GRASS: Graph-based Switching Models

• Graph-based Switching Models (GRASS) are mathematical frameworks that use network topology to encode switching and resource redistribution processes across diverse systems.
• They integrate physical laws and learned interactions, exemplified by quantum memristive device simulations and graph neural network-driven SAT solver selection.
• By linking local transitioning rules to global emergent behaviors, GRASS offers unified insights into phase transitions, hysteresis, and optimization in complex networks.

Graph-based Switching Models (GRASS) comprise a class of mathematical and computational frameworks that encode switching, mode-transition, or resource redistribution processes on graphs, utilizing the discrete or continuous structure of a network for modeling, inference, and prediction. Across recent literature, "GRASS" appears as a specific acronym for distinct but related graph-based switching mechanisms in the context of quantum transport in nanoscale filaments, dynamical systems with interacting agents, resource exchange under nonequilibrium constraints, and algorithm selection using graph neural networks.

1. Quantum Graph Models for Filamentary Switching

Graph-based switching models for atomic-scale memristive devices formalize the physics of metallic nanofilament formation, dissolution, and quantum transport by representing the device geometry as a metric graph $\Gamma = (V, E, \ell)$. Here, nodes $V$ correspond to atomic sites, edges $E$ model hopping paths where atomic orbitals overlap, and edge lengths $\ell_{ij}$ encode geometric constraints. The dynamics are governed by a one-dimensional Schrödinger operator on each edge,

$$H_{ij} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + u_{ij}\,, \quad x \in [0, \ell_{ij}],$$

where $u_{ij}$ is an edge potential, $m$ is the electron mass, and $\hbar$ is Planck’s constant.

Boundary conditions at each vertex enforce wavefunction continuity and conservation of quantum current. The adjacency matrix $A_{ij}$ encodes graph connectivity. Scattering theory on such graphs proceeds through local unitary scattering matrices $\sigma_j(k)$ at nodes and a recursive construction of global transmission amplitudes $T_\Gamma(E)$ via linear systems involving the graph adjacency. The overall conductance emerges from the Landauer–Büttiker formula, aggregating transmission probabilities between electrodes at the Fermi energy.

Filament topologies ranging from chains to branched graphs are simulated with stochastic, voltage-driven growth and dissolution rules, inducing memristive I–V hysteresis and quantized conductance steps. Quantum interference in looped topologies leads to resonant and anti-resonant transport features. The approach links atomic network geometry directly to device-level observables, enabling computationally efficient evaluation ($\mathcal{O}(E^3)$ complexity) for network-scale neuromorphic applications (Silva et al., 2024).

2. SAT Solver Selection via Graph-based Switching

In portfolio-based algorithm selection, GRASS denotes a method whereby switching among candidate SAT solvers is learned using a heterogeneous graph neural network (GNN) framework. Boolean CNF formulae are encoded as tripartite graphs: nodes represent positive literals, negative literals, and clauses; edges either link literals to clauses (indicating literal participation) or pair complementary literals. Each node is enriched with domain-specific features, such as Horn clause statistics or literal occurrence ratios, and relative clause position via sinusoidal positional encodings reflecting empirically observed runtime sensitivity to clause order in CDCL solvers.

The GNN model propagates information over the heterogeneous graph, aggregates clause- and literal-level representations, then projects to a solver-probability distribution optimized under a runtime-aware regret loss,

$$L = \frac{1}{N} \sum_{i=1}^N \left( \sum_{k=1}^K p_i^k t_i^k - t_i^* \right)^2,$$

where $t_i^k$ is the runtime of solver $k$ on instance $i$, $p_i^k$ is the predicted probability for solver $k$, and $t_i^* = \min_k t_i^k$.

Integration of domain knowledge (e.g., Hornness indicators, clause size ratios) significantly improves performance, and ablation studies confirm sensitivity to these features. Empirical results on large industrial and competition benchmarks show lower average runtime and lower "cost of wrong prediction" compared with classical and neural baselines (Zhang et al., 2024).

3. Switching Dynamical Systems with Learned Interaction Graphs

Graph-based switching dynamical systems (GRASS) advance traditional switching dynamical systems by introducing a time-varying, learned graph of interactions among multiple objects, where each object's discrete mode-switching dynamics and continuous latent trajectory depend on both intra- and inter-object influences.

