Compartment identification

The search for compartments in complex networks is also often referred to as analysis of “community structure” (Newman and Girvan 2004) owing to origins in social science literature. Similarly, compartments are also referred to as “topological modules” in the general study of complex networks (Danon et al. 2005). A widely used approach to the identification of compartments in complex networks is to define a quality function, the modularity, that is high for “meaningful” partitions of a network into compartments and zero for typical (random) partitions (Newman and Girvan 2004). Despite some limitations (Fortunato and Barthélemy 2007, Sales‐Pardo et al. 2007), modularity maximization is the most accurate current method for identifying compartments. Here, we describe in detail the modularity functions we use for the identification of compartments in food webs.

The definition of a modularity function depends on how the network is represented. Food webs are directed networks in that trophic interactions occur between predators and prey that play distinct roles in the interaction. In a directed representation, the links can be thought of as arrows going from prey to predator in the direction of mass flow. As a simplification, however, one can disregard the direction of the interactions and create an alternate representation which is referred to as an undirected network. In a food web, the predators and prey are still connected but the links are no longer arrows; given the undirected representation of a food web it is not possible to determine which species are predators and which prey.

The modularity M_U(P) of a partition P of an undirected network is defined as the fraction of links within compartments minus the expected fraction of such links (Newman and Girvan 2004). The expected fraction of links within compartments is evaluated assuming that the probability that nodes i and j are connected is k_ik_j/2L, where k_i is the number of links of node i and L is the total number of links in the network. Therefore, the modularity is as follows (Newman and Girvan 2004):

The simplest generalization of Eq. 1 to directed networks is the following (Arenas et al. 2007, Leicht and Newman 2008):

An alternative approach for the directed case is to consider the outgoing (incoming) links of each node, and then build compartments comprising nodes with similar outgoing (incoming) links. The modularity M_O(P) of a partition P of a directed network, according to the outgoing links, is as follows (Guimerà et al. 2007):

The goal of a module identification algorithm is to find the partition P* that yields the largest modularity (note that the number N_C of compartments is only constrained to be N_C ≤ S, but is otherwise selected by the optimization algorithm). Due to the large number of possible partitions of a network and to the rugged structure of the modularity landscape, it is necessary to use heuristic algorithms to identify maximum‐modularity partitions. We use a spectral algorithm to maximize modularity Eq. 2 (Leicht and Newman 2008) and simulated annealing to maximize the modularity Eq. 3 (Guimerà and Amaral 2005, Guimera et al. 2007).

Compartment homogeneity

We quantify the homogeneity of compartments in three ways which we refer to as trophic‐level, sink, and niche homogeneity. Trophic‐level homogeneity measures the extent to which compartments comprise species with contiguous trophic levels. To calculate trophic‐level homogeneity within compartments, we start by sorting species according to their trophic level l_i, which we calculate as follows (Levine 1980):

We then count the number B* of boundaries, in that ordering, between the observed compartments (see Appendix A: Fig. A1). For example, if we have two compartments, one compartment comprising the species with lowest trophic levels and another comprising the species with the highest trophic levels, then B* = 1. In general, if compartments are perfectly homogeneous, the number of boundaries is B* = N_C − 1 (where N_C is, as before, the number of compartments). If the compartments are less homogeneous in terms of the trophic levels of the species they comprise, the number of boundaries will be larger.

We define trophic‐level homogeneity H_t as

Niche homogeneity measures to what extent compartments correspond to contiguous regions of the niche space. We define niche homogeneity H_n in the same way as trophic‐level homogeneity, but counting the boundaries in the niche space, rather than the trophic‐level space. (In principle, this can only be done with model networks, for which the niche value of each species is known; we discuss the issue of empirical proxies for niche value later.)

Sink homogeneity measures to what extent compartments correspond to sink food webs within the food web. For each species i, we first find the set T_i of species that belong to i's sink food web (which is made up of i's prey, the prey of i's prey, and so on until no new species are encountered). We then count the number of species that belong to both T_i and species i's compartment C_i. We define sink homogeneity H_s as

Food‐web models

We consider two models to generate food webs: the niche model (Williams and Martinez 2000) and the generalization of the niche model by Stouffer et al. (2006). To generate a food web using the niche model (Williams and Martinez 2000), one considers S species and assigns, to each of them, a niche value n_i drawn from a uniform distribution in the interval [0, 1]. Each predator j then preys on the species in a range r_j = n_jx of the niche axis, where x is drawn from a beta‐distribution p(x) = β(1 − x)^(β−1) and β = (S²/2L) − 1, where L is the total number of trophic interactions in the network. The center of the range r_j is selected uniformly at random in the interval [r_j/2, n_j]. All species i whose niche values n_i fall within this range are considered prey of species j. This implies that, in the niche model, all predators' diets are contiguous within the dimension provided by the niche axis.

