Circular–Directional Flow Decomposition (CDFD)

• CDFD is a mathematically rigorous framework that decomposes flows into cyclic (divergence-free) and acyclic (net flow) parts to reveal underlying structural dynamics.
• The methodology employs both continuous and discrete models, leveraging techniques like LP-based min-cost flow and Hodge–Kodaira decompositions to solve complex flow networks.
• Applications span optimal transport, urban mobility, and network clustering, demonstrating the framework’s value in both theoretical insights and practical implementations.

Circular–Directional Flow Decomposition (CDFD) is a set of mathematically rigorous frameworks for decomposing a flow—whether a vector field, a network flow, or a transition matrix—into a circular (cyclic, divergence-free) component and a directional (acyclic, net-flow) component. Across diverse contexts, this decomposition makes explicit the latent structure of cycles and directed “through-flow,” serving both theoretical analysis and practical applications. Both continuous and discrete versions are prominent, with connections to optimal transport, combinatorial Hodge theory, graph theory, and network science.

1. Fundamental Mathematical Structures

A recurring feature of CDFD frameworks is the splitting of a flow, typically represented as a measure, vector field, or matrix, into two complementary sub-components:

• Circular (cyclic) component: Satisfies a divergence-free or balance property (e.g., $\mathrm{div}\,X=0$ for vector fields, or net node inflow equals outflow for each node in a directed graph). This corresponds to flow that “recycles” or participates in cycles.
• Directional (acyclic) component: Satisfies the same boundary or supply-demand constraints as the original flow but supports no cycles; it is acyclic in discrete networks or represents pure source-to-sink transport in continuous settings.

In the continuous setting, consider a bounded domain $\Omega\subset\mathbb{R}^n$ and Radon measures $\mu_+, \mu_-$ with equal total mass. The space of admissible vector-valued measures $X\in\mathcal{M}^n(\Omega)$ is subject to

$$\mathrm{div}\,X = \mu_+ - \mu_-,\quad X\cdot n=0 \text{ on } \partial\Omega,$$

with cost given by a convex or subadditive functional $J(X)$. In this context, CDFD realizes a direct-sum decomposition

$X = X_c + X_d,$

where $X_c\in\mathcal{C} = \{V:\mathrm{div}\,V=0\}$ is cyclic and $X_d$ is purely directional, induced via the Dacorogna–Moser path-space construction and Smirnov's superposition theorem (Santambrogio, 2013).

In directed network models, a flow $w_{ij}$ on edges $\Omega\subset\mathbb{R}^n$0 is split as $\Omega\subset\mathbb{R}^n$1 where $\Omega\subset\mathbb{R}^n$2 is divergence-free ($\Omega\subset\mathbb{R}^n$3 for all $\Omega\subset\mathbb{R}^n$4) and $\Omega\subset\mathbb{R}^n$5 is acyclic (its support is a DAG) (Homs-Dones et al., 14 Jun 2025).

2. Decomposition Frameworks and Solution Spaces

CDFD is instantiated in multiple settings, each with explicit constraints and geometry:

• Continuous vector fields (e.g., traffic, Beckmann flows): The path-space decomposition of any admissible $\Omega\subset\mathbb{R}^n$6 yields a probability measure $\Omega\subset\mathbb{R}^n$7 on the space of Lipschitz curves, split into loops and non-loops. This yields $\Omega\subset\mathbb{R}^n$8 with $\Omega\subset\mathbb{R}^n$9 divergence-free, $\mu_+, \mu_-$0 carrying the net supply-demand (Santambrogio, 2013).
• Weighted directed networks: The CDFD of $\mu_+, \mu_-$1 is given by $\mu_+, \mu_-$2, with $\mu_+, \mu_-$3 balanced (divergence-free) and $\mu_+, \mu_-$4 acyclic. The set of all feasible $\mu_+, \mu_-$5 and $\mu_+, \mu_-$6 form the faces of a polytope-complex, i.e., the solution space is a contractible union of convex polytopes (Homs-Dones et al., 14 Jun 2025).

Key geometric facts:

• The set of all directional parts $\mu_+, \mu_-$7 satisfying $\mu_+, \mu_-$8 and node balancing $\mu_+, \mu_-$9 forms a polytope (or union thereof).
• The collection of decompositions projects to a closed interval $X\in\mathcal{M}^n(\Omega)$0 under the normalized circularity index $X\in\mathcal{M}^n(\Omega)$1 (Homs-Dones et al., 14 Jun 2025).

3. Benchmark Decompositions and Algorithms

Two canonical decompositions are distinguished for interpretability and computational tractability:

Decomposition	Objective	Algorithmic Approach	Complexity
Max-circularity	Maximize $X\in\mathcal{M}^n(\Omega)$2	Minimum-cost flow LP on flow polytope	Polynomial, via MCF solvers
BFF (Balanced Flow Forwarding)	Central, all cycles included in proportion	Recursive computation using SCCs, stationary distributions	Near-linear per SCC, polynomial overall

• Max-circularity: $X\in\mathcal{M}^n(\Omega)$3 is minimized in total flow (or $X\in\mathcal{M}^n(\Omega)$4 maximized), which reduces to a standard min-cost flow problem. The solution is always acyclic; cycles in $X\in\mathcal{M}^n(\Omega)$5 would reduce cost and thus are strictly avoided (Homs-Dones et al., 14 Jun 2025).
• BFF: Seeks a “central” decomposition by recursively allocating circular flow proportional to the original weights over all cycles, based on random-walk stationary distributions within each strongly connected component. The BFF is unique, uses every feasible cycle, and is computable via repeated stationary distribution solves (Homs-Dones et al., 14 Jun 2025).

