## Network Environ Analysis Primer

The details of NEA have been developed elsewhere (see Patten, 1978, 1981, 1982, 1985, 1991, 1992), so below we provide just a general overview for orientation to the discussion below. Ecosystem connections, such as flow of energy of nutrients, provide the framework for the conceptual network. The directed connections between ecosystem compartments provide necessary and sufficient information to construct a network diagram (technically referred to as a digraph) and its associated adjacency matrix—an n X n matrix with 1s or 0 s in each element depending on whether or not the compartments are adjacent. Using this information, structural analysis is possible, which is used to identify the number of indirect pathways and the rate at which these increase with increasing path length. With quantitative information regarding the storages and flows (internal and boundary) of the system compartments, additional functional analyses are possible—primarily referred to as flow, storage, and utility analyses (Table 5.1). The key to the analysis is using the direct adjacency matrix or non-dimensional, normalized matrices in the case of the functional analyses (gip py, and dy, respectively) to find indirect pathways or flow, storage, or utility contributions. The network parameters, gij, Py, and dj, in addition to having an important physical characterization in the network, control the integral network organization and structure within the system. Contributions along indirect pathways are revealed through powers of the direct matrix, for example, G has the direct flow intensities, G2 gives the flow contributions that have traveled 2-step pathways, G3 those on 3-step pathways, and Gm those on m-step pathways. Given the series constraints, higher order terms approach zero as m— 4, thereby making it possible to sum the direct and ALL indirect contributions (m > 2) produce an integral or holistic system evaluation (see Box 5.2). In the case of the functional analyses, integral flow, storage, or utility values are the summation of the direct plus all indirect contributions (N, Q, U, respectively). In this manner it is possible to quantify the total indirect contribution and compare it with the direct flows, the result being that often the direct contribution is less than the indirect, hence leading to the need for a holistic analysis that accounts for and quantifies wholeness and indirectness. This is the primary methodology for investigating system structure, function, and organization using NEA. Below we give two numerical examples that illustrates typical results of a NEA. The next section will give an overview of insights in the resulting, possible effects of networks.

Table 5.1 Overview of network environ analysis

Network Environ Analysis

Table 5.1 Overview of network environ analysis

Network Environ Analysis

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