Network optimization model implies strength of average mutual information in ascendency

Ecological Modelling 179 (2004) 373–392

Network optimization model implies strength of average mutual information in ascendency Luke G. Latham II a,∗ , Erik P. Scully b b

a The Agent Modeler Page, 600 Milford Court, Abingdon, MD 21009–2957, USA Department of Biological Sciences, College of Science and Mathematics, Towson University, 8000 York Road, Towson, MD 21252–0001, USA

Received 22 April 2003; received in revised form 30 September 2003; accepted 29 April 2004

Abstract Ulanowicz’s [J. Theor. Biol. 85 (1980) 223; Ulanowicz, R.E., 1997. Ecology, the ascendent perspective. In: Allen, T.F.H., Roberts, D.W. (Eds.), Complexity in Ecological Systems Series. Columbia University Press, New York, p. 201] ascendency (A) index of community growth and development is based, in part, upon the average mutual information (AMI) index of Rutledge et al. [J. Theor. Biol. 57 (1976) 355]. AMI is an average of mutual constraint on a quantum of material or energy in networks and is reputed to quantify development of ecological systems. Ascendency is the product of the AMI and the total system throughput (T). In published calculations of A, the magnitude of T dwarfs the magnitude of the AMI, and A is well correlated with some measures of analysis that are correlated with T [Ecol. Model. 79 (1995) 75]. Investigations have suggested that T is dominant in the calculation of A. Total system throughput could scale AMI in several ways (e.g., nth root, logx ), but AMI has been consistently scaled by T since its original formulation in [J. Theor. Biol. 85 (1980) 223]. We used a network optimization procedure to show that strict selection for networks with a high A produced food webs that were unlike networks selected for either high AMI or high T. The influence of AMI in the A-optimized systems is clearly discernible in a non-metric multidimensional scaling (NMDS) analysis based upon 54 indices that were calculated for the networks. These results suggest that the scaling of AMI by T in the original formulation of A yielded an index wherein the AMI plays an important role in quantifying dimensions of network structure not present when systems are merely optimized for T. © 2004 Elsevier B.V. All rights reserved. Keywords: Ascendency; Average mutual information; Network analysis; Information theory; Structural dynamic model

1. Introduction Trophic exchanges in ecological systems can be represented as networks. Interactions are modeled as transfers of material or energy between populations. In a symbolic representation of the system, each pop-

∗ Corresponding author. Tel.: +1-410-515-0946; fax: +1-410-704-2405. E-mail address: [email protected] (L.G. Latham II).

ulation appears as one node. The nodes are connected if there is a direct transfer of material or energy. Shannon (1948) and Shannon and Weaver (1949) founded communication theory. Further development of communication theory by Gallager (1968) led Rutledge et al. (1976) to apply an index of communication theory, the average mutual information (AMI), to ecological networks. The contribution of Rutledge et al. (1976) led Ulanowicz (1980) to develop an index, ascendency (A), which encompasses the natural growth and development of ecological sys-

0304-3800/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2004.04.017

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tems. Ascendency is a measure that seeks to explain mathematically much of the phenomena described by Odum’s (1969) paper The Strategy of Ecosystem Development. These phenomena include several trends in developing ecosystems, such as: increasing niche specialization, formation of a predominantly detrital food web, a reduction in nutrient exchange rates between populations and the environment, more developed internal symbiosis, better nutrient conservation, lower entropy, and higher information (Odum, 1969). Ulanowicz (1997) held that A increases during the development of an ecosystem from an early, undeveloped successional stage to a climax stage. The index is said to quantify feedback and system size, two important factors in development (Ulanowicz, 1980, 1986, 1989, 1997, 1998a,b; Ulanowicz and Wolff, 1990). The system development component of A is measured by the AMI. Ulanowicz (1980, 1986, 1997) has held that the AMI should increase with development, which would indicate a refining of the network to distinctly constrained flow patterns during the maturation phase of system development. Feedback (and autocatalytic cycling), possibly reflected by an increasing AMI, is seen as ‘constraint’ because the individuals of populations in material and energy cycles benefit from efficient channeling of flow along pathways that become less arbitrary as the system matures. [For a detailed explanation of the AMI, see Latham and Scully, 2002; for more information on the suggested behavior of AMI and T in development see Ulanowicz, 1997, Section 4.8, pp. 86–92.] Because T dwarfs the AMI numerically, several authors have commented or suggested that T dominates the calculation of A, while the AMI has received little attention. The following selection of comments was found in the recent literature. [Bracketed comments are ours.] Christensen (1994a, p. 142) Noteworthy is that with an emergy-based flow system the Ascendency becomes less a function of throughput, more a function of flow patterns. [Following his result that indicated an aggregation of lower trophic levels in sample systems from the literature where T was high had little effect on the Ascendency value. All sample systems had less than 35 groups of organisms (compartments).] Jørgensen (1994, p. 15)

The size term, T, is most dominant in most calculations. (A − T)/T = I − 1 is an intensive attribute and accounts to a certain extent only for the structure or information embodied in the network, i.e., the complexity of the network, independent of the energy through-flow. It might therefore be expected that the structure exergy is well correlated to A − T, while exergy would be better correlated to the size term, T, although probably also well to A, as the size term is most dominant. [The Exergy hypothesis can be found in Jørgensen (1992a) and Jørgensen et al. (1995).] Christensen (1994b, pp. 43–44) Ascendency is calculated as the product of throughput and information content, and is as such expected to reflect variations in both. Fig. 1 therefore shows the Ascendency as a function of throughput (Fig. 1a) and of information content of the links (Fig. 1b). [Fig. 1a is a graph of Ascendency by throughput with a clear linear relationship; Fig. 1b is a graph of Ascendency by average mutual information showing no relationship.] Throughput here varies over more than two orders of magnitude while information content only varies with a factor of two. Ascendency will therefore chiefly reflect the variation in throughput. . . . I conclude that Ascendency as presently calculated cannot reflect ecosystem growth and development as intended by Ulanowicz (1986). Only growth is reflected. Patten (1995, p. 77) Since the informational component, Eq. (1) [referring to AMI], of ASC [Ascendency] takes on only a narrow range of small real number values, 0 ≤ H ≤ ∼6, Ascendency tends to be dominated by the contribution of throughflow. Therefore, its maximization as a goal function is very akin to the maximum power principle. [The Maximum Power Principle (‘power’ is work per unit time) was first published by Lotka (1922). Odum and Pinkerton (1995) hypothesized that power output is maximized in ecosystem development.] Jørgensen et al. (2000, p. 262) As ascendency is dominated by its extensive variable, throughflow, if this is maximized then storage must be sacrificed accordingly in the steady-state