To generate a food web using the generalized niche model (see Appendix C: Fig. C1; Stouffer et al. 2006), one also assigns to each species a niche value drawn from a uniform distribution in the interval [0, 1], as in the niche model. The generalized niche model, however, allows for tunable prey contiguity. First, the reduced range for predator j is set to = c × r_j = c × n_jx, where c is a fixed parameter in the interval [0, 1]. Because species are distributed uniformly at random on the resource axis, a predator j with range r_j has on average r_jS prey. The same applies to the reduced range , and therefore a predator has Δk = (r_j − )S = (1 − c) r_jS anticipated prey unaccounted for after the range reduction. To account for this, Δk prey (rounded to the nearest integer value) are selected randomly from those species i with niche value n_i ≤ n_j that are not already a prey of species j. The parameter c is thus a measure of prey contiguity: for c = 0, all prey of j are selected randomly among species with n_i ≤ n_j and one recovers the generalized cascade model (Stouffer et al. 2005), whereas for c = 1, all prey are contiguous and one recovers the niche model. Diet contiguity c can be thought of as the degree to which a unidimensional niche axis is able to explain the properties of an empirical food web.

Parameter estimation for model food webs

For each empirical food web, we need to know three parameter values in order to generate food webs with the generalized niche model (Stouffer et al. 2006): the number of species S, the linkage density z, and the diet contiguity c. The number of species and the linkage density can be obtained directly from the empirical network. Therefore, only one parameter is estimated for each food web.

Estimating the diet contiguity c is involved since in principle there is no direct way to measure it from a given food web; rather, c needs to be estimated from the data (Stouffer et al. 2006). To do this, we use the following procedure for each food web:

1) Obtain 20 model food webs with each of three different values of c ∈ {0.5, c*, 1.0}, where c* is the best estimate of c obtained using the procedure proposed by Stouffer et al. (2006).

2) Estimate the z score for different compartmentalization properties (modularity, trophic and sink homogeneity, number of compartments, standard deviation of the compartment sizes, and average size of compartments with more than one species), for the prey and predator views. Estimate the log‐likelihood of each value of c as the sum of squares of the z scores.

3) Select the value of c with the highest log‐likelihood, generate new model food webs with values of c around it, and iterate the process until the likelihood does not improve.

Note that, since this procedure to estimate c is not exhaustive (exhaustive search is too expensive computationally given the large number of operations involved), we cannot rule out that there are slightly different values of c that better explain the data. We list our best estimates for the parameters in Table 1.

Table 1. Food webs and their properties from the literature.

Compartment definition and identification

As noted previously, compartmentalization refers to the existence of groups of species that have a higher probability of interacting with each another than with other species in the food web (May 1972, Girvan and Newman 2002, Krause et al. 2003). What constitutes an interaction, however, depends largely on the process under consideration. One possible interpretation would be that a group of species interacts strongly if there are many direct trophic interactions between them. A group of species can also interact strongly via competition, if they share a large number of prey, even though there might not be any direct trophic interactions between the species in the group.

With this in mind, we consider here three different definitions of a compartment in a food web (Methods): (1) a group of species that are densely connected to each other (density view); (2) a group of species that share a large number of prey (predator view); and (3) a group of species that share a large number of predators (prey view) (Fig. 1).

From a topological perspective, these are the only three sensible definitions of compartment since they correspond to measures of compartmentalization based on all links, the incoming links, and the outgoing links of each species, respectively. The density view is the appropriate generalization to directed networks (Arenas et al. 2007, Leicht and Newman 2008) of algorithms that have been used before to identify compartments in food webs disregarding the direction of trophic interactions (Girvan and Newman 2002, Krause et al. 2003, Rezende et al. 2009). The prey and predator views result in compartments that are closely related to the groups defined by Allesina and Pascual (2009) (Fig. 1D, H).

In the density view, the modularity function in Eq. 2 measures the difference between the fraction of interactions within compartments and the expected fraction of interactions within compartments in a random partition, taking into account the direction of the trophic interactions (Leicht and Newman 2008). In the predator (conversely, prey) view, we consider partitions based only on the prey (predators) of a species. In the predator (prey) view, the modularity in Eq. 3 is large when species in the same compartment share more prey (predators) than one would expect from chance alone (Guimerà et al. 2007).

We have analyzed the compartmentalized structure of 10 of the most complete food webs in the literature (Table 1 and Williams and Martinez [2008]) from each of these three viewpoints. Not surprisingly, the three views yield distinct compartmentalizations of a food web (Fig. 1A–D).

