Causal Dynamics for Complex Systems

A course in structural causal dynamics

Causal Dynamics for Complex Systems

All things are in all things, but in each appropriately.

— Proclus, Elements of Theology, Proposition 103 (c. 5th century)

How an actual entity becomes constitutes what that actual entity is. Its ‘being’ is constituted by its ‘becoming’.

— A. N. Whitehead, Process and Reality (1929)

…from so simple a beginning endless forms most beautiful and most wonderful have been, and are being, evolved.

— Charles Darwin, On the Origin of Species (1859)

Preface

Status: In Progress

v0.5

Causal Dynamics for Complex Systems joins causal reasoning with dynamical systems modelling. The object is a complex system (organism, ecosystem, clinical cohort, coupled socio-technical process) under intervention: how the process unfolds through time, what feedback sustains it, which equilibria (“attractor states”) it can settle into as its environment shifts, and what remains learnable when observation is partial and noisy.

The practical questions are these: can we predict how a process evolves under different interventions from imperfect data? What would it take for such a system to settle into a new equilibrium if conditions or treatments change? They arise wherever feedback, latent structure, and limited measurement meet (ecology, epidemiology, medicine, climate, and engineered systems among them).

The book organises that problem so that complex dynamical systems are described carefully enough, and their mechanisms understood deeply enough, that the observable effects of interventions become predictable in principle. The framing is general; the worked examples often draw on biology because that is where I work, but the logic applies universally.

The material is organised in three strata: Structural (causal mechanisms, formal constraints, and invariances), Dynamical (states and processes unfolding through time), and Observable (the measurement process through which aspects of actualisation become data). Pearl’s three levels of Reason (association, intervention, and counterfactual) cut across those strata. The strata describe modelling roles; the levels describe queries. How pedagogy (depth-first) relates to epistemology (surface-first) is spelled out under How to read this book.

Computation in these pages uses a Julia stack: CausalDynamics.jl (graphs, identification, SCM/CDM do), CausalTargeted.jl (cross-fitted LMTP / MTP grids), CausalMediation.jl (interventional and related mediation), and DAGMakie.jl (DAG figures). Identify → estimate → display, with typed certificates between steps. This book is not the documentation for that software; the packages implement the challenges the book describes. The concepts remain usable in other languages and toolkits. Julia is the working language here because its one-language workflow, mathematical syntax, multiple dispatch, and composability suit the task. For further detail, see Why Julia in the Introduction.

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Chapter Progress

• Introduction: in-progress, v0.5

Part I — Structural

• The Causal Hierarchy and Three Strata: in-progress, v0.4

• The Primary Unit: The Dyad: draft, v0.5

• Graph Theory and Causal Patterns: draft, v0.5

• Structural Causal Models as Executable Mechanisms: draft, v0.6

• Identification: When Can We Learn from Data?: draft, v0.6

• Causal Discovery: draft, v0.1

• Do-Calculus: Rules for Interventions: draft, v0.5

• Counterfactuals: Unit-Level Alternatives at Structural Level: draft, v0.6

• Transportability: Generalising Structural Claims: draft, v0.5

• From Structure to Time: FEP and Attractors: draft, v0.3

• Cyclic Causal Structures: draft, v0.1

Part II — Dynamical

• Deterministic Dynamics (ODEs) as Structural Assignments over Time: draft, v0.2

• Stochastic Dynamics (SDEs) and Process Noise: draft, v0.2

• State-Space Models: Inferring Structure from Observations: draft, v0.4

• Intervening in Deterministic Systems: draft, v0.4

• Intervening in Stochastic Systems: draft, v0.4

• Advanced Dynamics: Regimes, Networks, and Resilience: draft, v0.4

• Multi-Scale Causal Dynamics: in-progress, v0.2

• Counterfactual Dynamics: Alternative Trajectories: draft, v0.4

• Sensitivity Analysis and Robustness in Dynamics: draft, v0.4

• From Dynamical to Observable: Records and Actualisation: draft, v0.5

Part III — Observable

• Observational Methods: Learning from Data: draft, v0.4

• TMLE and Doubly Robust Estimation: draft, v0.5

• Model Validation with Observable Data: draft, v0.4

• Interventional Reasoning: Forecasting Under Interventions: draft, v0.4

• Policy Evaluation and Dynamic Treatment Strategies: draft, v0.5

• Causal Decision-Making: draft, v0.4

• Counterfactual Reasoning: Unit-Level Alternatives: draft, v0.4

• Hypothesis Generation from Counterfactuals: draft, v0.4

• Experimental Design: Optimal Measurements: draft, v0.4

Part IV — Synthesis

• CDMs: The Unified Framework: draft, v0.4

• Biological Case Studies: draft, v0.3

• Reporting Standards for Causal-Dynamical Claims: draft, v0.2

• Software Engineering Patterns for Causal-Dynamical Modelling: draft, v0.4

Appendices

• Glossary and Concept Reference: in-progress, v0.3