A unifying approach for food webs, phylogeny, social networks, and statistics

Abstract

A food web consists of nodes, each consisting of one or more species. The role of each node as predator or prey determines the trophic relations that weave the web. Much effort in trophic food web research is given to understand the connectivity structure, or the nature and degree of dependence among nodes. Social network analysis (SNA) techniques—quantitative methods commonly used in the social sciences to understand network relational structure—have been used for this purpose, although postanalysis effort or biological theory is still required to determine what natural factors contribute to the feeding behavior. Thus, a conventional SNA alone provides limited insight into trophic structure. Here we show that by using novel statistical modeling methodologies to express network links as the random response of within- and internode characteristics (predictors), we gain a much deeper understanding of food web structure and its contributing factors through a unified statistical SNA. We do so for eight empirical food webs: Phylogeny is shown to have nontrivial influence on trophic relations in many webs, and for each web trophic clustering based on feeding activity and on feeding preference can differ substantially. These and other conclusions about network features are purely empirical, based entirely on observed network attributes while accounting for biological information built directly into the model. Thus, statistical SNA techniques, through statistical inference for feeding activity and preference, provide an alternative perspective of trophic clustering to yield comprehensive insight into food web structure.

Fig. 1 depicts a simple food web. Food webs are network structures consisting of nodes, each containing one (e.g., Human) or various species (e.g., Ticks). For a trophic food web, nodes are interwoven by directed links that conventionally point from prey to predator (1). Certain patterns among trophic relations suggest the clustering of nodes; the ability to unveil these patterns can facilitate other aspects of food web research, such as the identification of functional groups, trophic levels, and keystone species. This in turn can provide information about the stability of the web under perturbations (e.g., species extinction). Conventional social network analysis (SNA) techniques have been applied to trophic research for this purpose (e.g., refs. 2 and 3). These methods typically seek optimal partitioning of the network into compartments of nodes subject to prespecified mathematical criteria (4, 5). After food web features have been identified, a natural question follows: What factors contributed to the feeding behavior among nodes that gave rise to those features? Conventional SNA frameworks provide no direct means to address this question.

Fig. 1.

Pictorial representation of feeding links pointing from prey to predator. Nodes (hypothetical) are shown in an arbitrary arrangement.

Recent advancement in statistical regression methodologies that model complex network structures (6–9) has allowed researchers, mostly from the social sciences, to unravel valuable information entwined in the relational links observed empirically among nodes. Standard regression regarding nodes as independent has been used to connect various within-node characteristics in a food web (1): Because predator-prey links were not part of the regression, they were used post hoc to qualitatively explain the relationship among nodal characteristics. Instead, a key feature of the more novel statistical SNA methods is one’s ability to utilize network dependency and explicitly express predator-prey links as a regression function of both within- and internode characteristics—e.g., biomass of the node, the role of the node as predator or prey, and phylogenetic similarity between nodes. This allows direct inference for what makes a given node a predator or prey and for dependence features including the tendency for predator-prey role reversal between a given pair of nodes. A fundamental principle of regression modeling is the appropriate use of available predictor variables (here, nodal characteristics) and the dependency among data to improve the accuracy and precision of inference drawn for the behavior of the response variable (here, the feeding links, which exhibit complex dependencies). This type of statistical modeling of trophic (feeding) relations is not to be confused with that in ref. 10, which uses compartment membership to explain between-node similarity, nor with that in ref. 11, which employs Bayesian melding (12) to model intercompartmental energy-matter flows subject to mass balance.

We apply an existing statistical SNA framework (13) known as latent space modeling (7, 8) to analyze eight empirical trophic food webs that have been previously studied (14). These webs were observed from Goose Creek Bay in the St. Marks National Wildlife Refuge in the southeastern United States (15), the Benguela marine ecosystem off the South African coast (16), the grasslands in England and Wales (17), the Caribbean coral reefs with a reduced set of nodes (18), the northeast US continental shelf (19), a pond on Skipwith Common in England (20), the Caribbean island of St. Martin (21), and Ythan Estuary in Scotland (22). Henceforth, we refer to them respectively as Bay, Benguela, Grass, Reef, Shelf, Skipwith, St. Martin, and Ythan. These webs correspond to various types of aquatic and terrestrial communities, through which we demonstrate the practicality of statistical SNA in a rather general context of food web ecology. Our analyses illustrate the new perspectives of trophic patterns, presented through three configurations of the food web graph, according to the statistical inference for node activity and feeding preference. We also demonstrate that the statistical framework can provide rigorous quantitative evidence for the contribution of phylogenetic information to predator-prey relations. We discuss how our findings reflect the recent debate over the mean trophic level (MTL) in ref. 23.

