The spatial scaling of mutualistic network diversity

How species richness scales spatially is a foundational concept of community ecology, but how biotic interactions scale spatially is poorly known. Previous studies have proposed interactions-area relationships (IARs) based on two competing relationships for how the number of interactions scale with the number of species, the ‘link-species scaling law’ and the ‘constant connectance hypothesis.’ The link-species scaling law posits that the number of interactions per species remains constant as the size of the network increases. The constant connectance hypothesis says that the proportion of realized interactions remains constant with network size. While few tests of these IARs exist, evidence for the original interactions-species relationships are mixed. We propose a novel IAR and test it against the two existing IARs. We first present a general theory for how interactions scale spatially and the mathematical relationship between the IAR and the species richness-area curve. We then provide a new mathematical formulation of the IAR, accounting for connectance varying with area. Employing data from three mutualistic networks (i.e. a network which specifies interconnected and mutually-beneficial interactions between two groups of species), we evaluate three competing models of how interactions scale spatially: two previously published IAR models and our proposed IAR. We find the new IAR described by our theory-based equation fits the empirical datasets equally as well as the previously proposed IAR based on the link-species scaling law in one out of three cases and better than the previously-proposed models in two out of three cases. Our novel IAR improves upon previous models and quantifies mutualist interactions across space, which is paramount to understanding biodiversity and preventing its loss.


Introduction
Species richness varies across space (Whittaker 1960) and with spatial scale (Preston 1962).The species-area relationship (SAR) is a foundational idea in ecology and conservation biology, with support from numerous ecosystems worldwide (Drakare et al 2006, Field et al 2009, Udy et al 2021).The SAR states that successively larger spatial areas contain increasing numbers of species following a power function (Connor and McCoy 1979).In conservation literature, SARs have principally been used to predict the number of species that will be lost with habitat destruction (Simberloff 1992, Pimm and Askins 1995, Pimm and Raven 2000)-or more recently, with contemporary anthropogenic climate change (Thomas et al 2004, Urban 2015).One limitation of these predictions of species loss given area reductions, however, is that they fail to account for the loss of biological interactions and associated knock-on secondary extinctions (Lewis 2006, Aizen et al 2012, Urban et al 2013, Brodie et al 2014, Sandor et al 2022).
Interaction networks are comprised of species which interact with at least one other species within the same network.Interaction networks can be trophic, or they can describe additional biotic interactions between two or more taxa groups.Two-taxa group networks, such as networks of mutually-beneficial interactions like those of plants and their pollinators, are often referred to as bipartite networks (Memmott 1999, Jordano et al 2003, Thébault and Fontaine 2010).Because species richness and composition vary across space and spatial scale (Wiens 1989), and since this variation applies to both sets of interacting species within a network, the structures of bipartite networks must also vary across both space (e.g.Burkle and Alarcón 2011, Trøjelsgaard et al 2015, Guimarães 2020) and spatial scale (Sabatino et al 2010, Sugiura 2010, Burkle and Knight 2012, Galiana et al 2018, Guimarães 2020).Interactions such as mutualisms are essential to community structure (Stachowicz 2001, Galiana et al 2018), but our understanding of them in a spatial context is still lacking (Galiana et al 2019).How might the structure of such networks change across space and scale?