The generative model comprises:

• $N$ objects, each with discrete mode $z_t^n \in \{1, \ldots, K\}$ and continuous state $x_t^n \in \mathbb{R}^d$,
• Adjacency matrix $A_t \in \{0,1\}^{N \times N}$ encoding current interaction structure,
• Joint latent–data factorization: $$p(z_{1:T}, A_{1:T}, x_{1:T}, y_{1:T}) = \left[ \prod_n p(z_1^n) p(x_1^n|z_1^n) p(y_1^n|x_1^n) \right] \times \prod_{t=2}^T \bigl( p(A_t|A_{t-1}, z_{t-1}, x_{t-1})\, p(z_t|z_{t-1}, x_{t-1}, A_{t-1})\, p(x_t|x_{t-1}, z_t)\, p(y_t|x_t) \bigr).$$ Interaction-dependent mode transitions are mediated by influence-weighted softmaxes over incoming edges, while edge variables are inferred by neural relational inference (GNNs with Gumbel-softmax for discrete edges). Variational inference combines amortized encoders for continuous states and edges with exact forward–backward recursions for mode inference.

Benchmarks on ODE-driven synthetic particles and human motion datasets show absolute gains of 5–10% in clustering and prediction metrics relative to SDS baselines. Interpretability arises from the correspondence between inferred interaction edges and true events, such as collisions (Liu et al., 2023).

4. Nonequilibrium Resource Exchange: Switching Transitions on Graphs

In statistical mechanics, graph-based switching models (abbreviated as GRASS in certain contexts) capture phase-like switching between resource-rich and resource-poor states on large networks. The system consists of a graph $G=(V,E)$ of $N$ nodes, each holding a non-negative integer $n_i$ of tokens drawn from a globally conserved reserve. At each time step, with probability $(1-p)$, one node receives one token from the reserve if available; with probability $p$, it and all its neighbors each return a token to the reserve if possible. This asymmetric, locally many-to-one update protocol explicitly breaks detailed balance and results in a nonequilibrium steady state.

The master equation for the probability $P_i(n,t)$ of node $i$ holding $n$ tokens is: $$\frac{dP_i(n)}{dt} = (1-p)[P_i(n-1) - P_i(n)] + p(1+k_i)[P_i(n+1) - P_i(n)],$$ where $k_i$ is the degree of node $i$.

At steady state, for $p > p_c = 1/(k_{\min} + 2)$ with $k_{\min}$ the graph’s minimum degree, $P_i(n)$ converges to a geometric distribution with mean $\langle n_i \rangle = \frac{1-p}{(k_i + 2)p - 1}$. For $p \leq p_c$, a macroscopic fraction of tokens condenses onto the lowest-degree nodes, with the threshold position controlled solely by $k_{\min}$. Scaling near threshold reveals a universal $(p - p_c)^{-1}$ divergence in the fraction of tokens on the graph, independent of network ensemble (regular, Erdős–Rényi, Barabási–Albert) (Auromahima et al., 28 Dec 2025).

5. Structural, Computational, and Physical Implications

Across domains, graph-based switching models (GRASS) enable a direct mapping between network topology and emergent switching behavior—whether quantum conductance, mode-sequence patterns, macroscopic state transitions, or combinatorial optimization performance. In quantum device modeling, GRASS provides an explicit (Hamiltonian plus boundary conditions) formulation suitable for integration into large-scale memristive network simulators, scaling efficiently for network use and capturing key physical observables such as quantized conductance and hysteresis.

In dynamical systems, GRASS models are the first to support both learning and inference of time-varying graphs whose edge structure gates both discrete and continuous transition dynamics, providing gains in fit and interpretability in multi-object settings.

For resource exchange, the GRASS abstraction yields a mathematically tractable model revealing how a single topological property—minimum vertex degree—fixes the phase transition threshold, providing insight into condensation phenomena on complex networks.

Lastly, in ML-driven SAT solver selection, graph-based switching leverages domain structure and sequential decision regret to outperform feature-based selectors, especially by minimizing the cost of selection errors rather than merely raw switching accuracy.

6. Commonalities, Contrasts, and Outlook

While the "GRASS" acronym is overloaded across regimes, all frameworks share a reliance on network structure to encode both local transition mechanisms (e.g., quantum boundary conditions, inter-object influences, neighbor-driven resource loss) and global emergent phenomena (e.g., phase-like transitions, collective switching modes, globally optimal selection policies). A plausible implication is that further generalization of GRASS frameworks may enable unified treatment of predictive, descriptive, and control-theoretic problems on complex networks, especially where switches, thresholds, or discrete transitions mediate system behavior.

Bibliographic references: (Silva et al., 2024, Zhang et al., 2024, Liu et al., 2023, Auromahima et al., 28 Dec 2025).