SWAP-Score: Permutation Optimality

• The paper introduces SWAP-Score as a quantitative index defined by normalizing the difference between the chance and minimum average swap distances on a permutohedron.
• It derives optimality using a quadratic assignment formulation, showing that while exact computation is NP-hard for large permutations, efficient solutions exist for small n.
• Empirical studies on word and gesture orders reveal high levels of optimality and contiguous probability structures, indicating strong alignment with swap-distance minimization principles.

to=arxiv_search.search 全民彩票 оЄjson {"query":"id:(Ferrer-i-Cancho, 2 Apr 2026) OR id:(Franco-Sánchez et al., 2024) OR id:(Peng et al., 2024) OR id:(Vats et al., 2013)","max_results":10,"sort_by":"relevance"} to=arxiv_search.search 微信的天天中彩票്ലjson {"query":"(Ferrer-i-Cancho, 2 Apr 2026)","max_results":5,"sort_by":"relevance"} to=arxiv_search.search 凤凰大参考 天天中彩票可以json {"query":"How to measure the optimality of word or gesture order with respect to the principle of swap distance minimization","max_results":5,"sort_by":"relevance"} to=arxiv_search.search 老时时彩 大发快三是json {"query":"SWAP-NAS: Sample-Wise Activation Patterns for Ultra-fast NAS","max_results":5,"sort_by":"relevance"} SWAP-Score is a quantitative index of how “optimal” a given ordering of elements is with respect to the principle that variants should minimize the number of adjacent swaps needed to reach a reference order. In the formulation developed for word order and gesture order, its theoretical foundations rest on viewing all orderings as vertices of a permutohedron graph and defining cost as the average swap distance from that source (Ferrer-i-Cancho, 2 Apr 2026). Closely related work formalizes the underlying cost as average swap distance and studies its behavior under several null models (Franco-Sánchez et al., 2024). The label “SWAP-Score” also appears in unrelated senses in sparse regression and in training-free neural architecture search, where it denotes a local loss-reduction gain and a count of unique sample-wise activation patterns, respectively [(Vats et al., 2013); (Peng et al., 2024)].

1. Permutohedron model and swap distance

Given a sequence of $n$ distinct elements, there are $N=n!$ possible permutations. The permutohedron of order $n$ is the graph whose vertices are these permutations $\pi$, and where two vertices $\pi$ and $\pi'$ are connected by an edge exactly when $\pi'$ is obtained from $\pi$ by swapping two adjacent elements (Ferrer-i-Cancho, 2 Apr 2026).

The swap distance $d(\pi,\sigma)$ between two permutations is the length of the shortest path between them in the permutohedron, that is, the minimum number of adjacent-element swaps needed to transform one into the other. If all permutations are indexed by $i,j\in[1\ldots N]$, then

$N=n!$0

with diameter

$N=n!$1

For $N=n!$2, the six permutations of $N=n!$3, $N=n!$4, and $N=n!$5 form a hexagon. $N=n!$6 differ by swapping $N=n!$7 and therefore have distance $N=n!$8, whereas $N=n!$9 require two adjacent swaps and therefore have distance $n$0. Here $n$1 (Ferrer-i-Cancho, 2 Apr 2026).

This construction supplies the graph-theoretic substrate for treating order variation as a geometric distribution over permutations rather than as an unordered frequency table. A plausible implication is that optimality is not attached to a single order alone, but to the alignment between a probability distribution over orders and the topology of the permutohedron.

2. Formal derivation of the degree of optimality

Assume a probability distribution $n$2 over the $n$3 orderings. The average swap distance of this distribution is

$n$4

It measures expected cost if one randomly selects two orders according to $n$5 and then counts how many adjacent swaps separate them (Ferrer-i-Cancho, 2 Apr 2026).

A raw $n$6 is difficult to interpret because it depends on $n$7 and on the concentration of the probability mass. The normalization therefore uses two benchmarks. The first is a chance baseline obtained by shuffling the assignment of the $n$8's to the permutohedron vertices uniformly at random. If $n$9 denotes the expected $\pi$0 under this random-permutation null model, then

$\pi$1

where $\pi$2 is the Simpson index of $\pi$3, equivalent to probability-mass at distance zero (Ferrer-i-Cancho, 2 Apr 2026).