Previous work on the topic of network-area relationships (NARs) has led to two hypotheses for the way that links (i.e.interactions within the network) scale with network size: the 'link-species scaling law' and the 'constant connectance hypothesis.'Both of these hypotheses were formulated for food webs, but they can be adapted to bipartite networks.The link-species scaling law is a theoretical lower bound for realized interactions within the network (versus all possible linkages) because it assumes that the number of links per species remains constant as the size of the network increases (Cohen andBriand 1984, Dallas andJordano 2021).The constant connectance hypothesis is a theoretical upper bound for realized interactions within the network because it assumes that the proportion of realized links remains constant with network size (Martinez 1992, Dallas andJordano 2021).Brose et al (2004) proposed equations that united SARs with the link-species scaling law and the constant connectance hypothesis, creating the first NARs.Further work by Galiana et al (2018) proposed more testable hypotheses about how different community assembly mechanisms result in NARs: (1) link scaling with SARs ('trophic sampling model,' akin to the theory laid out by Brose et al 2004), (2) a proportional relationship between area and the colonization-extinction ratio where extinction rates decrease with increasing area ('trophic theory of island biogeography model'), and (3) colonization-extinction processes along with dispersal and increased spatial heterogeneity as a driving force behind changes in network structure with area ('trophic meta-community model').However, only the first of these specifically relates links within the network to area.Most recently, using a world-wide helminth parasite-host dataset, Dallas and Jordano (2021) found that links per species increased with area without reaching saturation, as predicted in the 'trophic sampling model' (Galiana et al 2018).While few tests exist of NARs as defined by Brose et al (2004), evidence for the original forms of both the link-species scaling law and constant connectance hypothesis is mixed (Montoya and Solé 2003, Riede et al 2010, MacDonald et al 2020).Here, we extend SARs and NARs by proposing an interactions-area relationship (IAR) for bipartite networks that allows the number of links per species, and so the proportion of realized links, to vary with increasing network size.
In order to extend understanding of SARs, we first present theory governing how scaling should proceed as we expand from a species-area framework to a network-area one.We begin by exploring how species interactions scale with space, specifically the shape and limits of an IAR for bipartite networks.We then provide mathematical relationships between IARs and SARs.Third, we model how the accumulation of species interactions with area relates to bipartite network connectance, a commonly-used measure of the number of realized links per species within a network (May 1972, Jordano 1987) that serves as a proxy for network stability and strongly covaries with other network properties such as nestedness and modularity (Poisot and Gavel 2014, Chagnon 2015, Delmas et al 2019).As some evidence exists that connectance (C) of bipartite webs changes across area (e.g.Aizen et al 2012, Wood et al 2015, Dallas and Jordano 2021, Wang et al 2023, but see Dáttilo et al 2019) and declines with increasing network size (Jordano 1987, Olesen and Jordano 2002, Devoto et al 2005, Olesen et al 2006, Thébault and Fontaine 2008, Vázquez et al 2009, Dallas and Jordano 2021), our mathematical formulation of the IAR specifically accounts for connectance and allows it to either remain constant or vary with area.We justify this approach to connectance within our equation by determining how connectance changes with area for three bipartite network datasets.While bipartite networks can consist of groups of species with antagonistic, neutral, or beneficial interactions, we specifically use mutualistic networks, which describe the mutually-beneficial interactions between two groups of species, e.g.plants and their animal pollinators.We use these bipartite network datasets to illustrate how our theory fits empirical data, how it compares to previously-published scaling models, and how interactions within a mutualistic network scale with space.
1.1.A new scaling model: the IAR Ecologists currently have little empirical evidence with which to describe the IAR, and expectations for the shape and bounds of this curve do not consider network complexity (Wood et al 2015).Previous models for the relationship between area and biological interactions range from simple (Brose et al 2004) to complex (Brose et al 2004, Valiente-Banuet et al 2015, Galiana et al 2018), although increasing complexity has not necessarily translated to more accurate representations of empirical data (e.g.Galiana et al 2018).Here, we generate a general mathematical expectation for the number of interactions in a bipartite network per area by combining SARs (Connor and McCoy 1979) with variation in connectance (Jordano 1987).This combination of species richness and connectance accounts for most variation in network properties across spatial scales (Wood et al 2015, Galiana et al 2018).
Our derivation of the IAR rests on the following assumptions.First, that the number of interactions must always increase as area increases.In the case of continuous, nested habitat, once an interaction between two species occurs, the interaction is always present in larger areas.In the case of independent, isolated habitat patches, larger-sized habitat patches must have more interactions than smaller ones, even if species turnover between isolated habitat patches is high, because larger patches have more species based on the species-area curve.Second, as new species are added to the network, more interactions are also added, simply because a species with no interactions cannot be part of a network.
