Connectivity of Soft Random Geometric Graphs over Annuli

Nodes are randomly distributed within an annulus (and then a shell) to form a point pattern of communication terminals which are linked stochastically according to the Rayleigh fading of radio-frequency data signals. We then present analytic formulas for the connection probability of these spatially embedded graphs, describing the connectivity behaviour as a dense-network limit is approached. This extends recent work modelling ad hoc networks in non-convex domains.


I: Introduction
Soft random geometric graphs [1] are network structures [2] consisting of a set of nodes placed according to a point process in some domain V ⊆ R d mutually coupled with a probability dependent on their Euclidean separation.Examples of their current application include modelling the collective behavior of multi-robot swarms [3], disease surveillance [4], catastrophic wildfire prevention [5] and electrical smart grid engineering [6], alongside our focus, communication theory [7], where these graphs have recently been used to model ad hoc wireless networks [8][9][10][11][12][13] sharing information over stochastic channels [9,14,15].
Though studying the behaviour of randomly generated networks goes back to Erdős and Rényi [16] in the 1960's, the discussion of random graphs embedded in a metric space [17,18] is due to Gilbert [19] during his time at Bell Labs.While he was there he considered 'random plane networks' as a model for a set of parties trying to communicate, consisting of nodes placed in the infinite plane according to a Poisson point process of density λ, linked if their Euclidean separation d < ν, ν ∈ R + : the nodes corresponded to point-like 'stations' of transmission range ν, while the links corresponded to two-way communication channels for the exchange of electromagnetic data signals.Today these conceptual networks are called 'unit disk graphs' [20], studied in contexts outside communication theory (for example ecology [21]) with numerous published results, including those by Penrose, Gupta, Kumar, Mao and Anderson [17,[22][23][24][25][26][27][28][29].
The introduction of probabilistic or 'soft' connection [1,8,14] by a variety of authors (again motivated by problems in communication theory [8,10]) results in an entirely different behaviour in the graph ensemble: results are not as abundant, since percolation theory [30] has dealt from the beginning with 'hard' connection, but recently a number of papers have appeared providing results and insights into network connectivity within bounded domains where the data channels are considered to stochastically corrupt signals such that no connection is ever guaranteed (or forbidden) at any transmission range.Soft connection, where two nodes i and j (forming one of the N (N − 1)/2 node pairs in the random point pattern) connect with a probability H(r ij ) when separated by a Euclidean distance r ij , models this type of more realistic environment with enough flexibility to incorporate any kind of signal fading [15], an essential feature, since the modern band-limited world of wireless communication continues to be squeezed ever further for greater efficiency and performance towards the upcoming 5G standard.
Connectivity [8,14,[31][32][33] is the native observable displayed by all graphs, and its study is a natural part of the mathematics of the ensemble.For example in [14] (using a cluster expansion technique from statistical physics), at high node density ρ the connection probability of a soft random geometric graph formed within V ⊆ R d of volume V is approximated as the complement of the probability that, after link formation, only one isolated node appears beside an otherwise connected graph.Justified by a conjecture of Penrose [1] (asserting that within a soft random geometric graph G, the number of isolated nodes N 0 (G) follows a Poisson distribution of intensity ρ V dr i exp −ρ V dr j H(r ij ) , which quickly decays as ρ → ∞), the crucial impact of boundaries [8,14,34] takes centre stage, since node isolation is more likely near these boundaries where the solid angle available to nodes is restricted.
The impact of obstacles placed within V is however not understood.Obviously, connectivity will be hampered, but analytically quantifying the extent to which this occurs has not yet been undertaken.Constituting an extension of recent work on connectivity within non-convex domains [31,32,[35][36][37] (such as over regions containing internal walls [31] or a complex, fractal boundary [36]), in this paper we therefore derive analytic formulas for the connection probability P f c of a soft random geometric graph as a function of node density ρ within the annulus (or 'obstructed disk', see Fig. 1), quantifying how circular obstacles of radii r affect connectivity.Specifically, we consider the situation where the node-to-node connection probability H(r ij ) decays exponentially with Euclidean node separation r ij (known as 'Rayleigh fading'), allowing estimation of the necessary node-to-node transmission range parameter r 0 (ρ) required to ensure connectivity above a specific threshold probability (in a similar manner to Gupta and Kumar's results [28]).
This paper is structured as follows: In section two we describe our model and use it to derive the connection probability in a simple disk domain D of radius R. In section three we evaluate P f c for the annulus domain A, incorporating both small and large circular obstacles into D, continuing towards the 3-dimensional analogue of the annulus known as the spherical shell S in section four.We conclude in section five, where we discuss our results and some open problems pertaining to obstructed random graphs.

