Good Nodes Set Initialization

• Good Nodes Set Initialization is a method for computing an optimal subset of nodes that ensures effective control, stability, and rapid convergence across various networked systems.
• It employs deterministic algorithms and domain-specific constraints in applications such as wireless communications, network controllability, and neural architectures.
• The approach integrates combinatorial, spectral, and optimization techniques to enhance clustering quality, influence maximization, and polynomial interpolation performance.

A good nodes set initialization refers to the systematic selection or computation of a distinguished subset of nodes in a network or combinatorial structure that serves as an optimal starting point for control, optimization, propagation, clustering, or analytical algorithms. This concept encompasses initialization protocols in wireless networks, network controllability, influence maximization, neural architectures, clustering, graph learning, and algebraic interpolation. The initialization strategy often directly impacts convergence, objective optimality, and stability of the downstream process.

1. Fundamental Principles of Good Nodes Set Initialization

In diverse networked systems, the initialization of a "good nodes set" typically adheres to several principles:

• Coverage/Controllability: The selected set should dominate or control the relevant topology (e.g., connected dominating set (Kowalski et al., 2017), minimum input node set (Zhang et al., 2017)).
• Optimality and Efficiency: Initialization procedures are often designed to minimize size, maximize control impact, or accelerate convergence.
• Determinism and Stability: Elimination of randomness in selection leads to reproducible, stable outcomes, especially in clustering and label propagation contexts (chandran et al., 2022, Nie et al., 2023).
• Incorporation of Domain-Specific Constraints: The definition of "goodness" is adapted according to the problem: actuator limits in control systems (Mahia et al., 2014), propagation thresholds in social networks (Cordasco et al., 2016), or polynomial interpolation properties in algebraic geometry (Hakopian et al., 18 Aug 2025).

These principles manifest in explicit algorithms, often exploiting combinatorial, spectral, or optimization-theoretic guarantees.

2. Deterministic Initialization in Wireless and Communication Networks

In the SINR wireless communication framework, a backbone overlays fast communication via connected dominating sets. The initialization protocol for the good nodes set—i.e., leader nodes—must operate under minimal assumptions (no physical coordinates, only label and neighbor-label knowledge). The deterministic procedure involves:

• Bucketing nodes by degree,
• Running degree-based selector rounds using (k,m,N)-selectors and strongly-selective families (SSF),
• Guaranteeing that in each pivotal grid-box (square of side $r/\sqrt{2}$), at most one leader is elected,
• Achieving a 1-hop dominating set where every node is either a leader or adjacent to one,
• Ensuring round complexity $O(\Delta \log^2 N)$ (where $\Delta$ is maximum degree and $N$ is label range).

The resulting good nodes set forms the seed for backbone construction and subsequent connectivity phases (Kowalski et al., 2017).

3. Initialization for Network Controllability and Influence Maximization

Controllability in complex networks and influence maximization in social/intervention networks rely critically on good nodes set selection:

• Structural Controllability: Initialization computes the union $U$ of all possible minimum input node sets (MISs) using a single maximum matching followed by alternating-path BFS. This exploits the duality between maximum matchings and control node sets, with $O(|E|\sqrt{N})$ complexity, and enables further refinement via centrality or motif-based heuristics (Zhang et al., 2017).
• Linear Threshold Influence: The minimum target set (MTS) initialization uses a threshold-aware peeling algorithm to efficiently find a near-optimal activating seed set. It maintains sets for seeds, undecided nodes, and "limbo" nodes, with provable optimality on trees, cycles, cliques, DAGs, and produces seed sizes matching tight theoretical bounds (Cordasco et al., 2016).

Both frameworks demonstrate that a tailored combinatorial initialization produces high-quality, minimal, and well-distributed control/seed sets that enhance performance in influence spread, containment, or controllability.

