The shape of shortest paths in random spatial networks

TL;DR

This paper investigates the scaling laws of shortest paths in various spatial network models, revealing two classes with distinct exponents and Gaussian travel time fluctuations, and confirms the KPZ relation across models.

Contribution

It identifies two distinct classes of spatial networks with different wandering and fluctuation exponents, and demonstrates Gaussian travel time fluctuations in these models.

Findings

Two classes with different exponents: (ξ=3/5, χ=1/5) and (ξ=7/10, χ=2/5).

Gaussian travel time fluctuations observed, not Tracy-Widom.

KPZ relation χ=2ξ−1 holds for all models.

Abstract

In the classic model of first passage percolation, for pairs of vertices separated by a Euclidean distance $L$, geodesics exhibit deviations from their mean length $L$ that are of order $L^\chi$, while the transversal fluctuations, known as wandering, grow as $L^\xi$. We find that when weighting edges directly with their Euclidean span in various spatial network models, we have two distinct classes defined by different exponents $\xi=3/5$ and $\chi = 1/5$, or $\xi=7/10$ and $\chi = 2/5$, depending only on coarse details of the specific connectivity laws used. Also, the travel time fluctuations are Gaussian, rather than Tracy-Widom, which is rarely seen in first passage models. The first class contains proximity graphs such as the hard and soft random geometric graph, and the $k$-nearest neighbour random geometric graphs, where via Monte Carlo simulations we find $\xi=0.60\pm 0.01$ and $\chi = 0.20\pm 0.01$, showing a theoretical minimal wandering. The second class contains graphs based on excluded regions such as $\beta$-skeletons and the Delaunay triangulation and are characterised by the values $\xi=0.70\pm 0.01$ and $\chi = 0.40\pm 0.01$, with a nearly theoretically maximal wandering exponent. We also show numerically that the KPZ relation $\chi = 2\xi -1$ is satisfied for all these models. These results shed some light on the Euclidean first passage process, but also raise some theoretical questions about the scaling laws and the derivation of the exponent values, and also whether a model can be constructed with maximal wandering, or non-Gaussian travel fluctuations, while embedded in space.

Tables

Table 1. Table 1: Exponents ξ 𝜉 \xi and χ 𝜒 \chi , and passage time distribution for the various networks considered.

Network	$\xi$	$\chi$	Distribution of $T$
Proximity graphs
Hard RGG	3/5	1/5	Normal (Conj.)
Soft RGG with Rayleigh fading	3/5	1/5	Normal (Conj.)
$k$-NNG	3/5	1/5	Normal
Excluded region graphs
DT	7/10	2/5	Normal
GG	7/10	2/5	Normal
$\beta -$skeletons	7/10	2/5	Normal
RNG	7/10	2/5	Normal
Euclidean FPP
With $\alpha =3/2$	7/10	2/5	Normal
With $\alpha =5/2$	7/10	2/5	Normal

Equations

$$T(x,y)=\underset{P}{\min }\sum _P\tau (e)$$

$$\mathbb{E}\left(T\left(x,y\right)\right)\sim |x-y|$$

$$\text{Var}\left(T\left(x,y\right)\right)\sim |x-y{|}^{2\chi },$$

$$\mathbb{E}(D(x,y))\sim |x-y{|}^{\xi }$$

$$\text{Var}\left(T\left(x,y\right)\right)\le C|x-y|/\log |x-y|.$$

$$\chi =2\xi -1$$

$$\chi \ge (1-d\xi )/2$$

$$\varphi (x)=x^{\alpha },\alpha >1$$

$$\xi (2)\ge 3/5$$

$$\frac{1}{d+1}\le \xi \le 3/4$$

$$\chi \ge \frac{1-(d-1)\xi }{2}.$$

$$1/8\le \chi$$

$$1/3\le \xi \le 3/4$$

$$\xi \ge \frac{3}{3+d}$$

$$RNG\subset GG\subset DT$$

$${\xi }_{RNG}\ge {\xi }_{GG}\ge {\xi }_{DT}$$

$$H(|x-y|)=\exp \left(-\gamma |x-y{|}^{\eta }\right)$$

$$\mathcal{V}=[-w/2-ϵ/2,w/2+ϵ/2]\times [-h/2,h/2],$$

$$n\sim \text{Pois}\left(h\left(w+ϵ\right)\rho \right)$$

$${\chi }_{RGG,NNG}=0.20\pm 0.01$$

$${\xi }_{RGG,NNG}=0.60\pm 0.01$$

$${\chi }_{DT,\beta \text{-skel},\text{EFPP}}=0.40\pm 0.01$$

$${\xi }_{DT,\beta \text{-skel},\text{EFPP}}=0.70\pm 0.01$$

$$\text{Var}\left(T\left(x,y\right)\right)\sim |x-y{|}^{4/5},$$

$$\mathbb{E}\left(D\left(x,y\right)\right)\sim |x-y{|}^{7/10}$$

$$\text{Var}\left(T\left(x,y\right)\right)\sim |x-y{|}^{2/5},$$

$$\mathbb{E}\left(D\left(x,y\right)\right)\sim |x-y{|}^{3/5},$$

$$\frac{3}{3+d}\le \xi \le \frac{3}{4}$$

$$g(z)=\sum _{i\ne j\ne k}\frac{{\sigma }_{ij}(z)}{{\sigma }_{ij}}$$

$$g(z)=\frac{1}{{\int }_{{\mathcal{V}}^2}{\chi }_{xy}(0)\mathrm{d}x\mathrm{d}y}{\int }_{{\mathcal{V}}^2}{\chi }_{xy}(z)\mathrm{d}x\mathrm{d}y$$

$$g(ϵ)=\frac{2}{\pi }\left(1-{ϵ}^2\right)E\left(ϵ\right)$$

$$\mathbb{E}D(x,y)=C|x-y{|}^{\xi }$$

$${\chi }_{xy}(z)=\theta \left(D(x,y)-z_{\perp }\right)$$

Full text

I Introduction

Many complex systems assume the form of a spatial network barthelemy2011 ; barthelemy2018 . Transport networks, neural networks, communication and wireless sensor networks, power and energy networks, and ecological interaction networks are all important examples where the characteristics of a spatial network structure are key to understanding the corresponding emergent dynamics.