II: Our Random Graph Model and the Disk Domain D
Consider N nodes placed uniformly at random inside a bounded subset V ⊆ R d of volume V at density ρ = N/V (using the Lebesgue measure) and at positions r i , i ∈ {1 . . .N }.We define the characteristic function χ ij equal to unity whenever the two nodes i and j are mutually visible (such that the straight line joining them contains no point lying outside V), and as zero otherwise.i and j (at r i and r j ) possess Euclidean separation r ij and are connected (through a 'link') with probability χ ij H(r ij ).
Generate a connected cluster of N − 1 nodes at r 1 . . .r N −1 ∈ V.The probability that a node then placed at r N remains isolated is given by After taking the expectation P 0 over all possible N − 1 cluster configurations and all possible r N ∈ V [14], we approximate the full connection probability P f c as the complement of the probability that there exists exactly one completely isolated node in an otherwise connected graph [1, 14] This first order approximation to P f c was first given by Mao and Anderson in [8] and later in [14] (which provides a more complex second order approximation based on the appearance of two isolated nodes or a small two-node cluster).
We also write where V(r j ) ⊆ V represents the portion of the domain visible to some node at r j .This model can incorporate any choice of H (and so any type of connection environment), but our particular choice comes from a Rayleigh fading link model of free-space signal propagation in the Earth's atmosphere, within which the information outage probability P out is given, via the Shannon-Hartley theorem, by where x represents the minimum outage rate threshold, h the modulus of the channel transfer coefficient (a characteristic of the channel through which transmission occurs) and SNR the signal-to-noise ratio (the power ratio between the received signal and the long term noise average).Since we consider stochastic channels, we take h 2 to be a random variable drawn from the exponential distribution, and since the received signal intensity is inversely proportional to the square of the node-node separation SNR = c/r 2 ij the probability of link success is given by where β = (2 x − 1)/c.In cluttered environments (e.g.dense urban areas) the exponent of r ij in (6) (called the path loss exponent) is experimentally found to be greater than 2, however for mathematical tractability we will consider only free space propagation where the path loss exponent is exactly equal to two.Finally note that we define the constant r 0 = 1/ √ β as the 'typical' range over which nodes are likely to connect; should r ij > r 0 we consider connection to be non-typical (though obviously still possible).

II.1 -Connectivity in D
We first take V to be the disk D of radius R [14].We begin the analysis by introducing the function M (r j ), which is related to the probability I N −1 (r j ) that a node at r j ∈ D will fail to bind to a cluster of N − 1 nodes configured randomly around it If we partition D (left panel, Fig. 2) such that dy dx (7) and then take H(r ij ) from Eq. 5, we get since the integral over D 1 cancels.Note that we make the assumption R √ β 1 since we consider the typical connection range r 0 = 1/ √ β to be significantly smaller than the disk radius R, which is realistic for large scale dense networks.Now consider two regimes for the distance : in the first, where 1, we can make the approximation e −βy 2 ≈ 1, since the distances y from the horizontal to the lower semi-circle will be small, giving where we use after Taylor expanding Eq. 10 for 1 (distinguishing periphery locations).For the other regime (where 1) we can use M( β due to the exponential decay of the connectivity function, allowing us to evaluate Eq. 2. Thus, using = R − and setting L + as the solution of M boundary ( ) = π β such that we integrate in a piecewise fashion over all ∈ D): we have have the full connection probability in D. This approaches equation Eq. 38 of reference [14] as R √ β → ∞, where the second term in the exponent of the final term in Eq. 12 is a 'curvature correction' to the disk result in [14].
Monte-Carlo simulations (where graphs are picked form the ensemble and enumerated should they connect), presented in Fig. 3 alongisde our approximation in Eq. 12, show a significant improvement on the disk result in [14].The discrepancy at low density is expected, since we base our calculations on the assumption that nothing other than a single isolated node causes disconnection or 'outage' (due to the Poisson-like decay of the k−isolated node probabilities, recently proven by Penrose [1]).

