bioRxiv preprint doi: https://doi.org/10.1101/2020.04.15.042697; this version posted April 15, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY 4.0 International license. Socio-Ecological Network Structures from Process Graphs Angelyn Lao1*, Heriberto Cabezas2, Akos´ Orosz2, Ferenc Friedler2, Raymond Tan3 1 Mathematics and Statistics Department, De La Salle University, 2401 Taft Avenue, 0922 Manila, Philippines 2 Institute for Process Systems Engineering and Sustainability, P´azm´any P´eterCatholic University, Szentkir´alyi utca 28, 1088 Budapest, Hungary 3 Chemical Engineering Department, De La Salle University, 2401 Taft Avenue, 0922 Manila, Philippines *Corresponding author E-mail: [email protected] Abstract We propose a process graph (P-graph) approach to develop ecosystem networks from knowledge of the properties of the component species. Originally developed as a process engineering tool for designing industrial plants, the P-graph framework has key advantages over conventional ecological network analysis (ENA) techniques. A P-graph is a bipartite graph consisting of two types of nodes, which we propose to represent components of an ecosystem. Compartments within ecosystems (e.g., organism species) are represented by one class of nodes, while the roles or functions that they play relative to other compartments are represented by a second class of nodes. This bipartite graph representation enables a powerful, unambiguous representation of relationships among ecosystem compartments, which can come in tangible (e.g., mass flow in predation) or intangible form (e.g., symbiosis). For example, within a P-graph, the distinct roles of bees as pollinators for some plants and as prey for some animals can be explicitly represented, which would not otherwise be possible using conventional ENA. After a discussion of the mapping of ecosystems into P-graph, we also discuss how this April 10, 2020 1/18 bioRxiv preprint doi: https://doi.org/10.1101/2020.04.15.042697; this version posted April 15, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY 4.0 International license. framework can be used to guide understanding of complex networks that exist in nature. Two component algorithms of P-graph, namely maximal structure generation (MSG) and solution structure generation (SSG), are shown to be particularly useful for ENA. This method can be used to determine the (a) effects of loss of specific ecosystem compartments due to extinction, (b) potential efficacy of ecosystem reconstruction efforts, and (c) maximum sustainable exploitation of human ecosystem services by humans. We illustrate the use of P-graph for the analysis of ecosystem compartment loss using a small-scale stylized case study, and further propose a new criticality index that can be easily derived from SSG results. Author summary In this study, we propose the novel application of the process graph (P-graph) methodology to the analysis of ecological networks. P-graph was originally developed for engineering design problems; in our work, we show how its five axioms and two algorithms - maximal structure generation (MSG) and solution structure generation (SSG) can be adapted to the problem of understanding complex interactions in ecosystems. The methodology allows multiple types of interactions among ecosystem components to be handled simultaneously based on representation as a bipartite graph. Complete network structures can be deduced from knowledge of local interactions of components using MSG. Finally, all structurally feasible networks of viable ecosystems can be identified with SSG. We illustrate the features of the P-graph methodology with a stylized illustrative example. Introduction 1 Mathematical models have proven to be valuable and useful tools for the analysis of 2 ecological networks and their emergent properties. Early examples include input-output 3 models similar to those used to describe economic structures [1]. Metrics to describe the 4 structure of ecological networks naturally flowed from the use of such quantitative tools 5 (e.g., [2]). These tools provide a lens for the analysis of complex interactions that arise 6 from interactions among ecosystem components. In many cases, specialists only fully 7 April 10, 2020 2/18 bioRxiv preprint doi: https://doi.org/10.1101/2020.04.15.042697; this version posted April 15, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY 4.0 International license. understand local interactions of ecosystem components, and thus need modelling 8 techniques such as ecological network analysis (ENA) to deduce high-level interactions 9 that occur through direct and indirect linkages. 10 Searching the Scopus database using \ecological network analysis" as a search term 11 yields 429 published documents, over half of which were published from 2016 to the 12 present. Despite the broad array of techniques already used in ENA, according to Poisot 13 et al. [3], \ecology will probably continue to benefit from those tools, metrics and models 14 developed in other fields." Thus, in this paper, we discuss a potential new tool for ENA. 15 The use of network-based techniques for the analysis of social-ecological 16 interdependencies remains a challenge [4]. Such models are extended from ENA 17 methods through linkage with a network model of a human community at an 18 appropriate scale. Network techniques are useful for understanding emergent behavior 19 that arises from complex interactions among system components [5]. Such system-level 20 behavior is often not immediately evident from the local properties of individual 21 components, and failure to account for them can often lead to unexpected results [3]. 22 On the other hand, judicious use of ENA and extensions that link them to 23 man-made systems can provide useful insights for sustainable use or resources and 24 ecosystem services [6{9]. These services refer to the conditions and processes by which 25 ecosystems sustain and fulfill human life [10], which can be achieved for example 26 through the provision of shelter, nectar, alternative prey/host and/or pollen for natural 27 enemies which can be deployed by humans [11, 12]. The insights drawn can be used to 28 guide decisions on ecosystem conservation or restoration measures. By incorporating 29 ecological-economic interactions, the models can also be used to estimate the limits of 30 exploitation of ecosystem products and services. 31 Two challenges are apparent in the current literature on social-ecological models. 32 First, existing techniques must assume that a single type of interaction predominates in 33 the system. For example, trophic linkages in food webs are the most commonly 34 represented type of relationship in ecological network models. In order to better 35 understand the behavior of real ecosystems, the capability to represent the existence of 36 multiple simultaneous interdependencies is needed [13]. The current approach relies on 37 multiplex network modelling approaches { i.e., the use of multiple linked network 38 models, each representing one type of interdependence [4,14]. 39 April 10, 2020 3/18 bioRxiv preprint doi: https://doi.org/10.1101/2020.04.15.042697; this version posted April 15, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY 4.0 International license. The second challenge is network assembly. Typically, model developers use 40 knowledge of local interactions of system components, coupled with heuristic network 41 assembly rules, to deduce the structure of the network [15]. While this technique 42 appears to work reasonably well, a mathematically rigorous approach to network 43 assembly can improve ecological modelling by eliminating the potential for human error 44 and biases that always exists when a heuristic is used. 45 The structure of even relatively small ecosystems can be rather complex. Any 46 interactions that could not be observed will, therefore, often not be included. One 47 should also note that frequently more is known about the individual species making up 48 the ecosystem rather than the structure of the ecosystem itself. For example, zoologists 49 are usually able to identify staple food sources of animal species with good accuracy. 50 The reason is that it is far easier to study one species at a time than many species 51 simultaneously. Most ENA work then gives fragmented or one-dimensional 52 representations of real ecosystems. Whereas in reality, ecosystem components play 53 multiple roles relative to each other; thus, there is a need for a framework that allows 54 concurrent modelling of these multiple roles and their resulting complex 55 interactions [16]. 56 While there has been a concerted effort by a small number of researchers to be able 57 to track and measure the complexity of ecological systems, the common approaches in 58 systems ecology still limit their focus to biomass (or energy, nutrients, etc.) fluxes 59 between species (nodes), recycling of material, decomposition, or production. The 60 species are treated as compartments that are interconnected by transaction of the 61 energy{matter substance flowing between them. More subtle interactions such as, for 62 example, the provision of shelter are difficult to include within this framework. 63 We propose the use of a class of models known as process graphs to deal with these 64 difficulties. The process graph (or P-graph) framework was originally developed as a 65 graph theoretic technique for handling combinatorial challenges in industrial plant 66 design [17, 18].