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relationship. . . . Maximization of ascendency, a measure heavily dominated by throughflow, can thus be taken as generally consistent with the exergy-storage hypothesis. Fath et al. (2001, p. 503) The AMI scales T according to the system organization, but the [ascendency] measure is primarily dominated by the throughflow because the contribution of average mutual information is usually small in relation to that of throughflow. Thus, this goal function [ascendency] is consistent and highly correlated with the first reference condition [referring to ‘maximum power’ or generally, maximum T] (also see Jørgensen, 1994). In regard to Christensen’s (1994a,b, 1995) static models, they were representations of systems at some unknown state in their developments; therefore, it is difficult to draw conclusions about a relationship between A and maturity in each system’s development (Mageau et al., 1998). Christensen’s work most appropriately addresses the issue of A’s ability to assess maturity given a single static position of the system, but does not answer the more fundamental question whether A tracks development within individual systems. It seemed that Christensen (1995, p. 4) was acknowledging this point when he wrote: Both Ulanowicz (1986) and Jørgensen (1992b) assumed that system development can be captured with a single goal function. One problem with this is that we do not intuitively know if we have the right answers. We cannot study two systems and conclusively say which is the more developed or more mature. To address this issue, we wished to determine if a network optimization routine that built networks under the direction of these indices would suggest that networks optimized for high A were similar to networks optimized for high T or high AMI. We built a structurally dynamic model that manipulates networks under the direction of an index itself, which we term a “selection regime.” A structurally dynamic model is one in which the structure of the model can change during simulation in response to a goal function (Jørgensen, 1999).

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We used the process of artificial selection to build networks by: (1) generating a set of networks, each with random changes; (2) mass balancing the networks; and (3) selecting the network with the highest value of some index of interest to be the basis for another round of progeny networks. This process of creating variation followed by selection allowed us to study the structure of networks that were highly correlated with the index under study. We also had the opportunity to randomly select networks over time starting with the same base system used for each selection regime replicate. This allowed us to determine if changes the in network structure in experimental runs were different from networks selected without a goal function. Many indices were tracked and used to compare the selection regimes and their controls with an appropriate multivariate statistical method (non-metric multidimensional scaling (NMDS)). Analyzing the indices of thousands of randomly derived networks does not reveal that any index rises consistently during succession, so we chose to use an index-driven approach. Our philosophy is not unlike that suggested by Straškraba (1980) and Mauersberger and Straškraba (1987) and put into practice by several researchers (Nielsen, 1992, 1994, 1995, 1997; Coffaro et al., 1997; Jørgensen, 1999): structurally dynamic models change in response to a control function over time. The control function serves as a goal to shape the network; consequently, providing information about network structure that is correlated with the index of interest. For an excellent review of these concepts, including references to previous examples of structurally dynamic models and goal function theory, see Bendoricchio and Jørgensen (1997) and Jørgensen (1999). Salomonsen (1992, p. 172) also suggested this method: It may be a better approach to make a model with a flexible set of parameters. In ecological modelling it is not possible to reflect every detail of the system, and most often it is only desirable to simulate whole system characteristics, e.g., Secchi depth or total production. In such cases it would be sufficient to know the ranges of a chosen set of parameters representing the gene pool of the ecosystem, and to have an algorithm operating on these parameters, to find the values in best accordance with the intrinsic

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goal of the system. Such an algorithm is also referred to as a goalfunction [sic] (Straškraba, 1980). Ulanowicz does not believe that A is optimized by changes in a network over time, as we have assumed here, but rather that there is a propensity for A to increase over time during growth and development (Ulanowicz, 1997; Ulanowicz and Abarca-Arenas, 1997). Ulanowicz suggested to us in the planning stages of this project that it might be more realistic to randomly select networks from the top tier of those that have a high A value, not by the highest value of the index among the progeny systems (personal communication). In a test run under that scenario and running the simulation for the same period of time, the differences among the final configurations of the networks was reduced; consequently, the model required more time in order to see how the selection regimes influenced overall network structure. The trajectory of the indices was not different, only the speed at which the indices changed (data not shown). We decided to proceed with the optimization model and present those results here, because this method was the most efficient (i.e., required the least amount of time) for determining if the AMI has leverage on the configurations of networks that are under pressure to increase A and how each of the selection regimes was different from the controls (i.e., random selection). We are not attempting to model real systems with the structurally dynamic approach here. We are seeking to use the approach as a tool to assess the relative contribution of the AMI in networks that are forced (via artificial selection) to improve A generally.

2. Total system throughput (T ) The total system throughput is the sum of the total link flows in the system (Eq. (1)). Total links are a measure of maximum power in a system (Fath et al., 2001). T =

n+2 n



Tij ,

(1)

i=0 j=1

where Tij is the flow from compartment i to compartment j, and n the number of compartments.

We use the Hirata and Ulanowicz (1984) convention: Compartment 0 (zero) is the source of inputs to the system. Compartment n + 1 is the flow of usable exports from the system. Compartment n + 2 is the destination of unusable exports (dissipation) from the system. 3. The average mutual information The average mutual information is the average constraint placed upon a single unit of flow in the network (Rutledge et al., 1976; Ulanowicz, 1997; described in detail in Latham and Scully, 2002) (Eq. (2)). AMI = k

n+2 n

Tij i=0 j=1

T

log2

Tij T , Ti Tj

(2)

where Tij and T are as noted in Eq. (1), Ti is the total flow out of compartment i, Tj the total flow into compartment j, and k a scalar constant. 4. Ascendency Ulanowicz (1980, 1986, 1997) describes the growth and development of natural systems as the product of the AMI and T. By substituting the total system throughput, T, for the scalar constant, k, in the AMI equation (Eq. (2)), we obtain the equation for ascendency (A, Eq. (3)). The index is said to quantify rising autocatalytic activity among components; consequently, it reflects development of positive feedback in systems (Weber et al., 1989; Ulanowicz, 1997). Note that Ulanowicz and Norden (1990) provided corrections to earlier publications dealing with the derivation of A, e.g., Ulanowicz (1980, 1986). We use the Ulanowicz and Norden (1990) equations for A and its related indices throughout. A=

n+2 n

i=0 j=1

Tij log2

Tij T Ti Tj

(3)

