Random Graph Architecture (RGA)

• Random Graph Architecture (RGA) is a framework that uses stochastic models like ER, WS, and motif-based approaches to define probabilistic wiring schemes in neural and computational systems.
• RGAs are implemented in modern network architectures such as GNNs, NAS, and SNNs, enhancing expressivity, adaptability, and performance across various tasks.
• Analytic tools and innovative learning strategies, including generating functions and critical path-based continual learning, enable precise performance evaluation and dynamic network optimization.

Random Graph Architecture (RGA) formalizes the use of random graph models as foundational wiring schemes for complex networks, including neural, computational, and spiking systems. RGAs encode connections between computational nodes using stochastic graph-theoretic principles, replacing conventional deterministic layer-wise topologies with probabilistically sampled directed acyclic graphs (DAGs), small-world networks, or ensembles of subgraph motifs. Architectures built on RGAs demonstrate enhanced expressivity, neuroplasticity, and adaptability, and underpin recent advances in graph neural networks (GNNs), neural architecture search (NAS), spiking neural networks (SNNs), and ensemble learning.

1. Random Graph Model Foundations

The core principle of RGA is the selection of connectivity according to random graph models, with the Erdős–Rényi (ER) G(n, p), Watts–Strogatz (WS), and—in some contexts—Barabási–Albert (BA) algorithms as canonical choices.

• ER Model: Given n nodes, each potential edge is included independently with probability p. For DAGs used in neural architectures, edges are assigned only from i < j to enforce acyclicity. The adjacency matrix $A$ satisfies $A_{ij}\sim\mathrm{Bernoulli}(p)$ for $i<j$, zero otherwise (Mao et al., 2021, Valsesia et al., 2021, Li et al., 2022, Huang et al., 12 Dec 2025).
• WS Model: Starting from a regular ring lattice (each node linked to its k nearest neighbors), each edge is “rewired” with probability β to another randomly chosen node, generating high clustering and low average path length (Huang et al., 12 Dec 2025, Huang et al., 9 May 2025).
• Motif-Based Models: The ensemble of RGAs can be generalized further by using prescribed motif subgraphs. One specifies a collection S of motifs (α = 1…M), their roles/orbits, and role-distributions (see below) (Karrer et al., 2010).

Graph-theoretic properties relevant to RGA analysis include the degree distribution $P(d)$, clustering coefficient C, average path length L, existence and size of the giant component, and centrality measures such as betweenness.

2. Formal Definitions: Role-Vectors, Subgraphs, and Wiring

A key formalism for RGAs is prescribed through “role-vectors” and motif-distributions. Each vertex is assigned a vector $d^{(i)}=(d_1,\ldots,d_c)$, counting the number of stubs for each possible role (Karrer et al., 2010). The probability distribution $p(d)$ over role-vectors defines the system's microscopic structure, subject to graphicality consistency conditions across motifs and their automorphism group orbits:

$$\forall\alpha,\;\forall r,s\in\text{roles}(\alpha):\quad \frac{\langle d_r\rangle}{n_r} = \frac{\langle d_s\rangle}{n_s}$$

The degree distribution generating function is

$$H(z) = G_0(z_1, \ldots, z_c) = \sum_d p(d) \prod_{r=1}^c z_r^{d_r}$$

Random stub-matching yields macro structures with prescribed statistics (cliques, loops, biconnected motifs). In recent SNNs, the backbone DAG wiring is sampled via ER or WS, and each node (“ResNode”) further implements local spike-dynamics and residual connections (Huang et al., 9 May 2025, Huang et al., 12 Dec 2025).

3. RGA Instantiation in Modern Network Architectures

RGAs have been operationalized across deep learning paradigms:

• Graph Neural Networks (GNNs): RAN-GNNs replace layerwise architecture with random DAG wiring among L nodes using ER(G(L, p)). The resulting ensemble of network paths enables multi-scale receptive fields. Each edge receives a trainable weight and the signal at node i is aggregated over its predecessors, modulated by edge weights (Valsesia et al., 2021). RR-GCN further randomizes relation-specific transformations, demonstrating competitive performance in unsupervised settings (Degraeve et al., 2022).
• Neural Architecture Search (NAS): The RandWire space defines cells as DAGs with adjacency matrix $A\in\{0,1\}^{N\times N}$, subject to acyclicity (strictly lower-triangular). Operator assignment is catalogued by $X\in\{1,\ldots,D\}^N$. GraphPNAS learns a deep generative model for P(A, X), using message-passing GNNs and reinforcement learning for architecture optimization (Li et al., 2022).
• Spiking Neural Networks (CogniSNN): Nodes correspond to spiking residual units; edges carry weighted spike-signals, and the architecture’s topology allows efficient signal routing and neuroplasticity. The “pure spike” OR gate residual mechanism enables deep path propagation without degradation (Huang et al., 9 May 2025, Huang et al., 12 Dec 2025).
• Random Graph-Based Neuro Networks (RGNN): Block-wise random graphs define “Layer-Weaken” structures where neurons perform random Fourier feature mapping and are grouped via randomly sampled connectivity. The final pattern matrix aggregates outputs for classification, with performance comparable to monolithic CNNs (Mao et al., 2021).