Degree of compartmentalization of empirical food webs

Recent studies have observed correlations between species characteristics and compartmentalization in food webs: species within compartments seem to have similar habitats (Girvan and Newman 2002, Krause et al. 2003) and seem to be related phylogenetically (Rezende et al. 2009). However, none of these studies has evaluated quantitatively the extent to which the various factors can actually explain the observed compartmentalization. In fact, quantitative investigations of compartmentalization have only been able to establish that real food webs are more compartmentalized than would be expected for a randomly assembled food web (Krause et al. 2003, Rezende et al. 2009).

Here, we take a more systematic approach and proceed by investigating empirical compartmentalization vis‐à‐vis that of two of the simplest models known to generate realistic food webs (rather than purely random networks): the niche model of Williams and Martinez (2000) and its generalization by Stouffer et al. (2006). By understanding the cause of compartmentalization in the models, we gain insights into the causes of compartmentalization in real food webs.

We present here the results for the generalized niche model while those for the niche model are shown in Appendix E. Specifically, we compare modularity values for the real food webs to those obtained from the generalized niche model, for each of the three views. As we show in Fig. 2A–C, the modularity of empirical food webs is within the 99% expectation interval of the models in 28 out of 30 cases, regardless of the compartment definition we use. In the two remaining cases, the modularity of the empirical food web is only slightly below the model predictions. Thus, real food webs are never more compartmentalized than one would expect from the generalized niche model.

Beyond modularity, we have also measured a variety of other properties of the compartments (Appendix D: Fig. D1): the number of compartments, the standard deviation of the number of species per compartment, and the mean size of compartments containing more than one species. For all of these properties and, again, for each definition of compartment, almost all the points (88 out of 90) fall within the 99% expectation intervals.

In general, the original niche model performs worse than the generalized niche model (see Appendix E). This is largely due to its tendency to overestimate the modularity of empirical networks.

Trophic structure of empirical and model compartments

In the previous section we have shown that the degree of compartmentalization of empirical food webs is never larger than expected from the generalized niche model. Moreover, we have shown that other properties of real compartments (the number of compartments, the typical compartment size, and the standard deviation of the size of compartments) are also compatible with the predictions of the generalized niche model.

Ecologically, however, a far more important consideration is whether the trophic structure of real compartments is also the same as that of model compartments. To investigate this question, we examine two measures of compartment homogeneity (see Methods for details): trophic‐level homogeneity and sink homogeneity. Trophic homogeneity measures to what extent species in the same compartment occupy similar trophic levels. Sink homogeneity measures to what extent species in the same compartment belong to the same set of sink food webs.

In Fig. 2D–I, we show the homogeneity of compartments identified in our model networks using the three compartment identification approaches. For all views, mean trophic‐level homogeneity is typically higher than expected from chance alone, suggesting that trophic level is a strong driver of compartmentalization. Mean sink homogeneity, by contrast, is significantly positive only in the density view, and random or even negative for the predator and prey views (since compartments in these two views cut across food chains).

Importantly, Fig. 2 also shows that, like for the number of compartments and their sizes, the generalized niche model accounts for the trophic structure of the compartments: in only two cases out of 60 is the observed homogeneity smaller than expected from the model. As before, the original niche model tends to less accurately explain the trophic structure of empirical compartments (see Appendix E).

Empirical compartmentalization and niche space

The previous sections suggest that the compartmentalization observed in real food webs can be accounted for by some mechanism that is already encoded in the generalized niche model. This result is counterintuitive because the generalized niche model does not incorporate attributes—such as habitat fragmentation or phylogenetics—that would explicitly give rise to compartments. Conceptually, the only relevant property of a species in the model is its niche value: other than the niche value of each species, the only parameters of the model are global parameters such as the number of species, the average number of prey (or predators) per species, and the diet contiguity. Therefore, compartments in generalized niche model food webs seem to arise from the underlying niche structure of the food web.

This conjecture has two important implications, which we investigate using the generalized niche model. First, if compartments arise from the niche structure underpinning the food web, compartmentalization should increase with increasing diet contiguity c (see Methods) because the niche values of species become more relevant when diets are continuous. The increased explanatory power of niche values is due to the fact that large contiguity of diets in the model induces strong correlations because two species with similar niche values are also expected to have similar predators (and, to a lesser extent, prey). Indeed, we observe that, regardless of the compartment definition, increasing diet contiguity in the generalized niche model results in greater compartmentalization (Fig. 3 and Appendix F). The original niche model's overestimation of compartmentalization appears to stem from this fact as diet contiguity in this model is maximal.

Second, if compartments arise from the niche structure, then one would expect the niche to be a strong driver for compartmentalization and compartments to have a strong “niche signal.” To investigate this, we examine a niche homogeneity that quantifies to what extent species in the same compartment have similar niche values (see Methods for details). In Fig. 4, we show that compartments in all views are typically homogeneous in terms of the niches of the species they comprise, in agreement with our expectation.