Compositional Modeling

Regulatory networks as signed graphs, with signed functors describing when one network occurs inside another – which is what a network motif is. From there, functors relate regulatory networks to reaction networks, making precise when a reaction network is a mechanism for a regulatory one, and to Lotka-Volterra dynamics, for open systems as well as closed ones. It is the project’s biochemistry arm, and its clearest example of the same pattern used three ways: motifs, mechanisms and dynamics all become functors out of one category of networks.

Read it for The motif-as-functor definition, then the Lotka-Volterra functor. The open systems extension is the technical heart.

A complete classification of the additive invariants of open Petri nets – the natural-number-valued quantities that add under both sequential and parallel composition. Two theorems: for open Petri nets the invariants are fixed by their values on single-transition nets, and for monically open nets by their values on transitionless nets together with all single-transition nets. The modeling project composes Petri nets constantly, and this says exactly which numerical quantities survive that composition – which ones can be computed on the parts and added, instead of recomputed on the whole.

Read it for The two classification theorems, and the definition of what it means for an invariant of an open system to be additive.

Deterministic dynamical systems, discrete and continuous, compose hierarchically: the paper reformulates existing operads of wiring diagrams and introduces new ones in the language of C-sets, establishes dynamical systems as algebras of those operads, and shows Euler’s method is functorial for undirected systems as well as directed ones. It is where the lab’s dynamical systems machinery is defined, and the functoriality of Euler’s method is the reason a composite system’s simulator can be assembled from its parts’ simulators.

Read it for The operads and the functoriality result. The AlgebraicDynamics code follows the paper closely enough to read alongside it.

A compositional framework for epidemiological modeling in which the structure of a model – its submodels and how they interact – is explicit and algebraic rather than implicit in code. Stratification, composition, analysis and calibration all become operations on that structure, so a local change to a component no longer means a global edit to a codebase. It is the modeling project’s flagship paper, and the clearest statement of the whole lab’s motivation: the gap between how a scientist thinks about a model and the program that implements it is the thing being closed.

Read it for The stratification example. It is the operation that makes the case on its own.

The construction behind stratification. One Petri net is chosen as a type system, a typed model is then an object of the category of Petri nets sliced over it, and the product in that slice category is the stratified model – so stratifying by age, region or host and vector is a categorical operation rather than a hand edit to a reaction network. The flagship paper says what stratification buys; this is the four-page account of the construction that AlgebraicPetri.jl actually implements, and of how it reproduces stratified models already in the epidemiological literature.

Read it for The slice-category definition and the host/vector example. It is a technical demonstration, so there is little to skip.

Read first An algebraic framework for structured epidemic modelling

A space of models is represented as a diagram over a category of models, and limits and colimits of those diagrams build larger model spaces out of smaller ones. The paper implements the computer algebra of finitely presented categories and diagram categories needed to do this, with strategies for picking a model out of the space, demonstrated on mass-action kinetic models fitted to data. Composition builds one model; this is the lab’s answer to searching the space of models you could have built, which is what a modeler actually does.

Read it for The model space construction and the epidemiology case study.

Model augmentation, combination and comparison are treated as metamodeling tasks, and static and dynamic program analysis is used to extract enough semantics from executable scientific models to perform them – metamodeling as metaprogramming, with a categorical account of what the tasks are. It is where the lab first tried to recover a model’s meaning from its code. The later work inverts the problem, specifying structure up front instead, and this paper is the reason why.

Read it for The definition of the metamodeling tasks, and the case study.

An early statement of the programme: applied category theory supplies reusable software components for scientific computing, with Catlab.jl as the categorical infrastructure and SemanticModels.jl as the modeling layer on top, composing systems as cospan algebras. It is the origin point for Catlab and, through it, for most of what the lab has built since. Read it as history rather than as current practice.

Read it for The framing in the introduction. The software described has been superseded.

The software half of the SemanticModels work: a Julia package that automates model augmentation and creation by metamodeling and metaprogramming. The argument for Julia is the substance – its type system, its reachable syntax tree, and the embedded domain-specific languages that multiple dispatch makes possible let a model be manipulated at run time and still compile to efficient code. It is the implementation the rest of the SemanticModels papers describe, and the lab’s first attempt to treat model manipulation as a programming-language problem rather than a modeling one.

Read it for The argument for Julia as the host language. The package itself has been superseded by AlgebraicPetri.jl.

Read first A Compositional Framework for Scientific Model Augmentation

SemanticModels.jl builds a knowledge graph linking elements of scientific code – variables, values, functions, expressions – to elements of scientific understanding, and reasons over it to augment, synthesize and validate epidemiological models. The earliest paper in the project, and the clearest statement of the extraction approach the lab later moved away from.

Read it for The knowledge graph construction. Historical interest unless you work on model extraction.