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Mercury additions to the study catchmentMETAALICUS was conducted on the Lake 658 catchment at the Experimental Lakes Area (ELA; now IISD-ELA), a remote area in the Precambrian Shield of northwestern Ontario, Canada (49° 43′ 95″ N, 93° 44′ 20″ W) set aside for whole-ecosystem research31. The Lake 658 catchment includes upland (41.2 ha), wetland (1.7 ha) and lake surface (8.4 ha) areas. Lake 658 is a double basin (13 m depth), circumneutral, headwater lake, with a fish community consisting of forage (yellow perch (P. flavescens) and blacknose shiner (Notropis heterolepis)), benthivorous (lake whitefish (C. clupeaformis) and white sucker (Catostomus commersonii)), and piscivorous (northern pike (E. lucius)) fishes. The lake is closed to fishing.Hg addition methods used in METAALICUS have been described in detail elsewhere19,32,33. In brief, three Hg spikes, each enriched with a different stable Hg isotope, were applied separately to the lake surface, upland and wetland areas. Upland and wetland spikes were applied once per year (when possible; Fig. 1a) by fixed-wing aircraft (Cessna 188 AGtruck). Mercury spikes (as HgNO3) were diluted in acidified water (pH 4) in a 500 l fiberglass tank and sprayed with a stainless-steel boom on upland (approximately 79.9% 200Hg) and wetland (approximately 90.1% 198Hg) areas. Spraying was completed during or immediately before a rain event, with wind speeds less than 15 km h−1 to minimize drift of spike Hg outside of target areas. Aerial spraying of upland and wetland areas left a 20-m buffer to the shoreline, which was sprayed by hand with a gas-powered pump and fire hose to within about 5 m of the lake32. Average net application rates of isotopically labelled Hg to the upland and wetland areas were 18.5 μg m−2 yr−1 and 17.8 μg m−2 yr−1, respectively.The average net application rate for lake spike Hg was 22.0 μg m−2 yr−1. For each lake addition, inorganic Hg enriched with approximately 89.7% 202Hg was added as HgNO3 from four 20-l carboys filled with acidified lake water (pH 4). Nine lake additions were conducted bi-weekly at dusk over an 18-week (wk) period during the open-water season of each year (2001–2007) by injecting at 70-cm depth into the propeller wash of trolling electric motors of two boats crisscrossing each basin of the lake32,33. It was previously demonstrated with 14C additions to an ELA lake that this approach evenly distributed spike added in the evening by the next morning34.We did not attempt to simulate Hg in rainfall for isotopic lake additions because it is impossible to simulate natural rainfall concentrations (about 10 ng l−1) in the 20-l carboys used for additions. Instead, our starting point for the experiment was to ensure that the spike was behaving as closely as possible to ambient surface water Hg very soon after it entered the lake. Several factors support this assertion. By the next morning each spike addition had increased epilimnetic Hg concentrations by only 1 ng l−1 202Hg. Average ambient concentrations were 2 ng l−1. Thus, while the Hg concentrations in the carboys were high (2.6 mg l−1), the receiving waters were soon at trace levels. Furthermore, we investigated if the additions altered the degree of bioavailability or photoreactivity of Hg(ii) in the receiving surface water. We examined the bioavailability of spike Hg(ii) as compared to ambient Hg in the lake itself using a genetically engineered bioreporter bacterium35. On seven occasions, epilimnetic samples were collected on the day before and within 12 h of spike additions. The spike was added to the lake as Hg(NO3)2, which is bioavailable to the bioreporter bacterium (detection limit = 0.1 ng Hg(ii) l−1), but we never saw bioavailable ambient or spike Hg(ii) in the lake, presumably because it was quickly bound to dissolved organic carbon (DOC). This indicates that, in terms of bioavailability, the spike Hg was behaving like ambient Hg soon after additions. Photoreactivity in the surface water was examined on seven occasions, by measuring the % of total Hg(ii) that was dissolved gaseous Hg for spike and ambient Hg, either 24 h or 48 h after the lake was spiked36. There was no significant difference (paired t-test, P > 0.05), demonstrating that by then the lake spike was behaving in the same way as ambient Hg during gaseous Hg production.Lake, food web and fish samplingWater samples were collected from May to October every four weeks at the deepest point of Lake 658. Water was pumped from six depths through acid-cleaned Teflon tubing into acid-cleaned Teflon or glass bottles. Water samples were filtered in-line using