Structure of pathways in ecological networks: relationships between length and number

In ecosystems network, structure determines adjacent (direct) and non-adjacent (indirect) pathways over which energy, matter, and information can flow. The more pathways, the more possible ways the conservative substance can move in zero-sum transactions between network nodes that the pathways interconnect, and the more possible non-conservative, nonzero-sum relations can be secondarily derived from these. Structural analysis is a tool we employ, from a family of input–output methods for exploring zero- and nonzero-sum attributes of environmental networks, to count pathways of varying length between network nodes. In this paper, we examine the relationship between pathway length (k) and number (P_k) as determined by system size (n, number of nodes) and extent and pattern of connectance (C). We develop a measure (m_a) of pathway growth in numbers with increasing length, and then normalize this to the maximum rate possible (m_a/m_c) for a given system size. These measures apply to two pathway types—paths, m_a(0) and m_a(0)/m_c(0), which forbid adjacent node repetitions, and walks, m_a(1) and m_a(1)/m_c(1), which allow them. We find that network size has a curvilinear effect on the pathway number versus length relationship, and extent and pattern of connectance are convolved. Values computed for the paths and walks of three ecosystem models (oyster reef, freshwater marsh, and reservoir cove) are used to compare their pathway structure.