Ordered Topological Deep Learning (OrdGCCN)

• OrdGCCN is a neural network framework that operates on higher-order topological domains by integrating order-awareness into neighborhood aggregation.
• It extends standard GNNs using sequential aggregators to capture directional and sequential dependencies, thereby enhancing model expressivity.
• Empirical validation via RouteNet shows OrdGCCN achieves near-simulator fidelity in delay prediction with significantly reduced runtimes in network modeling tasks.

Ordered Topological Deep Learning (OrdGCCN) is a neural network framework operating on discrete higher-order topological domains, extending topological deep learning methods by incorporating order-awareness into neighborhood aggregation. OrdGCCN generalizes prior combinatorial and graph-based neural architectures by modeling collections of cells, or higher-order discrete entities, with explicitly defined ordered relationships, enabling fine-grained learning of directional or sequential dependencies. The framework was introduced as the theoretical foundation underlying the RouteNet architecture, establishing the first large-scale, real-world application of ordered topological learning for network modeling (Bernárdez et al., 20 Mar 2025).

1. Mathematical Foundations: Ordered Discrete Topological Spaces

The fundamental domain of OrdGCCN is a combinatorial complex $(V, C, \operatorname{rk})$, where $V$ is a finite set of vertices, $C \subseteq \mathcal{P}(V) \setminus \{\emptyset\}$ (the set of nonempty subsets, or cells), and $\operatorname{rk}:C\rightarrow\mathbb{Z}_{\geq0}$ is a rank function. The axioms require that each $v\in V$ yields a singleton $0$-cell of rank $0$, and that every subset $\tau\subseteq\sigma\in C$ inherits cell status with non-increasing rank.

Cells are assigned neighborhoods via a function $N:\ C\rightarrow 2^C$ and organized into augmented Hasse graphs $G_N$, whose nodes are nonempty neighborhoods and whose edges represent ordered inclusion. Two canonical neighborhoods, defined for $V$0, are the $V$1-faces and $V$2-cofaces:

• $V$3
• $V$4

Neighborhoods can be further restricted by a (partial or total) order $V$5 imposed on $V$6 for certain cells $V$7. This yields an ordered neighbor list $V$8, where $V$9 denotes the $C \subseteq \mathcal{P}(V) \setminus \{\emptyset\}$0th ordered neighbor, and $C \subseteq \mathcal{P}(V) \setminus \{\emptyset\}$1 denotes the $C \subseteq \mathcal{P}(V) \setminus \{\emptyset\}$2th $C \subseteq \mathcal{P}(V) \setminus \{\emptyset\}$3-chain (i.e., the prefix subchain up to and including index $C \subseteq \mathcal{P}(V) \setminus \{\emptyset\}$4).

Signals, or features, are modeled as maps $C \subseteq \mathcal{P}(V) \setminus \{\emptyset\}$5, where $C \subseteq \mathcal{P}(V) \setminus \{\emptyset\}$6 is the feature vector associated with cell $C \subseteq \mathcal{P}(V) \setminus \{\emptyset\}$7.

2. Network Architecture and Update Mechanisms

OrdGCCN generalizes Generalized Combinatorial Complex Networks (GCCNs) by integrating order-awareness into neighborhood aggregation. Each layer of the OrdGCCN features two principal update stages per cell:

1. Intra-Neighborhood (Ordered) Update: Given an ordered neighborhood $C \subseteq \mathcal{P}(V) \setminus \{\emptyset\}$8, sequential embeddings are computed as follows:

$C \subseteq \mathcal{P}(V) \setminus \{\emptyset\}$9

where $\operatorname{rk}:C\rightarrow\mathbb{Z}_{\geq0}$0 is a learnable update function (typically an MLP or RNN cell), $\operatorname{rk}:C\rightarrow\mathbb{Z}_{\geq0}$1 processes cell pairs $\operatorname{rk}:C\rightarrow\mathbb{Z}_{\geq0}$2, and the aggregator $\operatorname{rk}:C\rightarrow\mathbb{Z}_{\geq0}$3 is typically an RNN scanning the ordered neighbor list.

2. Inter-Neighborhood Aggregation: The final embedding for cell $\operatorname{rk}:C\rightarrow\mathbb{Z}_{\geq0}$4 aggregates across all neighborhoods and their processed neighbors:

$\operatorname{rk}:C\rightarrow\mathbb{Z}_{\geq0}$5

where $\operatorname{rk}:C\rightarrow\mathbb{Z}_{\geq0}$6 maps learned intermediate embeddings, and $\operatorname{rk}:C\rightarrow\mathbb{Z}_{\geq0}$7 (possibly distinct from $\operatorname{rk}:C\rightarrow\mathbb{Z}_{\geq0}$8) is a permutation-invariant aggregator.

Stacking $\operatorname{rk}:C\rightarrow\mathbb{Z}_{\geq0}$9 such layers yields $v\in V$0.