Shortest paths form an important aspect of their study. Consider for example the appearance of bottlenecks impeding traffic flow in a city barthelemy2013 ; kirkley2018 , the emergence of spatial small worlds amaral2000 ; hirsch2019 , bounds on the diameter of spatial preferential attachment graphs flaxman2004 ; jordan2010 ; jacob2015 , the random connection model kartungiles2018 ; kog105 ; knight2017 ; privault2019 , or in spatial networks generally aldous2009 ; aldous2010 , as well as geometric effects on betweenness centrality measures in complex networks kog105 ; crucitti2006 , and navigability boguna2008 .

First passage percolation (FPP) hammersley1965 attempts to capture these features with a probabilistic model. As with percolation grimmettbook , the effect of spatial disorder is crucial. Compare this to the elementary random graph erdos1959 . In FPP one usually considers a deterministic lattice such as $\mathbb{Z}^{d}$ with independent, identically distributed weights, known as local passage times, on the edges. With a fluid flowing outward from a point, the question is, what is the minimum passage time over all possible routes between this and another distant point, where routing is quicker along lower weighted edges?

More than 50 years of intensive study of FPP has been carried out auffinger2017 . This has lead to many results such as the subadditive ergodic theorem, a key tool in probability theory, but also a number of insights in crystal and interface growth takeuchi2010 , the statistical physics of traffic jams grimmettbook , and key ideas of universality and scale invariance in the shape of shortest paths takeuchi2011 . As an important intersection between probability and geometry, it is also part of the mathematical aspects of a theory of gravity beyond general relativity, and perhaps in the foundations of quantum mechanics, since it displays fundamental links to complexity, emergent phenomena, and randomness in physics berry1978 ; bianconi2016 .

A particular case of FPP is the topic of this article, known as Euclidean first passage percolation (EFPP). This is a probabilistic model of fluid flow between points of a $d$-dimensional Euclidean space, such as the surface of a hypersphere. One studies optimal routes from a source node to each possible destination node in a spatial network built either randomly or deterministically on the points. Introduced by Howard and Newman much later in 1997 howard1997 and originally a weighted complete graph, we adopt a different perspective by considering edge weights given deterministically by the Euclidean distances between the spatial points themselves. This is in sharp contrast with the usual FPP problem, where weights are i.i.d. random variables.

Howard’s model is defined on the complete graph constructed on a point process. Long paths are discouraged by taking powers of interpoint distances as edge weights. The variant of EFPP we study is instead defined on a Poisson point process in an unbounded region (by definition, the number of points in a bounded region is a Poisson random variable, see for example chiubook ), but with links added between pairs of points according to given rules bollobas2001 ; penrosebook , rather than the totality of the weighted complete graph. More precisely, the model we study in this paper is defined as follows. We take a random spatial network such as the random geometric graph constructed over a simple Poisson point process on a flat torus, and weight the edges with their Euclidean length (see Fig. 1).

We then study the random length and transversal deviation of the shortest paths between two nodes in the network, denoted $x$ and $y$, conditioned to lie at mutual Euclidean separation $|x-y|$, as a function of the point process density and other parameters of the model used (here and in the following $|x|$ denotes the usual norm in euclidean space). The study of the scaling with $|x-y|$ of the length and the deviation allow to define the fluctuation and wandering exponents (see precise definitions below). We will consider a variety of networks such as the random geometric graph with unit disk and Rayleigh fading connection functions, the $k$-nearest neighbour graph, the Delaunay triangulation, the relative neighbourhood graph, the Gabriel graph, and the complete graph with (in this case only) the edge weights raised to the power $\alpha>1$. We describe these models in more detail in Section III.

To expand on two examples, the random geometric graph (RGG) is a spatial network in which links are made between all pair of points with mutual separation up to a threshold. This has applications in e.g. wireless network theory, complex engineering systems such as smart grid, granular materials, neuroscience, spatial statistics, topological data analyis, and complex networks last2017 ; penrose20162 ; hirsch2015 ; giles2016 ; kartungiles2018 ; kartungiles2019 ; kartungiles20192 ; ostilli2015 ; cunningham2019 ; mulder2018 .