The second benchmark is the minimum attainable cost over all assignments of the same multiset of probabilities $\pi$4: $\pi$5 For $\pi$6, a closed-form expression is reported as

$\pi$7

The SWAP-Score, or degree of optimality $\pi$8, is then

$\pi$9

Interpretatively, $\pi$0 means $\pi$1, so the arrangement is globally optimal; $\pi$2 means $\pi$3, so the observed cost is what one would expect by chance; and $\pi$4 indicates worse-than-chance ordering (Ferrer-i-Cancho, 2 Apr 2026).

3. Reduction to the Quadratic Assignment Problem

The optimization problem for $\pi$5 can be written in matrix terms. Let $\pi$6 be the $\pi$7 matrix with entries $\pi$8, and let $\pi$9 be the permutohedron distance matrix $\pi'$0. Then

$\pi'$1

where $\pi'$2 is a permutation matrix. Finding $\pi'$3 is therefore a Koopmans–Beckmann quadratic assignment problem: $\pi'$4 Since general QAP is NP-hard, exact solution is infeasible for large $\pi'$5 (Ferrer-i-Cancho, 2 Apr 2026).

The computational regimes stated for this formulation are threefold. Brute-force enumeration is only possible up to $\pi'$6. Beyond that, one must turn to specialized QAP solvers such as QAPLIB or to integer-programming formulations, or else to approximation or heuristics such as greedy local improvements (Ferrer-i-Cancho, 2 Apr 2026).

For $\pi'$7, the paper gives an efficient simplification. The two globally optimal assignments arrange probabilities in breadth-first traversal of the permutohedron, assigning $\pi'$8 to one vertex, $\pi'$9 to an adjacent vertex, $\pi'$0 to the other neighbour, $\pi'$1 to a distance-$\pi'$2 vertex, $\pi'$3 to the other distance-$\pi'$4 vertex, and $\pi'$5 to the antipode (Ferrer-i-Cancho, 2 Apr 2026). This suggests that, in the smallest nontrivial case, global optimality is tightly coupled to the local metric structure of the graph.

4. Worked example and empirical gesture results

A concrete example is given for English gestures in reversible events. The reported relative frequencies for the six orders are

$\pi'$6

Using these values, the paper reports $\pi'$7, $\pi'$8, $\pi'$9, $\pi$0, and finally

$\pi$1

(Ferrer-i-Cancho, 2 Apr 2026).

The broader empirical application concerns a dataset of unconventional gestures across four languages—English, Russian, Irish, and Tagalog—and two verb-reversibility conditions. The reported results are that, in every sub-case, $\pi$2, so the distributions are at least $\pi$3 optimal; half of the eight conditions attain $\pi$4, that is, perfect global optimality; Wilcoxon tests confirm $\pi$5 with $\pi$6; and a Poisson-binomial test gives $\pi$7 for the number of perfect cases (Ferrer-i-Cancho, 2 Apr 2026).

Moreover, all eight empirical probability arrangements are contiguous on the permutohedron, meaning that the non-zero orders form a single path. The paper characterizes this as a structural epiphenomenon of swap-distance minimization and reports $\pi$8 against chance (Ferrer-i-Cancho, 2 Apr 2026). A plausible implication is that swap-distance minimization constrains not only aggregate cost but also the support geometry of the observed distribution.

5. Relation to average swap distance and entropy minimization

Ferrer-i-Cancho et al. formalize the underlying quantity as average swap distance,

$\pi$9

or equivalently $d(\pi,\sigma)$0, where $d(\pi,\sigma)$1 is the probability mass at distance $d(\pi,\sigma)$2 (Franco-Sánchez et al., 2024). In that treatment, $d(\pi,\sigma)$3, with equality precisely when all mass is on a single permutation. For $d(\pi,\sigma)$4, and conjecturally for all $d(\pi,\sigma)$5,

$d(\pi,\sigma)$6

and equality holds, for instance, when $d(\pi,\sigma)$7 is uniform or when $d(\pi,\sigma)$8 puts mass $d(\pi,\sigma)$9 on two permutations at maximal distance $i,j\in[1\ldots N]$0 (Franco-Sánchez et al., 2024).