3. Expressivity and Distinction from Standard GNNs

Conventional Graph Neural Networks (GNNs) and standard GCCNs operate under strict permutation invariance: neighborhood information is aggregated via symmetric set functions (e.g., sum, mean), which prevent differentiation between structurally similar graphs differing only by the order of neighboring elements. This constraint limits representational power, particularly when modeling systems where interactions are not inherently exchangeable.

By contrast, OrdGCCN explicitly encodes and leverages order in neighborhoods. Sequential aggregators such as RNNs process neighbors according to their imposed order, thereby capturing sequential, directional, or flow-like effects not accessible to unordered models. The expressivity of OrdGCCN strictly exceeds that of standard GCCNs and GNNs; a higher-order Weisfeiler-Lehman test (Ord-CCWL) using tuples—rather than multisets—for color refinement can be simulated by OrdGCCN, making it strictly more powerful for distinguishing topological patterns sensitive to order.

4. Empirical Validation: Experimental Configuration

The primary empirical validation for OrdGCCN occurs in the domain of computer-network modeling. Extensive experiments reinterpret RouteNet—a high-performing network modeling tool—as a full OrdGCCN architecture, and benchmark its predictive performance and efficiency against both traditional (analytical and simulator-based) and modern machine learning baselines.

Experimental setup includes:

• Topologies: Real-world Internet configurations (NSFNET, GEANT, GBN; 12–24 nodes) and synthetic scale-free graphs (up to 300 nodes).
• Traffic models: Poisson, deterministic, On–Off bursts, autocorrelated exponentials, modulated exponentials, and real traffic (MAWI traces).
• Queueing: 1–3 queues per port, with FIFO, Strict Priority, WFQ, and DRR policies.
• Testbed: 8-router hardware, using optical taps and synthetic/real traffic generation (GAN + TREX/Tcpreplay).
• Baselines: Packet-level simulator (ns-3), queueing-theory models, MLPs/RNNs, a variety of GNNs (GCN, GAT, GIN, MPNN), unordered GCCN, OrdGCCN without sequential aggregator, and RouteNet as full OrdGCCN.
• Metrics: Mean Absolute Percentage Error (MAPE), Mean Absolute Error (MAE), Mean Squared Error (MSE) for per-flow delay prediction. Inference time measured for simulating 1 second of network activity.

5. Performance Outcomes and Comparative Analysis

Delay prediction results, measured in MAPE (lower is better), are presented as follows:

Model	Simulated	Diff. Routing	Real Traffic
Packet Simulator	1.2%	N/A	56%
Queuing Theory (QT)	12.6%	22.4%	35.1%
MLP/RNN	12.3%	1150%	24.7%
GCN/GAT/MPNN	2.9%	15.0%	23.5%
GCCN (topology)	5.8%	14.2%	22.3%
OrdGCCN (no sequential)	2.1%	1.7%	90.4%
RouteNet (OrdGCCN)	1.7%	1.2%	2.3%

Inference time for simulating $v\in V$1 second on a $v\in V$2-node, $v\in V$3Gbps topology: Packet simulator requires $v\in V$4 hour, RouteNet/OrdGCCN $v\in V$5 second, and queuing-theory $v\in V$6 ms.

The minimal error rates achieved by RouteNet (OrdGCCN) demonstrate robust generalization and significant gains over both conventional ML and unordered GNN models, at orders-of-magnitude reduced runtimes compared to packet simulation. The performance remains strong across synthetic and real network data (Bernárdez et al., 20 Mar 2025).

6. Applications, Scalability, and Usage Guidelines

OrdGCCN achieves a favorable balance between computational cost and accuracy. Packet-level simulators deliver maximal fidelity at prohibitive computational expense; queueing-theory is computationally inexpensive but inaccurate under realistic, non-idealized traffic. OrdGCCN (RouteNet) delivers near-simulator fidelity with inference-time performance suitable for deployment ($v\in V$7 second versus $v\in V$8 hour per simulated second).

Recommended deployment domains include any setting with higher-order relationships admitting a natural order—such as traffic flows, chemical reaction paths, assembly sequences, or time-ordered events. Sequential aggregators (RNN, LSTM, GRU, or transformer) are advocated when the order of neighbor interactions carries meaning; orderless OrdGCCN variations are suitable only if task performance is demonstrably unaffected by permutation.

Scalability is achieved through size-agnostic model construction: training on 10–50 node topologies yields generalizable models for much larger graphs (up to 300 nodes) with controlled losses in accuracy. Key design steps include identification of the relevant cell structure and selection of minimal, semantically-meaningful ordered neighborhoods. Care must be taken to select sequential aggregators for neighborhoods where sequence data is informationally significant.

In summary, OrdGCCN provides a principled, expressive, and scalable topological deep learning mechanism for modeling complex, higher-order, and order-sensitive domains, validated through demonstrated superiority in real-world network modeling tasks (Bernárdez et al., 20 Mar 2025).