In SARs, S = cA z , where S is species richness, A is area and c and z are constants; log 10 (S) increases linearly with log 10 (A), and z represents the slope of the resulting line (hereafter log 10 () is simplified to log(); Connor and McCoy 1979).Using this equation, we can first define the minimum and maximum number of interactions for any given area (Martinez 1992).If we assume that a mutualist network of area A is made up of S 1 species from group 1 (e.g.plants) and S 2 species from group 2 (e.g.seed dispersers or pollinators), then the minimum number of interactions is the case in which each species in the more speciose group interacts with only one species in the other group.Thus, if S 1 > S 2 , the mathematically minimum number of interactions is S 1 .More generally, for a given area, the minimum number of interactions is max {S 1 , S 2 }, and the maximum number of interactions for a given area is the situation in which each species is connected to every species in the other group, equal to S 1 × S 2 .
Determining the minimum and maximum number of interactions, or interaction richness, given the size of two species groups (Martinez 1992) sets the outer limits for a potential IAR.Like SARs, we assume that the size of each species group-and thus the number of interactions-changes with area.Because the minimum number of interactions (I) is max {S 1 , S 2 }, the mathematical lower bound of the IAR is simply the species-area curve of the larger of the two species groups: where c 1 and z 1 , and c 2 and z 2 , are the constants described by the power function model of SARs for groups 1 and 2, respectively (Connor and McCoy 1979).A reformulation of equation ( 1) gives: For every other possible IAR, the number of interactions is set by the number of species within both groups.To this end, if we assume that S 1 = c 1 A z1 and S 2 = c 2 A z2 , then the mathematical upper bound to the number of interactions is: (3) By taking the log of this equation, we get the linear relationship: Because the upper bound for the IAR is set by the product of the number of species in group 1 and group 2, increasing either z 1 or z 2 increases the slope of the curve with respect to area.
The IAR of any given network can occupy the space between the upper and lower bounds, such that I min ⩽ I ⩽ I max .We note, however, that no natural network has ever been described with the minimum, nor maximum, potential interaction richness (Blüthgen et al 2007).
Connectance is the fraction of realized links between species within the network and is measured on a scale of 0-1, where zero indicates no connections and 1 indicates that every species in one group interacts with every species in the other (Bascompte 2008).Connectance mathematically relates to the number of species within a network, but it is not dependent on the number of species, nor is the number of species dependent on connectance.Using equation (3) as a starting point, we can determine the equation for any curve between the lower and upper bound as a function of a network's connectance, defined as C = I S1×S2 (Jordano 1987), such that: (5) Equation ( 1), which defines the lower bound of the IAR, is the special case where connectance is equal to I max{S1,S2} .Equation ( 3), which defines the upper bound, is the special case where C = 1.Thus, equation ( 5) can be rewritten as: Equations ( 6) and ( 7) both assume that connectance is constant across all areas.When this is the case, the intercept of equation ( 7) is defined by the constant log (C) + log (c 1 ) + log (c 2 ), such that the intercept of the IAR is modified by the connectance of the network.Similarly, the slope of the IAR is the sum of z 1 and z 2 , which we call β.Because empirical estimates of z generally lie between 0.15 and 0.35 (Lomolino 1989), we can expect empirical estimates of β to be between 0.3 and 0.7 when C is constant across area.For C to be constant across area, each species added to the network must have the same general number of interactions with other members of the network as the average.As a simple example, if C a = 0.5 and, when moving from A a to A b , every species added to group 1 and group 2 interacts with approximately half of the opposite group, C b will still = 0.5.
Connectance may change as a function of area, however (Martinez 1992, Vázquez et al 2009, Sugiura 2010, Wood et al 2015, Galiana et al 2018).If connectance increases with area, the slope of the IAR will be greater than β, and if connectance decreases with area, the slope will be less than β.If we assume, for simplicity (also see figure 1), that log-transformed connectance changes linearly with log-transformed area such that, then the full equation for the IAR, given changing connectance, is: If species are added that have more interactions than the bipartite network's average, connectance increases as does the slope of the IAR.As another a simple example, if C a = 0.5 and, when moving from A a to A b , every species added to group 1 and group 2 interacts with more than half of the opposite group, C b will be > 0.5.The most extreme case of this, and thus greatest IAR slope possible, is moving from C a = 1 S2 (where S 2 < S 1 ) to C b = 1, and can be calculated for any given system with equation ( 9).If species are added that have fewer interactions than the bipartite network's average, connectance decreases (figure 1).How much the slope of the IAR increases or decreases depends on the magnitude of the connectance decrease, relative to β.Because we allow connectance to vary over area in this way, its addition to the IAR theoretically allows for a more precise linear fit of the number of ecological network interactions by area.If connectance is unchanging with area, the intercept of the linear fit of the IAR is adjusted by the connectance value (equation ( 9), where δ = 0 and α = C).If connectance increases or decreases with area, the slope of the IAR additionally accounts for the slope of connectance with area (equation ( 9)).

