Balanced Classification (BACL) Theory

• Balanced Classification (BACL) is a framework that defines and quantifies phase balanced states in networks, emphasizing non-synchronous, globally constrained configurations.
• It employs topological modifications including symmetric loop additions and connection edges to construct balanced planar graphs supporting specific winding numbers.
• The balancing ratio, an order parameter quantifying basin stability of balanced states, provides actionable insights for designing networks under global constraints.

Balanced Classification (BACL) refers to a suite of theoretical constructs, methodologies, and quantitative tools developed to understand, quantify, and engineer the emergence, stability, and classification of balanced states—specifically “phase balanced states”—in networks of coupled oscillators modeled by the Kuramoto framework, with a primary focus on planar graphs that exhibit nontrivial symmetries and topological structure (Kaiser et al., 2018). This paradigm stands in contrast to the classical perspective that emphasizes global synchrony (i.e., states with maximal phase alignment) and instead characterizes and distinguishes network structures that allow for the existence and stability of globally balanced dynamical configurations.

1. Phase Balanced States in the Kuramoto Model

In the Kuramoto model, the traditional focus is on phase-synchronized states characterized by a large Kuramoto order parameter, i.e., $|p(\theta)| \approx 1$, where %%%%1%%%%. The balanced state is defined by the vanishing of the complex order parameter: $\sum_{j=1}^N \exp(i\theta_j) = 0.$ This yields phase arrangements where the contributions of all oscillators destructively interfere in the complex plane, generalizing the concept of splay or incoherent states. Unlike synchronization, global constraints (e.g., fluid volume conservation) can forbid synchrony, rendering these balanced fixed points the only relevant stable attractors in the network dynamics.

Balanced states are typified by configurations where phases are evenly distributed around the unit circle (e.g., roots of unity) but, crucially, can also manifest in more topologically intricate graphs beyond simple cycles.

2. Topological Construction and Classification of Balanced Planar Graphs

The foundational graph class for balanced states is the cycle graph $C_N$, a circulant graph admitting analytic solutions for balanced states linked to integer winding numbers. The extension to non-circulant, planar graphs is nontrivial; the work establishes rules for systematic construction:

• Symmetric Loop Addition: Loops are appended to $C_N$ at equidistant vertices, with the condition that the number of loops $k$ divides $N$. The cyclic symmetry associated with roots of unity must be preserved, ensuring phase assignments in the looped structures correspond to compatible subgroup decompositions.
• Connection Edges: Additional edges can be drawn between vertices with identical phase in the balanced state. This operation preserves balance and can "divide" the principal cycle into multiple smaller cycles sharing the winding number appropriately.

These graph manipulation rules allow identification and “classification” of planar graphs (both circulant and a family of non-circulant types) that support stable, balanced fixed points. The existence and stability of such points depend sensitively on the preservation of specific phase symmetries in the altered topology.

3. The Balancing Ratio: Order Parameter and Its Analytical Properties

To rigorously quantify the propensity of a graph to realize balanced dynamics, the balancing ratio $b(G)$ is introduced as an order parameter. It is defined by the cumulative basin stability of all balanced fixed points: $$b(G) = \sum_{\theta \in \text{balanced fixed points}} S_B(\theta)$$ where $S_B(\theta)$ denotes the probability (under randomized initial conditions) that the system evolves toward the balanced state $\theta$. This construction distinguishes graphs not just by the possibility but also by the prevalence of balanced states in their phase space.

For cycle graphs, the variance in the basin stability of winding number classes empirically scales linearly with $N$ (nodes), and analytical expressions for $b(C_N)$ (and analogous expressions for more complicated graphs) are derived, confirming and quantifying the impact of topological size and structure on balanced state accessibility.

Graph topology	Analytical balancing ratio scaling
Cycle $C_N$	$b(C_N) \approx 1 - (\sqrt{2\pi}\sigma(N))^{-1}$, $\sigma^2 \approx 0.032N$
Prism or central edge graph	Similar form with distinct $\sigma^2$ scaling based on topology

This quantitative criterion enables classification and direct comparison of diverse planar graphs’ capacities to support balanced dynamics.

4. Analytical Insights: Basin Stability, Winding Numbers, and Topological Constraints

In balanced cycle graphs, basin stability as a function of winding number is governed by a Gaussian distribution whose variance grows linearly in $N$. This analytic insight is extended to non-circulant planar graphs constructed via the prescribed structural modifications, where careful accounting of winding number “redistribution” among cycles is required.

For non-circulant topologies, the existence of balanced states is contingent on “topological compatibility.” Loops must be integrated so their winding structure matches cyclic subgroups of the underlying graph; added connection edges must strictly join vertices of identical phase. Violating these requirements eliminates stable balance, offering sharp necessary topological criteria for BACL.

5. Practical and Theoretical Implications in Network Theory

Balanced fixed point theory, and by extension BACL, is crucial in physical/biological/engineered networks where global constraints preclude complete synchrony—such as volume-conserving oscillator networks or distributed control of robotic swarms. Here, the classical order parameter is insufficient, and the balancing ratio emerges as the discriminative metric.

BACL unifies and extends earlier work restricted to circulant graphs, showing that a wide class of planar graphs—provided symmetry and balance-respecting modifications—can robustly host balanced states. This supports the rational design of networks where required operational modes are non-synchronous but globally constrained.

In broader network theory, deriving analytical scaling laws for basin stability (and thus for $b(G)$) with respect to both graph size and topological features provides actionable insight into how underlying connectivity structures govern accessible dynamical regimes. This informs both the paper and engineering of complex oscillator systems under nontrivial constraints.

6. Extension and Generalization: Beyond Synchrony and Toward Systematic Network Synthesis

The systematic construction approach developed in this context suggests generalizable procedures for extending BACL principles:

• Given a target dynamical property (e.g., balanced fixed points), begin with a base topology with known analytic characteristics.
• Apply only symmetry- and balance-preserving modifications guided by algebraic/topological criteria.
• Evaluate the balancing ratio as a quantitative classification tool to compare alternative topologies or design choices.

This methodology enables exploration of the “design space” of oscillator networks, offering both predictive theoretical understanding and prescriptive design hints for applications where global constraints must be strictly observed.

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Balanced Classification (BACL) as derived in the context of planar graphs and Kuramoto-type networks thus provides a mathematically grounded, analytically tractable framework that links network topology, symmetry, and dynamical landscape. The balancing ratio as an order parameter, the preservation (or violation) of phase symmetry in graph construction, and the explicit analytic scaling relations offer a rigorous classification theory for the existence and prevalence of balanced states across complex networks subject to global constraints (Kaiser et al., 2018).