5. Model description and parameters We wrote a computer program that built ecosystem models, which had the capacity for random interspecific changes and guild membership. A guild

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represents a group of compartments that have similar feeding relationships. We allowed each parent network to undergo random changes in the formation of progeny systems, followed by selection of a progeny system using the information index as a selection criterion to become the new parent system for further development in the next iteration. We propose that A, AMI, and T were tested for their ability to shape resultant networks and that these selection regimes can be compared and contrasted by looking at a suite of indices. The null hypothesis for our project is that systems selected by A will not be different from systems selected by T. We assumed that the non-metric multidimensional scaling statistical procedure would indicate that the networks were significantly different when the average of 12 replicate runs for each selection

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regime lay outside of one standard error in a NMDS plot. 5.1. Program overview Our experimental design in shown in Fig. 1. The computer program generated a random base system, which served as the starting point for a control run and an index-selected run. The control run was initiated first. Each iteration of the model, ten progeny networks were produced, each with random changes from the parent system. The number of modifications to make for each progeny was set to 10% of the number of compartments in the parent network. Six network changes could occur through time: the addition or deletion of compartments, the addition or deletion of links, and the strengthening or weakening

Fig. 1. Our experimental design involved artificially selecting networks by ascendency (A), average mutual information (AMI), and total system throughput (T), which we refer to as “selection regimes.” There were 12 replicate selection regimes for each index with controls (controls were produced by randomly selecting networks). The final configurations of the networks were assessed by 54 indices. The similarity of the final configurations was determined with a standard multivariate statistical procedure (non-metric multidimensional scaling).

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of links. One of the progeny was randomly chosen to become the parent for the next iteration. During the run, indices were recorded for the randomly selected systems. After 40 iterations, the control run ended. Halting the simulation at 40 iterations was an arbitrary decision based upon test runs of the model. Forty iterations were sufficient to make clear distinctions among the selected systems when they were present. The optimization run started with the same base system, and progeny were produced the same way. The only difference was that the progeny system with the highest value of the index under scrutiny was the system that was chosen to be the new parent system and give rise to new progeny. All indices were recorded, and the selection run continued until 40 iterations were reached. We divided each index by the maximum value of that index to standardize the data. [The rationale for standardization is explained in Section 6: Statistics.] The standardized index scores were entered into a non-metric multidimensional scaling analysis to compare selection regimes (Kruskal and Wish, 1978; Mather, 1976; Manly, 1994). The control/selected run pairs were duplicated (i.e., a new, randomly generated base network was generated after each pair of runs) until we had 12 control/selection pairs for each selection regime. We performed three selection regimes (A selection, T selection, and AMI selection) with and without limiting system size to the base network. Limiting system size to the base network would provide an opportunity to compare the rate that AMI can increase when T is held constant under A selection with the rate that AMI can increase when T can take on large values under A selection. It was logical to limit size to that of the base system for the control/treatment pair. The results of runs with and without a network size limit will appear in two separate NMDS plots. Nielsen (1992) suggested the following categories for structurally dynamic models: experimentalistic (developed in close association with experiments in real ecosystems), empiristic (using any type of empirical information), trophic (contains and describes the tropic structure or web of a system), and holistic (knowledge from the other three categories is used in the model and a goal function shapes some or all of the results of simulation). Our model has clearly taken into account trophic structure by including feeding guilds, and we artificially-select networks with a goal

function to learn about associated changes in other indices; therefore, our model is experimentalistic and partially holistic. 5.2. Hierarchically-bound rules A guild hierarchy was imposed in the model (Figs. 2–5). The trophic structure was defined by trophic level and by interspecific interaction and conceptualized as a lake or stream community. Note that flow can move both up the trophic levels (Fig. 2) and back to bacteria and nutrient pools via direct flow to bacteria (Fig. 3), through fungal/invertebrate decomposers (Fig. 4), and via parasites (Fig. 5)—cyclic flow is accommodated by this scheme. The trophic structure imposed the following restrictions on compartment and link additions. (‘L’ in the following description refers to ‘level.’ Refer to Figs. 2–5, which portray the relationships that follow graphically.) L0-Nutrient Pools are non-biotic pools of material/energy for the network. Pools could be added at any time. Nutrient Pools could receive input from outside the system, L1-Bacteria, L1-Primary Producer, or L6-Invertebrate/Fungal Decomposer. L1-Primary Producers and L1-Bacteria were the base of the food web. L1-Bacteria consumed any L0-Nutrient Pool, any L1-Primary Producer, or any Level 2–7 compartments. L1-Primary Producers consumed any L0-Nutrient Pool. Populations of L2-Zooplankton Bacteria Feeders, L2-Zooplankton Herbivores, L2-Invertebrate Bacteria Feeders, L2-Invertebrate Herbivores, and L2-Fish Herbivores could be added. L2-Zooplankton Bacteria Feeders consumed L1-Bacteria. L2-Zooplankton Herbivores consumed L1-Primary Producer. L2Invertebrate Bacteria Feeders consumed L1-Bacteria. L2-Invertebrate Herbivores consumed L1-Primary Producer. L2-Fish Herbivores consumed L1-Primary Producer. Populations of L3-Invertebrate Zooplankton Specialists, L3-Invertebrate Invertebrate Specialists, L3Invertebrate Generalists, L3-Fish Zooplankton Specialists, L3-Fish Invertebrate Specialists, L3-Fish Fish Herbivores Specialists, and L3-Fish Generalists could be added. L3-Invertebrate Zooplankton Specialists consumed any L2-Zooplankton. L3-Invertebrate Invertebrate Specialists consumed any L2-Invertebrate or any L6-Invertebrate Decomposer population. L3-

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Fig. 2. Diagram of trophic compartments and potential linkages among compartments in the food web. The model could build networks with up to 50 of each compartment type shown. All compartments could have input from outside the system, dissipations (unusable exports), usable exports, or self-links.