Modeling Framework

For each food web, associated with any pair of nodes is a numerical value y_ij representing trophic linkage in the form of sending activity, or the activity of being consumed, directed from the ith node to the jth. The dataset takes on one of two forms. The first is presence-absence data, where y_ij = 1 if the link i → j is present, and y_ij = 0 otherwise. For example, in Fig. 1, Ticks are observed to predate on Human, but the converse is not true, and thus y_HT = 1 and y_TH = 0. The other form is weighted data, where y_ij is the magnitude or weight of some consumption measure (e.g., volume) for the predation of i by j (24). The observed y_ijs can be displayed in a square matrix called the diet matrix.

Not all eight food webs being analyzed are associated with weights. Thus, we only consider the presence and absence of predation between pairs, using logistic regression. That is, the odds p_ij/(1 - p_ij) represent the underlying predation behavior of j on i, where p_ij is the probability of the event {y_ij = 1}. For predicting predation, taxonomy is the sole nonfeeding information that is readily available for all eight of our food webs. We define two measures of phylogenetic similarity, x_ij and z_ij, to quantify the taxonomic likeness between i and j (Materials and Methods). Then, to describe internodal relations, we consider the mixed-effects model

[1]

where the log-odds is expressed as a fixed mean μ_ij (= β₀ + β₁x_ij, β₀ + β₁z_ij, or simply β₀), plus random deviation from the mean. The total random deviation is decomposed into four mean-zero components: s_i due to i in the role of the sender, r_j due to j in the role of the receiver, inner product due to the interaction between i and j, and ε_ij, which is the remainder not attributable to the former three components. For example, Human’s activity level as prey and as predator is represented by s_H and r_H, respectively. For Human and Ticks, u_H (v_H) being close to u_T (v_T) in the latent two-dimensional u space (v space) would indicate that Human and Ticks are similarly preferred as prey by other nodes (have similar preference for prey nodes). Network dependence not addressed by parameters in Eq. 1 is modeled through ρ_sr = Correlation(s_i,r_i) for all i, and through ρ = Correlation(ε_ij,ε_ji) for all i ≠ j. Phylogenetic similarity is relevant to feeding when the regression coefficient β₁ ≠ 0. Another way to view the influence of this predictor is the points of reference it provides when interpreting model parameters. For example, without predictors in Eq. 1, having s_i > s_j is equivalent to i being consumed by more nodes in the food web than j. With x_ij or z_ij, the interpretation changes: i is more actively consumed than j when (i,j) is compared against those pairs of nodes sharing the same phylogenetic similarity. See Materials and Methods for detailed interpretation of all model parameters.

“The largest gains in estimating regression coefficients often come from specifying structure in the model” (25). To demonstrate the informational gain in the food web inference from specifying network structure, we compare model goodness-of-fit between model 1 and simple logistic regression, which naively ignores all dependence among y_ijs. Model 1 and its complex correlation structure can be expressed in a Bayesian hierarchical framework (13), which was implemented as “gbme.asym.r” (http://www.stat.washington.edu/hoff/Code/GBME/) to extend earlier models in ref. 7. We employed this software to perform Bayesian statistical inference (25) for all model parameters. Invariance of under orthogonal transformation of u_i or v_j prompted us to work out a suitable Procrustes transformation of u_i and v_j so that their estimates produced by “gbme.asym.r” were interpretable (7, 8). See Materials and Methods for details.

Results

Visual Representation of Trophic Features.

