A comparative study of ecological networks using spectral projections of normalized graph Laplacian

Abstract: Ecological networks originating as a result of three different ecological processes are examined and cross-compared to assess if the underlying ecological processes in these systems produce considerable difference in the structure of the networks. Absence of any significant difference in the structure of the networks may indicate towards the possibility of a universal structural pattern in these ecological networks. The underlying graphs of the networks derived by the ecological processes, namely host-parasite interaction, plant pollination and seed dispersion are all bipartite graphs and thus several algebraic structural measures fail to distinguish between the structure of these networks. In this work we use weighted spectral distribution (WSD) of normalized graph Laplacian, which have been effectively used earlier to discriminate graphs with different topologies, to investigate the possibility of existence of structural dissimilarity in these networks. Graph spectrum is often considered a signature of the graph and WSD of the graph Laplacian is shown to be related to the distribution of some small subgraphs in a graph and hence represent the global structure of a network.We use random projections of WSD to $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$ and establish that the structure of plant pollinator networks is significantly different as compared to host-parasite and seed dispersal networks. The structures of host parasite networks and seed dispersal networks are found to be identical. Furthermore, we use some algebraic structural measures in order to quantify the differences as well as similarities observed in the structure of the three kinds of networks. We thus infer that our work suggests an absence of a universal structural pattern in these three different kinds of networks.