III: The Annulus Domain A
The domain under consideration is now the annulus A of inner radius r and outer radius R, depicted in the left panel of Fig. 1.
where υ(θ) = (r + ) cos(θ) − r 2 − (r + ) 2 sin 2 (θ) represents the polar equation of the lower semicircle from the middle panel of Fig. 2 This remains valid as long as is small enough for nodes not to 'notice' the outer boundary, in the sense that the connectivity mass more than one typical connection range r 0 = 1/ √ β from the periphery is approximately independent of (or doesn't notice) its geometry.
Evaluating eq. 13 gives To the author's knowledge, this integral has no standard form in terms of elementary or special functions, apart from (where F 1 is the F 1 Appell hypergeometric series), which appears after Taylor expanding the integrand and using the binomial theorem to achieve two integrals in terms of sums of powers of the cosine function (which then integrate into the hypergeometric functions in Eq. 15).In order to simplify the problem we approximate within two obstacle-size regimes, the first where r r 0 , and the second where r r 0 (where r 0 = 1/ √ β, the typical distance over which nodes connect).In each regime we can make some vital assumptions about the shape of the region visible to a node, hopefully yielding tractable formulas for the connectivity mass in terms of powers of .We will start by considering the small-obstacle regime r r 0 .

III.1 -Small Obstacles
The first approximation we are able to make is that the small, purple, cone-like domain in the middle panel of Fig. 2 (called V c ) is only significantly contributing to the connectivity mass at small obstacle displacements , since at larger displacements it thins and the wedge-like region composing the rest of A(r j ) dominates.Practically, it is V c that presents the main integration difficulties, so we approximate for the integral over this region where the radial coordinate r 1, using e −βr 2 ≈ 1, leading to and so For small we now have leaving us to integrate over the annulus where we take using our result from the unobstructed disk.We approximate expanding the lower orders of the exponential.After taking L − as the solution to M O ( ) = π β , we obtain after losing the upper limit term, which falls into the bulk term of ζ (ρ) but does not appear explicitly since we evaluate each integral to leading order.We now finally have the full connection probability Clearly, the obstacle term is valid for all domain geometries that contain circular obstacles such as the Sinai-like domain in Fig. 1 (right panel).We numerically justify Eq. 23 in Fig. 4. using Monte Carlo simulations in a similar manner to Fig. 3, achieving good agreement within the high node density regime.

III.2 -Large Obstacles
When the regime r r 0 is considered, we can no longer consider the purple cone-like region in Fig. 2 small when the node r j of A(r j ) is close to the obstacle.We can, however, replace this approximation with a 'small obstacle-curvature' approximation, expanding υ(θ) in Cartesian coordinates therefore simplifying the semicircle equation (hence the curved boundary of A(r j ) in the right panel of Fig. 2), but keeping the same geometry within the typical connection range r 0 .This allows us to approximate A(r j ) such that for the large obstacle case where after expanding the error function at r = ∞, which should be valid for large obstructions.Expanding this in powers of after performing the integral gives which is M O ( 1).This implies the connectivity mass is scaling in the same way as for the outer boundary, but where the curvature correction is of opposite sign.Since we already have the other mass approximations, we have the full connection probability which is successfully numerically justified in Fig. 5 using Monte Carlo simulations.

IV -The Spherical Shell Domain S
Consider now the spherical shell domain S of inner radius r and outer radius R, which is the three-dimensional analogue of the annulus.
Thus consider the three-dimensional version of V c , called S c , and again make the approximation e −βr 2 ≈ 1, such that the mass over this region M S C ( ) is given by Break S c up into the area of the cone of radius λ, height h and apex angle 2θ (with the apex at a distance from the obstacle), and the spherical segment, which, on removal from the cone, creates the necessary shape Adding the mass over S(r j ) \ S c M S(rj )\Sc ( ) where the full solid angle available to a 3D-bulk node is 4π and the angle ω ≤ Ω available to the node at r j is such that we have M S O (the connectivity mass near the spherical obstacle) becomes which is M S O ( 1).This implies that small spherical obstacles reduce the connection probability within the unobstructed sphere domain S by Extending Eq. 25 into the third dimension, M S O ( ) becomes where ν (x, z) = + 1 2r x 2 + z 2 , yielding (which is M S O for the large spherical obstacle case), implying a reduction in the connection probability of We now have the connection probability in a spherical shell of inner radius r r 0 and inner radius r r 0 We numerically justify Eqs.43 and 44 in Fig. 6.