We first summarize the statistical inference by three graphs in Fig. 2. Unlike that of Fig. 1, the arrangement of nodes in the graphs labeled SR (s_i vs. r_i), U (u_i vectors), and V (v_i vectors) is due to the fitting of model 1. Respectively, the graphs reflect connectivity structure from the perspectives of sender-receiver activity, sending preference, and receiving preference. The graphs allow immediate visual assessment of predator-prey connectedness in the web, and the extent of trophic clustering (tight or loose clusters, and how many). For Bay, SR shows a fair number of feeding links, although not particularly dense. It also suggests roughly four clusters, comprising nodes that are (i) most actively consumed but least active as consumers (Halodule wrightii: “43,” micro epiphytes: “44,” phytoplankton: “47,” detritus: “48”), (ii) the most active consumers but average on the scale of being consumed (ominivorous crabs: “10,” predatory shrimps: “16,” predatory worms: “34”), (iii) least actively consumed but are very active consumers (benthos-eating birds: “37,” herbivorous ducks: “42”), and (iv) the rest of the web, possibly divided further depending on the clustering resolution. Clusters i to iii appear tighter than iv, but the majority of nodes are evenly scattered. Thus, from the perspective of feeding activity, some trophic levels (TLs) are not clearly distinguishable from each other. For Skipwith, U and V each shows a very tight cluster made up of numerous nodes, with the remaining nodes forming isolated small clusters. Thus, overall, some TLs from the perspective of feeding preference as prey or as predator can be clearly distinguished. For node- or cluster-specific insight, we see that the large clusters in U and V are made up of different sets of nodes; thus, trophic clusters depend on the perspective of feeding behavior from which clustering is viewed. In U, detritus (“37”) is greatly isolated from the rest of the web—i.e., consumers of detritus differ substantially from those of any other node in the web. Particularly, consumers of Notonecta glauca (“13”), Hydroporus erythrocephalus (“22”), and Sialis lutaria (“28”) differ the most. Indeed, network links or arrows that originate from “37” land in very different parts of the web compared to arrows originating from “13,” “22,” or “28.” Similarly, the consumers of Chydorus latus (“7”), Corynoneura scutellata (“32”), and Tanytarsus bruchonius (“35”) differ the most from those of Lumbriculus variegatus (“3”). Graph V can be similarly interpreted but with respect to prey items. For example, the clusters and arrows indicate that Acanthocyclops vernalis (“8”) consumes very different items than do the diving bell spider (“5”) and great diving beetle (“27”). Descriptions of SR, U, and V for all eight webs appear in Table S1. As can be seen in at least one of the three graphs, the notion of “trophic levels” is consistently ambiguous for all eight webs. This agrees with the findings in ref. 23 that TL estimates have substantial uncertainty and that catch MTL is a poor biodiversity indicator for marine ecosystems. Our results extend the argument beyond marine environments and are conveniently available from a single empirical analysis applied to each of a small number of webs.

Fig. 2.

Food web graphs displaying trophic structure from fitting model 1 with phylogeny measure x_ij as the predictor. (SR) Feeding activity in Bay. The s axis refers to activity as prey, and the r axis, as predator. Label of node i is located at the mean of the bivariate posterior distribution (an “estimate”) of [s_i,r_i]; this distribution describes the probability of the position of [s_i,r_i] on the sr plane, given the knowledge of the observed food web. Distribution density appears as heat map for benthos-eating birds (“37”) and detritus (“48”). Legend for node labels appears in SI Text. (U, V) Preference of being consumed/consuming in Skipwith. They are similarly interpreted as (SR), but referring to u_i vectors for L. variegatus (“3”), S. lutaria (“28”), C. scutellata (“32”), and detritus (“37”), and v_i vectors for diving bell spider (“5”), A. vernalis (“8”), S. scoticum (“12”), and detritus. Nodes far apart in the latent u or v space differ substantially with respect to feeding preference.

Insight into other aspects of network dependency is also available from the statistical inference. For Bay, the negative trend in SR provides evidence for ρ_sr < 0—i.e., active predators are unlikely to be active prey, and vice versa. Inference summary statistics for various model parameters provide complementary information to the graphs.

Numerical Summaries of Inference and Goodness-of-Fit.

