Scaling Laws for Search Factor

• Scaling Law for Search Factor is a principle that relates search efficiency metrics to system size and model parameters through power-law relationships.
• Empirical analyses in neural language models reveal that RBP scales with model size, allowing practitioners to predict top-k decoding accuracy without exhaustive parameter searches.
• In quantum search, scaling laws connect optimal search time and success probabilities to network topology, highlighting the practical limits of speedup in realistic systems.

A scaling law for the search factor describes how the computational, inferential, or success probability cost of a search-related operation changes as the underlying system size, model, or structural parameter varies. Scaling laws provide concrete, often power-law, relationships between the search factor—a metric quantifying the efficiency or probability of search processes—and quantities such as model parameters, problem size, or structural complexity. These laws connect analysis of neural LLM decoding, quantum search in networks, and eigenvector search for ranking and underpin practical decisions around computational scaling, model architecture, and algorithm selection.

1. Formal Definitions of Search Factor Metrics

In the context of neural LLMs, the search factor is quantified by the Relative-Based Probability (RBP). For a LLM with $S$ parameters, a fixed vocabulary $V$, and a ground-truth token $t$, the rank $R$ of $t$ is defined as:

$R = \sum_{v\in V} 1\{p(v) \geq p(t)\}$

The RBP at threshold $k$ is:

$\mathrm{RBP}_k(S) = \Pr(R \leq k)$

This is the probability that the correct token is among the top-$k$ predictions. Empirically, $\mathrm{RBP}_k$ is estimated as the fraction of positions where the true token achieves rank $V$0 on held-out data (Yue et al., 23 Oct 2025).

In quantum network search (e.g., for quantum spatial search or adiabatic PageRank), the search factor typically refers to physical or computational metrics such as optimal search time ($V$1), maximal finding probability ($V$2), or run-time $V$3 for eigenvector preparation, expressed as functions of system size and network properties (Frees et al., 2012, Sato et al., 2024).

2. Empirical Forms and Exponents of Search Factor Scaling Laws

Neural LLMs (RBP Scaling)

Across contemporary LLMs (e.g., Pythia, GPT-2, OPT, Qwen) and mainstream datasets, RBP scaling demonstrates a robust power-law relationship:

• For small-to-moderate $V$4:

$V$5

or inverted,

$V$6

• Alternatively, for sufficiently large $V$7:

$V$8

where $V$9, $t$0 are scale factors, $t$1 the scaling exponents, and $t$2 an offset negligible in practical small-$t$3 regimes.

Table: Empirical Exponents for RBP Scaling (Yue et al., 23 Oct 2025)

Dataset	k=1 $t$4	k=10 $t$5	k=100 $t$6
Wiki	(0.079, 0.992)	(0.138, 0.993)	(0.196, 0.995)
HotpotQA	(0.061, 0.990)	(0.085, 0.987)	(0.103, 0.995)
AusLegal	(0.071, 0.994)	(0.115, 0.994)	(0.147, 0.994)
HumanEval	(0.091, 0.985)	(0.165, 0.984)	(0.193, 0.994)

Quantum Spatial Search

For quantum spatial search on complex networks, the key scaling relationships are expressed as power laws in the normalized average path length $t$7:

$t$8

$t$9

$R$0

This data collapse persists across small-world and small-world-regime scale-free networks when plotted against $R$1, regardless of edge weighting (Sato et al., 2024).

Adiabatic Quantum PageRank

The adiabatic PageRank runtime is governed by the minimum spectral gap $R$2:

$R$3

Empirically, for scale-free networks with realistic degree exponents $R$4, the minimum gap scales as

$R$5

with $R$6 for less realistic networks, but $R$7 for Web-like topologies, leading to

$R$8

This result rules out general exponential speedup; the scaling remains polynomial, mirroring the best known classical algorithms (Frees et al., 2012).

3. Theoretical Underpinnings and Derivation Sketches

Neural LLMs

The RBP scaling law derivation is built on two assumptions:

1. For $R$9, top-$t$0 ranking is dictated by the high-rank (tail) statistics of the model’s token-score distribution.
2. Empirical token-rank distributions across model sizes are approximately log-normal:

$t$1

with $t$2 and $t$3 slowly varying in $t$4.

Under this ansatz, the RBP is computed by

$t$5

Expanding $t$6 yields the observed power law in $t$7 (Yue et al., 23 Oct 2025).

Quantum Spatial Search

The search amplitude is related to the sum over all multi-step paths of the network, with path weights dominated by those of length near the average path length $t$8. The peak search probability and duration are determined by the abundance and structure of such paths, leading to the scaling laws for $t$9, $R = \sum_{v\in V} 1\{p(v) \geq p(t)\}$0, and $R = \sum_{v\in V} 1\{p(v) \geq p(t)\}$1 as functions of $R = \sum_{v\in V} 1\{p(v) \geq p(t)\}$2 (Sato et al., 2024).

