### Summary

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgements
- References
- Supporting Information

**1.** Understanding the structure of ecological networks is a crucial task for interpreting community and ecosystem responses to global change.

**2.** Despite the recent interest in this subject, almost all studies have focused exclusively on one specific network property. The question remains as to what extent different network properties are related and how understanding this relationship can advance our comprehension of the mechanisms behind these patterns.

**3.** Here, we analysed the relationship between nestedness and modularity, two frequently studied network properties, for a large data set of 95 ecological communities including both plant–animal mutualistic and host–parasite networks.

**4.** We found that the correlation between nestedness and modularity for a population of random matrices generated from the real communities decreases significantly in magnitude and sign with increasing connectance independent of the network type. At low connectivities, networks that are highly nested also tend to be highly modular; the reverse happens at high connectivities.

**5.** The above result is qualitatively robust when different null models are used to infer network structure, but, at a finer scale, quantitative differences exist. We observed an important interaction between the network structure pattern and the null model used to detect it.

**6.** A better understanding of the relationship between nestedness and modularity is important given their potential implications on the dynamics and stability of ecological communities.

### Introduction

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgements
- References
- Supporting Information

One well-studied type of ecological network is that of mutual dependences between plants and their pollinators or seed dispersers. It is well established that interactions in these mutualistic networks are heterogeneously distributed among species. That is, the bulk of species have a few interactions, but a few species are much more connected than expected by chance (Jordano, Bascompte & Olesen 2003). This heterogeneity describes a species-level property. If we look at the identity of who interacts with whom at a community-wide level, these networks tend to show a significantly nested pattern wherein specialists interact with proper subsets of the species interacting with generalists (Bascompte *et al.* 2003). These patterns have also been found in host–parasite networks (Vázquez *et al.* 2005). More recently, a significantly modular pattern characterized by the existence of densely connected, non-overlapping subsets of species – called modules – has also been identified. In this case, modules are composed of species having many interactions among themselves as well as very few with species in other modules (Jordano 1987; Dicks, Corbet & Pywell 2002; Olesen *et al.* 2007; Dupont & Olesen 2009).

The dynamical implications of one of the two community-level patterns, nestedness, have begun to be explored. Recent theoretical studies have shown that a nested structure minimizes competition and increases the number of coexisting species (Bastolla *et al.* 2009), and also makes the community more robust to both random extinctions (Memmott, Waser & Price 2004; Burgos *et al.* 2007) and habitat loss (Fortuna & Bascompte 2006). On the other hand, there are fewer studies which investigate the dynamical consequences of the modular structure for mutualistic networks. Nevertheless, since the seminal work of May (1972), it has been considered that modular or compartmentalized patterns described in food webs increase network stability, retaining the impacts of a perturbation within a single module and minimizing impacts on other modules (Krause *et al.* 2003; Teng & McCann 2004; see, however, Pimm 1979).

In spite of the relevance of nestedness and modularity for the stability and persistence of communities, the relationship between these structural patterns remains unknown (see Fig. 1). Only Olesen *et al.* (2007) explored both nestedness and modularity and found coexistence of these patterns in some pollination networks (see also Ramos-Jiliberto *et al.* 2009; Valdovinos *et al.* 2009). However, these authors looked at these two network patterns independently using different null models. Recent warnings suggest that network patterns should be addressed jointly instead of addressing one network pattern at a time (Lewinsohn *et al.* 2006). Similarly, understanding the relationship between several network patterns will help us to accurately determine the relevant and redundant aspects of network structure (Vermaat, Dunne & Gilbert 2009).

In order to better understand the relationship between the two network properties, we have explored nestedness and modularity for a large collection of mutualistic and host–parasite networks using a rigorous comparative framework.

### Results

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgements
- References
- Supporting Information

We found that there is a significant correlation between nestedness and modularity for plant–pollinator communities (*r*=0·363, *P*=0·035) but not for plant–seed disperser (*r*=0·151, P=0·503) and host–parasite (*r*=−0·066, *P*=0·689) networks. However, we note that real communities differ among themselves with regard to both the number of species and of interactions which presents a possible confounding effect. By narrowing our focus on the values of nestedness and modularity calculated for the population of random matrices generated by the two null models, we were able to eliminate this confounding effect.

