LINX: A topology based methodology to rank the importance of flow measurements in compartmental systems

https://doi.org/10.1016/j.envsoft.2020.104796Get rights and content

Highlights

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    Quantification of flows between environmental model compartments can be costly.

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    Value of some flows can be estimated using other known flows in the system.

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    Flows are not equal in terms of their predictive ability of estimating other flows.

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    Link Importance iNdeX (LINX) quantifies each flow’s predictive ability.

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    LINX helps minimize model building effort and increase model accuracy.

Abstract

In ecological and other transactional energy–matter flow networks, accurate quantification of flows between compartments can be difficult and costly. For models at steady state or undergoing linear change, energy–matter conservation together with the steady-state condition can be exploited to estimate unknown flows from known ones. In compartmental network models, some flows are more important than others in terms of their connections to other flows, participation in cycles, geodesic distance to the environment (in the graph theoretical sense), and other topological features. In respect to estimating unknown flows, such importance differences also come into play. Pursuing this, we formulate a Link Importance iNdeX (LINX) that quantifies each flow’s importance in a model. This index identifies and quantifies the redundancy imposed by network topology and mathematical conservation rules. We anticipate that it will find use in minimizing the cost and effort of data collection while also increasing model accuracy.

Section snippets

Software availability

Software name: LINX

Developer: Caner Kazanci, Kelly J. Black

System requirements: Linux, Mac OS, Windows

Program language: Matlab, C++

Availability: Matlab file exchange (https://www.mathworks.com/matlabcentral/fileexchange/72143-linx), GitHub (https://github.com/KellyBlack/LINX)

License: GPL-3.0

Flow importance index: The idea

In a network model at steady-state, not every flow needs to be quantified empirically, because the steady-state assumption introduces constraints. In Fig. 1, for example, determining any one of the four flows is sufficient to quantify the remaining three flows because the steady-state condition forces all the flows to be equal. Their importance values are also equal because the amount of information each provides about the others is identical.

Because each flow determines all the others, a chain

Notation and the steady-state assumption

It is common to represent quantified flows in steady-state multi-compartment models by matrices. The representation of the flow orientation differs in literature. For instance, Patten (1978) employs a columns(j)-to-rows(i) flow orientation, fji, to represent the flow from compartment i to j. Ulanowicz (1986) represents the same flow using a rows(i)-to-columns(j) flow orientation, Tij. Since the environment is not represented as a compartment in standard network analyses (e.g. Patten and

Flow importance index: A simple example

The three compartment food chain model (Fig. 1) is too simple an example to demonstrate how a link importance index is to be formulated in general. Material covered in Section 3 enables us to demonstrate how the flow importance measure is computed for the simple three compartment model shown in Fig. 2. Unlike the previous example, no single flow value is enough to determine all five flows in this model. We need to find the minimum number of flows that need to be quantified in order to compute

LINX: Preliminary general formulation

To construct a preliminary general formulation for LINX, we need first to find out the minimum number of flows needed to determine all flows in multi-compartment models in general. For an n-compartment model, which always contains k>n flows, including environmental inputs and outputs, at least k−n flows must be quantified to determine all flows. This is because the steady-state condition forms a linear system with k variables and n equations, leaving k−n degrees of freedom (see Appendix A for

LINX: Improved formulation

Computation of unknown flows based on quantified flows requires the solution of a linear system of equations, Ux=v, where v is a vector based on quantified flows, U is a matrix based on the model’s network structure, and x represents the unknown flows to be determined. For the three-compartment model shown in Fig. 2, one of these linear equations is −10011−10−10︸U=S(F∖A)f2f4f5︸x=f3−f10−f3︸v.One issue with the solution of such equations is the propagation of error from the quantified flows (f1,f3

Practical considerations: Data availability

Quantifying a flow not only informs about its value, but values of other flows as well, because of the steady-state assumption. LINX simply computes the amount of this additional information to determine the importance of each link. This information cannot be accurately computed for flows about which prior information exists from previous observations, experiments, or literature. Such empirical information will render LINX information partially useless. Also, known flow values will provide

Practical considerations: Computational feasibility

In this section, we describe how to compute the LINX values using Matlab, and then discuss issues of performance and feasibility. A Matlab code that automatically computes the LINX values given a model’s stoichiometric matrix is available at GitHub (Kazanci and Black, 2020) and Matlab Central/File Exchange (Kazanci, 2020). This code is compatible with the freely available GNU Octave software (Eaton et al., 2014) as well as Matlab. For instance, one can compute the LINX values for the

Conclusion

Computational methods serve as an essential part of compartmental network-flow modeling. Their utilization, however, usually occurs after parameters have been determined through empirical data collection, missing data methods and literature search. The computational method of this paper is applied early in the parametrization phase of modeling, after some flow data, but not all, have been acquired. Exploiting conservation laws and network topology, LINX enables modelers to make informed

CRediT authorship contribution statement

Caner Kazanci: Conceived the ideas, Developed and refined the formulation, Software, Writing - original draft. Malcolm R. Adams: Conceived the ideas, Developed and refined the formulation, Writing - original draft. Aladeen Al Basheer: Conceived the ideas, Developed and refined the formulation, Writing - original draft. Kelly J. Black: Conceived the ideas, Developed and refined the formulation, Software, Writing - original draft. Nicholas Lindell: Contribution to the proof of the theorem,

Acknowledgment

All authors participated in the preparation of the publication, contributed critically to the drafts and gave final approval for publication.

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