This paper is structured as follows. We first recap known results obtained for both the FPP and Euclidean FPP in Section II. We also discuss previous literature for the FPP in non-typical settings such as random graphs and tessellations. The reader eager to view the results can skip this section at first reading, apart from the definitions of IIA, however the remaining background is very helpful for appreciating the later discussion. In the Section III we introduce the various spatial networks studied here, and in Section IV we present the numerical method and our new results on the EFPP model on random graphs. In particular, due to arguments based on scale invariance, we expect the appearance of power laws and universal exponents (takeuchi2011, , Section 1). We reveal the scaling exponents of the geodesics for the complete graph and for the network models studied here, and also show numerical results about the travel time and transversal deviation distribution. In particular, we find distinct exponents from the KPZ class (see for example halpin1995 and references therein) which has wandering and fluctuation exponents $\xi=2/3$ and $\chi=1/3$, respectively. Importantly, we conjecture and numerically corroborate a Gaussian central limit theorem for the travel time fluctuations, on the scale $t^{1/5}$ for the RGG and the other proximity graphs, and $t^{2/5}$ for the Delaunay triangulation and other excluded region graphs, which is also distinct from KPZ where the Tracy-Widom distribution, and the scale $t^{1/3}$, is the famous outcome. Finally, in Section V we present some analytic ideas which help explain the distinction between universality classes. We then conclude and discuss some open questions in Section VI.

II Background: FPP and EFPP

In EFPP, we first construct a Poisson point process in $\mathbb{R}^{d}$ which forms the basis of an undirected graph. A fluid or current then flows outward from a single source at a constant speed with a travel time along an edge given by a power $\alpha\geq 1$ of the Euclidean length of the edge along which it travels howard1997 . See Fig. 2, where the model is shown on six different random spatial network models.

Euclidean FPP on a large family of connected random geometric graphs has been studied in detail by Hirsch, Neuhäuser, Gloaguen and Schmidt in neuhauser2012 ; hirsch2015 ; hirsch20152 and the closely related works neuhauser2012 ; brereton2014 ; neuhauser2015 ; hirsch2015 ; hirsch2016 ; hirsch20152 ; coupier2018 , and references therein. Developing FPP in this setting, Santalla et al santalla2015 recently studied the model on spatial networks, as we do here. Instead of EFPP, they weight the edges of the Delaunay triangulation, and also the square lattice, with i.i.d. variates, for example $\text{U}[a,b]$ for $a,b>0$, and proceed to numerically verify the existence of the KPZ class for the geodesics, see e.g. chatterjee2013 , and the earlier work of Pimentel pimentel2011 giving the asymptotic first passage times for the Delaunay triangulation with i.i.d weights. Moreover, FPP on small-world networks and Erdős-Renyi random graphs are studied by Bhamidi, van der Hofstad and Hooghiemstra in bhamidi2010 , who discuss applications in diverse fields such as magnetism abraham1995 , wireless ad hoc networks beyme2014 ; knight2017 ; kartungiles2018 , competition in ecological systems khordzhakia2005 , and molecular biology bundschuh2000 . See also their work specifically on random graphs hofstad2001 . Optimal paths in disordered complex networks, where disorder is weighting the edges with i.i.d. random variables, is studied by Braunstein et al. in braunstein2003 , and later by Chen et al. in chen2006 . We also point to the recent analytic results of Bakhtin and Wu, who have provided a good lower bound rate of growth of geodesic wandering, which in fact we find to be met with equality in the random geometric graph bakhtin2019 .

To highlight the difference between these results and our own, we have edge weights which are not independent random variables, but interpoint distances. As far as we are aware, this has not been addressed directly in the literature.

II.1 First passage percolation

Given i.i.d weights, paths are sums of i.i.d. random variables. The lengths of paths between pairs of points can be considered to be a random perturbation of the plane metric. In fact, these lengths, and the corresponding transversal deviations of the geodesics, have been the focus of in-depth research over the last 50 years auffinger2017 . They exist as minima over collections of correlated random variables. The travel times are conjectured (in the i.i.d.) case to converge to the Tracy-Widom distribution (TW), found throughout various models of statistical physics, see e.g. (santalla2015, , Section 1). This links the model to random matrix theory, where $\beta$-TW appears as the limiting distribution of the largest eigenvalue of a random matrix in the $\beta$-Hermite ensemble, where the parameter $\beta$ is 1,2 or 4 keating2000 .

The original FPP model is defined as follows. We consider vertices in the $d$-dimensional lattice $\mathbb{L}^{d}=(\mathbb{Z}^{d},E^{d})$ where $E^{d}$ is the set of edges. We then construct the function $\tau:E^{d}\to(0,\infty)$ which gives a weight for each edge and are usually assumed to be identically independently distributed random variables. The passage time from vertices $x$ to $y$ is the random variable given as the minimum of the sum of the $\tau$’s over all possible paths $P$ on the lattice connecting these points,

[TABLE]

This minimum path is a geodesic, and it is almost surely unique when the edge weights are continuous.

The average travel time is proportional to the distance

[TABLE]

where here and in the following we denote the average of a quantity by $\mathbb{E}\left(\cdot\right)$, and where $a\sim b$ means $a$ converges to $Cb$ with $C$ a constant independent of $x,y$, as $|x-y|\to\infty$. More generally, if the ratio of the geodesic length and the Euclidean distance is less than a finite number $t$ (the maximum value of this ratio is called the stretch), the network is a $t$-spanner narasimhan2007 . Many important networks are $t$-spanners, including the Delaunay triangulation of a Poisson point process, which has $\pi/2<t<1.998$ bose2011 ; xia2011 . The variance of the passage time over some distance $|x-y|$ is also important, and scales as

[TABLE]

The maximum deviation $D(x,y)$ of the geodesic from the straight line from $x$ to $y$ is characterised by the wandering exponent $\xi$, i.e.

[TABLE]

for large $|x-y|$. Knowing $\xi$ informs us about the geometry of geodesics between two distant points. See Fig. 3 for an illustration of wandering on different networks.