Using a population of randomizations for each real matrix and calculating the correlation between the two structural properties in each of these populations of randomizations, we observed that there is a change in the sign of the correlation between nestedness and modularity as a function of the connectance (see Fig. 2). For communities with low connectances, the higher the nestedness, the higher the modularity. By contrast, the higher the nestedness, the lower the modularity for communities with high connectances. Thus, an increase in the number of interactions for a fixed number of species reduces the possibility that the interaction matrix would be both nested and modular. This implies that only communities with low connectances are likely to simultaneously present nested and modular patterns. This confirms but constrains the results of Olesen *et al.* (2007) for pollination networks.

The above general result is modulated quantitatively by the type of community (seed dispersal, pollination and host–parasite) and the type of null model used to infer statistical significance. Specifically, at a finer scale, the correlation coefficient between nestedness and modularity decreases significantly with network connectance (Fig. 2) for seed dispersal and host–parasite communities (*r*=−0·801, *P*<0·001; *r*=−0·625, *P*<0·001 for the fixed model and *r*=−0·710, *P*<0·001; *r*=−0·768, *P*<0·001 for the probabilistic one). For pollination communities, this relationship is significant only for matrices created with the probabilistic model (*r*=−0·718, *P*<0·001). Therefore, there is also a change in the magnitude of this relationship depending on the null model used for creating the random matrices. The probabilistic model exhibits a smaller range of coexistence for the two structural patterns. Additionally, the negative correlation between nestedness and modularity is observed at smaller values of connectance for the probabilistic model than for the fixed model (see Fig. 2).

The differences between null models also affect the detection of the nested and modular patterns in real communities. When compared to the probabilistic model, 77%, 88% and 69% of seed dispersal, pollination and host–parasite communities are significantly nested, respectively, compared to just 0%, 15% and 0%, when compared to the fixed model (see Appendix S1). When examining modularity, 9%, 29% and 38% of seed dispersal, pollination and host–parasite communities, respectively, are significantly modular when compared to the probabilistic model, in contrast to 23%, 59% and 67% when compared to the fixed model respectively. The results for nestedness do not change qualitatively when using the analytical measure (Bastolla *et al.* 2009). The primary difference is that when compared to the probabilistic model, the analytical measure less frequently indicates significant nestedness. Importantly, there is a significant positive correlation between the tendency to detect significant nestedness, as measured by the *z*-score, for the metric based on nestedness temperature and the analytic measure for the two null models (*r*=0·930, *P*<0·001 and *r*=0·747, *P*<0·001 respectively).

It is worth noting that the five communities which are significantly nested according to the fixed model are also significantly modular according to that model and significantly nested when compared to the probabilistic model. In the same way, 23 of the 27 significantly modular communities according to the probabilistic model are also significantly nested according to that model and significantly modular according to the fixed model. This implies that both null models are very conservative for the specific patterns they tend to detect as significant. That is, if a community is nested according to the fixed model, that community should also be nested compared to the probabilistic model. Similarly, if a community is modular according to the probabilistic model, that community should also be modular when compared to the fixed model because the probabilistic model is more conservative in detection of significant modularity. It appears that there is strong interplay between the structural patterns of the networks and the null models which detect them. This result builds on the interaction between null model and index of nestedness found by Ulrich & Gotelli (2007).

One factor which helps explain why the probabilistic model tends to detect nestedness more frequently than the fixed one is that the fixed null model exhibits high type II error when detecting nestedness, i.e. it incorrectly accepts the null hypothesis more frequently (Ulrich & Gotelli 2007). As noted above, the trial swap reduces the type II error through an improvement of the sampling of the parameter space (Miklós & Podani 2004). Accordingly, Joppa *et al.* (2010) have found that the fraction of mutualistic communities significantly nested increases to about one of three when using a version of the trial swap and the nestedness temperature used here. This, however, is not incompatible with contributions from additional factors. Among others, there could be correlations between nestedness and degree distribution (see Medan *et al.* 2007).