Fig. 3. Diagram of potential trophic links through the Bacteria compartments and forward to the Nutrient Pools.

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Fig. 4. Diagram of potential trophic links through the Invertebrate or Fungal Decomposer compartments and forward to the Nutrient Pools.

Fig. 5. Diagram of potential trophic links through the Parasite compartments and forward to Bacteria compartments and Nutrient Pools. Networks could have up to 15 parasite compartments.

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Invertebrate Generalists consumed any L2-population or any L6-Invertebrate Decomposer population. L3-Fish Zooplankton Specialists consumed any L2-Zooplankton. L3-Fish Invertebrate Specialists consumed any L2-Invertebrate or any L6-Invertebrate Decomposer population. L3-Fish Fish Herbivores Specialists consumed any L2-Fish Herbivore population. L3-Fish Generalists consumed any L2-population or any L6-Invertebrate Decomposer population. Populations of L4-Fish Invertebrate Generalists, L4-Fish Fish Generalists, and L4-Fish Generalists could be added. L4-Fish Invertebrate Generalists consumed any L3-Invertebrate. L4-Fish Fish Generalists consumed any L3-Fish. L4-Fish Generalists consumed any L3-population. The top level could have L5-Fish Generalists. L5-Fish Generalists consumed any L4-Fish population. Invertebrate/Fungal Decomposer compartments could have a strong influence on cycling in the networks, due to their capacity to redirect flow to the lower trophic levels. L6-Invertebrate/Fungal Decomposer consumed any Invertebrate or Fish compartment (not L1-Bacteria). Parasite compartments could become associated with any invertebrate, fish, or decomposer compartment. L7-Parasite consumed any single Invertebrate or Fish compartment (not L1-Bacteria, not L1-Primary Producer).

1–100,000 units per L1-Bacteria and L1-Primary Producer, and 1–10,000 units for each L2-level compartment. Units of flow among compartments were somewhat arbitrary and not meant to reflect energy or carbon explicitly, although we decided to bind unusable dissipation (i.e., respiration) from each compartment at 90% of each compartment’s outflow (discussed further in Section 9). Our decision probably entails considering the networks as energy flow networks. Ten percent of L0-Pool input was directed to export from the system. Ninety percent of input to other compartments was directed to dissipation (i.e., respiration and other destinations that can not be used by the system or other systems), while 1% was directed to export. At this point, each biotic compartment had 9% more input than output. A subroutine moved through the network systematically compartment by compartment adding links between compartments until the 9% difference between input and output was distributed through the network. The approach of systematically balancing compartments is clearly a bottom-up approach. Other approaches (e.g., random, top-down) were not tested, and we do not know if our choice for linking or mass balancing compartments introduced bias. However, we did run randomly selected control networks for comparison starting the same base systems that were used for the treatment selection regimes.

5.3. Base network generation

5.4. Progeny modifications

Base networks consisted of 25 L0-Pool compartments, 16–25 L1-Bacteria compartments, 16–25 L1-Primary Producer compartments, 4–6 of each of the L2 compartment types (except L2-Fish Herbivore compartments = 1–3). The model could build networks with up to 50 of each type of compartment (except L-7 Parasites = 15 maximum). Compartment 0 was the source of exogenous inputs. We used the scheme of Hirata and Ulanowicz (1984), where usable exports (e.g., organic matter) are directed to compartment n + 1 and unusable exports (i.e., dissipations/respiration) from the system are directed to compartment n + 2. Compartment 1016 was the destination of usable exports. Compartment 1017 was the destination of unusable exports. Input from outside the system was: 1–1,000,000 units per L0-Pool,

A description of the six alterations that could occur for generated progeny networks follows. Each alteration conformed to the hierarchy/trophic rules outlined above. For the removal of a compartment, associated links to other compartments became 0 (zero). For the addition of a compartment, a maximum of 1015 network compartments could be built into a network. Each compartment addition could have up to five sources of inflow, which were randomly determined from those available. The quantity of the inflow(s) to the new compartment could be 1 to the total outflow of each source compartment. The outflow from the added compartment was set equal to its newly established inflow and was split between export (10%) and dissipation (90%).

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For the removal of a link, a link between two compartments was randomly selected for removal. For the addition of a link, if no link existed between two randomly selected compartments, a link was generated. The additional inflow to the destination compartment was split between export (10%) and dissipation (90%). For a decrease in a link magnitude, any randomly chosen link could be decreased 1–100%, rounded to

the nearest natural number. In an increase to a link magnitude, any randomly chosen link could be adjusted upward 1–100%, rounded to the nearest natural number of arbitrary units. The size of the adjusted link would not exceed the total inflow of the source compartment. After a modification was made to the network, any compartments with a zero inflow were purged. The

Table 1 Indices tracked during simulation and used to characterize the networks Total system throughput (T, Eq. (1))a Average mutual information (AMI, Eq. 2)a Ascendency (A, Eq. (3))a

Gross production to respiration ratio (PG /R, Eq. (17)) Net production (PN , Eq. (18))b

Ascendency/capacity (A/C, Eq. (6))a ,b

Proportion of total flow that is input (pinput , Eq. (19))b Proportion of total flow that is export (pexport , Eq. (20))b Proportion of total flow that is dissipation (pdissipation , Eq. (21))b,c Compartment diversity (Diversity, Eq. (22))b

Rutledge diversity (HR , Eq. (7))b

Link density (LinkDen, Eq. (23))b

Rutledge stability (DR , Eq. (8))b

Average link strength (AvLinkSt, Eq. (24))b

Total uncertainty (Hmax , Eq. (9))b

Average compartment throughflow (AvgCompT, Eq. (26))b Number of compartment removals (RmC)

Overhead (O, Eq. (4))a ,b Capacity (C, Eq. (5))a ,b

Network structure uncertainty (Hsys , Eq. (10))b Constraint information (Hc , Eq. (11))b Constraint efficiency (CE, Eq. (12))b

Number of compartment additions (AdC) Number of removed links (RmL)

Number of links (#links)

Number of added links (AdL)

