Quantifying and estimating ecological network diversity based on incomplete sampling data

An ecological network refers to the ecological interactions among sets of species. Quantification of ecological network diversity and related sampling/estimation challenges have explicit analogues in species diversity research. A unified framework based on Hill numbers and their generalizations was developed to quantify taxonomic, phylogenetic, and functional diversity. Drawing on this unified framework, we propose three dimensions of network diversity that incorporate the frequency (or strength) of interactions, species’ phylogenies and traits. As with surveys in species inventories, nearly all network studies are based on sampling data and thus also suffer from under-sampling effects. Adapting the sampling/estimation theory and the iNEXT (interpolation/extrapolation) standardization developed for species diversity research, we propose the iNEXT.link method to analyze network sampling data. The proposed method integrates the following four inference procedures: (i) Assessment of sample completeness of networks, (ii) asymptotic analysis via estimating the true network diversity, (iii) non-asymptotic analysis based on standardizing sample completeness via rarefaction and extrapolation with network diversity, and (iv) estimation of the degree of unevenness or specialization in networks based on standardized diversity. Interaction data between European trees and saproxylic beetles are used for illustrating the proposed procedures. The software iNEXT.link is developed to facilitate all computations and graphics.

Methods
The experiment underlying the data was conducted by Vogel and colleagues in the Steigerwald forest located in northern Bavaria, Germany (49°32' N; 10°23' E). The Steigerwald forest covers around 16,500 ha of forests from colline up to sub-montane mountain range with a mean annual temperature around 8 °C and an annual precipitation around 750 to 800 mm (forest climate station Ebrach). Temperatures higher than 10°C are reached on about 160 days per year. 

In 2015, six study sites (Plot A−F) were established. We focus on the data based on two subplots within each plot: (1) open habitat (sun-exposed on a forest meadow), and (2) closed habitat (canopy-shaded within a closed forest stand). Freshly cut logs and branch bundles of six tree species (Abies alba, Carpinus betulus, Fagus sylvatica, Pinus sylvestris, Populus tremula and Quercus petraea) were exposed on each subplot. Deadwood objects were sampled for saproxylic beetles using stem emergence traps. With this sampling technique, it is possible to collect emerging individuals, developing in deadwood, while excluding other occasional visitors. Traps were filled with saturated saline solution as sampling fluid and emptied monthly between April and September in 2016, 2017, 2018, and 2019. Beetles were classified as saproxylic according to Schmidl and Bussler (2004). 

Within each plot and each habitat, data from the four years were pooled. Taxonomic data were adapted from Vogel et al. (2020); a summary table of statistics is provided in Table S3.1 of Appendix S3. The phylogeny and traits for beetle species were based on Hagge et al. (2021) and Seibold et al. (2015) respectively. All species were characterized by a set of 10 functional traits (Chao et al. 2019); species pairwise distances were computed from the Gower distance.