Computer simulations suggest that complex networks can be designed to show two features at once: a nontrivial topological marker in the bulk, or interior, and transport that favors one direction. In an overcoupled multiport router, the modeled chirality, a measure of directional preference, peaked near a target energy of 1, while regular and small-world networks retained higher chirality than random networks around that point.
The evidence comes from a preprint study of finite numerical models, not from a measured device. Its central question was whether inverse design could produce directional transport while jointly retaining nontrivial topology and robustness.
Building the model
Each model was a magnetic graph: a network whose links carry phase values, with reverse links assigned the model's complex-conjugate counterpart. To examine topology locally, the researchers used a spectral localizer, a single Hermitian operator that combines energy and spatial shifts into one calculation.
The optimization was staged. Gradient descent first targeted the local Chern index, a numerical marker for the network's local topological state. Jacobian descent then sought a larger local gap while at least maintaining that index. The researchers did not treat spectral, localization and flux diagnostics as sufficient on their own to establish protected transport, so they added chirality and disorder-robustness tests.
More links, larger markers
One part of the study compared longer-range connections in a high-degree Hofstadter extension. Node degree was 4 for nearest-neighbor connections, 8 for next-nearest-neighbor connections, 12 for third-neighbor connections and 20 for fourth-neighbor connections. In the optimized degree-20 design, the bulk local Chern index was about 3. Longer-range designs were reported with larger local gaps and higher indices, a pattern consistent with greater modeled channel capacity. The ensemble summaries used 50 realizations per condition and showed one-standard-deviation error bars.
Rewiring the graph
The study then moved to genuine network architectures. It started with 100-node k-regular rings and randomized them with degree-preserving double-edge swaps. The rewiring parameter classified the resulting graphs as regular at zero, small-world near 0.1 and random near 10. Because the swaps preserve degree, the comparison focused on how rewiring changed the architecture.
At sufficiently high node degree, optimized regular, small-world and random networks all showed an ensemble-averaged bulk local Chern index of 1. Edge confinement, meaning how tightly the modeled states stayed near the edge, tracked a sizable local gap across those results. The joint pattern appeared within the tested high-degree regime.
A different pattern at longer range
The flux statistics also differed by connection range. At shorter ranges, they were described as uniform or weakly disordered and Hofstadter-like. At longer ranges, they became strongly disordered and Haldane-like. The distinction was statistical rather than an exact cancellation of fluxes.
The stress test was selective
Directional transport was tested separately from the topology measures. In the overcoupled multiport router, chirality peaked near target energy 1. Regular and small-world designs remained more chiral than random designs around that energy, showing that the simulated routing varied across network architectures.
The disorder tests revealed a different split. Perturbing onsite terms was more damaging at peripheral nodes. Perturbing hopping amplitudes, and especially hopping phases, suppressed chirality more strongly at hubs. The researchers drew independent perturbations uniformly between negative delta and positive delta and averaged chirality at energy 1 over 40 realizations for each perturbation strength.
Across the tested architectures, the authors report stronger chiral robustness in small-world and random networks than in regular networks. They highlight small-world designs for combining high chirality, robustness and reconfigurability. The comparison therefore applies to the generated network families and selected perturbations, rather than to every possible rewired or noisy network.
A computational result with a practical gap
The study remains a computational result. Its evidence comes from finite magnetic-graph and network calculations, inverse-design simulations and scattering-router tests, with no physical implementation or experimental measurement reported. The authors say that putting the idea into practice would require large-scale magnetic graphs with controlled hopping amplitudes and phases.
That boundary matters because the paper's spectral, localization and flux measures were not considered enough by themselves to establish protected transport. The added chirality and disorder tests strengthen the modeled case, but they still cover selected finite networks, rewiring values and perturbation types.
The authors state that data can be requested from the corresponding authors and that all original code is publicly available through the shared Zenodo deposit.
Paper data and sources
Original title: Topological insulators on complex networks
Authors: Sunkyu Yu, Xianji Piao, Namkyoo Park
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-10-02
DOI: Not available
Original paper · Full text