Circuit-Traversal Fairness Algorithm

• Circuit-traversal-based fairness algorithms are frameworks for dynamically evaluating fairness metrics across network circuits or logical gates.
• They integrate real-time state, such as queues and travel conditions, with structural contexts like node features and sensitive subgroup assignments to optimize equitable schedules.
• These algorithms have been applied in diverse settings—from polling systems and probabilistic circuits to social networks, GNNs, and Tor relays—ensuring compositional fairness.

A circuit-traversal-based fairness algorithm refers to any fairness-enforcing scheduling, auditing, or resource allocation framework in which decisions are made based on traversing a network “circuit” (graph, sequence, DAG, or Boolean circuit), dynamically evaluating fairness metrics across nodes, edges, schedules, or subpopulations. Multiple recent works have operationalized this paradigm in both combinatorial network settings and probabilistic/Boolean circuit contexts. The core unifying feature is that fairness is assessed, enforced, or certified by explicit traversal (real or virtual) over network elements or logical gates, combining dynamic state (queues, allocations, predictions) with structural context (travel conditions, feature assignments, sensitive subgroups) (Singh et al., 2022, Selvam et al., 2022, Liu et al., 8 Dec 2025, Pal, 19 Jan 2026, Basyoni et al., 2020).

1. System Model and Definitions

Circuit-traversal-based fairness algorithms operate over structures such as directed or undirected graphs, probabilistic circuits, loss-prone polling systems, Boolean circuits, or multiplexed connections. Examples include:

• Single-server circuit polling: A server visits $m$ stations arranged on a fixed circuit, processing local queues with Poisson arrivals and bounded buffers. At each epoch, the scheduler selects the next station based on queue lengths, historical utilities, and instantaneous travel conditions (good/bad) to balance efficiency and fairness (Singh et al., 2022).
• Probabilistic circuits for fairness auditing: Smooth, decomposable DAGs encoding joint distributions over features and outcomes, where circuit traversal enables mining of discrimination patterns in exponentially large partial assignment lattices (Selvam et al., 2022).
• Social network edge recommendation: Addition of edges to a graph based on resistance-distance structure, where traversal is used to embed, scan, and optimize over node pairs for improved group-wise information access (Liu et al., 8 Dec 2025).
• Boolean function circuits in GNN fairness: Traversal of gates in a Boolean circuit representing complex subpopulations; fairness is enforced at every gate for expressivity and compositional guarantees (Pal, 19 Jan 2026).
• Multiplexed relay circuits in Tor: Resources are allocated proportionally among connection-sharing circuits via continuous per-circuit rate history, traversing all circuits at each schedule (Basyoni et al., 2020).

Circuit-traversal is characterized by sequential or parallel evaluation of elements (nodes, gates, edges, assignments) to compute fairness-relevant scores, select updates, or certify satisfaction of constraints.

2. Algorithmic Frameworks

Distinct algorithmic instantiations exist across application domains:

• Fair Opportunistic Polling Schedulers (FoPS$(\alpha)$): At each scheduling epoch, FoPS$(\alpha)$ computes for each station $j$ a net fair-weighted utility

$$O_j(s,N,C) = \frac{\bar u_j(s,N,C)}{\max\{\delta,U_{j,k-1}\}^\alpha} - \sum_{i=1}^m \frac{l_i(s\to j|N,C)}{\max\{\delta,U_{i,k-1}\}^\alpha}$$

selecting the next station(s) with maximum $O_j$. Recursive average-utility updates ensure convergence to stationary fairness solutions (Singh et al., 2022).

• Branch-and-Bound Traversal of Probabilistic Circuits: Algorithm 1 in (Selvam et al., 2022) explores the assignment lattice $(x,y)$ in a circuit, using pruning based on upper bounds on discrimination score

$$UB(x,y,E)=\max\left\{ \max_{u\in(R)}P(d|x,y,u)-\min_{u\in(R)}P(d|y,u),\ \min_{u\in(R)}P(d|x,y,u)-\max_{u\in(R)}P(d|y,u) \right\}$$

and dynamic programming for maximizing predictive ratio under deterministic, compatible circuits.

• Resistance-Distance-Based Greedy Traversal: Algorithm Exact and Fast in (Liu et al., 8 Dec 2025) traverse all possible edge additions via effective resistance, utilizing high-dimensional JL-sketched embeddings and convex-hull diameter heuristics for scalable selection.
• Gate Traversal in Boolean Circuits (FairSBF): For each gate $g_j$ in the defining circuit $C_f$, compute per-gate fairness gap

$$\Delta_j = \left| \frac{1}{|S_j|} \sum_{v \in S_j} \hat y_v - \mu \right|$$

and sum over gates to penalize all subpopulation disparities during GNN training (Pal, 19 Jan 2026).

• Proportional Fairness over Relayed Circuits: AR-PF enforces for each circuit the condition $\gamma_i r_i/R_i$ equalized with $\gamma_k r_k/R_k$ via simple closed-form scheduling, traversing all buffered circuits per connection (Basyoni et al., 2020).