That paper studies three null hypotheses. Under the die-rolling null hypothesis,

$i,j\in[1\ldots N]$1

Under the Polya-urn null hypothesis, no closed form is given and $i,j\in[1\ldots N]$2 is estimated by Monte Carlo. Under random permutation of the empirical $i,j\in[1\ldots N]$3-vector,

$i,j\in[1\ldots N]$4

where $i,j\in[1\ldots N]$5 is Simpson’s index (Franco-Sánchez et al., 2024).

The same work compares swap-distance minimization with entropy minimization, where block entropy is

$i,j\in[1\ldots N]$6

It reports strong evidence of entropy minimization and swap distance minimization with respect to a die rolling experiment, strong evidence for both forces with respect to a Polya urn process for $i,j\in[1\ldots N]$7 but weaker evidence for $i,j\in[1\ldots N]$8, and evidence of swap distance minimization when word order frequencies are shuffled, indicating that swap distance minimization effects are beyond pressure to minimize word order entropy (Franco-Sánchez et al., 2024). The bridge between the two perspectives is the Simpson index, which links quadratic diversity to the Rényi entropies through $i,j\in[1\ldots N]$9.

6. Extensions, limitations, and scope conditions

The framework generalizes conceptually to $N=n!$00 by redefining the permutohedron in higher dimensions, but this incurs the full QAP hardness for computing $N=n!$01. Approximate heuristics or problem-specific bounds are therefore required in larger cases (Ferrer-i-Cancho, 2 Apr 2026).

Tied probabilities $N=n!$02 slightly complicate ranking and formulas for $N=n!$03 but do not invalidate the approach. At the same time, the SWAP-Score is strictly comparative: it tells us how close an observed distribution is to the best possible under the swap-distance principle, but it does not identify why human or gestural systems approximate that optimum (Ferrer-i-Cancho, 2 Apr 2026).

The stated domain of extension goes beyond linguistic word order and gestures to any permutation-based phenomenon where cost is measured in adjacent exchanges, including dependency-distance minimization and biological sequence rearrangements (Ferrer-i-Cancho, 2 Apr 2026). This suggests that the framework is best understood as a general optimization template for distributions over permutations rather than as a language-specific statistic.

7. Other uses of the label in arXiv literature

In sparse regression, Vats and Baraniuk use the label for a different object. There, with response $N=n!$04, design matrix $N=n!$05, current support $N=n!$06, and residual loss $N=n!$07, a swap removes $N=n!$08 and inserts $N=n!$09, producing $N=n!$10. The SWAP-Score is the gain

$N=n!$11

and the algorithm selects the pair maximizing $N=n!$12. The global objective is $N=n!$13, the score is computed efficiently via rank-one updates of orthogonal projectors, and recovery guarantees are given under restricted eigenvalues and a generalized coherence parameter $N=n!$14 (Vats et al., 2013).

In training-free neural architecture search, the label denotes a zero-cost proxy based on Sample-Wise Activation Patterns. For a ReLU network $N=n!$15 with fixed random parameters $N=n!$16, mini-batch size $N=n!$17, and $N=n!$18 scalar pre-activations, one forms binary vectors $N=n!$19 by thresholding each pre-activation across the batch, collects the set $N=n!$20, and defines

$N=n!$21

A regularised version multiplies $N=n!$22 by $N=n!$23, where $N=n!$24 is the number of parameters. The paper reports Spearman’s rank correlation coefficient of $N=n!$25 between regularised SWAP-Score and CIFAR-100 validation accuracies on NAS-Bench-201 networks, and states that SWAP-NAS achieves competitive performance on CIFAR-10 and ImageNet in approximately $N=n!$26 minutes and $N=n!$27 minutes of GPU time respectively (Peng et al., 2024).

These usages are mathematically unrelated. In one case SWAP-Score is a degree of optimality on a permutohedron; in another it is a local improvement criterion for support swaps; in a third it is the cardinality of a set of activation patterns. This suggests that, in technical writing, the term is not self-identifying and must be interpreted through the paper-specific formal definition.