pre-ashed quartz fibre filters (Whatman GFQ, 0.7 µm). Subsequently, Hg species were measured in the filtered water samples (dissolved Hg and MeHg) and in particles collected on the quartz fibre filter (particulate Hg and MeHg).From 2001 to 2012, Lake 658 sediments were sampled at 4 fixed sites up to 5 times per year. Sampling frequency was highest in 2001, with monthly sampling from May to September, and declined over the course of the study. Fixed sites were located at depths of 0.5, 2, 3 and 7 m. A sediment survey of up to 12 additional sites was also conducted once or twice each year. Survey sites were selected to represent the full range of water depths in both basins. Cores were collected by hand by divers, or by subsampling sediments collected using a small box corer. Cores were capped and returned to the field station for processing within a few hours. For each site, three separate cores were sectioned and composited in zipper lock bags for a 0- to 2-cm depth sampling horizon, and then frozen at −20 °C.Bulk zooplankton and Chaoborus samples were collected from Lake 658 for MeHg analysis. Zooplankton were collected during the day from May to October (bi-weekly: 2001–2007; monthly: 2008–2015). A plankton net (150 μm, 0.5 m diameter) was towed vertically through the water column from 1 m above the lake bottom at the deepest point to the surface of the lake. Samples were frozen in plastic Whirl-Pak bags after removal of any Chaoborus using acid-washed tweezers. Dominant zooplankton taxa in Lake 658 included calanoid copepods (Diaptomus oregonensis) and Cladocera (Holopedium glacialis, Daphnia pulicaria and Daphnia mendotae). Chaoborus samples were collected monthly in the same manner at least 1 h after sunset. After collection, Chaoborus were picked from the sample using forceps and frozen in Whirl-Pak bags. Chaoborus were not separated by species for MeHg analyses, but both C. flavicans and C. punctipennis occur in the lake. Profundal chironomids were sampled at the deepest part of the lake using a standard Ekman grab sampler. Grab material was washed using water from a nearby lake and individual chironomids were picked by hand.All work with vertebrate animals was approved by Animal Care Committees (ACC) through the Canadian Council on Animal Care (Freshwater Institute ACC for Fisheries and Oceans Canada, 2001–2013; University of Manitoba ACC for IISD-ELA, 2014–2015). Licenses to Collect Fish for Scientific Purposes were granted annually by the Ontario Ministry of Natural Resources and Forestry. Prior to any Hg additions, a small-mesh fence was installed at the outlet of Lake 658 to the downstream lake to prevent movement of fish between lakes. Sampling for determination of MeHg concentrations (measured as total mercury (THg), see below) occurred each autumn (August–October; that is, the end of the growing season in north temperate lakes) for all fish species in Lake 658, and for northern pike and yellow perch in nearby reference Lake 240 (Extended Data Tables 2, 3). Fish collections occurred randomly throughout the lakes. Forage fish (YOY and 1+ yellow perch, and blacknose shiner) were captured using small mesh gillnets (6–10 mm) set for 90% of the Hg in muscle tissue from yellow perch in Lake 658 is MeHg40,41, here we report fish mercury data as MeHg.THg concentrations (ambient, lake spike, upland spike and wetland spike) in fish muscle samples were quantified by ICP-MS39. Samples were digested with HNO3/H2SO4 (7:3 v/v) and heated at 80 °C until brown NOx gases no longer formed. The THg in sample digests was reduced by SnCl2 to Hg0 which was then quantified by ICP-MS (Thermo-Finnigan Element2) using a continuous flow cold vapour generation technique41. To correct for procedural recoveries, all samples were spiked with 201HgCl2 prior to sample analysis. Samples of CRMs (DORM2 (2001–2011), DORM3 (2012–2013), DORM4 (2014–2015); National Research Council of Canada) were submitted to the same procedures; measured THg concentrations in the reference materials were not statistically different from certified values (P > 0.05). Detection limit for each of the spikes was 0.5% of ambient Hg.Calculations and statistical methodsAnalyses were completed with Statistica (6.1, Statsoft) and Sigmaplot (11.0, Systat Software). We present wet weight (w.w.) MeHg concentrations for all samples, except sediments which are dry weight (d.w.) concentrations. For zooplankton, Chaoborus, and profundal chironomids, d.w. MeHg concentrations were multiplied by a standard proportion (0.15) to yield w.w. concentrations for each sample42. The resulting w.w. concentrations were averaged over each open water season to