The other exponent, $\chi$, informs us about the variance of their random length. Another way to see this exponent is to consider a ball of radius $R$ around any point. For large $R$, the ball has an average radius proportional to $R$ and the fluctuations around this average grow as $R^{\chi}$ santalla2015 . With $\chi<1$ the fluctuations die away $R\to\infty$, leading to the shape theorem, see e.g. (auffinger2017, , Section 1).

II.1.1 Sublinear variance in FPP

According to Benjamini, Kalai and Schramm, $\text{Var}\left(T(x,y)\right)$ grows sub-linearly with $|x-y|$ benjamini2002 , a major theoretical step in characterising their scaling behaviour. With $C$ some constant which depends only on the distribution of edge weights and the dimension $d$, they prove that

[TABLE]

The numerical evidence, in fact, shows this follows the non-typical scaling law $|x-y|^{2/3}$. Transversal fluctuations also scale as $|x-y|^{2/3}$ auffinger2017 . In this case, the fluctuations of $T$ are asymptotic to the TW distribution. According to recent results of Santalla et al. santalla2017 , curved spaces lead to similar fluctuations of a subtly different kind: if we embed the graph on the surface of a cylinder, the distribution changes from the largest eigenvalue of the GUE, to GOE, ensembles of random matrix theory.

When we see a sum of random variables, it is natural to conjecture a central limit theorem, where the fluctuations of the sum, after rescaling, converge to the standard normal distribution in some limit, in this case as $|x-y|\to\infty$. Durrett writes in a review that “…novice readers would expect a central limit theorem being proved,…however physicists tell us that in two dimensions, the standard deviation is of order $|x-y|^{1/3}$”, see (benjamini2002, , Section 1). This suggests that one does not have a Gaussian central limit theorem for the travel time fluctuations in the usual way. This has been rigourously proven johansson2000 ; baik1999 ; deuschel1999 .

II.1.2 Scaling exponents

A well-known result in the 2d lattice case kardar1986 is that $\chi=1/3$, $\xi=2/3$. Also, another belief is that $\chi=0$ for dimensions $d$ large enough. Many physicists, see for example huse1985 ; kardar1986 ; kardar1987 ; krug1987 ; krug1989 ; medina1989 ; krug1991 , also conjecture that independently from the dimension, one should have the so-called KPZ relation between these exponents

[TABLE]

This is connected with the KPZ universality class of random geometry, apparent in many physical situations, including traffic and data flows, and their respective models, including the corner growth model, ASEP, TASEP, etc deift2006 ; grimmettbook ; derrida1998 . In particular, FPP is in direct correspondence with important problems in statistical physics halpin1995 such as the directed polymer in random media (DPRM) and the KPZ equation, in which case the dynamical exponent $z$ corresponds to the wandering exponent $\xi$ defined for the FPP calabrese2011 ; santalla2015 .

II.1.3 Bounds on the exponents

The situation regarding exact results is more complex chatterjee2013 ; auffinger2017 . The majority of results are based on the model on $\mathbb{Z}^{d}$. Kesten kesten1993 proved that $\chi\leq 1/2$ in any dimension, and for the wandering exponent $\xi$, Licea et al. licea1996 gave some hints that possibly $\xi\geq 1/2$ in any dimension and $\xi\geq 3/5$ for $d=2$.

Concerning the KPZ relation, Wehr and Aizenman wehr1990 and Licea et al licea1996 proved the inequality

[TABLE]

in $d$ dimensions. Newman and Piza newman1995 gave some hints that possibly $\chi\geq 2\xi-1$. Finally, Chatterjee chatterjee2013 proved Eq. 6 for $\mathbb{Z}^{d}$ with independent and identically distributed random edge weights, with some restrictions on distributional properties of the weights. These were lifted by independent work of Auffinger and Damron auffinger2017 .

II.2 Euclidean first passage percolation

Euclidean first passage percolation howard1997 adopts a very different perspective from FPP by considering a fluid flowing along each of the links of the complete graph on the points at some weighted speed given by a function, usually a power, of the Euclidean length of the edge. We ask, between two points of the process at large Euclidean distance $|{x-y}|$, what is the minimum passage time over all possible routes.

More precisely, the original model of Howard and Newman goes as follows. Given a domain $\mathcal{V}$ such as the Euclidean plane, and $\mathrm{d}x$ Lesbegue measure on $\mathcal{V}$, consider a Poisson point process $\mathcal{X}\subset\mathcal{V}$ of intensity $\rho\mathrm{d}x$, and the function $\phi:\mathbb{R}^{+}\to\mathbb{R}^{+}$ satisfying $\phi(0)=0$, $\phi(1)=1$, and strict convexity. We denote by $K_{\mathcal{X}}$ the complete graph on $\mathcal{X}$. We assign to edges $e=\{q,q^{\prime}\}$ connecting points $q$ and $q^{\prime}$ the weights $\tau(e)=\phi(|{q-q^{\prime}}|)$, and a natural choice is

[TABLE]

The reason for $\alpha>1$ is that the shortest path is otherwise the direct link, so this introduces non-trivial geodesics.

The first work on a Euclidean model of FPP concerned the Poisson-Voronoi tessellation of the $d$-dimensional Euclidean space by Vahidi-Asl and Wierman in 1992 vahidiasl1992 . This sort of generalisation is a long term goal of FPP auffinger2017 . Much like the lattice model with i.i.d. weights, the model is rotationally invariant. The corresponding shape theorem, discussed in (auffinger2017, , Section 1), leads to a ball. The existence of bigeodesics (two paths, concatenated, which extend infinitely in two distinct directions from the origin, with the geodesic between the endpoints remaining unchanged), the linear rate of the local growth dynamics (the wetted region grows linearly with time), and the transversal fluctuations of the random path or surface are all summarised in auffinger2014 .

