One-Dimensional Random Geometric Graphs

• One-dimensional random geometric graphs are spatial models formed by randomly placing nodes on a line segment or circle and connecting pairs within a fixed distance.
• They exhibit sharp threshold laws for connectivity and isolated nodes, with distinct scaling behaviors on intervals versus circles due to boundary effects.
• Extensions include soft connectivity rules, intersections with Erdős–Rényi graphs, and exact combinatorial encodings that yield explicit results for component counts and path enumerations.

One-dimensional random geometric graphs are spatial random graphs obtained by placing vertices at random on a line segment or on a circle and connecting pairs whose separation satisfies a prescribed proximity rule. In the hard-threshold model, an edge is present exactly when the distance between two points is at most a radius $r$; in soft variants, the edge is present with a distance-dependent probability $H$. Because one dimension converts global connectivity into a problem about ordered gaps while preserving pronounced boundary effects on the interval, the model admits sharp threshold laws, exact formulas for component counts, explicit combinatorial encodings, and analytically tractable rare-event and spectral phenomena (0809.0918, Decreusefond et al., 2010, Wilsher et al., 2020, Badiu et al., 2021).

1. Canonical models and geometric setting

The standard binomial one-dimensional hard random geometric graph begins with $n$ i.i.d. node locations $X_1,\dots,X_n\sim \mathrm{Uniform}[0,1]$. Two metric choices are classical. On the unit interval,

$d(x,y)=|x-y|,$

while on the unit circle,

$d(x,y)=\min\{|x-y|,\,1-|x-y|\}.$

Fixing a range $r>0$, one joins $i<j$ if and only if $d(X_i,X_j)\le r$. The resulting graphs are denoted $G^{(L)}(n;r)$ in the linear case and $H$0 in the circular case. A basic observable is the number of isolated nodes,

$H$1

This interval/circle dichotomy is structurally central, since the circle removes boundary nodes whereas the interval does not (0809.0918).

A second canonical formulation replaces the fixed sample size by a Poisson point process. On $H$2, with intensity $H$3, edges are drawn between points $H$4 exactly when $H$5. In one dimension, connected components then become maximal chains of points whose consecutive gaps do not exceed $H$6. If the ordered points are $H$7, a cluster boundary occurs whenever $H$8. This gap representation makes one-dimensional component structure particularly explicit (Decreusefond et al., 2010).

Soft one-dimensional random geometric graphs keep the spatial input but randomize adjacency conditionally on distance. On $H$9, in either a binomial or Poisson placement, distinct nodes at $n$0 are connected independently with probability $n$1, or on the torus with probability $n$2, where $n$3 is circular distance and $n$4 is a scale parameter. The hard-threshold graph is recovered by taking $n$5, while examples such as $n$6 interpolate to genuinely soft regimes (Wilsher et al., 2020).

2. Isolated vertices, ordered gaps, and threshold phenomena

For the hard model, the absence of isolated vertices has the same first-order threshold on the unit interval and on the unit circle. Writing

$n$7

a critical scaling is

$n$8

equivalently

$n$9

Then

$X_1,\dots,X_n\sim \mathrm{Uniform}[0,1]$0

At this scale, $X_1,\dots,X_n\sim \mathrm{Uniform}[0,1]$1, and the interval boundary effect is only $X_1,\dots,X_n\sim \mathrm{Uniform}[0,1]$2-small, so the same zero-one law holds for $X_1,\dots,X_n\sim \mathrm{Uniform}[0,1]$3 and $X_1,\dots,X_n\sim \mathrm{Uniform}[0,1]$4 (0809.0918).

Connectivity on the unit interval is governed by a different obstruction: an ordered gap larger than the connection radius. If $X_1,\dots,X_n\sim \mathrm{Uniform}[0,1]$5 and the spacings are $X_1,\dots,X_n\sim \mathrm{Uniform}[0,1]$6, $X_1,\dots,X_n\sim \mathrm{Uniform}[0,1]$7, and $X_1,\dots,X_n\sim \mathrm{Uniform}[0,1]$8, then the graph is connected if and only if $X_1,\dots,X_n\sim \mathrm{Uniform}[0,1]$9. This yields the exact finite-$d(x,y)=|x-y|,$0 formula

$d(x,y)=|x-y|,$1

and under the scaling

$d(x,y)=|x-y|,$2

one has

$d(x,y)=|x-y|,$3

Thus the connectivity threshold is

$d(x,y)=|x-y|,$4

up to $d(x,y)=|x-y|,$5 corrections (Kartun-Giles et al., 2021).

The interval model therefore exhibits a genuine separation between the radius needed to eliminate isolated vertices and the radius needed for full connectivity. The threshold for no isolated nodes is

$d(x,y)=|x-y|,$6

whereas the threshold for connectivity is

$d(x,y)=|x-y|,$7

For any fixed $d(x,y)=|x-y|,$8, choosing $d(x,y)=|x-y|,$9 produces a graph with no isolated nodes with high probability but which is disconnected with high probability. This is the “curious gap” of the one-dimensional interval model, and it contrasts with the asymptotic coincidence of these thresholds in many other random graph models, including higher-dimensional geometric random graphs and Erdős–Rényi graphs (Zhao et al., 2015).