determine annual means. For fish muscle biopsies, d.w. MeHg concentrations were multiplied by individual d.w. proportions to yield w.w. MeHg concentrations for each sample. To avoid any size-related biases, we calculated standardized annual MeHg concentrations (ambient and lake spike) for northern pike and lake whitefish by determining best-fit relationships between FL and MeHg concentrations for each year (quadratic polynomial, except for a linear fit for lake whitefish in 2004), and using the resulting regression equations to estimate MeHg concentrations at a standard FL43 (the mean FL of all fish sampled for each species: northern pike, 475 mm; lake whitefish, 530 mm). Square root transformation of raw northern pike data was required to satisfy assumptions of normality and homoscedasticity prior to standardization. The resulting data represent standardized concentrations of lake spike and ambient MeHg for each species each year.We used the ratio of lake spike and ambient Hg in each sample as a measure of the amount by which Hg concentrations were changed with the addition of isotopically enriched Hg:$${rm{P}}{rm{e}}{rm{r}}{rm{c}}{rm{e}}{rm{n}}{rm{t}},{rm{i}}{rm{n}}{rm{c}}{rm{r}}{rm{e}}{rm{a}}{rm{s}}{rm{e}}={[{rm{l}}{rm{a}}{rm{k}}{rm{e}}{rm{s}}{rm{p}}{rm{i}}{rm{k}}{rm{e}}{rm{H}}{rm{g}}]}_{i}/{[{rm{a}}{rm{m}}{rm{b}}{rm{i}}{rm{e}}{rm{n}}{rm{t}}{rm{H}}{rm{g}}]}_{i}times 100$$
(1)
where [lake spike Hg]i is the concentration of lake spike MeHg in sample i, and [ambient Hg]i is the concentration of ambient MeHg in sample i. For northern pike and lake whitefish, we calculated the mean annual relative increase from all individuals (not the size-standardized concentration data).Biomagnification factors (BMF) were calculated to describe differences in Hg concentrations between predator and prey5:$${rm{BMF}}={log }_{10}({[{rm{MeHg}}]}_{{rm{p}}{rm{r}}{rm{e}}{rm{d}}{rm{a}}{rm{t}}{rm{o}}{rm{r}}}/{[{rm{MeHg}}]}_{{rm{p}}{rm{r}}{rm{e}}{rm{y}}})$$
(2)
where [MeHg]predator is the mean (forage fish) or standardized (large-bodied fish) concentration of MeHg in the predator (ng g−1 w.w.) and [MeHg]prey is the mean concentration of MeHg in the prey (ng g−1 w.w.). MeHg concentration of prey items were averaged from samples collected throughout the open-water season immediately prior to autumn sampling of fish species to represent an integrated exposure for calculation of BMF. We used a dominant prey item to represent the diet of each fish species. For age 1+ yellow perch, northern pike, and lake whitefish, dominant prey items were zooplankton, forage fishes (YOY and 1+ yellow perch, and blacknose shiner) and Chaoborus, respectively.To assess loss of lake spike MeHg by northern pike during the recovery period (2008–2015), we calculated28 whole body burdens (in μg) of lake spike MeHg for the standardized population and for individuals that had been sampled in autumn 2007 (t0 is the final time spike Hg was added to the lake) and again in at least one subsequent year during annual autumn sampling (n = 16 fish, of which 1–9 individuals were recaptured annually from 2008–2015). This calculation of MeHg burden is a relative measure of whole fish Hg content because MeHg is higher in muscle tissue than in other tissue types28,40. For the standardized population data, we used best-fit relationships between FL (in mm) and body weight (in g; quadratic polynomial) to determine body weight at the standard FL. We multiplied this body weight by standard ambient and spike MeHg concentrations (in ng g−1 w.w.) in muscle tissue for each year to determine body burdens over time (in ng). For individual fish, we multiplied spike MeHg concentration (in ng g−1 w.w.) by body weight (in g) to yield individual body burdens (in ng). To account for differences among individuals and between individuals and the population, we normalized the data to examine the mean proportion of original (t0) lake spike MeHg burden present in northern pike each year of the recovery period (2008–2015).$${rm{change}},{rm{in}},{rm{burden}},{rm{from}},{t}_{0}={{rm{burden}}}_{{rm{tx}}}/{{rm{burden}}}_{{rm{t}}0}$$
(3)
We used a best fit regression (exponential decay, beginning in the second year of recovery) to estimate the half-life (50% of original burden) of lake spike MeHg for the population.Northern pike and lake whitefish ages were determined by cleithra and otoliths, respectively, if mortality had occurred, but most ages were quantified using fin rays collected from live fish44 (K. H. Mills, DFO or North/South Consultants). Northern pike of the sizes selected for biopsy sampling had a median age of 3 years (range: 2–12 years; n = 305); the median age of lake whitefish was 17 years (range: 3–38 years; n = 86).Reporting summaryFurther information on research design is available in the Nature Research Reporting Summary linked to this paper. More