II.2.1 Bounds on the exponents

Licea et al licea1996 showed that for the standard first-passage percolation on $\mathbb{Z}^{d}$ with $d\geq 2$, the wandering exponent satisfies $\xi(d)\geq 1/2$ and specifically

[TABLE]

In Euclidean FPP, however, these bounds do not hold, and we have howard2000 ; howard2001

[TABLE]

and, for the wandering exponent,

[TABLE]

Combining these different results then yields, for $d=2$

[TABLE]

Since the KPZ relation of Eq. 6 apparently holds in our setting, the lower bound for $\chi$ implies then a better bound for $\xi$, namely

[TABLE]

which in the two dimensional case leads to $\xi\geq 3/5$, the same result as in the standard FPP.

Also, the ‘rotational invariance’ of the Poisson point process implies the KPZ relation Eq. 6 is satisfied in each spatial network we study. We numerically corroborate this in Section IV. See for example (auffinger2017, , Section 4.3) for a discussion of the generality of the relation, and the notion of rotational invariance.

II.3 EFPP on a spatial network

This is the model that we are considering here. Instead of taking as in the usual EFPP into account all possible edges with an exponent $\alpha>1$ in Eq. 8, we allow only some edges between the points and take the weight proportional to their length (ie. $\alpha=1$ here). This leads to a different model, but apparently universal properties of the geodesics. We therefore move beyond the weighted complete graph of Howard and Newman, and consider a large class of spatial networks, including the random geometric graph (RGG), the $k$-nearest neighbour graph (NNG), the $\beta$-skeleton (BS), and the Delaunay triangulation (DT). We introduce them in Section III.

III Random spatial networks

We consider in this study spatial networks constructed over a set of random points. We focus on the most straightforward case, and consider a stationary Poisson point process in the $d$-dimensional Euclidean space, taking $d=2$. This constitutes a Poisson random number of points, with expectation given by $\rho$ per unit area, distributed uniformly at random. We do not discuss here typical generalisations, such as to the Gibbs process, or Papangelou intensities last2017 .

First, we will consider the complete graph as in the usual EFPP, with edges weighted according to the details of Sec. II.3. We will then consider the four distinct excluded region graphs defined below. Note that some of these networks actually obey inclusion relations, see for example aldous2010 . We have

[TABLE]

where RNG stands for the relative neighborhood graph, GG for the Gabriel graph, and DT for the Delaunay triangulation. This nested relation trivially implies the following inequality

[TABLE]

as adding links can only decrease the wandering exponent. We are not aware of a similar relation for $\chi$. We will also consider three distinct proximity graphs such as the hard and soft RGG, and the $k$-nearest neighbour graph.

III.1 Proximity graphs

The main idea for constructing these graphs is that two nodes have to be sufficiently near in order to be connected.

III.1.1 Random geometric graph

The usual random geometric graph is defined in penrosebook and was introduced by Gilbert gilbert1961 who assumes that points are randomly located in the plane and have each a communication range $r$. Two nodes are connected by an edge if they are separated by a distance less than $r$.

We also have the following variant: the soft random geometric graph coon2012 ; penrose2016 ; kartungiles2018 . This is a graph formed on $\mathcal{X}\subset\mathbb{R}^{d}$ by adding an edge between distinct pairs of $\mathcal{X}$ with probability $H(|x-y|)$ where $H:\mathbb{R}^{+}\to\left[0,1\right]$ is called the connection function, and $|x-y|$ is Euclidean distance.

We focus on the case of Rayleigh fading where, with $\gamma>0$ a parameter and $\eta>0$ the path loss exponent, the connection function, with $|x-y|>0$, is given by

[TABLE]

and is otherwise zero. This choice is discussed in (giles2016, , Section 2.3).

This graph is connected with high probability when the mean degree is proportional to the logarithm of the number of nodes in the graph. For the hard RGG, this is given by $\rho\pi r^{2}$, and otherwise the integral of the connectivity function over the region visible to a node in the domain, scaled by $\rho$ penrose2016 . As such, the graph must have a very large typical degree to connect.

III.1.2 $k$-Nearest Neighbour Graph

For this graph, we connect points to their $k\in\mathbb{N}$ nearest neighbours. When $k=1$, we obtain the nearest neighbour graph ($1$-NNG), see e.g. (waltersreview, , Section 3). The model is notably different from the RGG because local fluctuations in the density of nodes do not lead to local fluctuations in the degrees. The typical degree is much lower than the RGG when connected waltersreview , though still remains disconnected on a random, countably infinite subset of the $d$-dimensional Euclidean space, since isolated subgraph exist. For large enough $k$, the graph contains the RGG as a subgraph. See Section V.2 for further discussion.

III.2 Excluded region graphs

The main idea here for constructing these graphs is that two nodes will be connected if some region between them is empty of points. See Fig. 4 for a depiction of the geometry of the lens regions for $\beta-$skeletons.

III.2.1 Delaunay triangulation

The Delaunay triangulation of a set of points is the dual graph of their Voronoi tessellation. One builds the graph by trying to match disks to pairs of points, sitting just on the perimeter, without capturing other points of the process within their bulk. If and only if this can be done, those points are joined by an edge. The triangular distance Delaunay graph can be similarly constructed with a triangle, rather than a disk, but we expect universal exponents.

