Network Quiver: Unified Framework

• Network quiver is a unifying formalism based on directed graphs that encodes structural features, symmetries, and data flows in complex systems.
• It supports rigorous methodologies such as Lyapunov–Schmidt and center manifold reduction while ensuring lossless model compression and symmetry preservation.
• The framework underpins analysis in neural networks, dynamical systems, and gauge theories, unifying disparate representations and enhancing functional geometry.

A network quiver is a unifying formalism for representing and analyzing network-structured systems across mathematics, physics, and machine learning. The concept leverages quiver theory—finite directed graphs together with their representations—to encode structural features, symmetry, and data flow in dynamical systems and neural networks. Network quivers provide an intrinsic, coordinate-free viewpoint that captures both classical and hidden symmetries, subnetwork/quotient relations, and parameter redundancies, and underpins rigorous model-reduction, moduli space identification, and lossless compressibility results in feed-forward architectures (Nijholt et al., 2020, Armenta et al., 2021, Ganev et al., 2022).

1. Mathematical Definition of Network Quivers

A quiver $Q = (A \rightrightarrows V)$ consists of a finite set of vertices $V$, a finite set of arrows (edges) $A$, and two maps $s: A \to V$, $t: A \to V$ assigning each arrow $a$ its source $s(a)$ and target $t(a)$. A representation of $Q$ comprises, for each vertex $v \in V$, a finite-dimensional real or complex vector space $V$0, and for each arrow $V$1 a linear map $V$2. In the context of dynamical systems or networks, these representations are further endowed with structure, typically smooth maps or vector fields satisfying quiver-equivariance:

• A family of smooth maps $V$3, for $V$4, is $V$5-equivariant if for every arrow $V$6,

$V$7

This expresses compatibility: each $V$8 sends the orbit of $V$9 to the orbit of $A$0 (Nijholt et al., 2020).

This formalism generalizes numerous symmetry and structural phenomena: classical group actions (quivers with one vertex and loop arrows), monoid actions, feed-forward or skip connections (as directed paths), subnetworks, and quotient networks.

2. Quivers in Dynamical Systems and Equivariant Model Reduction

Network quivers encode structured dynamical systems by assigning each subsystem or "view" a vertex and each inter-subsystem relation an arrow. Standard bifurcation and dimension-reduction techniques—Lyapunov–Schmidt reduction, center manifold reduction, and normal form reduction—are intrinsically compatible with the quiver framework. For a system of Q-equivariant vector fields $A$1, local reduction procedures yield reduced vector fields $A$2 that are again Q-equivariant and live on induced subrepresentations (Nijholt et al., 2020):

• Lyapunov–Schmidt reduction: The Quiver Lyapunov–Schmidt Theorem ensures $A$3 on the reduced kernels.
• Center manifold reduction: The induced center manifold projections, and the reduced dynamics, again commute with the $A$4.
• Normal form reduction: One can choose the near-identity transformations and normal forms themselves to be Q-equivariant.

The preservation of quiver structure throughout these procedures guarantees that reduced models inherit all network symmetries, feed-forward structures, and sub/quotient network behaviors.

3. Network Quivers in Neural Network Architectures

Feed-forward and more general neural network topologies are naturally modeled as quivers:

• Vertices correspond to input, hidden, and output layers (possibly with additional bias nodes).
• Arrows encode affine or linear connections (weight matrices).
• The representation assigns each vertex a feature (activation) space, and arrows become the parameter matrices between spaces.

A generalized recursion formalism computes the partial feedforward at each node. For complex architectures—with skip connections, width splits, aggregations—this enables a uniform, recursive framework for forward propagation. The formalism naturally incorporates "rescaling" activations, i.e.,

$A$5

where $A$6 (the radial case is $A$7 for some scalar function $A$8). Such activations commute with orthogonal transformations, facilitating significant symmetry and reduction results (Ganev et al., 2022).

4. Symmetry, Orbits, and Moduli Spaces of Network Functions

Network quivers, when interpreted as parameter spaces for neural architectures, reveal large symmetry groups—typically products of general linear or orthogonal groups acting by basis change in hidden layers. The orbits of these symmetry groups parametrize equivalence classes of network parameterizations that realize the same function. For thin representations (dimension one per hidden node), the moduli space is the geometric invariant theory (GIT) quotient of the parameter space by this group action (Armenta et al., 2021). Explicitly, for a thin, double-framed quiver with specified input and output framings, the space of network functions is

$A$9

and each point corresponds to a network function, not a raw tuple of parameters. The network category $s: A \to V$0 formalizes this as isomorphism classes of thin Q-representations with chosen activations.

A table summarizing this linkage:

Object	Mathematical Formalism	Interpretative Role
Quiver $s: A \to V$1	Directed graph	Layer-and-connection layout
Representation	$s: A \to V$2, $s: A \to V$3	Feature spaces, weight maps
Symmetry group $s: A \to V$4	Product of $s: A \to V$5	Hidden-layer basis changes
Moduli space $s: A \to V$6	$s: A \to V$7 quotient of reps	Space of network functions

This moduli-centric view clarifies parameter redundancy, functional invariance, and the geometry of the function space.

5. Lossless Compression and Gradient Descent Dynamics

The symmetry structure of network quivers enables exact, lossless model compression for networks with rescaling activations. The main result is that if the incoming-weight span at a hidden vertex is $s: A \to V$8, the feature space can be compressed to $s: A \to V$9 via a QR decomposition, reducing parameter count with no loss of expressiveness (Ganev et al., 2022). The QR-Compress algorithm performs this reduction layer by layer. More formally, the original and compressed networks yield identical forward maps, and in the radial activation case, gradient descent on the reduced model imitates projected gradient descent on the original, due to the commutation of radial activations with orthogonal symmetry.

6. Applications: Gauge Theory, Holomorphic Blocks, and Algebraic Structures

In high-energy physics, the network quiver formalism generalizes to encode the structure of 3d quiver gauge theories, especially in the computation of holomorphic blocks and partition functions. Networks of intertwiner operators for the Ding-Iohara-Miki algebra (DIM) provide direct algebraic representations of these blocks as vacuum matrix elements, Dotsenko-Fateev integrals, and kernel functions of supersymmetric Ruijsenaars-type Hamiltonians (Zenkevich, 2018). Superalgebra quivers with fermionic screenings extend the framework to "superquivers" and generalized W-algebras.

7. Unifying Perspective and Structural Implications

Network quivers unify disparate mathematical descriptions of networked systems—encoding symmetry, feed-forward chain architectures, subnetwork and quotient relations, and parameter redundancies—under a common categorical and representation-theoretic umbrella (Nijholt et al., 2020, Armenta et al., 2021, Ganev et al., 2022, Zenkevich, 2018). Their intrinsic structure guarantees that all model reduction, compression, and coordinate transformations respect the internal symmetries of the system, systematically tracking submodel and quotient behaviors. This deepens the theoretical understanding of neural networks, dynamical systems, and quantum gauge theories, and enables rigorous control over equivalence, optimization, and functional geometry in high-dimensional settings.