Tables 1 and 2 feature Bay, Reef, Skipwith, and St. Martin. They represent a range of scenarios on (a) the importance of phylogenetic information, (b) the extent of predator-prey reciprocity, and (c) the association between an individual’s sending and receiving activity. Based on Table 1, Bay/Skipwith is high/low for all a–c, and Reef/St. Martin is high/low for a but low/high for b–c. Through Table 2, these webs showcase the superior goodness-of-fit of model 1 relative to the naive model. Inference for model parameters appears in Tables 1 and 2 and Table S2: All eight webs show evidence that (i) phylogeny is noticeably relevant to feeding [moderate to strong evidence that β₁ ≠ 0 from at least one perspective of feeding (see Materials and Methods)], and (ii) ρ_sr < 0, although weakly for Grass (x_ij), Reef, or Skipwith. Bay, Benguela, Reef, and Shelf show evidence of varying strength that phylogenetically similar nodes are more likely to yield feeding interaction (β₁ > 0). Grass and Ythan show the opposite tendency with reasonable evidence; both webs exhibit minimal connectance (), thus the prevalence of y_ij = 0 may explain the evidence for β₁ < 0. Moderate to strong evidence that predation is unlikely to reciprocate between nodes (ρ < 0) is seen in all webs except Grass and Skipwith, whose weak ρ and ρ_sr may reflect their heavy dominance by insects with similar taxonomy. A weak ρ_sr also among Reef nodes can be due to their unusually coarse taxonomic classification.

Table 1.

Bayesian inference numerical summaries for selected food webs from fitting statistical social network model 1, with phylogenetic similarity x_ij as the predictor

			Credible interval
Food web*	Parameter	Posterior median†	Lower limit	Upper limit	Credibility‡
Bay	β1	5.10	1.20	10.46	0.95
	ρsr	−0.46	−0.75	−0.07	0.95
	ρ	−0.82	−0.95	−0.11	0.75
Reef	β1	2.16	0.06	4.21	0.90
	ρsr	−0.08	(interval includes 0)		0.50
	ρ	−0.29	−0.52	−0.02	0.60
Skipwith	β1	−1.07	(interval includes 0)		0.50
	ρsr	−0.04	(interval includes 0)		0.50
	ρ	−0.39	−0.67	−0.04	0.50
St. Martin	β1	0.39	(interval includes 0)		0.50
	ρsr	−0.52	−0.80	−0.07	0.95
	ρ	−0.98	−1.00	−0.24	0.99

^*Other food webs and z_ij appear in Table S2.

^†The posterior median can be considered a parameter “estimate.”

^‡Credible intervals presented have approximately the highest credibility without including 0. High credibility for an interval excluding 0 indicates statistical importance of the corresponding parameter to feeding potential (p_ij).

Table 2.

Goodness-of-fit summaries for selected food webs (others in Table S2)

Model	Predictor	GoF*	Model	Predictor	GoF
Bay			Reef
1	xij	367	1	xij	524
1	zij	386	1	zij	526
1	—	394	1	—	533
naive†	xij	719	naive	xij	1273
naive	zij	719	naive	zij	1287
Skipwith			St. Martin
1	xij	138	1	xij	252
1	zij	138	1	zij	204
1	—	142	1	—	287
naive	xij	319	naive	xij	482
naive	zij	321	naive	zij	481

^*Derived from the Bayes factor on the model’s ability to predict the act of feeding (y_ij). When comparing between models, noticeably smaller GoF values suggest better fit.

^†Simple logistic regression ignoring network dependence—i.e., naively setting for all i,j in model 1.

When analyzing a social network with n nodes, it is common to provide descriptive indices of network features (e.g., refs. 26 and 27) such as the connectance index, C, for trophic food webs. Popular indices for our eight food webs appear in ref. 14. Our statistical SNA can provide probabilistic interpretations associated with these indices by accounting for the natural variability inherent in feeding behavior, via the posterior predictive distribution (25) of the index (Fig. 3 and Table S3). This distribution provides information about the uncertainty in the index under plausible scenarios that may alter the links in the food web. It can further be used for model validation (Materials and Methods). In our case, validation of model 1 suggested that the model is indeed consistent with the observed food webs.

Fig. 3.