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PreliminariesA bipartite network captures connections between nodes of one type (agents) and nodes of a second type (artifacts). Throughout this section, we use the ecological case of Darwin’s Finches to provide a concrete example24,25. On his voyage to the Galapagos Islands on the H.M.S. Beagle, Darwin observed that only some species of finches lived on each island. These patterns can be represented as a bipartite network in which finch species (the agent nodes) are connected to the islands (the artifact nodes) where they are found26. A bipartite network can be represented as a binary matrix in which the agents are arrayed as rows, and the artifacts are arrayed as columns. We use ({mathbf {B}}) to denote a bipartite network’s representation as a matrix, where (B_{ik}=1) if agent i is connected to artifact k, and otherwise is 0. The sequence of row sums and the sequence of column sums of ({mathbf {B}}) are called the agent and artifact degrees sequences, respectively. These sequences are among the bipartite network’s most significant features and are known to have implications for bipartite projections and backbones15,27,28. In the ecological case, the agent degree sequence captures the number of islands where each species is found, while the artifact degree sequence captures the number of species found on each island.The projection of a bipartite network is a weighted unipartite co-occurrence network in which a pair of agents is connected by an edge with a weight equal to their number of shared artifacts. For example, the bipartite projection of Darwin’s finch network is a species co-occurrence network in which a pair of finch species is connected by an edge with a weight equal to the number of islands where they are both found. We use ({mathbf {P}}) to denote the matrix representation of a bipartite projection, which is computed as ({{mathbf {B}}}{{mathbf {B}}}^T), where ({mathbf {B}}^T) indicates the transpose of ({mathbf {B}}). In a projection ({mathbf {P}}), (P_{ij}) indicates the number of times agents i and j were connected to the same artifact k in ({mathbf {B}}). The diagonal entries of ({mathbf {P}}), (P_{ii}), are equal to the agent degrees, but in practice are ignored.The backbone of a bipartite projection is a binary representation of ({mathbf {P}}) that contains only the most ‘important’ or ‘significant’ edges. For example, the backbone of a species co-occurrence network connects pairs of species if they are found on a significant number of the same islands, which might be interpreted as evidence that the two species do not compete for resources and perhaps are symbiotic. We use ({mathbf {P}}’) to denote the matrix representation of the backbone of ({mathbf {P}}). Because multiple methods exist for deciding when an edge is significant and thus should be preserved in the backbone, we use (mathbf{P }^{‘{text {M}}}) denote a backbone extracted using method M. It is important to note that for a given bipartite projection, there is no ‘true’ backbone, but only backbones corresponding to specific backbone methods M. The backbone extracted using FDSM (i.e. (mathbf{P }^{‘{text{FDSM}}})) may be similar or different from a backbone extracted using another method such as SDSM (i.e. (mathbf{P }^{‘{text {SDSM}}})), and these similarities and differences depend on the information that is considered by the respective methods when determining whether edges’ weights are significant. It is these similarities and differences that we explore in the four studies below.Backbone extraction methods that were originally developed for non-projection weighted networks are often applied to weighted bipartite projections. One simple method preserves an edge in the backbone if its weight in the projection exceeds some global threshold T. However, when (T = 0), which is common, the backbone will be dense and have a high clustering coefficient because each artifact of degree d induces (d(d-1)/2) edges in the backbone29. Using (T > 0) can yield a sparser and less clustered backbone30,31,32, but still yields highly clustered networks in which low-degree nodes are excluded while high-degree nodes are preserved19. More sophisticated methods, including the disparity filter19 and likelihood filter20, aim to overcome these limitations of the global threshold method by using a different threshold for each edge based on a null model. However, all methods that were developed for