Emergent Loop Geometry in Gauge-Equivariant Graph Neural Networks

Ali Rayat; Gia-Wei Chern

cond-mat.stat-mechOct 13, 2026

Emergent Loop Geometry in Gauge-Equivariant Graph Neural Networks

Authors: Ali Rayat, Gia-Wei Chern

Organizations: Department of Physics, University of Virginia, Charlottesville, VA 22904, USA

Abstract

Gauge-equivariant graph neural networks generate ordered products of gauge links through local message passing, but the geometry of these internal structures is not captured by the receptive field alone. We investigate the statistical ensemble of closed lattice walks generated by recursive node- and edge-feature operations, independently of network training. Analytical and numerical results reveal a separation between rapid growth of contour length and much slower spatial expansion: deeper compositions generate increasingly long paths through repeated traversal of relatively compact regions. Loop expansions of fermionic observables provide a physical interpretation of this hierarchy, showing how spatial locality remains compatible with high-order dependence on nearby gauge fields. These findings identify complementary roles of network depth in extending spatial reach and enriching the representations constructed within that reach, providing a geometrical perspective on learning lattice gauge observables.

Figures & tables

Figure 1: Schematic of the graph neural network (GNN) and message-passing architecture. The input lattice is represented as a graph with information carried by both nodes and edges. Starting from the input edge variables

U_{ij}

and local node features

\mathbf{V}_{i}^{\rm input}

, successive message-passing layers generate hidden node features

\mathbf{V}_{i}

and edge features

\mathbf{E}_{ij}

. At each layer, an edge feature

\mathbf{E}_{ij}

is updated using information from the edge itself and the two nodes at its endpoints (top), while a node feature

\mathbf{V}_{i}

is updated by combining its own features with information aggregated from its four nearest-neighbor nodes (bottom). Repeated message passing therefore propagates information over increasingly extended regions of the lattice, producing the final node and edge representations at the output layer. Different colors schematically distinguish feature channels and are not intended to represent their gauge-transformation properties.

Figure 2: Examples of closed lattice paths generated by gauge-equivariant message passing and represented by node features

\mathbf{V}_{i}

. The green circle marks the reference node

i

at which each loop is anchored, while the red arrows indicate the ordered sequence of lattice links forming the path. Starting from elementary plaquette loops, successive node and edge message-passing operations generate increasingly extended and geometrically complex closed paths, including paths that retrace links or revisit lattice sites. Panels (a)–(i) show representative short loops generated at different stages of this recursive construction. The corresponding message-gathering operations for these examples are discussed in the main text.

Figure 3: Representative closed paths generated by the recursive loop-sampling algorithm. Rows (a)–(c) show realizations at recursion depths $n=4$ , $5$ , and $6$ , respectively; $\ell$ denotes the full contour length of each path. The green star marks the reference site where the path begins and ends. Color indicates the traversal step $k$ , progressing from the beginning to the end of each path. Unlike the short examples in Fig. 2 , these realizations are generated by stochastic selection of node and edge message operations, without prescribing their shapes. Paths generated at the same depth vary substantially in length and geometry and can traverse links repeatedly, revisit sites, and contain nested or overlapping subloops.

Figure 4: Loop-length statistics of recursively generated closed paths. (a) Semilog plot of the loop-length distribution $P_{n}(\ell)$ in $d=2$ for recursion depths $n=2,\ldots,6$ . The tails bend downward, indicating that their decay steepens with $\ell$ and is faster than a single exponential over the displayed range. (b) Mean loop length $\langle\ell\rangle$ and (c) maximum observed loop length $\ell_{\max}$ as functions of recursion depth for $d=2,3,4$ . Both increase rapidly with depth; the maximum is the largest length in the sampled loops, rather than the maximum allowed by the recursion.

Figure 5: Geometric characterization of generated loops in dimensions

d=2

3

, and

4

as a function of recursion depth

n

. Shown are the ensemble-averaged (a) radius of gyration

\langle R_{g}\rangle

, (b) maximum distance from the loop center of mass

\langle R_{\max}\rangle

, and (c) effective dimension

\langle D_{\mathrm{eff}}\rangle

. The radius of gyration is nearly independent of

d

, while the maximum excursion shows a modest dimensional dependence. Both spatial measures grow much more slowly than the mean contour length in Fig. 4 (b). The effective dimension remains nearly constant for

d=2

but increases with depth for

d=3

and

4

Figure 6: Site revisitation and loop-length diversity as functions of recursion depth

n

for dimensions

d=2

3

, and

4

. (a) Mean number of distinct lattice sites visited by a loop. (b) Mean revisitation fraction

\langle P_{\mathrm{revisit}}\rangle

, where

P_{\mathrm{revisit}}=1-N_{\mathrm{sites}}^{\mathrm{distinct}}/\ell

for a closed path of length

\ell

, with the repeated final endpoint omitted. (c) Shannon entropy of the loop-length distribution,

S_{\ell}(n)=-\sum_{\ell}P_{n}(\ell)\ln P_{n}(\ell)

. Although the number of distinct sites grows with depth, the increasing revisitation fraction shows that many additional steps return to previously visited sites. The entropy curves nearly coincide across dimensions, consistent with the same length-generation rules in each dimension.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 7: Recursive generation of closed paths by gauge-equivariant message passing. The tree traces the possible origins of a node feature $V$ at the output layer through successive message-passing layers. The $X_{1}$ , $X_{2}$ , and $X_{3}$ branches carry forward a node feature, multiply two node features, and transport a neighboring node feature between two edge features, respectively [Eqs. ( 4 )–( 6 )]. The $Y_{1}$ , $Y_{2}$ , and $Y_{3}$ branches carry forward an edge feature $E$ or dress it with node features at one or both endpoints [Eqs. ( 8 )–( 10 )]. Choosing an operation at each branch, together with all factors required by that operation, specifies an ordered product of input features. The resulting node feature represents a closed lattice path, whose trace is a gauge-invariant loop observable. The dots indicate that the recursion continues to the input links and plaquettes.

Figure 8: Correlations between loop length and spatial geometry. Columns show the radius of gyration $R_{g}$ (left), number of distinct visited sites $N_{\mathrm{sites}}^{\mathrm{distinct}}$ (middle), and effective dimension $D_{\mathrm{eff}}$ (right) as functions of the full contour length $\ell$ . Rows correspond to spatial dimensions $d=2$ , $3$ , and $4$ , from top to bottom. Each point represents one generated loop, with colors indicating recursion depths $n=2,\ldots,6$ . Longer loops generally visit more distinct sites, but their spatial extent increases only modestly, with substantial variation in geometry at a given length. The distinct-site count grows more slowly than $\ell$ , indicating repeated traversal of previously visited regions. Larger $D_{\mathrm{eff}}$ corresponds to a more isotropic distribution of the visited positions.

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