Posterior predictive distribution of the connectance index C, from fitting model 1 for the Bay food web. For the presently observed food web, C = 0.10 (dashed line).

Discussion

Using statistical SNA methods to study trophic relations, we have demonstrated a new direction for quantitative analysis of food webs. The statistical inference framework provides a common thread to the understanding of feeding patterns and broad-sense connectivity, the relevance of nodal attributes (e.g., phylogenetic similarity) to feeding behavior, trophic clustering from various perspectives of feeding behavior, and the uncertainty of food web descriptive measures (e.g., MTL, C). In general, food web features are subject to natural variability (uncertainty due to natural events in which feeding behavior may vary) and to observation error such as in the identification of predator’s stomach content. Under this framework, we can conveniently visualize numerous food web features through a small set of graphical displays and use probabilistic statements about quantitative parameters in the regression model to summarize these features and assess the extent of the influence of nodal attributes on the trophic structure. It facilitates a comprehensive understanding of trophic structure that is readily achievable with a single, unified quantitative food web analysis.

Materials and Methods

Data.

The eight food webs that we analyzed represent a variety of aquatic and terrestrial commmunities (14) (SI Text). Each of the corresponding eight source articles (15–22) contains information from which the diet matrix for the presence (1) and absence (0) of feeding linkage between nodes can be deduced. Cannibalism data (y_ii) are not modeled by Eq. 1; in this context, of interest is the structure of codependence among the different nodes rather than a node’s self influence. Taxonomic information of nodes accompanies each article, although their formats differ. For some webs, this information contains common names only, but for others it contains Latin names at various levels of the phylogenetic tree. For convenience, we converted all eight sets of taxonomic information into Linnaean trees of a common format, with the ranks of (i) domain, (ii) kingdom, (iii) phylum, (iv) class, (v) order, (vi) family, (vii) genus, and (viii) species. Thus, information about subclass, suborder, etc., was only used to determine missing information about superceding ranks. Six of the eight webs comprise nodes (e.g., detritus) that entirely consist of organic but inanimate matter. For these six, we appended the Linnaean tree to an artificial superceding rank of “animacy,” which took on one of two values, animate or inanimate. We consulted the online resource Integrated Taxonomic Information System (ITIS, http://www.itis.gov) to make the conversions; various other online sources (via Internet search engines) were used only when the original taxonomic information contained Latin names that were not found within the ITIS. Based on the resulting phylogenetic trees, we computed two different measures of phylogenetic similarity for each food web: similarity x that addresses missing taxonomic topology, and a more conservative version z that regards missing topology as implying different phylogeny.

Constructing Phylogenetic Similarity Measures.

The taxonomic classification (e.g., Linnaean) of an organism is a set of polychotomous qualitative characters. In the absence of a universal measure of phylogenetic similarity between two organisms according to their taxonomic classification, Gower’s general coefficient of similarity and its variants (28) may be used to quantify the comparison. In the case of complete topological information on the Linnaean tree, the path length between two species (29) is a special case of Gower’s measure. However, the nodes in each of our eight food webs result from aggregation of species at uneven taxonomic resolutions, so that full topology is unavailable/inapplicable.

Uneven aggregation is common in food web studies (15, 22, 30). For example, consider the following four nodes in the Benguela food web: Node “3” is identified as “bacteria”; “4,” as “benthic carnivores”; “23,” as “kob”; and “26,” as “whales and dolphins.” If mapped to the eight-character Linnaean classification of ranks i–viii from above, then these nodes are identified at four different resolutions. Specifically, let “NA” denote “not applicable” or missing. Then, listed in the order of i–viii, “3” is Bacteria-NA-NA-NA-NA-NA-NA-NA; “4” is Eukaryota-Animalia-NA-NA-NA-NA-NA-NA; “23” is Eukaryota-Animalia-Chordata-Actinopterygii-Perciformes-Sciaenidae-Argyrosomus-A. hololepidotus; and “26” is Eukaryota-Animalia-Chordata-Mammalia-Cetacean-NA-NA-NA. To quantify the phylogenetic similarity between any two nodes, let a_ijk = 1 if nodes i and j match on character k, and a_ijk = 0 otherwise. Also let w_ijk = 1 if information exists for the kth character for both i and j, and w_ijk = 0 otherwise. Typical use of Gower’s measure removes from consideration any character that involves one or more NAs. Thus, a common variant of Gower’s similarity coefficient between i and j is