non-projection weighted networks have the same shortcoming when applied to weighted bipartite projections: they ignore information about the artifacts, which is lost when generating the projection18. In the ecological case, the global threshold, disparity filter, and likelihood filter methods all decide whether two species should be connected in the backbone only by examining how many islands these two species are both found on, but do not consider the characteristics of those islands, including how many other species are found there, or even how many islands there are. Therefore, although these methods are promising for extracting the backbone from non-projection weighted networks, different methods are required for extracting the backbone from a bipartite projection.Bipartite ensemble backbone modelsBipartite ensemble backbone models decide whether an edge’s observed weight (P_{ij}) is significantly large, and thus whether a corresponding edge should be included in the backbone by comparing it to an ensemble of random bipartite networks. Let ({mathscr {B}}) be the set of all bipartite networks (mathbf {B^*}) having the same number of agents and artifacts as ({mathbf {B}}). In the ecological case, (mathbf {B^*}) might be viewed as representing a possible world containing the same species and islands, but in which locations of species on islands is different, and likewise ({mathscr {B}}) is the set of all such possible worlds. The bipartite ensembles used in backbone models take a subset ({mathscr{B}}^{text{M}}) of ({mathscr {B}}), subject to certain constraints M, and impose a probability distribution on it. In all models except the SDSM, the uniform probability distribution is imposed on ({mathscr{B}}^{text{M}}), that is, each element of the ensemble is equally likely. The backbone is then extracted from the projection of ({mathbf {B}}) by using the distribution of edge weights arising from projections of members of the ensemble to evaluate their statistical significance.We use (P^*_{ij}) to denote a random variable equal to ((mathbf {B^*}mathbf {B^*}^T)_{ij}) for (mathbf {B^*}~in ~{mathscr {B}}^{text {M}}). That is, (P^*_{ij}) is the number of artifacts shared by i and j in a bipartite network randomly drawn from ({mathscr {B}}^{text {M}}). In the ecological case, (P^*_{ij}) represents the number of islands that are home to both species i and j in a possible world, while the distribution of (P^*_{ij}) is the distribution of the number of islands shared by species i and j in all possible worlds.Decisions about which edges should appear in a backbone extracted at the statistical significance level (alpha) are made by comparing (P_{ij}) to (P^*_{ij})$$begin{aligned} P_{ij}’= {left{ begin{array}{ll} 1 &{} quad {text { if }} Pr (P^*_{ij} ge P_{ij}) < frac{alpha }{2},\ 0 &{} quad {text {otherwise.}} end{array}right. } end{aligned}$$This test includes edge (P'_{ij}) in the backbone if its weight in the observed projection (P_{ij}) is uncommonly large compared to its weight in projections of members of the ensemble (P^*_{ij}). We use a two-tailed significance test in the studies below because, in principle, an edge’s weight in the observed projection could be uncommonly larger or uncommonly smaller than its weight in projections of members of the ensemble, however a one-tailed test may also be used. In the ecological case, two species are connected in the backbone if their number of shared islands in the observed world is uncommonly large compared to their number of shared islands in all possible worlds.There are many ways that ({mathscr {B}}) can be constrained33, with each set of constraints describing a particular ensemble ({mathscr {B}}^{text {M}}), which is used in a particular ensemble backbone model M to yield a particular backbone ({mathbf {P}}^{'M}). In the case of ensembles used to extract the backbone of bipartite projections, our focus in this paper, two broad types of constraints are common23. First, ensembles can be distinguished by what they constrain: only the number of edges, the degrees of the agent nodes, the degrees of the artifact nodes, or the degrees of both the agent and artifact nodes. Second, ensembles can be distinguished by how they impose these constraints: the constraints can be satisfied exactly, or only on average. In statistical physics, ensembles that impose exact or ‘hard’ constraints are known as microcanonical, while ensembles that satisfy constraints on average or impose ‘soft’ constraints are known as canonical9.Prior work on these ensembles generally adopts either a theoretical focus on the ensembles themselves, or an applied focus on the consequences of ensemble