Proportion of total flow in detrital pathways (pdetrital , Eq. (13))b Proportion of specialists in the third trophic level (pspecialists , Eq. (14))b Number of compartments (#comp)

Number of decreased links (DeL)

Gross primary production (PG , Eq. (15))b Respiration (dissipation) (R, Eq. (16))b

Number of increased links (InL)

Proportion of compartment type L0-Pools (pL0 ) Proportion of compartment type L1-Bacteria (pL1-Bact ) Proportion of compartment type L1-Primary Producers (pL1-PP )

Proportion of compartment type L2-Consumers (pL2 ) Proportion of compartment type L3-Consumers (pL3 ) Proportion of compartment type L4-Consumers (pL4 ) Proportion of compartment type L5-Consumers (pL5 ) Proportion of compartment type L6-Decomposers (pdecomp ) Proportion of compartment type L7-Parasites (pparasit ) Proportion of total flow through L0-Pools (pfL0 ) Proportion of total flow through L1-Bacteria (pfL1-Bact ) Proportion of total flow through L1-Primary Producers (pfL1-PP ) Proportion of total flow through L2-Consumers (pfL2 ) Proportion of total flow through L3-Consumers (pfL3 ) Proportion of total flow through L4-Consumers (pfL4 ) [0] Proportion of total flow through L5-Consumers (pfL5 ) Proportion of total flow through L6-Decomposers (pfdecomp ) Proportion of total flow through L7-Parasites (pfparasit ) Proportion of self-linked compartments (pselfLin )b Connectance (Connect)b Compartmentalization (Compart)b

Variable names with lowercase ‘p’ or ‘pf’ mean ‘proportion’ or ‘proportion flow’ and refer to either proportion of compartments or proportion of flow. a Ulanowicz and Norden (1990) provided corrections to ascendency calculations described in Ulanowicz (1980, 1986). b Equation in Appendix A. c Non-biotic pools transfer large quantities of throughflow, but they do not respire. System-wide respiration is calculated as the total respiration of biotic compartments divided by total system throughput. Although all biotic compartments respire approximately 90% of their input, the total network respiration (dissipation) is a smaller fraction of total flow due to the presence of pools.

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system was mass balanced (i.e., the total flow into each compartment was equal to the outflow from that compartment and the total flow into the system was equal to the total flow out of the system). The mass balance was performed in an iterative fashion: the system was searched for compartments that either had input less than output or input greater than output. When a compartment was found with total inflow exceeding total outflow, 90% of the difference was added to that compartment’s dissipation (respiration), while the remaining 10% was randomly redistributed among existing outflows. If no outflows were found, the difference was directed to dissipation. When a compartment was found with total outflow exceeding total inflow, the excess outflow was randomly trimmed from the existing outflows. The trimming was carried out such that dissipation did not fall below 90% of total inflow. This process was repeated until no out-of-balance compartments were found. 5.5. Indices tracked during simulations Fifty-four indices were recorded during the simulations (Table 1). These indices were either common in the network analysis literature or obvious to us from the network we modelled. There was no simple way to decide which indices to use in the analysis. We believe that these 54 indices are sufficient to decide if the AMI had a controlling interest in the configurations of networks selected by A. Other investigators may have added or deleted indices from this list, but those choices would probably be just as arbitrary as ours. We comment more on this point in the discussion. We denote indices in Table 1 where we have shown an equation in the Appendix A.

6. Statistics We required a multivariate statistical procedure to show us graphically the relative distances of the final network configurations among A-selected, AMI-selected, T-selected, and randomly selected networks. We recorded 54 indices on the final networks, so we required a procedure to get those 54 dimensions down to 2 or 3 that could be easily graphed. The raw index scores from the simulations were standardized by dividing each score by the maximum

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value obtained for that index for all runs and entered into a non-metric multidimensional scaling analysis in PC-ORD® (McCune and Mefford, 1999). NMDS is an ordination method that iteratively searches for a ranking and placement of each observation (i.e., a selection regime run; control or selected) on k axes that minimizes the “stress” of the configuration. Stress is a measure of the dissimilarity between the distances among the simulation runs in the original matrix and the distance among the simulation runs in the reduced k-dimensional space. NMDS was used because it is insensitive to nonnormal data and data that are on arbitrary, discontinuous, or other scales (McCune and Mefford, 1999; for a general description, see Manly, 1994). Each axis in the ordination space is unitless and arbitrary; therefore, the NMDS plot is only valuable to separate the selection regimes by their relative distances on the plot. We used one standard error with our averages to assess variation in the results and decide if selection regimes produced similarly structured networks. The calculations and procedures required to perform an NMDS analysis are lengthy and best left to the computer. PC-ORD was used on “autopilot mode” for the initial run. The autopilot mode completes preliminary runs to six dimensions with the real data and randomized data (i.e., uses random starting points for the ordination). Based on the lowest final stress in each dimension from a real run, the best solutions are recorded. PC-ORD selects the dimensionality from among the best solutions. The final stress must be lower than that for 95% of the randomized runs (Monte Carlo test). We used the results of the preliminary analysis as the starting point for the final analysis using the number of dimensions suggested by the software. Proper interpretation of NMDS requires: (1) for each number of dimensions produced, the solution for a given axis is unique; and (2) the coordinates on axes are arbitrary. The first item is of utmost importance here, because unlike principal component analysis (PCA), each axis is not related in a sequence of decreasing variance. The plot is only useful as a means to separate relative distances of the selection regimes. NMDS assumes that a unit of change is equivalent for each index, but the raw index values do not conform to this assumption. A small change in one index might be just as important as a large change in another index. Because testing showed that the NMDS analysis

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was sensitive to the variation in magnitude of each index across the simulation runs, we divided each simulation run index by the maximum value of that index from all runs. Standardization removed the problem where a small numerical change in one index, within a small numerical range for that index in all simulation runs, did not receive equal value in the analysis. Because the output of NMDS includes a point in ordination space for each replicate simulation run, the plot coordinates of a given regime (control or index-selected) were averaged. Standard errors were calculated and appear on the plots around the averages as error bars (±1 S.E. above and below/left and right of the points on each axis of the 2D plot).

