A graph-theoretic framework for the systematic representation and generation of conceptual hydrological model structures

The development of transparent and trustworthy hydrological modes requires careful attention to the choices made throughout the modeling process, including the assumptions made about the model structure. However, the structures of conceptual models are often defined indirectly through equations and the code implementation. This limits reproducibility, comparability and transparency of its structural assumptions. Moreover, the is currently no consistent unifying framework that represents the multitude of conceptual model structures found in hydrological modeling literature, making systematic analysis and comparison difficult. 

We introduce a graph-theory based framework for explicitly and coherently representing conceptual model structures, where model compartments are represented as nodes and fluxes as directed edges. This allows mode structures to be defined independently of specific process formulations, while mass balance equations are derived directly from the graphs topology, ensuring consistent balances across all model compartments.

Representing conceptual model structures in an algebraic graph form, also allows to compare, analyze and manipulate in a ways that is difficult to achieve with pure equation-based representations. Graph and matrix encoding allows us to enumerate, compare and modify model structures in a controlled way, enforcing explicit constraints like hydrological plausibility, connectivity and closure. This representation forms a theoretical foundation for flexible and multi-model hydrological frameworks, allowing for the construction, testing and communication of different model hypotheses in a consistent way. Additionally, the graph-based representation support harmonious and unambiguous visual depictions of conceptual model structures, strengthening communication of modelling assumptions alongside their mathematical formulation. 

Using examples of watershed models, we illustrate how conceptual models correspond to specific graph and matrix configurations and how structural differences are reflected in the resulting system of ordinary differential equations. In particular, we show that the incidence matrix provides a direct algebraic mapping between hydrologic model structure and the governing system of ordinary differential equations, where state derivatives are obtained as the balance of incoming and outgoing fluxes associated with each node. Moreover, we demonstrate how the graph–matrix representation can be used to systematically sample a space of candidate model structures by permuting adjacency matrices under predefined structural constraints. Invalid or implausible structures are excluded through rule-based filtering, yielding a structured yet unconstrained exploration of the admissible model space.