Networks have become a routine tool for representing the complex systems that pervade biology, technology and society. While the development of methods for inferring network topology remains a dominant focus, ^{18 , 19 , 21 - 24} numerous methods are now also being advanced for quantifying the weights (fluxes and interaction strengths) of the edges between pairs of nodes (genes, metabolites, species) in different network types. ^{1 , 5 , 7 , 9 , 10} Motivating advances is the desire to predict how systems respond to perturbations wherein the concentration or process rates of a subset of nodes is chronically altered by external forces. ^{2 , 28} Examples of such perturbations include gene knock-downs, fisheries harvest, and the administering of antibiotics. ^{17 , 20 , 25 , 27}

Predictive insight into perturbation effects nevertheless remains hard to obtain. This is true even when a network’s topology is fixed and precisely specified because edge weights are always subject to estimation error and empirical variation. Because of the many, often counteracting pathways through which perturbations travel, these uncertainties rapidly compound. ^{16 , 26} In many cases, therefore, predicting even just the qualitative sign pattern by which nodes respond to perturbations (i.e. whether concentrations increase, decrease, or remain unchanged) is of desirable interest. ^{6 , 14 , 17}

Here we introduce PressPurt – a collection of computational tools designed to shed light on the qualitative and quantitative response of a network to press perturbations when there is uncertainty in the magnitude of edge weights. The PressPurt package is written to implement in a usable form the theoretical results established in Ref. 11. PressPurt is designed to identify the most sensitive interactions within the network which must be estimated most accurately to produce robust predictions of press perturbation responses.

For input, PressPurt requires only a single CSV file representing either a quantitative or qualitatively-specified Jacobian matrix and the specification of a desired error distribution for the edge weight uncertainties. The three steps of an analysis entail: (1) preprocessing the Jacobian matrix, (2) computing the entry-wise or multi-entry perturbation expectations for either qualitative or quantitative sensitivities, and (3) visualizing the results. In R, for example, determining and visualizing the qualitative entry-wise sensitivities of a four-species intraguild predation module (whose Jacobian matrix ¹¹ − 0.237 − 1.000 0.000 0.000 0.100 − 0.015 − 1.000 − 1.000 0.000 0.100 − 0.015 − 1.000 0.000 0.045 0.100 − 0.015 is provided in PressPurt as IGP.csv) involves three commands. First,

infile <- system.file("extdata", "Modules", "IGP.csv",               package = "PressPurt") PreProsMatrix <- PreprocessMatrix (input_file = infile                      max_bound = 10,                      zero_perturb = FALSE,                      threads = 2)

which returns information that includes, for each non-zero edge, their asymptotic stability intervals and values that would lead to a sign switch in the net effects matrix. Second,

Entrywise <- ComputeEntryWisePerturbationExpectation (                PreProsMatrix = PreProsMatrix,                distribution_type = "truncnorm",                input_a = 0, input_b = -2, threads = 1)

which specifies a truncated normal distribution of edge weight uncertainties with mean 0 and variance 2 (a negative value indicates that the standard deviation is to be the scaled to each edge weight) and whose output includes a matrix containing the consequent expected number of sign switches in the net effects matrix expressed as a percentage of all 16 possible mis-predictions, 0.03 0.06 0.00 0.00 0.06 0.12 0.05 0.02 0.00 0.03 0.16 0.01 0.00 0.05 0.08 0.17 .

Third,

GenerateEntryWiseFigures (EntryWise = Entrywise,                 all_numswitch_plots = TRUE)

which produces Figure 2 that overlays visualizations of the assumed uncertainty distribution associated with each edge weight and the consequent number of qualitative mispredictions (sign switches) that a given magnitude of edge weight uncertainty will incur in the net effects matrix as a whole.

The measures of sensitivity that PressPurt implements are exact and may be computed with relative ease and computational efficiency. PressPurt thereby obviates the need for what are typically computationally expensive simulations whose results can be difficult to interpret when assessing the sources of mispredictions in complex networks. Underlying these advances is the separation of uncertainty magnitudes from their frequency distributions ( Figure 1b). Moreover, the theorems of Ref. 11 show that the analysis of qualitative models via Loop Analysis ¹³ – the most commonly applied tool in context of fisheries management and conservation science wherein the values of A are specified as either −1, 0, or 1 – are a special case of the methods implemented in PressPurt. In this regard, PressPurt complements the functionality of other network-based packages such as Refs. 4, 8, 15.

Future extensions of PressPurt may include reformulating the code to: (1) compute which predictions in the net effects matrix are most sensitive to errors in focal (or multiple) entries of the Jacobian, (2) determine which interactions are most sensitive with respect to a specific net effects prediction, and (3) allow for different kinds of distributions or parameterizations to be specified for each interaction strength uncertainty in both the single and multiple edge computations. Lastly, other theorems contained in Ref. 11, including those relating to the characterization of quantitative mispredictions, may also be implemented.

All data underlying the results are available as part of the article and no additional source data are required.

PressPurt is available from: https://cran.r-project.org/web/packages/PressPurt/

Source code available from: https://github.com/dkoslicki/PressPurt

Archived source code as at time of publication: https://doi.org/10.5281/zenodo.5661173 ²⁹

License: MIT