Localization Number: Theory and Applications

• Localization Number is a multifaceted concept that denotes the minimum number of probes, anchors, or effective support size needed for identifying targets or system states.
• In graph theory, it represents the least cops required in a localization game and is bounded by key parameters like metric dimension, pathwidth, and chromatic number.
• In sensing and wave analysis, it guides optimal sensor placement and quantifies source counts or mode support, enhancing accuracy in diverse applications.

“Localization number” is not a uniform term across the literature. In graph theory it denotes the minimum number of probes needed to force identification of an invisible adversary in a localization game; in finite geometry it is studied for incidence, polarity, and Kneser graphs; in wireless sensing it is the number of anchors; in sound source localization it is the inferred number of active sources; and in wave localization it can be operationalized by the participation ratio or its inverse. Several papers also connect the term to counting quantities associated with trapping, topology, or multiplicity of global solutions rather than to a single universal invariant (Bonato et al., 2021, Kumar et al., 2016, Fu et al., 2022, Negro et al., 2022, Ongay-Valverde et al., 2018).

1. Graph-theoretic localization number

In the localization game on a graph $G$, $k$ cops probe vertices while an invisible robber moves with speed one or stays put. If the cops probe $u_1^{(t)},\dots,u_k^{(t)}$ at round $t$, they receive the distance vector $$d^{(t)}=(d_G(u_1^{(t)},r_t),\dots,d_G(u_k^{(t)},r_t))$$. The localization number is the minimum $k$ that guarantees eventual identification of the robber’s exact position: $\ell(G)=\min\{k\in \mathbb N:\text{ %%%%6%%%% cops have a winning strategy}\}.$ The same parameter is denoted $\zeta(G)$ in the cited papers (Bonato et al., 2021).

This parameter is weaker than metric dimension because the cops may use multiple rounds. Accordingly,

$\ell(G)\le \beta(G),$

where $\beta(G)$ is the minimum size of a resolving set. Several general structural bounds are known. Every graph satisfies

$k$0

and if $k$1, then

$k$2

There is also a pathwidth bound,

$k$3

and for hypercubes,

$k$4

(Bonato et al., 2018, Bonato et al., 2021).

Exact values are known for several elementary families. If $k$5 is a tree, then $k$6, with $k$7 exactly when $k$8 contains the graph $k$9 as a subgraph; otherwise $u_1^{(t)},\dots,u_k^{(t)}$0. For the star $u_1^{(t)},\dots,u_k^{(t)}$1, $u_1^{(t)},\dots,u_k^{(t)}$2 while $u_1^{(t)},\dots,u_k^{(t)}$3, which is the standard illustration that localization number can be much smaller than metric dimension. For $u_1^{(t)},\dots,u_k^{(t)}$4, one cop is insufficient and two cops suffice (Bonato et al., 2021).

The no-backtrack variant replaces $u_1^{(t)},\dots,u_k^{(t)}$5 by $u_1^{(t)},\dots,u_k^{(t)}$6, forbidding the robber from moving onto a previously probed vertex. This is a distinct parameter and should not be conflated with the standard localization number (Bonato et al., 2021).

2. Incidence graphs, polarity graphs, and other diameter-2 families

A substantial part of the literature studies localization number on highly structured graphs from finite geometry and extremal graph theory. For the incidence graph $u_1^{(t)},\dots,u_k^{(t)}$7 of a projective plane of order $u_1^{(t)},\dots,u_k^{(t)}$8,

$u_1^{(t)},\dots,u_k^{(t)}$9

For affine planes of order $t$0,

$t$1

More generally, if $t$2 is the incidence graph of a symmetric $t$3, then

$t$4

and for a $t$5,

$t$6

For Steiner triple systems $t$7 with $t$8,

$t$9

while asymptotically $$d^{(t)}=(d_G(u_1^{(t)},r_t),\dots,d_G(u_k^{(t)},r_t))$$0 (Bonato et al., 2021, Bonato et al., 2020).

Diameter-2 graphs without $$d^{(t)}=(d_G(u_1^{(t)},r_t),\dots,d_G(u_k^{(t)},r_t))$$1-cycles provide another major class. For polarity graphs of order $$d^{(t)}=(d_G(u_1^{(t)},r_t),\dots,d_G(u_k^{(t)},r_t))$$2,

$$d^{(t)}=(d_G(u_1^{(t)},r_t),\dots,d_G(u_k^{(t)},r_t))$$3

The corresponding metric-dimension bounds are $$d^{(t)}=(d_G(u_1^{(t)},r_t),\dots,d_G(u_k^{(t)},r_t))$$4, so $$d^{(t)}=(d_G(u_1^{(t)},r_t),\dots,d_G(u_k^{(t)},r_t))$$5 remains explicit at the level of sharp asymptotic constants (Bonato et al., 2020, Bonato et al., 2021).