Mutualist datasets and how connectance scales with area
To fit real data to our IAR equation, we collected occurrence data for an avian seed-dispersal network in the central Sierra Nevada Mountains, California, USA (figure S1).We focused on angiosperm shrubs or understory trees with fleshy or semi-fleshy bird-dispersed fruits (table S1).We employed a combination of on-the-ground plant surveys, citizen scientist bird surveys from the eBird platform (Sullivan et al 2009; downloaded 2 September 18), and our previously-compiled network (Sandor et al 2022).
Because the frugivore data are used in conjunction with the shrub data in calculations of number of interactions, we standardized survey effort across habitat patches for both groups of species (see Supplement 1 Methods for additional details).Using plant and bird species lists for each patch, we reduced the full central Sierra Nevada network to only those species found in each patch, retaining all interactions observed at the regional scale.Fortuna et al (2020) found that mutualists had high partner fidelity at biogeographical scales, which bolsters our assumption that interactions at a regional scale translate to interactions on a local scale.We summed the total number of bird-shrub interactions within each network patch and calculated connectance (Jordano 1987; table S2).
We pulled additional data from the literature for two other mutualistic networks that supplied patch size (area), number of species in each of two groups, and number of realized and observed interactions per patch: a pollination network in Germany (Grass et al 2018) and an ant-plant mutualist network in Japan (Sugiura 2010; Supplement 1 Methods, table S3).These three datasets are not intended to be representative of all possible bipartite networks, or even all possible mutualistic networks, but to provide evaluations of how connectance scales with area and how equation ( 9) fits empirical data.

Testing IAR fit and the connectance-area relationship
We performed an explicit comparison of model fit for equation ( 9) versus two previously proposed interaction scaling equations: the link-species scaling law and the constant connectance hypothesis, as modified into NARs by Brose et al (2004).Both of these prior formulations relate the number of links (equivalent to interactions) in a network (L) to the total species richness of the network, where b and u are constants that come from the relationship L = bS u .In this relationship that describes both the link-species scaling law and the constant connectance hypothesis in its general form, S is the total species richness of the network (Brose et  , and u is either set to 1 or 2, depending on whether the link-species scaling law (u = 1) or the constant connectance hypothesis (u = 2) is being used (Brose et al 2004).The b parameter is either the slope of the relationship between L and S (the link-species scaling law) or an unchanging approximation of connectance (the constant connectance hypothesis) with realized values of b ≈ 2 and 0 < b < 1, respectively, estimated across many datasets (e.g.Montoya and Solé 2003, MacDonald et al 2020, Gibert and Wieczynski 2021).The variables z and A come from SARs, assuming that the species richness of the network (S) scales with area, S = cA z .We did not compare our fit to other NARs such as the 'trophic theory of island biogeography model' or the 'trophic meta-community model' proposed by Galiana et al (2018).The community-assembly processes that these models simulate rely on variables relating to colonization, extinction, and dispersal between patches that are not recorded in any of the three empirical datasets.For all three test datasets, we first fit models for trophic-specific SARs, total SAR, and the connectance-area equation (equation ( 8)).We fit all models in a Bayesian framework, using vague normal priors for slope and intercept terms and vague uniform priors on standard deviations, using the program 'JAGS' (Plummer 2003) and 'rjags' (Plummer et al 2021).We ran three MCMC chains for each analysis, and used the Gelman-Rubin convergence statistic to check for convergence between and within chains by ensuring its values were between 1 and 1.1 (Gelman and Rubin 1992;table S4).We performed posterior checks of normality on our residuals.