7. Simulations There were 12 replicate selection regime runs, each of which included a control run (randomly-selected network) and a treatment run (index-selected network). The following simulation selection regimes were run: (1) average mutual information with no limit on system size [N = 12, 40 iterations per run]; (2) total system throughput with no limit on system size [N = 12, 40 iterations per run]; (3) ascendency with no limit on system size [N = 12, 40 iterations per run]; (4) average mutual information limited to the size of the base network [N = 12, 40 iterations per run]; (5) total system throughput limited to the size of the base network [N = 12, 40 iterations per run]; and (6) ascendency limited to the size of the base network [N = 12, 40 iterations per run].

Fig. 6. Time series plot of the AMI values of control and A-selected networks both with and without a system size limit (i.e., maximum T limited to that of the base network for each pair). Points are the average of 12 replicates, and error bars are ±1 S.E. Note that the rate of AMI increase was similar both with and without the T limit.

8. Results We graphed AMI, T, and A in time series plots for A-optimized networks, both with and without a limit on system size (Figs. 6–8). At five unit time intervals, we plotted the average of our 12 replicates with one standard error above and below the average. Fig. 6 shows that control networks showed a marked decline in AMI, which indicates a significant decrease in the average constraint on a given quantum of material or energy in the network over time. Both the size-limited and size unlimited systems showed an increase in AMI.

Fig. 7. Time series plot of the T values of control and A-selected networks both with and without a system size limit. Points are the average of 12 replicates, and error bars are ±1 S.E. The T limitation is apparent in the T-limited set of runs.

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Fig. 8. Time series plot of the A values of control and A-selected networks both with and without a system size limit. Points are the average of 12 replicates, and error bars are ±1 S.E. In A-selected networks, the rise in A is the product of both T and AMI gains. Under T-limitation, A rises at a slower rate due to the sole contribution of rising AMI.

Total system throughput is graphed in Fig. 7. The control selection produced smaller systems over time. The influence of limiting the size of a system is clearly evident in the graph, where those systems showed no increase in T over time. The size unlimited system showed a marked increase in T. Fig. 8 plots A for the selection regimes selected for high A values. It is no surprise based upon seeing Figs. 6 and 7 that the control networks show a large decline in A, due to the decline in both AMI and T over time. The unlimited systems benefited from a rise in both components of A, while the limited systems showed a much smaller elevation in A due to the sole influence of rising AMI. The multivariate NMDS analysis reordinates the many variables we recorded to provide a sense of distance among the final network configurations in the NMDS plots (Figs. 9 and 10). Ascendency-optimized networks had alternative structural elements to T-optimized networks according to the plots. Note that A-selected networks in both figures did not coincide with the T-selected networks. They were closer to T-selected networks than to AMI-selected networks, but this may only be due to our selection of indices or

385

Fig. 9. NMDS plot showing distances among controls (random selection), T-selected, A-selected, and AMI-selected networks based upon 54 indices when there was no T limit imposed during the model runs. Points represent the average of 12 replicates, and error bars are ±1 S.E. NMDS axes are unitless and arbitrary. The plot is only useful as a means to discern relative distances among the final configurations of the networks. Due to the influence of increasing AMI values during selection of networks for high A, the position of A-selected networks is displaced from the position of T-selected networks.

Fig. 10. NMDS plot showing distances among controls (random selection), T-selected, A-selected, and AMI-selected networks based upon 54 indices when T was limited to the T of the base network during the model runs. As in Fig. 9, the A-selected networks are displaced from the position of T-selected networks.

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Fig. 11. Ascendency regressed against T (a) and AMI (b) for 12 A-selected systems. The same pattern found by Christensen (1994b) emerges: The A values are well correlated with T, but not with the AMI.

other bias in the model. We discuss this issue further below. We regressed the A values against T and AMI from A-selected networks for Fig. 11. We see that A has a good correlation to T (r = 0.95; r2 = 0.90), but not to AMI (r = 0.60; r2 = 0.36). Although we found no colinearity structure between the T and AMI values, we chose not to enter the data into a two-way analysis of variance. Because the numerical value of A is the product of the two indices, they will both show up as significant effects under all but the most exaggerated circumstances (e.g., one or the other index is held constant or nearly constant).

9. Conclusions This paper was originally submitted with model runs and statistics based on unconstrained networks.

Respiration was not bounded to a minimum of 90% of the inflow to each compartment and took on low values for many biotic compartments during the runs (e.g., 20–40% of output; data not shown). High dissipation values are consistent with published respiratory quotients for organisms, and we decided to adopt an arbitrary energetic currency for these networks. After we received recommendations to explore more work with structurally dynamic models from Sven Jørgensen and two anonymous reviewers, we found a paper by Søren Nielsen, where he reported that biologically unrealistic values were obtained with unbounded parameters in his efforts to study exergy optimization with structurally dynamic models (Nielsen, 1995). We reprogrammed our model such that all biotic compartments would have a minimum respiration value of 90%. The results presented in this paper are from the revised model. Regardless of our decision to bind respiration values to this criterion, we found the same result with

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the revised model that we did with the unconstrained model (data not shown). Previous research outlined in the Introduction seems to suggest that T and A are well correlated, and the magnitude of A is strongly determined by high numerical values of T compared to the small numerical values of the AMI (e.g., in static models of 41 ecosystems studied by Christensen (1994b, 1995). We agree that this will be true for static calculations of A; however, A is an index reputed to quantify structure and size (T) over time in developing networks. Although the authors we quote were not saying that the AMI quantified little in networks that were optimizing A, it is not apparent from those comments that the AMI is a strong structural indicator in the calculation of A. This simulation demonstrates that networks selected by A were different from both AMI-selected and T-selected networks, landing between them on the NMDS plot of a comparison by all indices (Figs. 9 and 10). AMI made a strong contribution to network structure during the simulations under A selection (note the standard error bars are relatively small). This result implies that T does not dominate the connection between the growth of a system and the development of a system when a simple product of T and AMI are used to calculate A. Ulanowicz (1980, 1997) contention that A as the simple product of AMI and T will comprise both the organization and growth of networks is supported by our analysis. Of course, it remains to be seen with real networks (i.e., ecosystems) whether AMI rises with T and A during growth and development. From this model and the indices that we chose, we are left with the notion from the NMDS analysis that A-selected networks are similar to T-selected networks (i.e., appear clustered close to each other) and dissimilar to AMI-selected networks (Figs. 9 and 10). However, a strong word of caution is due on this point: many of the indices we chose to use contain the value of T in their calculation, and we arbitrarily chose indices from a group that was popular in the literature. These facts could lead to strong bias in where the A-selected networks appear on the NMDS plot relative to the positions of AMI-selected and T-selected networks. Our model did not simulate real systems; it only incorporated limits on feeding guild relationships without reference to biomass. We should properly conclude that the exact nature of changes in real systems that