4. Initialization in Learning Algorithms and Neural Networks

Initialization of nodes (weights, biases, virtual nodes) in learning systems is critical for facilitating effective training and inference:

• Deep Networks (AutoInit): Good initialization tunes per-layer scale factors via Jacobian norm targeting, ensuring criticality (e.g., $\mathcal{J}^{l,l+1} \approx 1$), convergence, and high training/validation accuracy. The method encompasses ReLU, BatchNorm, and residual blocks, employing SGD on auxiliary scales and Jacobian-log-loss objectives, and eliminates manual trial-and-error (He et al., 2022).
• Virtual Node Generation in Graph Learning: In sparsely labeled graphs, initialization synthesizes virtual labeled nodes to augment label propagation, maximizes classification confidence on uncertain nodes via a dual-objective loss, and is solved using greedy, submodular optimization techniques. This orthogonally enhances any graph learning or meta-learning pipeline (Cui et al., 2024).
• Neural Network-Guided Initialization for Combinatorial Optimization: In critical node detection for connectivity minimization, a GNN guides the GA's initial population toward promising critical nodes, as determined by centrality/statistical features, with ablation and empirical evidence showing accelerated convergence and improved solution quality over random initialization (Liu et al., 2024).

These initialization protocols exploit problem-adapted losses, feature-aware architecture, and approximate optimization criteria to identify highly informative nodes for training or search.

5. Initialization Strategies in Clustering and Label Propagation

Initialization directly impacts clustering quality and stability, particularly in spectral and propagation-based community detection:

• Similarity-Based Label Seeding (ILI-LPA): Initialization computes pairwise directional similarities and assigns identical labels to mutually "close" nodes, controlled by a threshold parameter $\beta$, significantly reducing randomness-induced instability in label propagation. This yields higher modularity and NMI, sustained performance under sparsity, and practical runtime scalability for large graphs (chandran et al., 2022).
• Hierarchical Nearest-Neighbor Initialization in N-Cut Solvers: N²HI produces deterministic, multi-level partitions via iterative 1-nearest neighbor clustering and graph coarsening, yielding an indicator matrix $Y^{(0)}$ aligned with density centers. This initialization accelerates coordinate-descent clustering, improves objective values (Normalized-Cut, NMI, ARI), and eliminates run-to-run variance compared to spectral $K$-means or random starts (Nie et al., 2023).

Such initializations are essential for stabilizing and accelerating iterative or non-convex clustering algorithms.

6. Algebraic and Geometric Initialization in Polynomial Interpolation

Good nodes set initialization also arises in algebraic settings, such as GCₙ (Carnicer-Gasca) sets for polynomial interpolation in bivariate settings:

• Construction of Prescribed 2-Node Lines in $n$-Correct Sets: Given a node $B$ and $n$ prescribed 2-node lines through $B$, the initialization extends to a full $GC_n$ set by constructing $n+1$ lines in general position, forming intersection points, and ensuring that each prescribed 2-node line is uniquely used by only one other node. The fundamental polynomials factor as products of linear forms corresponding to maximal lines, enabling unique interpolation and explicit combinatorial counting (Hakopian et al., 18 Aug 2025).

This type of initialization resolves structural constraints in algebraic multivariate interpolation theory.

7. Empirical Performance and Theoretical Guarantees

Across all domains, rigorous experimental evaluation and strong theoretical properties are central:

• Efficiency and Superiority: Methods such as All_Input for controllability (Zhang et al., 2017), MTS for influence (Cordasco et al., 2016), and N²HI for clustering (Nie et al., 2023) outperform classical baselines in both computational efficiency and final objective values.
• Optimality Bounds: Provable optimality for special graph classes and tight general bounds are established for combinatorial seed sets (Cordasco et al., 2016), and submodularity ensures monotonic improvement in virtual node generation (Cui et al., 2024).
• Stability and Robustness: Deterministic initializations eliminate variance and improve reproducibility (chandran et al., 2022, Nie et al., 2023).
• Parameter Sensitivity: Parameters such as the similarity threshold $\beta$ (ILI-LPA (chandran et al., 2022)) and time horizon $T$ in harmonic extension (Azad, 2022) are empirically tuned for best performance, with guidelines for their selection.

A plausible implication is that the initialization methodology should be regarded as a critical design stage in any algorithm that depends on node selection, as it can dominate downstream performance and robustness.

Overall, good nodes set initialization constitutes a foundational, domain-specific protocol in the design and analysis of networked systems, learning models, optimization schemes, and algebraic constructions, integrating combinatorial, spectral, statistical, and geometric principles for high-quality algorithmic starts.