Adiabatic PageRank

The adiabatic theorem dictates that the runtime $R = \sum_{v\in V} 1\{p(v) \geq p(t)\}$3 is inversely proportional to the minimum gap $R = \sum_{v\in V} 1\{p(v) \geq p(t)\}$4 of the interpolation Hamiltonian. The gap’s decay is controlled by the graph's structure; for realistic Web-like graphs, heavy-tailed degree distributions lead to $R = \sum_{v\in V} 1\{p(v) \geq p(t)\}$5 with $R = \sum_{v\in V} 1\{p(v) \geq p(t)\}$6 approaching $R = \sum_{v\in V} 1\{p(v) \geq p(t)\}$7. Distinct network-generation mechanisms with matched degree distributions can still yield different scaling exponents, indicating non-universality (Frees et al., 2012).

4. Concrete Illustrations and Comparative Tables

Representative growth of $R = \sum_{v\in V} 1\{p(v) \geq p(t)\}$8 and $R = \sum_{v\in V} 1\{p(v) \geq p(t)\}$9 with model size $k$0 for the Wikipedia dataset (Yue et al., 23 Oct 2025):

Model size $k$1	$k$2	$k$3
$k$4	$k$5	$k$6
$k$7	$k$8	$k$9
$\mathrm{RBP}_k(S) = \Pr(R \leq k)$0	$\mathrm{RBP}_k(S) = \Pr(R \leq k)$1	$\mathrm{RBP}_k(S) = \Pr(R \leq k)$2
$\mathrm{RBP}_k(S) = \Pr(R \leq k)$3	$\mathrm{RBP}_k(S) = \Pr(R \leq k)$4	$\mathrm{RBP}_k(S) = \Pr(R \leq k)$5
$\mathrm{RBP}_k(S) = \Pr(R \leq k)$6	$\mathrm{RBP}_k(S) = \Pr(R \leq k)$7	$\mathrm{RBP}_k(S) = \Pr(R \leq k)$8

Empirical exponents for adiabatic PageRank search on several graph classes (Frees et al., 2012):

Graph Model	Degree Exponents $\mathrm{RBP}_k(S) = \Pr(R \leq k)$9	Power-Law Exponent $k$0
GZL copying	$k$1	$k$2
$k$3-PA copying	$k$4	$k$5
GZL copying (Web-like)	$k$6	$k$7
$k$8-PA (Web-like)	$k$9	$\mathrm{RBP}_k$0

5. Practical Implications and Design Recommendations

In LLM decoding, greedy accuracy and top-$\mathrm{RBP}_k$1 hit rates can be forecast using the RBP scaling law, aiding practitioners in parameter budgeting without exhaustive size grid searches. For any required target accuracy, the needed model size $\mathrm{RBP}_k$2 can be directly deduced from the fitted scaling law. For tasks requiring high-probability sequence correctness over $\mathrm{RBP}_k$3 tokens, success scales as $\mathrm{RBP}_k$4, providing principled prediction of sequence-level “emergent” behavior. Top-$\mathrm{RBP}_k$5 or beam decoding benefits from observed increases in $\mathrm{RBP}_k$6 with $\mathrm{RBP}_k$7, suggesting compute-efficient alternatives to pure model scaling (Yue et al., 23 Oct 2025).

For quantum search on networks, the universal scaling collapse in normalized path length implies that, once trivial baseline dependencies are removed, algorithmic effort and success probability are solely functions of topological connectivity, supporting universality claims across network classes (Sato et al., 2024).

In quantum algorithms for ranking and graph eigenvector search, evidence indicates that degree distributions alone do not govern scaling exponents; the network generation method and heavy-tailedness are crucial determinants, and exponential quantum speedup is not realized in realistic web-like networks (Frees et al., 2012).

6. Comparison with Traditional Metrics and Broader Context

Traditional scaling studies in deep learning often use cross-entropy (CE) loss, yielding scaling laws of the form $\mathrm{RBP}_k$8 with $\mathrm{RBP}_k$9. Notably, for greedy decoding, the RBP exponent $V$00 closely matches $V$01, but RBP measures relative token ordering, directly predicting decoding hit rates rather than probability concentration. CE and RBP diverge in their behavior at small scales; CE may improve more rapidly, while RBP remains constrained until token-ranked ordering improves, underscoring the distinct operational significance of these metrics in model evaluation (Yue et al., 23 Oct 2025).

Quantum search regimes further highlight that system topology—especially average path length, tail distributions, and degree exponents—directly mediates scaling of practical search factors, with universal exponents for broad network classes but sensitivity to qualitative shifts in topology.

7. Summary Remarks

Scaling laws for search factor unify a spectrum of quantitative relationships for search processes in machine learning and quantum computation. In neural LLMs, the Relative-Based Scaling Law for RBP provides a principled methodology to anticipate gains in decoding reliability as a function of model size, distinct from traditional cross-entropy scaling, and offers a natural means to specify parameter investment according to application-level accuracy targets. In quantum search and spectral ranking in networks, analogous scaling principles reveal universal and non-universal law regimes, dictated by topology and spectral properties. Together, these results form a foundational toolkit for both theoretical analysis and practical resource planning in high-dimensional search- and ranking-driven systems (Yue et al., 23 Oct 2025, Frees et al., 2012, Sato et al., 2024).