For each simplex within the convex hull of the triangulation, the minimum angle is maximised, leading in general to more realistic graphs. It is also a t-spanner narasimhan2007 , such that with $d=2$ we have the geodesic between two points of the plane along edges of the triangulation to be no more than $t<1.998$ times the Euclidean separation xia2011 . The DT is necessarily connected.

III.2.2 $\beta$-skeleton

The lune-based $\beta$-skeleton is a way of naturally capturing the shape of points (deberg2008, , Chapter 9). see Fig. 4.

A lune is the intersection of two disks of equal radius, and has a midline joining the centres of the disks and two corners on its perpendicular bisector. For $\beta\leq 1$, we define the excluded region of each pair of points $(x,y)$ to be the lune of radius $|x-y|/2\beta$ with corners at $x$ and $y$. For $\beta\geq 1$ we use instead the lune of radius $\beta|x-y|/2$, with $x$ and $y$ on the midline. For each value of beta we construct an edge between each pair of points if and only if its excluded region is empty. For $\beta=1$, the excluded region is a disk and the beta-skeleton is called the Gabriel graph (GG), whilst for $\beta=2$ we have the relative neighbourhood graph (RNG).

For $\beta\leq 2$, the graph is necessarily connected. Otherwise, it is typically disconnected.

IV Numerical results

IV.1 Numerical setup

Given the models in the previous section, we numerically evaluate the scaling exponents $\chi$ and $\xi$, as well as the distribution of the travel time fluctuations. We now describe the numerical setup. With density of points $\rho>0$, and a small tolerance $\epsilon$, we consider the rectangle domain

[TABLE]

and place a

[TABLE]

points uniformly at random in $\mathcal{V}$. Then, on these random points, we build a spatial network by connecting pairs of points according to the rules of either the NNG, RGG, $\beta$-skeleton for $\beta=1,2$, the DT, or the weighted complete graph of EFPP.

Two extra points are fixed near the boundary arcs at $(-w/2,0)$ and $(w/2,0)$, and the Euclidean geodesic is then identified using a variant of Djikstra’s algorithm, implemented in Mathematica 11. The tolerance $\epsilon$ is important for the Soft RGG, since this graph can display geodesics which reach backwards from their starting point, or beyond their destination, before hopping back. We set $\epsilon=w/10$. This process is repeated for $N=2000$ graphs, each time taking only a single sample of the geodesic length over the span $w$ between the fixed points on the boundary. This act of taking only a single path is done to avoid any small correlations between their statistics, since the exponents are vulnerable to tiny errors given we need multiple significant figures of precision to draw fair conclusions. It also allows us to use smaller domains. The relatively small value for $N$ is sufficient to determine the exponents at the appropriate computational speed for the larger graphs.

The approach in santalla2015 involves rotating the point process before each sample is taken, which is valid alternative method, but we, instead, aim for maximium accuracy given the exponents are not previously conjectured, and therefore need to be determined with exceptional sensitivity, rather than at speed. Note that the fits that we are doing here are over the same typical range as in this work santalla2015 .

We then increase $w$, in steps of 3 units of distance, and repeat, until we have statistics of all $w$, up to the limit of computational feasibility. This varies slightly between models. The RGGs are more difficult to simulate due to their known connectivity constraint where vertex degrees must approach infinity, see e.g. (penrosebook, , Chapter 1). Thus we cannot simulate connected graphs to the same limits of Euclidean span as with the other models.

We are then able to relate the mean and standard deviation of the passage time, as well as the mean wandering, to $w$, at various $\rho$, and for each model. For example, the left hand plots in Fig. 5 show that the typical passage time $\mathbb{E}T(x,y)\sim w$, i.e. grows linearly with $w$, for all networks aldous2009 ; aldous2010 ; kartungiles2018 . The standard error is shown, but is here not clearly distinguishable from the symbols.

We ensure $h$ is large enough to stop the geodesics hitting the boundary, so we use a domain of height equal to the mean deviation $\mathbb{E}D(w)$, plus six standard deviations.

The key computational difficulty here is the memory requirement for large graphs, of which all $N$ are stored simultaneously, and mapped in parallel on a Linux cluster over a function which measures the path statistic. This parallel processing is used to speed up the computation of the geodesics lengths and wandering.

IV.2 Scaling exponents

The results are shown in Figs. 5. These plots, shown in loglog, reveal a power law behaviour of $T$ and $D$, and the linear growth of typical travel time with Euclidean span. We then compute the exponents to two significant figures using a nonlinear model fit, based on the model $a|x-y|^{b}$, and then determining the parameters $a,b$ using the quasi-Newton method in Mathematica 11.

Our numerical results suggest that we can distinguish two classes of spatial network models by the scaling exponents of their Euclidean geodesics. The proximity graphs (hard and soft RGG, and $k$-NNG) are in one class, with exponents

[TABLE]

whereas the excluded region graphs (the $\beta$-skeletons and Delaunay triangulation), and Howard’s EFPP model with $\alpha>1$, are in another class with

[TABLE]

Clearly, the KPZ relation of Eq. 6 is satisfied up to the numerical accuracy which we are able to achieve. We corroborate that this is independent of the density of points and connection range scaling, given the graphs are connected. The exponents hold asymptotically i.e. large inter-point distances. Thus we conjecture

[TABLE]

for the proximity graphs (the DT and the $\beta$-skeletons for all $\beta$), and, for the RGGs and the $k$-NNG,

[TABLE]

We summarize these new results in Table 1.