For Kneser graphs in the diameter-2 regime $$d^{(t)}=(d_G(u_1^{(t)},r_t),\dots,d_G(u_k^{(t)},r_t))$$6, the asymptotics depend on the parity of $$d^{(t)}=(d_G(u_1^{(t)},r_t),\dots,d_G(u_k^{(t)},r_t))$$7. If $$d^{(t)}=(d_G(u_1^{(t)},r_t),\dots,d_G(u_k^{(t)},r_t))$$8 is fixed and even, then

$$d^{(t)}=(d_G(u_1^{(t)},r_t),\dots,d_G(u_k^{(t)},r_t))$$9

If $k$0 is fixed and odd, then

$k$1

For metric dimension, the lower bound is $k$2, and for fixed even $k$3 there are infinitely many $k$4 for which

$k$5

These results are obtained through a hypergraph-detection reformulation of resolving sets (Bonato et al., 2020, Bonato et al., 2021).

Moore graphs of diameter $k$6 furnish a different extremal family. If $k$7 is a $k$8-regular Moore graph of diameter $k$9 with $\ell(G)=\min\{k\in \mathbb N:\text{ %%%%6%%%% cops have a winning strategy}\}.$0, then

$\ell(G)=\min\{k\in \mathbb N:\text{ %%%%6%%%% cops have a winning strategy}\}.$1

while

$\ell(G)=\min\{k\in \mathbb N:\text{ %%%%6%%%% cops have a winning strategy}\}.$2

For the Petersen graph, $\ell(G)=\min\{k\in \mathbb N:\text{ %%%%6%%%% cops have a winning strategy}\}.$3 (Bonato et al., 2020).

3. Random-graph asymptotics and effective localization numbers

In dense random graphs $\ell(G)=\min\{k\in \mathbb N:\text{ %%%%6%%%% cops have a winning strategy}\}.$4 in the diameter-2 regime, localization number admits explicit high-probability bounds. Writing $\ell(G)=\min\{k\in \mathbb N:\text{ %%%%6%%%% cops have a winning strategy}\}.$5, $\ell(G)=\min\{k\in \mathbb N:\text{ %%%%6%%%% cops have a winning strategy}\}.$6, and

$\ell(G)=\min\{k\in \mathbb N:\text{ %%%%6%%%% cops have a winning strategy}\}.$7

Theorem 2.2 gives, asymptotically almost surely,

$\ell(G)=\min\{k\in \mathbb N:\text{ %%%%6%%%% cops have a winning strategy}\}.$8

under the stated assumptions on $\ell(G)=\min\{k\in \mathbb N:\text{ %%%%6%%%% cops have a winning strategy}\}.$9. For constant $\zeta(G)$0, this yields

$\zeta(G)$1

When $\zeta(G)$2, $\zeta(G)$3, so

$\zeta(G)$4

The diameter-2 threshold is controlled by the common-neighbor condition $\zeta(G)$5 (Dudek et al., 2017).

A separate, set-theoretic tradition uses localization numbers $\zeta(G)$6 for coverings of $\zeta(G)$7 by $\zeta(G)$8-branching trees: $\zeta(G)$9 This is equivalent to localization by constant-width $\ell(G)\le \beta(G),$0-slaloms in the eventual sense. The basic monotonicity relation is

$\ell(G)\le \beta(G),$1

and for $\ell(G)\le \beta(G),$2,

$\ell(G)\le \beta(G),$3

The effective analogues replace cardinal characteristics by computability-theoretic highness notions. A function $\ell(G)\le \beta(G),$4 is $\ell(G)\le \beta(G),$5-surviving if it is not a path through any computable $\ell(G)\le \beta(G),$6-branching subtree of $\ell(G)\le \beta(G),$7, and a Turing degree is $\ell(G)\le \beta(G),$8-surviving if it computes such an $\ell(G)\le \beta(G),$9 (Ongay-Valverde et al., 2018).

These two strands share the word “localization,” but their formal objects are different: one is a finite graph-search parameter; the other is a covering cardinal, together with effective non-coverability notions.

4. Anchor counts and source counts in sensing

In wireless sensor networks, the localization number is the number of anchors whose RSSI measurements are used as inputs to the position estimator. If $\beta(G)$0 anchors participate, the input is an RSSI vector in $\beta(G)$1, and the network learns a map $\beta(G)$2. The cited system uses a $\beta(G)$3-$\beta(G)$4-$\beta(G)$5 feed-forward MLP with tansig hidden layers and a linear output layer, trained in MATLAB with Bayesian Regularization and implemented on an Arduino UNO. In this setting, increasing the localization number improves accuracy. With four anchors, the average $\beta(G)$6D localization error is $\beta(G)$7 m, and the five-anchor configuration yields the lowest average error among the tested setups. The paper recommends at least four anchors for sub-meter indoor $\beta(G)$8D accuracy, while noting the theoretical minimum of three anchors for ideal $\beta(G)$9D trilateration (Kumar et al., 2016).

