  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 there is a quantity that 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 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 interesting features in the behavior for the K-complexity and SVD entropy
with clear differences between pollinator-plant and host-parasite networks,
especially when the degree distribution is not preserved. Although SVD entropy
has been widely used to characterize network complexity, our analyses show that
K-complexity is a more reliable tool. Additionally, we show that degree
distribution and density are important drivers of network complexity and should
be accounted for in future studies.
