Horizontal networks, however, where interactions occur within the same level of organisation (for example interactions between plants belonging to the same food web) have been more neglected by network ecology ( Ellison 2019). There is now a rich body of work describing the structural properties of food webs, plant-pollinator networks, and host-parasite interactions (e.g. Because network theory can be used to characterise diversity, stability and other community-level properties emerging from species interactions, it has had a long and meaningful impact on our understanding of ecological communities. Network theory has been widely applied to investigate the structure of biological systems such as food webs and other types of multi-level ecological interaction networks. Biological networks have been observed to typically differ from randomly-assembled networks in important and varied ways, thus informing us on the biological processes structuring these systems ( Dunne, Williams, et al. Likewise, interactions or links can operate via many different mechanisms and have a wide range of effects on the nodes. These nodes can take on a wide array of identities, including cells, individuals, populations or species. In order to understand system dynamics when multiple system elements are involved, such complex systems can be represented as networks where the elements are nodes and linked by interactions ( Pimm and Lawton 1978). In many biological systems, interactions between system elements (be these species, individuals, etc.) affect population-level performance and together determine the dynamics of the whole system. The advantages of these features are illustrated with a case study on an annual wildflower community of 22 focal and 52 neighbouring species, and a discussion of potential applications of this framework extending well beyond plant community ecology. The resulting interaction matrices can include positive and negative effects, the effect of a species on itself, and are non-symmetrical. Our method allows us to directly estimate pairwise effects when they are statistically identifiable and approximate pairwise effects when they would otherwise be statistically unidentifiable. We present a novel modelling framework which estimates the strength of pairwise interactions in diverse horizontal systems, using measures of species performance in the presence of varying densities of their potential interaction partners. Quantifying the strength of these interactions from empirical data can be difficult, however, because the number of potential interactions increases non-linearly as more elements are included in the system, and not all interactions may be empirically observable when some elements are rare. Such networks are built from matrices which describe the effect of each element on all others. Network theory allows us to understand complex systems by evaluating how their constituent elements interact with one another.
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