3. Intersections and multiple adjacency constraints

One-dimensional geometry also appears as a component of more complex adjacency rules. A basic model with multiple constraints is the intersection

$d(x,y)=\min\{|x-y|,\,1-|x-y|\}.$0

where $d(x,y)=\min\{|x-y|,\,1-|x-y|\}.$1 is an independent Erdős–Rényi graph and $d(x,y)=\min\{|x-y|,\,1-|x-y|\}.$2 if and only if both

$d(x,y)=\min\{|x-y|,\,1-|x-y|\}.$3

hold. The graph therefore retains the spatial locality of the geometric model but imposes an additional independent Bernoulli gate on each candidate edge (0809.0918).

On the unit circle, the isolated-node threshold remains clean. If

$d(x,y)=\min\{|x-y|,\,1-|x-y|\}.$4

then

$d(x,y)=\min\{|x-y|,\,1-|x-y|\}.$5

so the unique critical scaling is

$d(x,y)=\min\{|x-y|,\,1-|x-y|\}.$6

The expected number of isolated vertices takes the form

$d(x,y)=\min\{|x-y|,\,1-|x-y|\}.$7

and first- and second-moment arguments match exactly as in the pure geometric case (0809.0918).

On the unit interval, by contrast, the zero-law and one-law are separated. The zero-law still holds under

$d(x,y)=\min\{|x-y|,\,1-|x-y|\}.$8

but the one-law requires the stronger scaling

$d(x,y)=\min\{|x-y|,\,1-|x-y|\}.$9

The source of this gap is an endpoint contribution in

$r>0$0

which yields an extra term of the form

$r>0$1

Suppressing that boundary term forces $r>0$2. The discrepancy is therefore a boundary-effect artifact of the first-moment method on the interval, not a change in the second-moment mechanism (0809.0918).

4. Exact component laws in Poisson one-dimensional graphs

In the Poisson hard-threshold model on $r>0$3, connected components can be described as a renewal process. If $r>0$4 and $r>0$5 are the start and end of cluster $r>0$6, then the cluster length $r>0$7 and the inter-cluster gap $r>0$8 satisfy:

• the gaps $r>0$9 are IID $i<j$0,
• the busy-period lengths $i<j$1 are IID,
• and $i<j$2 are mutually independent.

The model has a queueing interpretation as an $i<j$3 queue with preemption, where arrivals form a Poisson process, service times are deterministic of length $i<j$4, and each new arrival preempts the customer in service. Under this correspondence, each busy period is exactly one graph cluster (Decreusefond et al., 2010).

A direct conditioning argument gives the Laplace transform of a busy period $i<j$5: $i<j$6 If $i<j$7 is the waiting time between consecutive cluster starts, then

$i<j$8

This leads to an explicit Laplace transform for the number $i<j$9 of complete clusters with start in $d(X_i,X_j)\le r$0, and after inversion one obtains the exact probability mass function

$d(X_i,X_j)\le r$1

The formula is a closed-form law for the number of connected components in the one-dimensional Poisson random geometric graph (Decreusefond et al., 2010).

Several limiting regimes are explicit. As $d(X_i,X_j)\le r$2, every gap exceeds $d(X_i,X_j)\le r$3 almost surely and $d(X_i,X_j)\le r$4 degenerates to the total point count $d(X_i,X_j)\le r$5, recovering the Poisson law $d(X_i,X_j)\le r$6. The same analysis yields factorial moments through Stirling-number expansions. The one-dimensional Poisson case is described as the unique setting in which one obtains an explicit closed-form pmf for the number of connected components; in dimensions $d(X_i,X_j)\le r$7, very few exact results are known (Decreusefond et al., 2010).

5. Combinatorial structure, path enumeration, and entropy

The ensemble of unlabeled one-dimensional hard geometric graphs admits a precise combinatorial description. For $d(X_i,X_j)\le r$8 i.u.d. points on $d(X_i,X_j)\le r$9 and connection range $G^{(L)}(n;r)$0, let $G^{(L)}(n;r)$1 denote the set of unlabeled graph structures. Reordering the vertex labels by increasing position produces an ordered graph $G^{(L)}(n;r)$2; two connected ordered graphs are isomorphic only if one is a left-right reversal of the other. Every ordered graph decomposes uniquely into $G^{(L)}(n;r)$3 maximal-clique blocks of sizes $G^{(L)}(n;r)$4, with $G^{(L)}(n;r)$5. The generating function for ordered graphs coincides with that of Dyck paths of semilength $G^{(L)}(n;r)$6 and height at most $G^{(L)}(n;r)$7, giving

$G^{(L)}(n;r)$8

For fixed $G^{(L)}(n;r)$9,

$H$00

and therefore

$H$01

A universal encoding based on the left and right endpoints of maximal cliques uses exactly $H$02 bits and recovers the ordered graph uniquely (Badiu et al., 2021).