4. Combinatorial and Hodge–Kodaira Approaches

In the context of undirected or origin-destination data, CDFD is realized as a discrete Hodge–Kodaira decomposition: $X\in\mathcal{M}^n(\Omega)$6 where $X\in\mathcal{M}^n(\Omega)$7 is the net-flow, $X\in\mathcal{M}^n(\Omega)$8 the potential-induced flow (solved via Laplacian systems), and $X\in\mathcal{M}^n(\Omega)$9 is the pure curl component supported on cycles (usually triangles). This decomposition is orthogonal and underpins scalar-potential analysis of flows, such as quantifying the proportion of flow explainable by potential versus cycles via the Frobenius norm partition $$\mathrm{div}\,X = \mu_+ - \mu_-,\quad X\cdot n=0 \text{ on } \partial\Omega,$$0 (Aoki et al., 2021).

When the “harmonic” subspace is trivial (e.g., in complete graphs), the CDFD reduces to a two-part split.

5. Dynamic, Probabilistic, and Markovian Variants

In stochastic or dynamic settings, variants of CDFD decompose time-dependent or stationary flows into path-based and cycle-based components:

• Dynamic flows: Any finite-support dynamic edge $$\mathrm{div}\,X = \mu_+ - \mu_-,\quad X\cdot n=0 \text{ on } \partial\Omega,$$1–$$\mathrm{div}\,X = \mu_+ - \mu_-,\quad X\cdot n=0 \text{ on } \partial\Omega,$$2 flow admits a decomposition into a sum of $$\mathrm{div}\,X = \mu_+ - \mu_-,\quad X\cdot n=0 \text{ on } \partial\Omega,$$3–$$\mathrm{div}\,X = \mu_+ - \mu_-,\quad X\cdot n=0 \text{ on } \partial\Omega,$$4 walk inflows plus a circulation (cycle-flow with zero net outflow at every node, including source and sink). The decomposition is obtained by iterative extraction of maximal feasible walk inflows and converges to a circulation on cycles (Graf et al., 2024).
• Stationary Markov flows: For a Markov chain with stationary distribution $$\mathrm{div}\,X = \mu_+ - \mu_-,\quad X\cdot n=0 \text{ on } \partial\Omega,$$5 and transition matrix $$\mathrm{div}\,X = \mu_+ - \mu_-,\quad X\cdot n=0 \text{ on } \partial\Omega,$$6, the stationary flow $$\mathrm{div}\,X = \mu_+ - \mu_-,\quad X\cdot n=0 \text{ on } \partial\Omega,$$7 admits a unique cycle decomposition: $$\mathrm{div}\,X = \mu_+ - \mu_-,\quad X\cdot n=0 \text{ on } \partial\Omega,$$8 where $$\mathrm{div}\,X = \mu_+ - \mu_-,\quad X\cdot n=0 \text{ on } \partial\Omega,$$9 is the long-run frequency of cycle $J(X)$0. Communication measures $J(X)$1, derived from the cycle weights, inform module and community structure by capturing multi-step and bidirectional interaction, outperforming purely local (one-step) approaches (Banisch et al., 2014).

6. Interpretative Indices and Empirical Findings

The circularity index $J(X)$2 captures the fraction of total flow engaged in cycles. For urban mobility data:

• High $J(X)$3 ($J(X)$4 directionality) indicates scalar potential captures flows well; high $J(X)$5 indicates significant circulation, frequently observed in large, polycentric urban regions (Aoki et al., 2021). For example, over 930 U.S. metropolitan areas, mean $J(X)$6, with extremes from $J(X)$7 (New York) to $J(X)$8 (Zapata, TX).
• Empirical network studies reveal that random and empirical networks exhibit a spectrum of circularity; e.g., the Sarafu payment network (Kenya) achieves $J(X)$9, $X = X_c + X_d,$0 globally, with strong modular variation (Homs-Dones et al., 14 Jun 2025).

Alternative structural metrics (e.g., LM-circularity, trophic incoherence) correlate loosely with CDFD’s benchmark indices but fail to distinguish balanced cycle-induced flow from structural connectedness (Homs-Dones et al., 14 Jun 2025).

7. Applications and Analytical Significance

CDFD provides critical insights across diverse domains:

• Optimal transport and traffic: The decomposition exposes unnecessary cycling (wasteful cycles) in optimal flow solutions. In convex Beckmann-type problems, optimality enforces $X = X_c + X_d,$1—i.e., absence of circular flow—whereas non-convex or design-driven objectives (branched transport) may sustain cycles as an optimal structural feature (Santambrogio, 2013).
• Urban mobility and structure: Quantifies polycentricity, identifies functional hubs, diagnoses inefficiency or redundancy in commuting patterns (Aoki et al., 2021).
• Network analysis and clustering: Transforms directed/irreversible processes into symmetric “communication graphs” respecting multi-step cyclic connectivity—a foundation for robust community detection and system resilience studies (Banisch et al., 2014).
• Payment, energy, and transaction systems: Informs netting procedures, closure properties, assessment of feedback or redundancy, and supports efficient flow allocation and routing (Homs-Dones et al., 14 Jun 2025).
• Algorithmic and computational tools: Employs convex optimization (min-cost flow, Laplacian solvers), iterative stationary distribution procedures, and superposition principles, with polynomial or near-linear complexity in typical cases (Homs-Dones et al., 14 Jun 2025).

In sum, CDFD establishes a conceptual and computational foundation for dissecting the global structure of flows into its cyclic and acyclic constituents, with broad theoretical and practical ramifications for networked systems in mathematics, engineering, and the social sciences.