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reflect changes in AMI while optimizing A remain unknown. If AMI increases in developing systems, the increase may be attributed to the strengthening of chains, the strengthening of cycles, or both (Latham and Scully, 2002). This study is a useful exercise because it demonstrates that A-selected networks were unlike T-selected networks (note the small standard error bars on the points in Figs. 9 and 10)—and the A-selected networks were in the direction of the AMI-selected networks. Our results provide a small counterbalance to a number of comments in the literature that focus on the fact that A and T will probably be well correlated in real systems and either discount the role of AMI in A or not mention that that AMI could play an important role in quantifying network structure. Our method lacks experimental knowledge from a specific system, but we do generate an important consideration for future work with real systems related to the testable hypothesis for ascendency that both T and AMI will rise in a developing system: We caution that the ability to detect if real systems are maximizing AMI along with T as they develop will be largely dependent upon the choices of indices by the researcher. Some indices only seem to change in response to the maximization of T, while others are clearly shifted when a system is maximizing AMI along with T. To say which of the individual indices were most sensitive to the influence of AMI, we would need to run the model 1836 times (i.e., 12 replicates × 3 selection regimes per replicate × 51 indices; assuming we run the model to completely avoid potential repeated measures error) or even 180 times if we grouped the indices ten at a time and used a simultaneous inference adjustment (e.g., sequential Bonferroni procedure). We do not currently have the computer power (or labor) required to run the model enough times to say with statistical significance which indices were sensitive to AMI when selecting for A. We reason that if the AMI increases along with T in developing systems that the networks will likely contain structural elements different from networks where only T increased in development. This is a clear counterpoint to the notion that A only or chiefly reflects T in ascendency calculations if Ulanowicz (1980, 1986, 1997) is correct. Two possible reasons account for our results coming to light with the model that we built but not with previous network analysis: (1) previous research has relied

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upon static snapshots of systems (e.g., Christensen, 1994a,b, 1995; Fath et al., 2001; Patten, 1995), while the simulation method we employed optimized the network structure over time; and (2) much of the previous research using models often relied upon a handful of rigorously connected compartments, where links could not form and break among compartments nor could compartments appear and disappear from the network to influence the results of the analysis as a system was simulated through time (e.g., Christensen, 1994a; Jørgensen, 1994), while our approach involved a completely structurally dynamic model (e.g., Jørgensen, 1986, 1988, 1992c, 1999). Our results are consistent with the efforts of Mageau et al. (1998), where a simulation model of succession in a pelagic system indicated that A followed the successional trends expected (cf. Odum, 1969). Their research addressed the use of network analysis to assess ecosystem health, and their model was not as structurally dynamic as the one presented here, but their model implied that changes in AMI had a strong influence on values of A during phases of simulation when a refinement of the flow network occurred. One of the most interesting results of our analysis is found in Fig. 11, where we have regressed the A values of our A-selected networks against T and AMI of those networks (N = 12). Christensen (1994b) used graphs similar to our Fig. 11 with 41 static ecosystem models, where he concluded that A would chiefly reflect growth and not development. Our Fig. 11 is similar to his graphs for real systems: the regression is strong for T (r = 0.95; r2 = 0.90), but not for AMI (r = 0.60; r2 = 0.36). If we were to only have seen such graphs at the outset, we may have concluded that the AMI is much less consequential to the A value than T or to representing network structure. It seems clear from Christensen’s work that T is well correlated with A across different networks. Since the original formulation of the ascendency hypothesis in 1980, Ulanowicz (1980, 1986, 1997) has held that the AMI is an important indicator of network structure accounting non-arbitrary flow patterns seen during successional development, particularly the development of stronger cyclic flow in networks. Ulanowicz cast the ascendency measure as the product of the AMI, a number of small magnitude, and T, a number of large magnitude. Let us not be confused on the point of T dominating changes in A over time:

There is no way to determine simply from the magnitudes of these values or from static snapshots of systems that A primarily reflects system size in successional development. One could argue the reverse of published statements: Given a system size of a trillion arbitrary units, an AMI change in the system of just 0.1 units would result in a change in A on the order of 100 billion units! There has been a lack of attention to the fact that a product in this calculation bestows great power to the AMI to reflect structure (or the lack thereof) in spite of its small numerical size in the A calculation. Based on the fact that A-selected networks were structurally dissimilar from T-selected networks (and in the NMDS ordination space in the direction of the AMI-selected networks), we reject the null hypothesis. Although numerically small compared to T, we conclude that AMI has the capacity to be an important structural indicator in the calculation of A—consistent with the proposals of Ulanowicz.

Acknowledgements The authors thank Robert Ulanowicz, Bernard Patten, and Brian Fath for reviewing a copy of this manuscript and providing helpful comments. Bernard Patten was particularly helpful in pointing out weaknesses in our manuscript. Thanks to Susan Gresens, Ph.D. and Joel Snodgrass, Ph.D. for their input and encouragement. Thanks to Sven Jørgensen, Ph.D. and two anonymous reviewers for their helpful comments. Stuart Pimm, Ph.D. assisted with the Compartmentalization index algorithm. Appreciation is expressed to Ms. Darcel Cobb and the staff of the Interlibrary Loan Office at Towson University for their generous assistance. Thanks to Marshall Mathers, Karl Sims, and Arthur Weis, Ph.D. Much of this research was conducted while one of us (LL) was teaching for Loyola College in Maryland (Baltimore, Maryland, USA). Appreciation is expressed to the administration, faculty, staff, and students at Loyola College in Maryland. Thanks to the Sandra A. Pieper Foundation for financial support. One of us (LL) also received support from the Department of Ecology and Evolutionary Biology at the University of California at Irvine. Software: Two programming environments were utilized in this research: Chipmunk BASIC® was used

