## Methods and Sample Network

To demonstrate basic environ analysis, it is best to proceed with an example. Consider the network in Figure 1, which has five compartments or nodes (x, for i = 1-5). Compartments are connected by transaction of the energy-matter substance flowing between them. These pairwise couplings are the basis for the internal network structure. A structural connectance matrix, or adjacency matrix, A, is a binary representation of the connections such that aij = 1 if there is a connection from j to i, and a 0 otherwise (eqn [1]):

Storage and flows must have consistent units (although it is possible to consider multiunit networks). Typically, units for storages are given in amount of energy or biomass per given area or volume (e.g., gm-2), and units for flows are the same but as a rate (e.g., gm-2d-1). The intercompartmental flows for Figure 1 are given in the following flow matrix, F:

0 |
0 |
0 |
0 |
fl5 |

f21 |
0 |
0 |
0 |
0 |

f3l |
f32 |
0 |
0 |
0 |

0 |
f42 |
f43 |
0 |
0 |

0 |
0 |
f53 |
f54 |
0 |

Note that the orientation of flow from j to i is used because that makes the direction of ecological relation from i to j. For example, if i preys on j, the flow of energy is from j to i. All compartments experience dissipative flow losses (y, for i = 1-5), and here the first compartment receives external flow input, z1 (arrows not starting or ending on another compartment represent boundary flows). For this example, these can be given as y = [ yi yi y3 y4 ys]

Total throughflow of each compartment is an important variable, which is the sum of flows into, Tn = z + Pf, or out of, Tout = y, + Pf the ,th compartment. At steady state, compartmental inflows and outflows are equal such that dx/dt = 0, and therefore, incoming and outgoing throughflows are also equal: T™ = T°ut = Ti. In vector notation, compartmental throughflows are given by

This basic information regarding the storages, flows, and boundary flows provides all the necessary information to conduct environ analysis. Environ analysis has been classified into a structural analysis, dealing only with the network topology, and three functional analyses (flow, storage, and utility) - which requires the numerical values for flow and storage in the network (Table 1).

The technical aspects of environ analysis are explained in detail elsewhere, so rather than repeat those here, the remainder of the article highlights some of the important results from environ analysis. But first, one issue that must be covered is the way in which network analysis identifies and quantifies indirect pathways and flow contributions. Indirectness originates from transfers or interactions that occur nondirectly, and are mediated by other within-system compartments. These transfers could travel two, three, four, or many links before reaching the target destination. For example, the flow analysis starts with the calculation of the nondimensional flow intensity matrix, G, where gj=j Tj. The generalized G matrix corresponding to Figure 1 would look as follows:

These values represent the fraction of flow along each link normalized by the total throughflow at the donating compartment. These elements give the direct, measurable flow z

Table 1 Basic methodologies for network environ analysis

Structural analysis

Functional analyses

Path analysis

Enumerates pathways in a network (connectance, cyclicity, etc.)

Flow analysis: gj = fj/Tj

Identifies flow intensities along indirect pathways Storage analysis: cij = f/xj

Identifies storage intensities along indirect pathways Utility analysis: dj = (fj- j/T Identifies utility intensities along indirect pathways intensities (or probabilities) between any two nodes j to i. To identify the flow intensities along indirect paths (e.g., j! k ! i), one need only consider the matrix G raised to the power equal to the path length in question. For example, G2 gives the flow intensities along all paths of length 2, G3 along all paths of length 3, etc. This well-known matrix algebra result is the primary tool to uncover system indirectness. In fact, it turns out that due to the way in which the G matrix is constructed, all elements in Gm go to zero as m n. Therefore, it is possible to sum the terms of Gm to acquire an 'integral' flow matrix (called N), which gives the flow contribution from all path lengths:

where G0 = I, the identity matrix, G1 the direct flows, and Gm for m >1 are all the indirect flows' intensities. Note, that the elements of G and N are nondimensional; to retrieve back the actual throughflows, one need only multiply the integral matrix by the input vector: T = Nz. In other words, N redistributes the input, z, throughout each compartment to recover the total flow through that compartment. Similarly, one could acquire any of the direct or indirect flows by multiplying Gmz for any m.

A similar argument is made to develop integral storage and utility matrices:

overview is given for four of these properties: dominance of indirect effects (or nonlocality), network homogenization, network mutualism, and environs.

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