A related anchor-count notion appears in iterative localization on random geometric graphs. There the localization number $k$00 is the minimum number of initially localized nodes needed so that all nodes eventually localize with high probability under the rule that a node becomes localizable once it has at least three localized neighbors in range. The paper maps the problem to bootstrap percolation on a virtual grid and gives a sufficient condition in terms of the probability $k$01 that a virtual grid cell initially contains an anchor: $k$02 together with an occupancy condition,

$k$03

This yields a sufficient lower bound on the number of anchors $k$04 through the explicit formula for $k$05 in terms of $k$06, $k$07, and $k$08 (Vaze et al., 2012).

In multi-source acoustics, the localization number is instead an inferred source count. The ISSL framework estimates a $k$09-bin spatial spectrum $k$10 by SSNet, then iteratively extracts peaks while ASDNet decides whether any active source remains. If $k$11, the procedure stops and returns the DOA set $k$12 together with

$k$13

This threshold-free stopping rule replaces fixed spectrum thresholds. On VCTK-3mix, ISSL reaches DOA $k$14 and source-number accuracy $k$15; on VCTK-4mix, the corresponding values are $k$16 and $k$17 (Fu et al., 2022).

5. Participation-ratio localization numbers in number-theoretic wave systems

For tight-binding Schrödinger operators with on-site potentials drawn from the Liouville function, the Möbius function, or a Legendre quadratic-residue sequence, the cited paper does not explicitly define a quantity named “localization number.” A rigorous reinterpretation consistent with the paper is to identify localization number with the paper’s mode spatial extent (MSE), which for normalized eigenvectors coincides with the participation ratio: $k$18 Equivalently one may use the inverse participation ratio,

$k$19

with the opposite monotonicity convention. The PR interpretation aligns directly with the paper’s analysis because MSE and PR are identical for normalized states (Negro et al., 2022).

Under this interpretation, every eigenmode is localized across the full spectrum, with no mobility edges. For chains of size $k$20, Liouville and Legendre/QR potentials have maximum MSE values around $k$21, while Möbius reaches about $k$22. Roughly $k$23 of modes are single-peak and $k$24 are multi-peak, and these fractions are essentially independent of $k$25 up to $k$26. The MSE distribution is log-normal, level spacings are Poisson for all modes and for the multi-peak subpopulation, and the integrated density of states has a broad downward-concave multifractal spectrum $k$27 for all three sequences (Negro et al., 2022).

This use of localization number is operational rather than nominal. It measures the effective number of lattice sites supporting a mode, rather than a number of probes or sensors.

6. Stochastic, topological, and geometric counting interpretations

In a transient diffusion in a $k$28-drifted Brownian potential with $k$29, the natural counting variable accompanying localization is the number $k$30 of positive $k$31-valleys visited up to time $k$32, where $k$33. The spatial localization theorem states that there exists $k$34 such that

$k$35

At the same time, the visited-valley count has a non-Gaussian scaling limit: $k$36 where $k$37 has a Mittag–Leffler distribution of order $k$38. Here localization number is not a standard term of the paper, but the data explicitly associates the quantitative count $k$39 with the localization mechanism (Andreoletti et al., 2013).

In disorder-driven localization of a $k$40 quantum anomalous Hall system, the integer connected to the localization pathway is the Chern number $k$41. The system localizes from $k$42 to $k$43, and although a hidden $k$44 state appears in individual disorder configurations, it is too narrow and too sample-dependent to produce a robust Hall plateau after averaging. The renormalization-group flow has stable fixed points at $k$45 and $k$46, and an unstable saddle at $k$47 with $k$48. In this context, the data block identifies the Chern number as the integer “localization number” governing the route to localization (Song et al., 2015).

A different counting interpretation arises in two-dimensional GPS source localization under the objective

$k$49

The paper proves that the number of global minimizers is at most $k$50. In the isosceles case with $k$51, $k$52, and

$k$53

one has exactly five global minimizers: $k$54 This is not a localization number in the graph-theoretic or sensing sense; it is the multiplicity of globally optimal source locations under a specific noisy range objective (Kwon, 2024).

Across these domains, “localization number” therefore designates at least four distinct kinds of object: a minimum probe count, a minimum anchor count, an inferred source count, and an effective support size such as participation ratio. The supplied literature also extends the phrase to allied counting quantities—visited valleys, Chern numbers along localization pathways, and the number of global minimizers—when localization is studied as a dynamical, topological, or geometric phenomenon rather than as a single optimization parameter.