In 52% of the communities, there is a significant difference (*P*<0·001) in the degree distribution of both plants (hosts) and animals (parasites) in at least 50% of the random matrices. This fraction increases to 94% if we allow that only one of the two degree distributions needs to be significantly different. The correlation between these differences and the differences in nestedness values predicted by the two null models is also significant (*r*=0·521, *P*<0·001, see Fig. 3). Greater differences in the degree distribution are strongly associated with greater differences in nestedness. Hence, the less heterogeneous the degree distribution, the lower is the nestedness value for matrices generated by the probabilistic model. These differences in the degree distribution are not significantly correlated with network connectance (*r*=0·080, *P*=0·442), which is consistent with the fact that this null model probabilistically maintains the overall number of interactions. Therefore, it seems that the probabilistic model increases the chance to detect significantly nested communities because it directly reduces the heterogeneity of the degree distribution. This implies type I errors and further explains the differences in nestedness detection between the two null models.

### Discussion

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgements
- References
- Supporting Information

Our current study methodologically improves that of Olesen *et al.* (2007) in at least two ways. First, the randomization in the test for modularity preserves the bipartite character of these networks (plants can interact only with animals). Second, the randomization uses exactly the same null model for checking the significance of both nestedness and modularity. More importantly, from a conceptual point of view, our current study explores the correlation between these two network patterns; specifically, we address how such a correlation is mediated by network connectance and by the interaction between the network pattern and the null model used to detect it. Despite these differences, our extended and improved exploration confirms the results by Olesen *et al.* (2007) as at least 15% of the communities are significantly nested and modular for both null models, all of them plant–pollinator networks.

Both nestedness (Bastolla *et al.* 2009) and modularity (May 1972; Teng & McCann 2004) are thought to provide benefits for ecological communities. In the context of our study, the intricate relationship between nestedness and modularity has clear potential to temper or augment the different implications of the two patterns. Consequently, as previously suggested by Lewinsohn *et al.* (2006), simultaneously looking at several network patterns can substantially advance our understanding of the architecture of ecological networks.

Recently, Vermaat *et al.* (2009) analysed the covariance among structural properties of food webs. They observed that 20 distinct properties could largely be captured in three major dimensions related to connectance, species richness and net primary productivity respectively. By contrast, we find here that nestedness and modularity do not appear to provide overlapping or redundant information; in fact, the relationship between these two properties and connectance implies the existence of trade-offs in how densely connected communities can fruitfully organize their connections.

It will be interesting to see how an additional property of host–parasite interactions – intervality – fits into the picture we provide here (Mouillot, Krasnov & Poulin 2008). This could contribute to the fact that a large fraction of host–parasite networks are significantly modular when comparing with the two null models. Hence, it appears that antagonistic interactions may tend to be organized in compartments even when they are densely connected.

Apart from this fact, one might have expected larger differences between the different types of networks in our analysis, in particular as the interactions in some are mutualistic, while in the others they are antagonistic. However, we do not find qualitative differences in how the correlation between nestedness and modularity changes with network connectance which appear to arise as a consequence of network type. Nevertheless, we observe here that the most highly connected communities tend to exhibit only one of these two properties.

One potential explanation of this fact is simply that it is exceedingly difficult to organize a large number of interactions. By contrast, it is possible that early on in its assembly, a simple fluctuation makes a community tend toward one pattern over the other and it continues along the same path. Another exciting alternative is that communities dynamically rearrange their interactions, via successful and unsuccessful introductions, speciations, or via extinctions, in one form or the other due to locally relevant features which are not accounted for in our meta-analysis. In either case, the fundamental question of why some communities are more densely connected than others also remains to be fully explained.

### Supporting Information

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgements
- References
- Supporting Information

**Appendix S1**. Data sets analysed in this study.

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