The normalized structural entropy has distinct asymptotic regimes. If $H$03 and $H$04, then the upper bound is given by a function $H$05 involving Kummer’s confluent hypergeometric function, and $H$06 as $H$07 with $H$08. If $H$09 but $H$10, then

$H$11

If $H$12 is fixed, then

$H$13

This places one-dimensional random geometric graphs among the spatial graph families for which combinatorial support growth and entropy can be controlled uniformly in $H$14 and $H$15 (Badiu et al., 2021).

A complementary exact combinatorial problem concerns the number $H$16 of $H$17-hop paths between two distinguished endpoints. In the hard-threshold model on $H$18, with endpoints fixed at $H$19 and $H$20, the intermediate vertices must lie in a sequence of “lenses”

$H$21

Conditioned on the number of points in each lens, the count of $H$22-hop paths is in bijection with the volume under a directed lattice path in a $H$23-dimensional hyperrectangular lattice, or equivalently with a restricted integer partition. The paper gives a probability generating function and an exact pmf as sums over lattice paths. Low-order cases are explicit: for $H$24, $H$25 is Poisson with mean $H$26, and for $H$27 the conditional p.g.f. is a normalized $H$28-binomial coefficient. This places one-dimensional RGG path enumeration in direct contact with lattice-path combinatorics (Kartun-Giles et al., 2021).

6. Soft random geometric graphs on the line and torus

In the soft model, isolated nodes again organize the main threshold theory, but the asymptotics differ qualitatively from the hard interval case. On the torus of length $H$29, with Poisson intensity $H$30 and connection kernel $H$31, the number $H$32 of isolated nodes satisfies

$H$33

Under the rescaling

$H$34

with $H$35,

$H$36

and hence

$H$37

The threshold depends only on the first moment

$H$38

Under the same scaling, $H$39, and when $H$40 the coefficient of variation tends to zero (Wilsher et al., 2020).

A key distinction from the hard interval graph is that uncrossed gaps are asymptotically negligible. If $H$41 is monotone and has unbounded support, then for any fixed $H$42,

$H$43

so

$H$44

Consequently the connectivity probability obeys the same sharp transition as the no-isolated-node event: $H$45 The hard model on the interval behaves oppositely: there, uncrossed gaps are the dominant obstruction to connectivity (Wilsher et al., 2020).

A more refined Poisson approximation is available on the torus of circumference $H$46. With

$H$47

the number $H$48 of isolated vertices converges in total variation to $H$49: $H$50 Since connectivity implies the absence of isolated nodes,

$H$51

asymptotically. The result is proved under assumptions that $H$52 is bounded and measurable, $H$53, and $H$54 (Wilsher et al., 2022).

7. Rare events, simulation, and spectral behavior

One-dimensional geometry also enables explicit rare-event calculations for edge counts. On a Poisson point process $H$55 in $H$56 with intensity $H$57, taking $H$58, let

$H$59

denote the number of edges. For the rare event $H$60, conditioning on the left-most point $H$61 yields

$H$62

and iterated conditioning on the recursively defined points $H$63 gives

$H$64

Analogous formulas are derived for $H$65 and for the number of missing edges $H$66 (Hirsch et al., 2020).

These identities lead to conditional Monte Carlo estimators obtained by replacing the indicator $H$67 by its conditional expectation given a boundary $H$68-algebra. By the law of total variance,

$H$69

with strict inequality unless $H$70 is $H$71-measurable. In the reported simulations for $H$72, the variance-reduction factor grows from $H$73 at $H$74 to almost $H$75 at $H$76. The same work interprets exceptionally small edge counts as a global repulsion effect and exceptionally large edge counts as concentrated in a single cell, aligning the one-dimensional problem with heavy-tailed large-deviation heuristics (Hirsch et al., 2020).

Recent work has also examined eigenmode localization in one-dimensional random geometric graphs with periodic boundary conditions. For $H$77 uniform points on $H$78, with edges when $H$79, the expected degree is

$H$80

for $H$81, and one often writes $H$82 so that $H$83. For the adjacency matrix, a large fraction of eigenvalues sits at $H$84, associated with localized “Type-I orbits.” For the Laplacian, the density of states has a low-energy power law

$H$85

and integer eigenvalues arise from localized orbit modes. The participation ratio

$H$86

separates localized from delocalized modes: for Laplacian eigenvectors, $H$87 as $H$88 until saturation near $H$89, whereas adjacency-matrix modes show a nonmonotonic dependence on $H$90 and no true mobility edge in the $H$91 one-dimensional limit. These results connect one-dimensional RGG spectra to ordered lattices with $H$92 neighbors and to one-dimensional tight-binding models with disorder (Schaefer et al., 26 Aug 2025).