Contents lists available at ScienceDirect Ecological Indicators journal homepage: www.elsevier.com/locate/ecolind Walk partitions of flow in Ecological Network Analysis: Review and synthesis of methods and indicators Stuart R. Borrett a,b, ⁎ , Ursula M. Scharler c a Department of Biology & Marine Biology, University of North Carolina Wilmington, Wilmington, NC 28403, United States b Duke Network Analysis Center, Social Science Research Institute, Duke University, Durham, NC 27708, United States c School of Life Sciences, University of Kwazulu-Natal, Durban, South Africa ARTICLE INFO Keywords: Input-output analysis Food web Materials flow analysis Ecosystem network analysis Network science ABSTRACT Ecological Network Analysis (ENA) has provided insights into the structure, function, and transformation of ecosystems for more than forty years. Key insights from ENA focus on how the patterns of directed weighted transactions among system components (e.g., species, functional groups, economic sectors) create emergent and often unexpected relationships in ecosystems that affect system function and sustainability. Flow analysis, also called throughflow analysis, is one of several core techniques in ENA. Generally, it traces the flux of energy or matter through the network from inputs to outputs. During the forty-years of development, flow analysis has accreted multiple extensions and modifications. In this concept and synthesis paper, we review four flow ana- lyses and show how they are conceptually linked by partitioning flows across subsets of pathways within net- works. These flow analyses include: (1) the definition of throughflow, a measure of the total processing power of a network; (2) Leontief’s decomposition based on walk length, indicating the direction and distance of energy or matter flow; (3) Finn’s measure of recycling of matter in networks; and (4) five mode analysis, characterizing flows according to their origin and destination. Presenting these techniques side-by-side with a common con- ceptual framework reveals overlaps and distinctive elements among the analytic products. This synthesis clarifies the flow analyses tools and their applications to ecological and socio-economic networks and provides example applications. Further, new insights are presented by combining existing flow analyses to calculate novel indices that further characterize the flow structure of networks. For example, both indirect flows in networks and cycling are highly important features in networks. In order to determine the proportion of indirect flows gen- erated through cycling, we can use the ratio of Cycled Flow identified from Finn’s analysis and the indirect flows identified in the Leontief analysis. As ENA matures through additional analysis development and applications, it will continue to provide insights into ecosystems and contribute to the broader area of network science. 1. Introduction “…we find a fundamental change of metaphores: from seeing the world as a machine to understanding it as a network” (Capra, 2015). “I trust the flow of life. ” Markéta Irglová Ecosystem ecology has developed in part by focusing on practical concerns such as ecosystem degradation and the challenge of managing their services and resilience (Díaz et al., 2018; Odum, 1997; Golley, 1993; Van Dyne, 1966). This focus enabled substantial progress in ap- plied ecosystem ecology, but also in building theoretical understanding (Dakos et al., 2015; Hastings and Gross, 2012; Jørgensen et al., 1992). Network ecology (Proulx et al., 2005; Ings et al., 2009; Borrett et al., 2014) is a key theoretical ecology framework that lies at the intersection of ecology and network science (Brandes et al., 2013; Wasserman and Faust, 1994; Newman, 2010). Over its history, it has built a considerable body of network analysis methodologies, and contributed to the characterization of ecosystem evolution and state. More recently, these methods have been integrated into policy frame- works, such as within European countries concerned with the man- agement of terrestrial ecosystems (Creamer et al., 2016). The char- acterization of ecosystem development by using network analysis methodologies has also attracted the attention of policy-related en- deavours such as the Marine Strategy Framework Directive of the European Union (Lynam et al., 2016; de la Vega et al., 2018). Ecological network analysis (ENA) is a branch of network ecology that is used to describe an ecosystem’s state, resilience, and functioning (Patten et al., 1976; Ulanowicz, 1986; Borrett et al., 2018). ENA https://doi.org/10.1016/j.ecolind.2019.105451 Received 5 September 2018; Received in revised form 27 May 2019; Accepted 30 May 2019 ⁎ Corresponding author at: Department of Biology & Marine Biology, University of North Carolina Wilmington, Wilmington, NC 28403, United States. E-mail address: borretts@uncw.edu (S.R. Borrett). Ecological Indicators 106 (2019) 105451 1470-160X/ © 2019 Elsevier Ltd. All rights reserved. T