where K = 8 for this example. The measure ranges between 0 and 1. This definition gives A_3,4 = A_3,23 = A_3,26 = 0, A_4,23 = A_4,26 = 1 (maximum possible value), and A_23,26 = 3/5. However, A_23,26 < A_4,23 = A_4,26 appears counterintuitive and remains so as long as w_ijk = 0 for any k involving NAs. As an alternative similarity measure, we propose taking

where {i 's character k is not NA, j 's character k is not NA}. For example, m = 8 for computing A_4,23, but m = 5 for computing A_4,26. Then, A_3,4 = A_3,23 = A_3,26 = 0, A_4,23 = 2/8 = 0.25, A_4,26 = 2/5 = 0.4, and A_23,26 = 3/8 = 0.375.

Our measure attempts to address missing phylogenetic topology and is intended to yield less lopsided values than those based on the conventional practice of discrediting NAs. However, the occasional pair with an equal number of NAs may still be problematic. For example, with the above eight-character classification, the nodes “birds” and “sharks” are only identified to the class level (Aves and Chondrichthyes, respectively). Here, their similarity coefficient is 3/4 = 0.75 = 2 × A_23,26, which may be unrealistic. A more conservative alternative is to regard all characters involving NAs to be different between nodes. Thus, we would replace the above A_4,23 with 0.25, and the coefficient between birds and sharks would be 3/8 = 0.375. The flip side of this conservative alternative is an unrealistically small value for comparing, say, microzooplankton and mesozooplankton, both of which are only identified to the kingdom level (Animalia), so that their similarity coefficient is 0.25 (conservative) as opposed to 1, the latter of which is obtained using either our less conservative method or the conventional approach of discarding NAs altogether. While it may be of general interest, a more sophisticated similarity measure in the presence of “jagged” phylogenetic topology is not the focus of our work. Indeed, comparisons similar to that of birds and sharks, or of micro- and macrozooplankton, are in the minority for the food webs analyzed here.

For statistical SNA using model 1, we considered (a) the similarity coefficient as proposed above to address missing topology, and (b) the more conservative measure that regards NAs as implying difference. We took the logarithm of each measure (1 was added to all A_ij values prior to transformation to avoid taking the log of 0) and denoted them by x_ij and z_ij, respectively. Such transformation reduced skewness of the semiqualitative covariate, thus increasing its ability to distinguish among cases (pairs of nodes here) to help predict the response. Note that x_ij = x_ji and z_ij = z_ji.

Statistical Analyses and Model Validation.

Given either x_ij or z_ij in Eq. 1 for a food web with n nodes, we performed Bayesian estimation of the parameters β₀, β₁, ρ_sr, and ρ, as well as [s_i,r_i], u_i, and v_i for all i = 1,…,n. A food web with n nodes consists of n(n - 1) pairwise directed links y_ij for i ≠ j. In Eq. 1, the sender and receiver effects, s_i and r_j, can be interpreted as sending and receiving activity. Thus, s_i > s_j implies that node i is more active as prey than node j. Similarly, r_i > r_j implies that i is more active as predator than j. Given node i, its activity level in the food web as either prey or predator is conveniently described by the vector [s_i,r_i]. The interaction term in Eq. 1 is the inner product of k-dimensional vectors u_i = [u_i1,…,u_ik] and v_j = [v_j1,…,v_jk]. In our analyses, we took k = 2 for easy visualization of these latent spaces, although one could define criteria for selecting an optimal k (7, 13). In essence, if u_a and u_b are neighbors in the u space, then the sending behavior (other than sending activity) of a to c is similar to that of b to c, for all nodes c. The same interpretation applies to the receiving behavior (other than receiving activity) of neighbors v_a and v_b in the v space. In the food web context, the u space then refers to preference of being consumed, and v space, to preference of consuming. Eq. 1 alone constitutes a two-way analysis-of-variance model with random row, column, and row-column interaction effects (31). Dependence features within the network are represented by additionally specifying that the vectors [s_i,r_i] and [ε_ij,ε_ji] each has some nonzero covariance. For this, we assume [s₁,r₁],…,[s_n,r_n] are independent and identically distributed (iid) as bivariate mean-zero Gaussian vectors, with variance-covariance matrix Σ, and [ε_ij,ε_ji] for all i≠j are iid bivariate mean-zero Gaussian vectors, with variance-covariance matrix Ω, where

