranged from local bays to the open ocean, depending on the original studies archived in the EcoBase repository.If ecosystems are considered in a more open and seamless manner, for example by considering animal migration, resource subsidies and seasonality, the present analysis could be extended to the construction of a dynamic food-web model on a larger spatiotemporal scale.Finally, size-dependent diet preference or ontogenetic diet shift, which are known to alter TP, especially those of carnivorous species (Sánchez-

Figure S1 :
Figure S1: Distribution of biomass fractions (biomass at trophic level k, B(k) , divided by the total biomass, B T , on a log scale) across unfolded trophic levels (2 ≤ k ≤ 100).Color is same as Figure2.

Figure S4 :
Figure S4: Biomass spectra obtained from TP data without FNU in case where (a) all the species and their own biomass are regarded as nodes and (b) species belonging to a certain TP range are grouped and their biomass was summed.Color is same as Figure 2.

Figure S5 :
Figure S5: Shape parameter (A wb ) for the Weibull function fitted to the biomass distributions across trophic levels in each of the 158 food webs.

Figure S6 :
Figure S6: Correlation matrix for the three D indices, i.e., horizontal diversity (D H ), range diversity (D R ), and vertical diversity (D V ) and Shannon's diversity index H'.

Figure S7 :
Figure S7: Plots for (a) the exponential fitting parameter A ex reconstructed using the central difference equation vs. the observed integrated trophic position (iTP) − 1.5 and (b) the standardized residual χ of the central difference equation (χ cd ) vs. the iTP.Red line indicates 1:1.

Figure S8 :
Figure S8: Correlation between the partial integrated trophic position (iTP) calculated from biomass and trophic position (TP) of species with 2 ≤ TP ≤ 3 and the iTP cd calculated from the central difference equation using the unfolded biomass assigned to trophic level (TL) = 2 and TL = 3.