We calculated posterior predictions for the number of interactions (I) using equation ( 9) and both alternative hypotheses of equation ( 10).For equation ( 9), we used full posterior distributions of parameters (i.e.c 1 , c 2 , z 1 , z 2 , α, and δ) after fitting the SARs and connectance-area relationship (equation ( 8)) to predict the number of interactions at each patch.
Unlike our proposed IAR (equation ( 9)), the NARs described by the alternative hypotheses of equation ( 10) have a potential non-independence issue when fitting variables used in the equation.The SAR is intrinsically related to both the link-species scaling law and the constant connectance hypothesis in that the dependent variable of the SAR (S) is the independent variable of L = bS u .To bypass this non-independence issue, we did not fit L = bS u to get values for the b parameter.Instead, for the b parameter of equation ( 10) for all three datasets, we used the estimation b ≈ 2 for the link-species scaling law (MacDonald et al 2020) and for the constant connectance hypothesis set b equal to the mean connectance with estimated uncertainty for each dataset (seed dispersal mean connectance: 0.42, pollination: 0.15, ant-plant: 0.63; table S5).We used realized values of the b parameter because we wanted to give both alternative hypotheses of equation ( 10) the best chance against our proposed IAR.If we had wanted the link-species scaling law to serve as a theoretical lower bound and the constant connectance hypothesis to serve as a theoretical upper bound, we would have chosen values of <1 and 1, respectively, for the b parameter.For each dataset, we pulled from a normal distribution with mean of 2 and standard deviation of connectance for that dataset to get values of b for the NAR version of the link-species scaling law.Similarly, for each dataset we pulled from a normal distribution with a mean and standard deviation of connectance for that dataset to get values of b for the NAR version of the constant connectance hypothesis.
Using the c, z, and b parameters and their associated uncertainty, we predicted the number of interactions at each patch using a log-transformed version of equation ( 10) for both alternative hypotheses.To determine relative model fit, we calculated and compared the root mean square error (RMSE) as well as a visual assessment of the difference between predicted and observed interactions for equation ( 9) and both alternative hypotheses of equation ( 10).We assigned best fit to the lowest RMSE across equation ( 9) and both alternative hypotheses of equation ( 10).When 95% confidence intervals of RMSE overlapped across two of the three equations, we assumed an equal best fit for both equations.
We found generally better support for our proposed IAR than the NAR versions of the link-species scaling law and constant connectance hypothesis (Brose et al 2004).The predicted number of interactions was more accurate for our proposed IAR (figure 2), with a lower RMSE for equation ( 9) than for both alternative hypotheses of equation ( 10) in two out of the three datasets.The 95% bounds of the RMSE did not overlap across models for the pollination dataset, with our proposed IAR having the lowest RMSE (table 1).Only the 95% bounds of the NAR version of the link-species scaling law overlapped with our proposed IAR for the seed dispersal dataset (table 1).Even though the NAR version of the link-species scaling law had a lower mean RMSE than our proposed IAR for the ant-plant dataset, the 95% bounds for the IAR and both NAR versions overlapped (table 1).

Discussion
In a world of increasingly fragmented and shrinking areas of undeveloped habitats, understanding the effects of spatial scale on interaction networks may be critical to understanding network persistence and resilience.Network interactions increase with area in a predictable way in all three mutualistic network datasets we examined.In contrast to our proposed IAR where the scaling of both groups of species with area is modeled explicitly, as is the connectance of the network across space, the previously published NARs against which we compared our proposed IAR make the simplifying assumptions that interactions in the network scale with total species and that connectance does not modify the slope of the interactions-area curve.While the NAR version of the link-species scaling law produced similar predictions for the number of interactions with a given area in two out of the three datasets, our proposed IAR was clearly the best in the remaining dataset (table 1, figure 2).
Both the seed dispersal dataset and ant-plant dataset had fewer data points than the pollination dataset (n = 12, n = 5, n = 32 for seed dispersal, ant-plant, and pollination, respectively), where our proposed IAR was unambiguously the best predictor of the data.The small number of data points in the seed dispersal and ant-plant datasets results in higher uncertainty of the fit parameters used in calculating the IAR and both versions of the NAR, which results in larger 95% bounds of the RMSE.While further comparisons between the IAR and both versions of the NAR with larger datasets are of course necessary, we find our results promising in indicating that our proposed IAR describes the relationship between area and the number of interactions within a bipartite network as well as, and in one out of three cases clearly better than, the NAR version of the link-species scaling law.