For a food web, ρ_sr describes the tendency for any given node to be active as both prey and predator—e.g., ρ_sr < 0 implies that if a node is active as prey, then it is unlikely to be active as predator, and vice versa. The parameter ρ is the correlation between ε_ij and ε_ji. It describes the tendency of predator-prey role reversal within any given pair of nodes—e.g., ρ < 0 implies that predation is unlikely to be reciprocated within any given pair. Finally, writing the 2D preference vectors as u_i = [u_i1,u_i2] and v_j = [v_j1,v_j2], we assume that u_1q,…,u_nq are iid Gaussian with mean 0 and variance for q = 1,2, and v_1q,…,v_nq are similarly distributed but with variance . This is the default assumption that is implemented in “gbme.asym.r” for performing statistical SNA.

Bayesian estimation of the set of parameters then proceeded for each food web dataset by applying the default prior distributions built into the “gbme.asym.r” software to produce Markov chain Monte Carlo (MCMC) draws (25) from the joint posterior distribution (jpd) of these parameters. The underlying jpd is the basis of Bayesian inference; we approximated it empirically by the distribution of the MCMC draws. For location parameters [s_i,r_i], u_i, and v_i, the mean of the MCMC draws (after Procrustes transformation, if applicable—see next section) was taken as the parameter estimate and used to produce network graphs in Fig. 2. Nodes far apart in SR/U/V differ substantially with respect to feeding activity/being preferred as prey by others/preference of consuming others. Each heat map in Fig. 2 is the density of the approximate jpd marginalized over all parameters except those that correspond to the displayed bivariate plane for the particular node of interest. For example, the upper-left part of the SR heat map is the density of the jpd marginalized over all parameters except [s₃₇,r₃₇]; it describes the probability of the location of [s₃₇,r₃₇] on the sr plane, given the knowledge of the observed food web. Alternatively, quantiles of the jpd can be used as numerical summaries of the density and to assess uncertainty.

For nonlocation parameters, the median and upper and lower αth quantiles of the MCMC draws were used to summarize the Bayesian inference in the respective form of a parameter estimate and a 100(1–2α)% credible interval (Bayesian analog of the confidence interval). Credibility or credible level 1–2α is the probability, based on the observed food web, that the parameter lies inside the associated credible interval. The conventional practice is to present all intervals at an arbitrary but high credible level—e.g., 95%. Then, often a 95% credible interval for, say, β₁ may include 0, although a 90% credible interval may not. Because 90% is also high, one ought not to regard phylogenetic similarity as a statistically unimportant predictor for feeding behavior simply because the 95% credible interval for β₁ includes 0. The same argument applies to the credible intervals for ρ and ρ_sr, which, when excluding 0, imply the statistical importance, respectively, of the tendency for predator-prey role reversal and of the association between activity as prey and as predator. To avoid being misled by the arbitrary cutoff for credible levels, we computed credible intervals for each model parameter in Table 1 and Table S2 at credible levels of 50% and above, in 5% increments up to 95%, then finally at 99%. Among these intervals, the one that excluded 0 with the highest credibility is reported. If its associated credible level is above 50%, then there is some statistical evidence that the parameter is important; the higher the credible level, the stronger the evidence. We do not report the actual interval if it included 0 with 50% credibility: In this case, there is a high posterior probability that the parameter falls in a range that includes 0, indicating little statistical importance of the corresponding web feature with respect to p_ij. This, however, does not preclude a web feature’s importance with respect to y_ij. In SI Text, we explain the two perspectives of statistical evidence for β₁ ≠ 0. The former corresponds to the direct relationship, as given by Eq. 1, between phylogenetic information and feeding potential (p_ij). The latter, though, corresponds to the amount of statistical gain in describing the act of feeding (y_ij) by including the predictor. A gain is evidenced by a noticeable reduction in GoF in Table 2 and Table S2 between models. The tables demonstrate the relevance of x and/or z to the act of feeding (except for Ythan), even when the relevance to feeding potential is inconsequential (Reef: z; Skipwith, St. Martin: x,z). GoF and credible intervals for β₁ together provide evidence for all eight webs that phylogenetic similarity as the predictor in Eq. 1 yields statistical gain to the understanding of one or both forms of feeding behavior.