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in the early phase of the project for test models, and FutureBASIC® was employed for the final model. The Chipmunk BASIC® programming environment is a freeware BASIC interpreter written and distributed by Ronald H. Nicholson Jr. (http://www.nicholson.com/ rhn/basic). FutureBASIC® is an object-oriented, complied BASIC. FutureBASIC® is a product of STAZ Software Inc. (7582 Alakoko Drive, Diamondhead, MS 39525, 800-348-2623, http://www.stazsoftware. com). The program was developed and executed on a 400 MHz Apple® PowerMac® G4 computer and an 800 MHz iBook® (Apple Computer Inc., 1 Infinite Loop, Cupertino, CA 95014, 408-996-1010, http://www.apple.com). PC-ORD is a product of MjM Software Design (PO Box 129, Gleneden Beach, OR 97388, USA, 1-800-690-4499, http://www.pcord.com). Legal: Apple, Macintosh, PowerMac, and iBook are registered trademarks of Apple Computer Inc., Chipmunk BASIC is a trademark of Ronald H. Nicholson Jr., FutureBASIC is a registered trademark of STAZ Software Inc., PC-ORD is a registered trademark of MjM Software Design.

Appendix A Equations for indices tracked during simulation and used to characterize the networks (not shown within the text). Overhead (O, Eq. (4)) n+2 n

Tij2 O=− Tij log2 Ti Tj

(4)

i=0 j=1

Capacity (C, Eq. (5)) n+2 n

Tij C=− Tij log2 T

(5)

i=0 j=1

Ascendency/capacity ratio (A/C, Eq. (6)) A/C =

A C

(6)

Rutledge diversity (HR , Eq. (7)) n
Ti Ti log2 HR = − T T i=0

(7)

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Rutledge stability (DR , Eq. (8)) DR = HR − AMI

(8)

Total uncertainty (Hmax , Eq. (9)) Hmax = n log2 (n + 2),

(9)

where n is the total number of compartments in the network. Network structure uncertainty (Hsys , Eq. (10)) n n

Tij Tij Hsys = − log2 Ti Ti

(10)

i=1 j=1

Constraint information (Hc , Eq. (11)) Hc = Hmax − Hsys

(11)

Constraint efficiency (CE, Eq. (12)) CE =

Hc Hmax

(12)

Proportion of detrital flow (pdetrital , Eq. (13)) pdetrital =

Tdecomposer + Tparasite + Tbacteria , T

(13)

where Tdecomposer is the throughput of decomposer compartments, Tparasite the throughput of parasite compartments, and Tbacteria the throughput of bacteria compartments; these throughflows exclude input from outside the network and self-link flows. Proportion of specialists in the third trophic level (pspecialists , Eq. (14)) The number of specialist compartments in the 3rd trophic level was expressed as a ratio to the total number of 3rd trophic level compartments. pspecialists InZoSp + InInSp + FshZoSp + FshFshHerbSp , = number of 3rd trophic level compartments (14) where InZoSp is the number of Invertebrate Zooplankton Specialist compartments, InInSp the number of Invertebrate Invertebrate Specialist compartments, FshZoSp the number of Fish Zooplankton Specialist compartments, FshInSp the number of Fish Invertebrate Specialist compartments, and FshFshHerbSp the number of Fish Fish Herbivore Specialist compartments.

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Gross primary production (PG , Eq. (15)) PG = TPP,output ,

(15)

where TPP,output is the total primary producer output.Respiration (R, Eq. (16)) R = TR ,

(16)

where TR is the total network dissipation (respiration). Gross production to respiration ratio (PG /R, Eq. (17)) TPP,output PG = R TR

(17)

Net production (PN , Eq. (18)) PN = TPP,output − TPP,dissipation ,

(18)

where TPP,dissipation is the total primary producer dissipation (unusable exports). Proportion of input (pinput , Eq. (19)) pinput =

Tinput , T

(19)

where Tinput is the total flow into the system. Proportion of export (pexport , Eq. (20)) pexport =

Texport , T

(20)

where Texport is the flow of useable exports from the system. Proportion of dissipation (pdissipation , Eq. (21)) pdissipation =

Tdissipation , T

(21)

where Tdissipation is the flow of unusable exports from the system. Compartment diversity (Diversity, Eq. (22)): Compartment diversity was ascertained using the Shannon– Wiener diversity index (Shannon and Weaver, 1949). Compartment diversity indicated the probability that the next compartment was the same as the previous (Brewer, 1988). The index provided information on both variety and evenness (Mann et al., 1989). Diversity = −

n


pi logpi ,

(22)

i=1

where pi is the proportion of biotic compartment type i (i.e., the number of compartment type i divided by the total number of compartments in the network).

The “variety” component of Odum’s (1969) thesis encompasses the variety of species expressed as a species–number ratio or a species–area ratio. According to Odum, this value should increase during the early stages of ecosystem development. The Shannon–Wiener index used in these simulations tracked compartment diversity. Herein lies an important distinction between the compartment diversity and Odum’s species diversity. We offer an index of the diversity of guilds in the model systems built by the program, indicated by generalists, specialists, decomposers, etc. Odum called for species evenness. We were unable to specify different species in these simulations. Because the number of individuals in each compartment of the model was not known, we were unable to report on Odum’s equitability component, the component that reports evenness in individuals across species. Link density (LinkDen, Eq. (23)) #links , (23) LinkDen = n where #links is the total number of links in the network. Average link strength (AvLinkSt, Eq. (24)) T AvLinkSt = (24) #links Average compartment throughflow (AvgCompT, Eq. (25)) T AvCompT = (25) n Proportion of self-linked compartments (pselfLink , Eq. (26)) #selflinks pselfLink = , (26) n where #selflinks is the number of self-link flows in the network. Connectance (Connect, Eq. (27)) #links − #selflinks Connect = , (27) n(n − 1) where #links excludes inputs from outside the network, exports, and dissipations. Compartmentalization index (Compart, Eq. (28)) Compart =

n n

i=1 j=1

pij ,

(28)

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where

pij =

number of compartments that interact with i and j number of compartments that interact with i or j

.

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