Comparison of two statistical measures of complexity applied to ecological bipartite networks

Networks are a convenient way to represent many interactions among different entities as they provide an efficient and clear methodology to evaluate and organize relevant data. While there are many features for characterizing networks, a quantity seems rather elusive: Complexity. The quantification of the complexity of networks is nowadays a fundamental problem. Here, we present a novel tool for identifying the complexity of ecological networks. We compare the behavior of two relevant indices of complexity: K-complexity and Single Value Decomposition (SVD) entropy. For that, we use real data and two types of null models. Both null models consist of randomized networks built by swapping a controlled number of links of the original ones. We analyze 23 plant–pollinator and 19 host-parasite networks as case studies. Our results show that (a) it is necessary to calculate, not only the original network K-complexity and SVD entropy but also to calculate the corresponding indices of the randomized networks (b) the density and degree distribution are essential in the characterization of a network and the randomized networks are a suitable tool to detect the network complexity, and (c) plant–pollinator networks are more complex than host-parasite networks. We found that, for the first null model, K-complexity and SVD did not change with link swapping in both pollinator-plant and host-parasite networks. For the second null model, K-complexity for pollinator-plant networks generally decreased with an increasing number of links swapped (i.e. negative slope), showing that plant–pollinator networks lose complexity with increasing link swapping. In contrast, there was a positive slope between K-complexity and link swapping for host-parasite networks, showing that these networks are less complex than plant–pollinator networks. For both types of networks, in general, the slope between K-complexity and the number of links swapped became more positive with network density. Overall, SVD entropy was less responsive to link swapping. Our analyses show that although SVD entropy has been widely used to characterize network complexity, K-complexity is a more reliable tool. Additionally, they show that degree distribution and density are important drivers of network complexity and should be accounted for in future studies.