Similar to SARs (e.g.Thomas et al 2004), our proposed IAR equation could be used to predict the loss of interactions within a bipartite network given an amount of habitat loss.We have ample evidence that predictions of species loss fail to account for associated interactions loss (e.g.Lewis 2006, Aizen et al 2012, Urban et al 2013, Brodie et al 2014, Sandor et al 2022), which could result in the functional or secondary extinction of many species engaged in these interactions (Dunne et al 2002, McConkey and Drake 2006, Tylianakis et al 2010, Säterberg et al 2013, Brodie et al 2014, Grass et al 2018).Although SARs have been criticized for overestimating species extinctions (e.g.He and Hubbell 2013), and our IAR could be subject to the same problems if certain assumptions (e.g.how species are distributed in space) are not met, our IAR equation provides a realistic starting point for estimating the effects of habitat loss on mutualism communities.Table 1.Estimates of root mean square error (RMSE) and 95% confidence bounds for three linkage-area models each fit to three different mutualistic datasets.Numbers in bold in each column show the lowest mean RMSE among models for each dataset.Confidence intervals for RMSE overlap between the interaction-area relationship (IAR; equation ( 9)) and the link-species scaling law hypothesis of equation ( 10) for both the seed dispersal and ant-plant datasets, meaning that either could be considered best fit.Confidence intervals for RMSE do not overlap for the pollination dataset.Our inclusion of a dynamic value for connectance explicitly within the IAR model distinguishes our work from previously proposed NARs (NARs; Brose et al 2004, Galiana et al 2018).In this way, our IAR is similar to the non-spatial, link-species relationship presented by MacDonald et al (2020), who included connectance in their modification of the link-species scaling law and constant connectance hypothesis, and Gibert and Wieczynski (2021), who allowed for changing connectance across number of total species by subtracting links that can never be realized within the web ('forbidden' links).Both studies found that these reformulations of the link-species scaling law and constant connectance hypothesis provided a better estimation of interaction abundance within food webs.
Although species richness has been shown to be a good predictor of network structure (Dzekashu et al 2023), additional recent research has shown that processes such as local spatial overlap (Reverté et al 2019, Guimarães 2020), temporal overlap when interactions depend on life stages (Guimarães 2020, Peralta et al 2020), functional traits (Albrecht et al 2018, Bender et al 2018, Peralta et al 2020), and intraspecific variation (Arroyo-Correa et al 2023) dictate the number of realized interactions within bipartite networks.In the face of habitat loss, network connectance and network stability more generally can be further affected by additional ecological processes such as the potential for rewiring (when mutualist species compensate for lost interactions by switching partners; Kiers et al 2010, Barnum et al 2015, Valiente-Banuet et al 2015, Grass et al 2018), more abundant species compensating for lost interactions (Hagen et al 2012), and metapopulation and metacommunity dynamics (Fortuna and Bascompte 2006, Häussler et al 2020, Li et al 2023).While our equations could be used to estimate the loss of species interactions across large regions, much as the species-area curve has been used, our IAR likely becomes less useful at smaller spatial scales due to these ecological processes.Further development of IAR models should focus on estimating or incorporating these processes in order to make model estimates more accurate at smaller spatial scales.
Whether or not, and in what direction, connectance scales with area demonstrates the underlying ecological processes driving community assembly.The negative relationship between connectance and area demonstrated in our empirical datasets echoes a known relationship of lower connectance with greater species richness (Jordano 1987), but could be the result of multiple factors: increasing numbers of weak links with larger area, stability dictating that diversity has a negative correlation with complexity, increased diversity of the network resulting in decreased co-existence between species, increased specialization of animals with plant mutualists in more diverse networks, or simply a relic of sampling methods (Riede et al 2010).While our datasets do not provide any means to test these non-mutually exclusive explanations, further research on how and in what situations these ecological processes affect the scaling of connectance with area would be beneficial.Additionally, we assumed in our proposed IAR that connectance scaled linearly with area, but all of the listed processes could result in non-linear relationships between network connectance and area.Research on the shape of the connectance-area relationship, in conjunction with the ecological processes that produce them, would further improve the IAR.