It is surprising that these exponents are apparently rational numbers. In Bernoulli continuum percolation, for example, the threshold connection range for percolation is not known, but not thought to be rational, as it is with bond percolation on the integer lattice (penrosebook, , Chapter 10). Exact exponents are not necessarily expected in the continuum setting of this problem, which suggests there is more to be said about the classification of first passage process via this method.

IV.3 Travel time fluctuations

We see numerically that the travel time distribution is a normal for most cases (see Fig. 6). We summarise these results in the Table 1 and in Fig. 7 we show the skewness and kurtosis for the travel time fluctuations, computed for the different networks. For a Gaussian distribution, the skewness is [math] and the kurtosis equal to $3$, while the Tracy-Widom distribution displays other values.

We provide some detail of the distribution of $T$ for each model from the proximity class in Fig. 6. This is compared against four test distributions, the Gaussian orthogonal, unitary and symplectic Tracy Widom distributions, and also the standard normal distribution.

This makes the case of EFPP on spatial networks one of only a few special cases where Gaussian fluctuations in fact occur. Auffinger and Damron go into detail concerning each of the remaining cases in (auffinger2017, , Section 3.7). One example, reviewed extensively by Chaterjee and Dey chatterjee2013 , is when geodesics are constrained to lie within thin cylinders i.e. ignore paths which traverse too far, and thus select the minima from a subdomain. This result could shed some light on their questions, though in what way it is not clear.

We also highlight that Tracy-Widom is thought to occur in problems where matrices represent collections of totally uncorrelated random variables mopost . In the case of EFPP, we have the interpoint distances of a point process, which lead to spatially correlated interpoint distances, so the adjacency matrix does not contain i.i.d. values. This potentially leads to the loss of Tracy-Widom. However, we also see some cases of $N\times N$ large complex correlated Wishart matrices leading to TW for at least one of their eigenvalues, and with convergence at the scale $N^{2/3}$ hachem2016 .

IV.4 Transversal fluctuations

The transversal deviation distribution results appear beside our evaluation of the scaling exponents, in Fig. 8. All the models produce geodesics with the same transversal fluctuation distribution, despite distinct values of $\xi$. The fluctuations are also distinct from the Brownian bridge (a geometric brownian motion constrained to start and finish at two fixed position vectors in the plane), running between the midpoints of the boundary arcs grimmettbook . It is a key open question to provide some information about this distribution, as it is rarely studied in any FPP model, as far as we are aware of the literature. A key work is Kurt Johannson’s, where the wandering exponent is derived analytically in a variant of oriented first passage percolation. One might ask if a similar variant of EFPP might be possible johansson2000 .

V Discussion

The main results of our investigation are the new rational exponents $\chi$ and $\xi$ for the various spatial models, and the discovery of the unusual Gaussian fluctuations of the travel time. We found that for the different spatial networks the KPZ relation holds and known bounds are satisfied. Also, due to known relations and the the KPZ law we have

[TABLE]

It is surprising to find a large class of networks, in particular the Delaunay triangulation, that displays an exponent $\xi=7/10$ and points to the question of the existence of another class of graphs which display the theoretically maximal $\xi=3/4$.

Both immediately present a number of open questions and topics of further research which may shed light on the first passage process on spatial networks. We list below a number of questions that we think are important.

V.1 Gaussian travel time fluctuations

We are not able to conclude that all the models in the proximity graph class $\chi=3/5$, $\xi=1/5$ have Gaussian fluctuations in the travel time. This is for a technical reason. All the models we study are either connected with probability one, such as the DT or $\beta$-skeleton with $\beta\leq 2$, or have a connection probability which goes to one in some limit. We require connected graphs, or paths do not span the boundary arcs, and the exponents are not well defined.

Thus, the difficult models to simulate are the HRGG, SRGG and $k$-NNG, since these are in fact disconnected with probability one without infinite expected degrees i.e. the dense limit of Penrose, see (penrosebook, , Chapter 1), or with the fixed degree of the $k$-NNG $k=\Theta(\log n)$ and $n\to\infty$ in a domain with fixed density and infinite volume. Otherwise we have isolated vertices, or isolated subgraphs, respectively.

However, the $k$-NNG has typically shorter connection range i.e. in terms of the longest edge, and shortest non-edge, where the ‘length of a non-edge’ is the corresponding interpoint distance between the disconnected vertices (waltersreview, , Section 3). So the computations used to produce these graphs and then evaluate their statistical properties are significantly less demanding. Thus, the HRGG is computationally intractable in the necessary dense limit, so we are unable to verify the fluctuations of either $T$ or $D$. However, we can see a skewness and kurtosis for $T(|x-y|)$ which are monotonically decreasing with $|x-y|$, towards the hypothesised limiting Gaussian statistics, at least for the limited Euclidean span we can achieve.

Given the $k$-NNG is in the same class, we are left to conjecture Gaussian fluctuations hold throughout all the spatial models described in Section III. It remains an open question to identify any exceptional models where this does not hold.

V.2 Percolation and connectivity

If we choose two points at a fixed Euclidean distance, then simulate a Poisson point process in the rest of the $d$-dimensional plane, construct the relevant graph, and consider the probability that both points are in the giant component, this is effectively a positive constant for reasonable distances, assuming that we are above the percolation transition. At small distances, the two events are positively correlated. Thus, one can condition on this event, and therefore when simulating, discount results where the Euclidean geodesic does not exist. This defines FPP on the giant component of a random graph.