V: Discussion and Conclusions
We have derived semi-rigorous analytic formulas for the full connection probability of soft random geometric graphs drawn inside various annuli and shells (of inner radius r and outer radius R) where the link formation probability between two nodes displaced by r ij is H (r ij ) = e −βr 2 ij , based on a Rayleigh fading model of radio signal propagation within a wireless network (where nodes typically, though not necessarily, connect over a distance r 0 = 1/ √ β).Due to the probabilistic connection, at no transmission range can connectivity be guaranteed for finite N (in a similar manner to Gupta and Kumar's formulas [27]), but with these new results we can tailor the transmission range such that connectivity will be achieved with a specific, approximate probability.
The main contribution of this work is to extend the soft connection model into simple non-convex spaces based on circular or spherical obstacles (rather than fractal boundaries [36], internal walls [31] or fixed obstacles on a grid [35]).For example, the obstacle terms derived can be easily applied to domains containing potentially multiple randomly placed small and large circular obstructions (given the obstacles are separated from each other and the boundary by at least 2r 0 at their closest point, see Fig. 1) (based on equations 23 and 28) where N s and N l are the number of small and large obstacles present in the domain respectively, and r s,l i are the various radii of the small (s) and large (l) obstacles.This works because the effect of each obstacle combines in a linear fashion to reduce the connection probability by the sum of the respective obstacle terms.Moreover, these obstacle-effect approximations work for any outer boundary geometry, and have been numerically verified with Monte Carlo simulations (Fig. 3, 4, 5 and 6) as such.Note also that Eq. 45 is transcendental in r 0 , so we cannot solve to find the typical connection range as function of the disconnection probability threshold; a numerical technique is required.
Finally, it is worth briefly discussing possible attempts to mitigate the connectivity decline caused by geometric obstructions to data transmission channels.One technique is for nodes to somehow detect the location [38][39][40] of the obstacles, and then increase their transmit power accordingly to accommodate the increased likelihood of becoming isolated.This introduces transmit powers that vary over the domain, so the use of a simple distance based link formation function (here given by H(r ij )) is no longer appropriate; researching techniques to improve connectivity in this manner is therefore a topic of further study.If possible, one could design the domain such that the first two sums in Eq. 45 are at a minimum given a set of constraints, such as a minimum introduced perimeter or number of obstacles N s,l (perhaps using the method of Lagrange multipliers), maximising the connection probability in an obstructed domain and mitigating the non-convexity specific effects.
We have, therefore, made further progress towards modelling wireless ad hoc networks in the more general 'nonconvex' spaces.Continued study in this area may unveil deeper, richer and likely peculiar behaviour of random graphs living in these more complicated domains, which is certainly an essential task should we wish to extend the applicability of these beautiful graph formations as network science continues to flourish.

FIG. 1 :
FIG. 1: (Colour online) Randomly constructed soft geometric graphs (picked from the ensemble, with links formed based on the Rayleigh fading of electromagnetic data signals and nodes configured randomly) drawn inside A (left) and a square domain (right) containing two circular obstacles.Nodes with only a few connections (usually near the domain edges) are highlighted in purple, demonstrating the boundary effect phenomenon.

FIG. 2 :
FIG.2:(Colour online) A depiction of the integration regions used for the disk domain D (left panel) and annulus domain A with small obstruction (middle panel) and large obstruction (right panel, with the region visible to the node at r j approximated by the second order polynomial ν(x) displayed).The small, cone-like region in the middle domain (containing the small obstacle) is highlighted in purple.

FIG. 3 :
FIG. 3: (Colour online) Left: A comparison of Monte-Carlo simulations (lighter, grey line, 1, 000, 000 ensemble picks) with our analytic approximation (equation 13, the blue, darker line) for the disk domain D taking R = 9, β = 1.Right: 1-P f c (blue, darker line) for the same domain compared with Monte-Carlo simulations (lighter, grey line), plotted using a logarithmic scale.The square dots in the right hand plot represent the old result in [14].