choice. In the theoretical literature, some (primarily mathematicians) have aimed to characterize the properties of ensembles, such as estimating the cardinality of the ensemble of matrices with fixed rows and columns (below, we call this ensemble ({mathscr{B}}^{{text{FDSM}}}))34. Others (primarily physicists) have aimed to identify conditions under which ensembles are equivalent or non-equivalent, typically interpreting ensembles as representing thermodynamic systems35,36,37. In the applied literature, the focus is not on identifying fundamental properties of ensembles, but instead on understanding the implications of choosing a particular ensemble when detecting a particular pattern, such as nestedness38 or community structure23,27. The present work falls into this latter group: we are not directly concerned with identifying fundamental properties of ensembles, but instead on identifying the consequences of ensemble choice, with the ultimate goal of offering practical guidance to applied researchers wishing to extract the backbone of a bipartite projection.In the remaining subsections below, we first describe the FDSM in terms of its ensemble. We then present four potential alternative backbone models whose ensembles differ only slightly from FDSM, in terms of either what they constrain or how they impose constraints. We then turn to exploring the consequences of choosing one of these alternatives over FDSM when extracting a backbone.Fixed degree sequence model (FDSM)In the fixed degree sequence model (FDSM), (mathbf {B^*}~in ~{mathscr {B}}^{{text{FDSM}}}) are constrained to have the same agent and artifact degree sequences as ({mathbf {B}}). That is, FDSM constrains the degrees of both the agent and artifact nodes, and requires that these constraints are satisfied exactly, making it a tightly-constrained microcanonical ensemble. Adopting the FDSM implies, for example, that in all possible worlds a given species is found on exactly the same number of islands, and a given island is home to exactly the same number of species. The distribution of (P^*_{ij}) arising from ({mathscr {B}}^{{text{FDSM}}}) is unknown, but can be approximated by uniformly sampling (mathbf {B^*}) from ({mathscr {B}}^{text{FDSM}}), constructing (mathbf {P^*}), and saving the values (P^*_{ij}). In the studies below, we use 1000 samples of (mathbf {B^*}) generated using the ‘curveball’ algorithm, which is among the fastest methods to sample ({mathscr {B}}^{text{FDSM}}) uniformly at random39,40. The FDSM has been used to extract the backbone of bipartite projections of, for example, movies co-liked by viewers21 and conference panel co-participation by scholars41,42.The FDSM offers an intuitively appealing approach to extracting the backbone of bipartite projections because it fully controls for both bipartite degree sequences, which are known to be responsible for many of the projection’s structural characteristics15,16. However, because the distribution of (P^*_{ij}) must be computed via Monte Carlo sampling, it is computationally costly, making it impractical for all but relatively small bipartite projections. There are at least three distinct computational challenges. First, although the curveball algorithm is the fastest among existing methods for randomly sampling a bipartite graph with fixed degree sequences (i.e. for sampling (mathbf {B^*}) from ({mathscr {B}}^{text{FDSM}})), it still can require several seconds per sample for large graphs. Second, once a (mathbf {B^*}) has been sampled, constructing each (mathbf {P^*}) requires matrix multiplication, which must be performed repeatedly and has complexity of at least ({mathscr {O}}(n^{2.37}))43. Finally, computing an edge’s p value (i.e. (Pr (P^*_{ij} ge P_{ij}))) with sufficient precision to achieve a specified familywise error rate that controls for Type-I error inflation due to multiple testing22 can require these sampling and multiplication steps to be performed a very large number of times (see Supplementary Text S2).These computational challenges have led researchers to develop other backbone models3,9,18. Many such models exist, however here we are focused on identifying methods that yield backbones similar to what would be obtained using FDSM, and thus which may serve as computationally-feasible alternatives to FDSM. Therefore, we consider only those models whose ensembles involve at least one of the two types of constraints imposed by FDSM. That is, we consider models that either (1) impose exact constraints, or (2) impose constraints on both the agent and artifact degrees.Fixed fill model (FFM)In the fixed fill model (FFM), (mathbf {B^*}~in ~{mathscr {B}}^{{text {FFM}}}) are simply constrained to contain the same number of 1s as ({mathbf {B}}). That is, the FFM constrains only the number of edges, but requires that this constraint is satisfied exactly. Adopting the FFM implies, for example, that in all possible worlds only the total number of species-island pairs is fixed, but any given species may be found on a different number of islands and any given island may be home to a different number of species. The distribution of (P^*_{ij}) arising from ({mathscr {B}}^{{text {FFM}}}) has not been described before, but is derived in Supplementary Text S1.1. We call it a Jacobi distribution because it is related to Jacobi polynomials.Fixed row model (FRM)In the fixed row model (FRM), (mathbf {B^*}~in ~{mathscr {B}}^{{text {FRM}}}) are constrained to have the same agent degree sequence as ({mathbf {B}}), but have unconstrained artifact degree sequences. That is, the FRM constrains the degrees of the agent nodes, and requires that this constraint is satisfied exactly. A canonical variant of the FRM, the (hbox {BiPCM}_r), also constrains the degrees of the agent nodes, but only requires this constraint to be satisfied on average; we do not consider it here because it involves neither of FDSM’s constraints9. Adopting the FRM for backbone extraction implies, for example, that in all possible worlds a given species is found on the same number of islands, but a given island may be home to a different number of species. The distribution of (P^*_{ij}) arising from ({mathscr {B}}^{{text {FRM}}}) is hypergeometric (see Supplementary Text S1.2), and for this reason it is sometimes referred to as the hypergeometric model22,23,44. The FRM has been used to extract the backbone of bipartite projections of, for example, movies co-starring actors22, papers co-written by authors22, parties co-attended by women44, majority opinions joined by Supreme Court justices44, and microRNAs co-associated with diseases45.Fixed column model (FCM)In the fixed column model (FCM), (mathbf {B^*}~in ~{mathscr {B}}^{{text {FCM}}}) are constrained to have the same artifact degree sequence as ({mathbf {B}}), but have unconstrained agent degree sequences. That is, the FCM constrains the degrees of the artifact nodes, and requires that this constraint is satisfied exactly. A canonical variant of the FCM, the (hbox {BiPCM}_c), also constrains the degrees of the artifact nodes, but only requires this constraint to be satisfied on average; we do not consider it here because it involves neither of FDSM’s constraints9. Adopting the FCM for backbone extraction implies, for example, that in all possible worlds a given species may be found on a different number of islands, but a given island is home to the same number of species. The distribution of (P^*_{ij}) arising from ({mathscr {B}}^{{text {FCM}}}) has not been described before, but is derived in Supplementary Text S1.3, where we show it is Poisson-binomial.Stochastic degree sequence model (SDSM)Finally, the stochastic degree sequence model (SDSM) takes ({mathscr {B}}^{{text {SDSM}}}) to be all binary (m times n) matrices, but also gives a process for generating these matrices with different probabilities. Each (mathbf {B^*}) is generated by filling the cells (B^*_{ik}) with a 0 or 1 depending on the outcome of an independent Bernoulli trial with probability (p^*_{ik}). The distribution of the random variable (P^*_{ij}) arising from ({mathscr {B}}^{{text {SDSM}}}) is Poisson-binomial with parameters which can be computed using the (p^*_{ik}) (see Supplementary Text S1.4)27,46. There are many ways to choose (p^*_{ik}), but in the studies below we choose (p^*_{ik}) so that it approximates (Pr (B^*_{ik} = 1)) for (mathbf {B^*}~in ~{mathscr {B}}^{{text{FDSM}}}). This choice of (p^*_{ik}) ensures that the SDSM constrains the degrees of both the agent and artifact nodes, but only requires these constraints to be satisfied on average. Adopting such a version of SDSM implies, for example, that in each possible world a given species may be found on many or few islands and a given island may be home to many or few species, but the average number of islands on which a given species lives in all possible worlds and the average number of species that live on an given island in all possible worlds matches these values the observed world. The SDSM has been used to extract the backbone of bipartite projections of, for example, legislators co-sponsoring bills1,18,47,48,49, zebrafish (Danio rerio) sharing operational taxonomic units50, countries sharing exports3, and genes expressed in genesets51. More

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