3. Fairness Criteria and Metrics

Fairness objectives rely on domain-specific quantitative metrics:

Domain	Metric/Objective	Key Formula/Concept
Polling system	Generalized $\alpha$-fairness: maximize $\sum_i f_\alpha(u_i)$, with $f_\alpha$ as convex	$M(\alpha) = \max_{i,j} u_i/u_j - 1$, $P(\alpha)$
Probabilistic circuits	Discrimination score: $\Delta(x,y) = P(d|x,y) - P(d|y)$, with threshold $\delta$	Definitions: $\delta$-fairness, minimal/maximal/Pareto
Social networks	Unfairness gap: $U_G = I_T - I_S$ where $I_S$ depends on resistance distance	Joint scalarized objective $F(a)$
Boolean function-GNNs	Fairness gap at subpopulation: $$|\mathbb{E}[\hat y|S]-\mathbb{E}[\hat y]| \leq \epsilon$$	Circuit-traversal $L_{\rm fair}$ loss
Tor circuit multiplex	Jain’s fairness index, throughput, latency	$J=\frac{(\sum s_i)^2}{n\sum s_i^2}$

These criteria enable explicit trade-off analysis between efficiency (e.g. throughput, utility) and equity (dispersion of allocation, parity across groups).

4. Theoretical Guarantees and Complexity

• Convergence and Optimality: Under bounded conditions, recursive circuit-traversal algorithms (e.g. FoPS$(\alpha)$) converge almost surely to fairness-optimal stationary distributions (Singh et al., 2022). Branch-and-bound algorithms over probabilistic circuits guarantee soundness, completeness, and pruning efficiency, with worst-case scaling $O(2^n|C|^2)$ (Selvam et al., 2022).
• Hardness: Many circuit-traversal fairness optimization formulations (e.g. edge addition for resistance minimization) are NP-hard and not supermodular, precluding constant-factor greedy guarantees (Liu et al., 8 Dec 2025).
• Approximation and Scalability: Efficient circuit-embedding, JL-sketching, and convex-hull diameter methods allow nearly-linear time fairness optimization on multi-million-node social graphs, preserving optimal marginal gains within quantifiable error (Liu et al., 8 Dec 2025). Sampling-based probabilistic circuit traversal finds high-discrimination patterns orders of magnitude faster than exhaustive search, with substantial pruning (Selvam et al., 2022).
• Compositional Bounds: Enforcing fairness at each gate in a Boolean circuit ensures overall subpopulation fairness gap grows at most linearly with circuit size; i.e., $$\lvert \mathbb{E}[\hat y|f=1] - \mathbb{E}[\hat y] \rvert \leq m\epsilon$$ if each gate achieves gap $\epsilon$ (Pal, 19 Jan 2026).

5. Practical Implementation and Empirical Results

• Polling System: FoPS$(\alpha)$ rapidly reduces disparity as $\alpha$ increases and system size grows; efficiency loss (price of fairness) becomes negligible with larger circuits (Singh et al., 2022).
• Probabilistic Circuits: Branch-and-bound search with circuit traversal achieves high pruning ratios (10-20$\times$ speedup), and compact summaries (minimal/maximal/Pareto) lift interpretability in real datasets (e.g. COMPAS, UCI Adult) (Selvam et al., 2022).
• Social Networks: Circuit-traversal edge-recommendation improves fairness in information access by substantial margins on real and synthetic graphs, with the “Fast” algorithm scaling linearly and accuracy within 1–2% of brute-force optimum on small instances; on linkedin and synthetic graphs $n>10^6$ solutions remain within $<$5% of true resistance/unfairness (Liu et al., 8 Dec 2025).
• GNNs on Complex Subpopulations: FairSBF matches or exceeds state-of-the-art methods (FairGNN, UGE, FairSIN) in both accuracy and fairness gap on intersectional and parity-defined subpopulations that break prior approaches (Pal, 19 Jan 2026).
• Tor Scheduling: AR-PF achieves near-perfect Jain fairness and highest throughput; optimization-based scheduler trades slightly lower fairness for significantly reduced latency on web/stream circuits (Basyoni et al., 2020).

6. Applications and Generalizations

Circuit-traversal-based fairness design principles generalize across varied platforms:

• Networked systems: Resource allocation among spatially arranged queues, servers, or robotic patrols with variable transit conditions.
• Probabilistic models: Fairness certification in tractable probabilistic circuits, Bayesian networks, or structured prediction pipelines.
• Graph mining: Edge addition for equitable information dissemination, incorporating global and intersectional fairness constraints.
• Algorithmic auditing: Mining discrimination patterns, extracting summaries, and certifying absence of bias under partial observation.
• Machine learning: Compositional fairness in GNNs, enforcing parity on arbitrary Boolean-defined subpopulations, including high-degree, intersectional, and parity-breaking cases.
• Privacy-preserving systems: Fair scheduling of Tor circuits to mitigate resource disparity among concurrent connections.

A plausible implication is that models and systems whose fairness properties are structurally mediated by circuit traversal admit both principled optimization (through explicit fairness criteria) and provable compositional guarantees across circuit elements.

7. Limitations and Open Problems

• Scalability trade-offs: While pruning, sampling, and embedding accelerate large-scale traversal, exact fairness certification remains computationally hard in worst cases, especially for high-complexity Boolean circuits.
• Expressivity barriers: Shallow GNNs or simple circuit traversals may fail to capture high-degree or influence-heavy subpopulations, limiting fairness coverage in datasets with deep intersectionality (Pal, 19 Jan 2026).
• Supermodularity absence: Non-supermodular objectives in network augmentation preclude general greedy approximation guarantees, requiring heuristics, surrogates, or empirical validation (Liu et al., 8 Dec 2025).
• Fairness-efficiency trade-off tuning: Selection of fairness factors (e.g. $\alpha$ in FoPS, $\lambda$ in edge addition, loss weights in GNNs) requires careful calibration per application; price of fairness can be negligible for large systems, but non-trivial for small, heterogeneous ones.

Further research into multi-objective compositional optimization, scalable circuit-mining for complex models, and universal fairness guarantees in networked and learning systems will continue to advance the theory and practice of circuit-traversal-based fairness algorithms.