Connectance and degree distributions are important components of the structure of ecological networks. In this contribution, we use a statistical argument and simple network generating models to show that properties of the degree distribution are driven by network connectance. We discuss the consequences of this finding for (1) the generation of random networks in null-model analyses, and (2) the interpretation of network structure and ecosystem properties in relationship with degree distribution.

Ecologists developed a strong interest for network theory as it allows them to make sense of some of the complexity of ecological communities. In contrast to early approaches on “community modules” (groups of a few species within a large community,

Early in the ecological network literature, connectance, i.e., the proportion of realized ecological interactions among the potential ones (most often the squared species richness), has been recognized as a central network property (

However, it is worth asking if we were not too quick in focusing most of our research effort on degree distribution, in detriment to more fundamental work on connectance and its effects. A network, ecological or otherwise, can be viewed as a physical space that edges (interactions) occupy. The size of this space is limited by the number of nodes. This means that there are physical constraints on the filling of a network, due to the fact that placing the first edge will limit the number of ways to place the remaining edges, and so on. For example, there is only one way to have a fully connected network, and there are a limited number of ways to have a network with the lowest possible connectance. For this reason, and given the rising importance of degree distribution in the literature, it is important that we clearly understand how constrained this distribution actually is in relation to connectance. In this contribution, using an argument from combinatorial statistics and simulations of pseudo-random networks under two different models, we present strong evidence that degree distribution, along with other emerging network properties, are constrained (and can be predicted to a certain extent) by connectance. We discuss the consequences of our results for the comparison of different ecological networks, and for the generation of random networks in null-model analyses.

Assuming an ecological network made of _{n}. With this information in hand, it is possible to know the total number of possible networks given a number

If we term _{n} the set of all possible _{n} edges in a _{n, l} of possible networks with _{n}, i.e., how many possibilities are there to pick _{n}. Formally, this is expressed as _{n, l}. In addition, in a null-model context (_{n−1, l} possible networks configurations. Note that these networks will also include situations in which _{n−2, l} and so forth is not necessary (all networks with more than one node of null degree are within the set of the networks with at least one node of null degree). Calling _{n, l} the number of networks with

Because _{n}∕2. As in this context the maximal number of edges is _{n}, we define effective connectance as _{n}, so _{n, l}) is reached at _{n, l} is hard to find, but simulations show that it also occurs around

_{n, l} will become asymptotically closer to _{n, l} when _{n}. In other words, there is only one way to fill a network of _{n} interactions, and in this situation there is no possibility to have nodes with a degree of 0. In the situation in which _{n}, _{n} > _{n−1}, it comes that _{n, l} = _{n, l} = 1. Intuitively enough, this implies that ecological systems in which connectance is high will display very little variation from one another, as far as the distribution of emergent network properties (e.g., variance of the degree distribution, nestedness, …) is concerned.

We now illustrate these predictions using networks of 10 nodes, with a number of edges varying from 10 to _{10} (i.e., 45 edges). As illustrated in

As predicted in the main text, (1) the size of network spaces peaks at

In

In the previous part, we show mathematically that connectance (the number of realized

In the simulations below, we use networks of 30 nodes, filled with 35 to _{30} interactions, by steps of 10. We use two different routines to generate random networks that are contrasted in the way they distribute edges among nodes. First, we generate Erdős-Rényi (ER, undirected) graphs, meaning that every potential interaction has the same probability of being realized (

For each replicate, we measure the degree distribution and report its variance, coefficient of variation, kurtosis, and skewness. In addition, for each network, we fit a power-law distribution on the sorted degree distribution using the least-squares method; we report the power-law exponent.

These results show that central properties of the degree distribution are contingent upon connectance, at a given network size, and under a given network generation model. ER networks are in blue, niche-model networks are in red. Each point represent a single generated network.

Qualitatively, the random graphs and the niche networks behave exactly the same. With the exception of the kurtosis,

To quantify the impact of connectance on the different network properties, we measured the proportion of variance explained by the linear regression of a given property against connectance (in such cases as had a linear relationship between the two, i.e., all measures but variance). Kurtosis is independent of connectance (

The estimate of the power-law exponent increases when connectance increases (

Randomized null models are often used to estimate how much a given network property deviates from its random expectation (^{3} or 10^{4} replicates (typical values in null models analyses) is orders of magnitude smaller than the

Second, generating null models with a low connectance is a computationally intensive task. When connectance decreases, the _{n, l}∕_{n, l}) goes toward zero. For this reason, classical rejection sampling (accept the random network if no nodes have no edges, reject it if not) is bound to take an unreasonable amount of time in networks with low connectance. In addition, there is a risk of selecting some particular types of networks. It makes intuitive sense that networks with extremely skewed degree distributions have less chance of being generated this way, as when a few nodes collect most of the edges, the probability than the remaining nodes each have at least one edge decreases. To the best of our knowledge, this source of bias has not received important attention in the literature. For this reason, using a purely random matrix shuffling as a starting point, then swapping interactions until no free nodes remain, seems to be a promising way to address this problem. Given the important of null-model approaches in network analysis, the generation of efficient and unbiased algorithms is a fruitful research avenue.

Connectance is an extremely intuitive property of network, expressing how much of the potential interactions are realized. Through statistical reasoning and simple simulations using models of random networks, we show that for a given number of species, connectance drives (i) how many different networks exist, and (ii) some key elements of the degree distribution. We observed both among and between model quantitative changes in degree distribution along a connectance gradient. The niche model is a particularly striking example of this, with the variance in the degree distribution increasing 50-fold when connectance moves from 0.1 to 0.5. This result has practical implications for network comparisons. As descriptors of degree distribution vary with connectance, connectance should be factored out from all analyses. So as to avoid colinearity issues, this can be done by either working on the residuals of the degree distributions’ property of interest. To some extent, the impact of connectance is lesser in the 0.05–0.3 range where most empirical food webs lies (although bipartite networks can have much higher connectances), but the effect is high enough that it should not be ignored: at equal number of species, networks with different connectances are expected to have different degree distributions.

Finally, this analysis raises interesting ecological questions. Early analyses focusing on degree distribution argued that ecological mechanisms were responsible for the shape of the distribution (

We thank Luis Gilarranz, Miguel Lurgi, and Enrico Rezende for comments, and Amael LeSquin for discussions on algebra.

The authors declare that they have no competing interests.