Relatedly, Galiana et al (2018) put forth testable hypotheses of three community assembly processes: (1) the scaling of species and thus networks with area is as per the SAR (i.e.connectance remains constant with area; 'trophic sampling model'); (2) extinction-colonization dynamics affect the scaling of species and interactions with area (i.e.connectance declines sharply with increasing area; 'trophic theory of island biogeography model'); and (3) extinction-colonization and dispersal processes affect the scaling of species and interactions with area (i.e.connectance declines with increasing area, but less sharply than the 'trophic theory of island biogeography' model, unless dispersal is limited; 'trophic meta-community model').In the seed dispersal dataset, connectance does not significantly decrease with area, which indicates that these data likely come from fully connected metacommunities, where neither dispersal nor colonization-extinction dynamics are adding to areal scaling of species.Connectance in both the Grass et al (2018) pollination and Sugiura (2010) ant-plant datasets decrease with increasing area, with connectance of the ant-plant dataset declining faster than the pollination dataset (figure 1).This suggests that colonization-extinction processes, and possibly dispersal, are affecting the network communities along with areal scaling, and that a more complex model accounting for these community assembly processes is warranted.However, the mean difference in predicted and actual number of interactions for our proposed equation was low for the seed dispersal, pollination (Grass et al 2018), and ant-plant (Sugiura 2010) datasets (−3.67, −0.40, and 4.49, respectively;table S8), and as such, additional variables within the model are not warranted.The high agreement between predicted and actual number of interactions for these datasets echoes Galiana et al's (2018) findings that species richness and connectance will account for most of the variation in network complexity with increasing area.
Our theory provides a framework for understanding how biological interactions accumulate with area, and by extension, how these interactions could be lost with habitat loss.We show that the interaction-area curve is directly tied to SARs of both groups of species involved.The inclusion of a changing connectance function distinguishes our IAR from those previously hypothesized (Cohen and Briand 1984, Martinez 1992, Brose et al 2004) and improves upon them.Given the need to go beyond documenting species loss and to better understand how communities of interacting species will be impacted by habitat loss, our IAR provides potential to make more accurate predictions for mutualistic networks, and a firmer theoretical basis for linking the loss of species to the loss of interactions.

M E Sandor et al Figure 1 .
Figure 1.Network connectance versus area for three studies of mutualistic networks.Each point represents the logged connectance of a network within a habitat patch at a given logged habitat patch area.(A).The mean fitted relationship (solid line) and 95% prediction intervals (dashed lines) between logged area and logged connectance are shown for each study.For the seed dispersal study (green circles) this relationship was not significant.The relationships between connectance and area for the pollination study (orange squares) and the ant-plant study (purple diamonds) were significant.Network connectance was multiplied by 100 before log transformation to ensure positive values.Connectance versus area is displayed on the right for the (B).seed dispersal, (C).pollination, and (D).ant-plant mutualistic networks.Area, connectance, and fitted mean (solid line) and 95% prediction intervals (dashed lines) were back-transformed from logged values, and connectance was divided by 100.Data for the pollination and ant-plant mutualistic networks are from previously published studies(Sugiura 2010, Grass et al 2018).

Figure 2 .
Figure 2. Network interactions versus area for a seed dispersal ((A); circles), a pollination ((B); squares), and an ant-plant ((C); diamonds) mutualistic network.In all figures, points represent the empirical number of interactions for habitat patches of a given area.Lines represent the mean predicted relationship between area and interactions for each equation specified at the top of the figure, as calculated by inputting fitted parameters into equations (9) and (10).Equation (9) (red solid line) is our proposed interactions-area relationship (IAR), equation (10) (with u = 1) is the network-area relationship (NAR) version of the link-species scaling law (light gray dashed line), and equation (10) (with u = 2) is the NAR version of the constant connectance hypothesis (dark blue dashed line).