Because our statistical SNA provides probabilistic interpretations of trophic structure, such interpretations can be applied to any measure that is computed from the diet matrix to describe trophic relational patterns, which are subject to natural variability. Here, we focus on the connectance index, C; the same principles are applicable to other descriptive measures. The index C is the fraction of observed pairwise links out of all possible pairs in the food web. Although C alone is not designed to describe broad-sense connectivity, it formalizes the notion of network link density that can be visualized in Fig. 2. The posterior predictive distribution of C (Fig. 3) describes the likelihood for the values of C, given the same set of n nodes, under plausible scenarios (due to natural variability) in addition to those that gave rise to the presently observed food web. For example, while C = 0.10 for the current Bay web, the probability is 0.95 for C to fall in the credible interval between 0.07 and 0.13. This inference for C also provides an assessment of uncertainty: A high probability for a short credible interval indicates little uncertainty in C. Similar probability statements for the other seven food webs appear in Table S3. Fig. 3 can also be viewed as a tool for model validation: Model 1 predicts the Bay food web to yield a C that would most likely lie in the range surrounding the mode of this distribution; because the presently observed C indeed lies in this range, our model is consistent with the observed. This consistency was seen in all eight food webs.

Procrustes Transformation.

Due to the invariance of under certain reflection/rotation of the u space and v space, we worked out the mathematics for the special handling of u_i and v_j so that the inference was readily interpretable.

Bayesian inference for model 1 via MCMC involves the simulation of T draws from the jpd of . In what follows, we let the superscript “(t)” denote the tth MCMC sample, for t = 1,2,…,T. The inner product is a parameter in Eq. 1 and is associated with a specific orientation of the u space and v space. The set of MCMC inner products consisting of for t = 1,2,…,T forms the inference for , though our interest lies in the inference for u_i and v_j instead of their product. In this case, basing the inference for u_i directly on is problematic, and similarly for v_j. The reason is as follows. Each and arises from the u space and v space under the tth orientation. The tth orientation may drastically differ from the (t + 1)st, yet may be identical to . Thus, directly pooling the s from all T different orientations leads to uninterpretable inference for u_i, which resides in the u space under a specific orientation. The same argument applies to v_j. Sensible inference for u_i and v_j may be obtained by forcing each tth pair of and to take on a common orientation via a two-sets orthogonal Procrustes transformation (32). This generalizes the Procrustes transformation of refs. 7 and 8 used for the symmetric case where u_i = v_i for all nodes i. For our model under asymmetry, we take and as the arbitrary “default” orientations of the u space and v space, respectively, for each ith node. For each t, the objective is to find a 2 × 2 orthogonal transformation matrix, Q^(t), such that aligns with u_i0 as closely as possible and aligns with v_i0 as closely as possible, for all i. Temporarily drop the superscript “(t)” to reduce clutter, and define U as the 2 × n matrix whose ith column is u_i; define U₀, V, and V₀ similarly. Therefore, for each MCMC sample,

where LDM^′ is the singular value decomposition of , i.e., D is the diagonal matrix whose entries are the singular values of C, L is the orthogonal matrix whose columns are the eigenvectors of CC^′, and M is the orthogonal matrix whose columns are the eigenvectors of C^′C. Each tth MCMC sample has a unique U^(t) and V^(t), and hence, Q^(t) derived as above.

Due to the orthogonality of Q^(t), we have ; thus, the posterior distribution of (hence, inference for) is invariant under the transformation Q^(t). The sets and form the Bayesian statistical inference for u_i and v_j, respectively, and are the basis of (U) and (V) in Fig. 2.