It’s not clear from our experiments whether the rare isolated nodes, or occasionally larger isolated clusters, either in the RGGs or $k$-NNG, affect the exponents. One similar model system would be the Lorentz gas: put disks of constant radius in the plane, perhaps at very low density, and seek the shortest path between two points that does not intersect the disks. The exponents $\chi$ and $\xi$ for this setting are not known dettmann2012 ; grimmettbook .

An alternative to giant component FPP would be to condition on the two points being connected to each other. This would be almost identical for the almost connected regime, but weird below the percolation transition. In that case the event we condition on would have a probability decaying exponentially with distance, and the point process would end up being extremely special for the path to even exist. For example, an extremely low density RGG would be almost empty apart from a path of points connecting the end points, with a minimum number of hops.

V.3 Betweenness centrality

The extent to which nodes take part in shortest paths throughout a network is known as betweenness centrality barthelemy2011 ; kirkley2018 . We ask to what extent knowledge of wandering can lead to understanding the centrality of nodes. The variant node shortest path betweenness centrality is highest for nodes in bottlenecks. Can this centrality index be analytically understood in terms of the power law scaling of $D$? Is the exponent directly relevant to its large scale behaviour?

In order to illustrate more precisely this question, let $G$ be the graph formed on a point process $\mathcal{X}$ by joining pairs of points with probability $H(|x-y|)$. Consider $\sigma_{xy}$ to be the number of shortest paths in $G$ which join vertices $x$ and $y$ in $G$, and $\sigma_{xy}(z)$ to be the number of shortest paths which join $x$ to $y$ in $G$, but also run through $z$, then with $\neq$ indicating a sum over unordered pairs of vertices not including $z$, define the betweenness centrality $g(z)$ of some vertex $z$ in $G$ to be

[TABLE]

In the continuous limit for dense networks we can discuss the betweenness centrality and we recall some of the results of kog105 . More precisely, we define $\chi_{xy}(z)$ as the indicator which gives unity whenever $z$ intersects the shortest path connecting the $d$-dimensional positions $x,y\in\mathcal{V}$. Then the normalised betweenness $g(z)$ is given by

[TABLE]

Based on the assumption that there exists a single topological geodesic as $\rho\to\infty$, and that this limit also results in an infinitesimal wandering of the path from a straight line segment, an infinite number of points of the process lying on this line segment intersect the topological geodesic as $\rho\to\infty$, assuming the graph remains connected, and so $\chi_{xy}(z)$ can then written as a delta function of the transverse distance from $z$ to the straight line from $x$ to $y$. The betweenness can then be computed and we obtain kog105 (normalised by its maximum value at $g(0)$)

[TABLE]

where $E\left(k\right)=\int_{0}^{\pi/2}d\theta\left(1-k^{2}\sin^{2}\left(\theta\right)\right)^{1/2}$ is the complete elliptic integral of the second kind. We have also rescaled such that $\epsilon$ is in units of $R$.

Take $D(x,y)$ to be the maximum deviation from the horizontal of the Euclidean geodesic. Numerical simulations suggest that

[TABLE]

where the expectation is taken over all point sets $\mathcal{X}$. The ‘thin cylinders’ are given by a Heaviside Theta function, so assume that the characteristic function is no longer a delta spike, but a wider cylinder

[TABLE]

where $z_{\perp}$ is the magnitude of the perpendicular deviation of the position $z$ from $\text{hull}(x,y)$. We then have that

[TABLE]

(where $\mathbf{0}$ is the transverse vector computed for the origin). This quantity is certainly difficult to estimate, but provides a starting point for computing finite density corrections to the result of kog105 .

The boundary of the domain is crucial in varying the centrality, which is something we ignore here. Without an enclosing boundary, such as with networks embedded into spheres or tori, the typical centrality at a position in the domain is uniform, since no point is clearly distinguishible from any other. This is discussed in detail on kog105 . In fact, a significant amount of recent work on random geometric networks has highlighted the importance of the enclosing boundary giles2016 ; coon2012 .

VI Conclusions

We have shown numerically that there are two distinct universality classes in Euclidean first passage percolation on a large class of spatial networks. These two classes correspond to the following two broad classes of networks: firstly, based on joining vertices according to critical proximity, such as in the RGG and the NNG, and secondly, based on graphs which connect vertices based on excluded regions, as in the lune-based $\beta$-skeletons, or the DT. Heuristically, the most efficient way to connect two points is via the nearest neighbour, which suggests that $\xi$ for proximity graphs should on the whole be smaller than for exclusion-based graphs, which is in agreement with our numerical observations.

The passage times show Gaussian fluctuations in all models, which we are able to numerically verify. This is a clear distinction between EFPP and FPP. After similar results of Chaterjee and Dey chatterjee2013 , it remains an open question why this happens, and also of course how to rigorously prove it.

We also briefly discussed notions of the universality of betweenness centrality in spatial networks, which is influenced by the wandering of shortest paths. A number of open questions remain about the range of possible universal exponents which could exist on spatial networks, whose characterisation would shed light on the interplay between the statistical physics of random networks, and their spatial counterparts, in way which could reveal deep insights about universality and geometry more generally.

Acknowledgements

The authors wish to thank Márton Balázs and Bálint Tóth for a number of very helpful discussions, as well as Ginestra Bianconi at QMUL, Jürgen Jost at MPI Leipzig, and the School of Mathematics at the University of Bristol, who provided generous hosting for APKG while carrying out various parts of this research. This work was supported by the EPSRC project “Spatially Embedded Networks” (grant EP/N002458/1). APKG was partly supported by the EPSRC project “Random Walks on Random Geometric Networks” (grant EP/N508767/1). All underlying data are reproduced in full within the paper.