4. Analytic Tools and Performance Characterization

Analysis of RGAs utilizes generating functions and combinatorial enumeration for connectivity, percolation, and path statistics:

• Giant Component and Percolation: Calculated via recursive fixed points over motif and vertex generating functions:

$u = F(G(u)),\quad S = 1 - G_0(u)$

The linear stability threshold is governed by the Jacobian $M_{rs}$ whose largest eigenvalue crossing 1 marks the emergence of a giant component (Karrer et al., 2010).

• Path Ensemble Analysis: In RAN-GNNs, the expected number of source–sink paths of length l is

$\mathbb{E}[N_l^{(k)}]=\binom{L-k-1}{l-2}p^{l-1}$

Adjusting p modulates expected path lengths: as p → 1, the architecture approaches ResNet-like all-to-all skip connections with oversmoothing; for p < 1, width and depth mixtures are adaptively tuned (Valsesia et al., 2021).

• Empirical Evaluation: RGAs repeatedly match or outperform deterministic-layer networks in classification (DVS-Gesture, CIFAR10-DVS, N-Caltech101) and regression benchmarks (ZINC molecule property prediction), yielding favorable accuracy, reduced forgetting in continual learning, and improved robustness under noise or frame loss (Huang et al., 9 May 2025, Huang et al., 12 Dec 2025, Valsesia et al., 2021, Mao et al., 2021, Li et al., 2022).

5. Learning Algorithms and Topological Adaptivity

RGAs enable novel learning strategies leveraging their graph-theoretic flexibility:

• Critical Path-Based Continual Learning: Paths with maximal or minimal betweenness centrality are identified (via BC(p)), then selectively retrained during transfer (“Key Pathway-based Learning without Forgetting,” KP-LwF). For similar tasks, high-BC paths are tuned; for dissimilar tasks, low-BC paths minimize interference. This protocol decreases forgetting by 3–5% vs vanilla distillation (Huang et al., 9 May 2025, Huang et al., 12 Dec 2025).
• Dynamic Growth Learning (DGL): At training/inference timesteps t < T, only the first $q(t)=\lfloor t|P|/T \rfloor$ paths are activated; as t → T, all are used. This generates time-varying subgraphs, promoting robustness against noise and latency constraints (Huang et al., 12 Dec 2025).
• Majority Voting in RGNN Ensembles: Multiple RGNNs with distinct random-graph parameters form an ensemble where final predictions are produced by majority vote, yielding autonomous near-optimal architecture exploration (Mao et al., 2021).

6. Structural, Algorithmic, and Practical Implications

RGAs fundamentally alter conventional perspectives on architectural design:

• Expressivity: The mixture of path lengths and centrality statistics yields implicit dynamic skip-paths, multi-scale receptive fields, and flexible expansion without vanishing/exploding gradients.
• Neuroplasticity and Reusability: Selection or freezing of critical or peripheral paths mimics biological mechanisms of synaptic reuse and modularity during lifelong learning.
• Configurability and Deployment Flexibility: RGA-based networks, notably CogniSNN, degrade gracefully under reduced timesteps or corrupted inputs, and exhibit strong adaptability in neuromorphic hardware constraints (Huang et al., 12 Dec 2025).
• Model Parallelism and Efficiency: Block-wise or ensemble RGAs reduce manual architecture search, operate with low computational overhead (FRF mappings, shallow blocks), and achieve competitive or superior empirical performance (Mao et al., 2021, Li et al., 2022).

7. Limitations and Theoretical Boundaries

RGAs are subject to several structural and statistical constraints:

• Scalability on Large Motifs: Enumeration of motif-based generating functions $F_r$ can require traversal of $2^{N^{(\alpha)}}$ subgraphs, limiting explicit construction to small motifs.
• Role Correlations and High-Order Overlaps: The basic RGA ensemble assumes independent role-distributions; capturing high-order motif correlations or multipartite structures requires explicit extension of the sampling protocol (Karrer et al., 2010).
• Sparse Motif Overlap: Singly connected motifs with multiple vertex overlaps have vanishing density as n → ∞ unless separately incorporated (Karrer et al., 2010).

In conclusion, Random Graph Architectures provide a mathematically rigorous framework for generating highly flexible, robust, and expressively adaptive architectures across computational domains. They bridge the gap between statistical graph theory and high-performance neural computation, advancing the capacity, resilience, and biological plausibility of modern machine learning systems.