Network Connectivity: Stochastic vs. Deterministic Wireless Channels

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by  Orestis Georgiou

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Network Connectivity: Stochastic vs. Deterministic
Wireless Channels
Orestis Georgiou1,2, Carl P. Dettmann2, and Justin P. Coon3
1Toshiba Telecommunications Research Laboratory, 32 Queens Square, Bristol, BS1 4ND, UK
2School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, UK
3Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, UK
Abstract—We study the effect of stochastic wireless channel
4 models on the connectivity of ad hoc networks. Unlike in the
1 deterministic geometric disk model where nodes connect if they
0 are within a certain distance from each other, stochastic models
2 attempt to capture small-scale fading effects due to shadowing
and multipath received signals. Through analysis of local and
n global network observables, we present conclusive evidence sug-
a gestingthatnetworkbehaviourishighlydependentuponwhether
J
a stochastic or deterministic connection model is employed.

8 Specifically we show that the network mean degree is lower
2 (higher) for stochastic wireless channels than for deterministic
ones,ifthepathlossexponentisgreater(lesser)thanthespatial

] dimension. Similarly, the probability of forming isolated pairs
I
N of nodes in an otherwise dense random network is much less
for stochastic wireless channels than for deterministic ones. The
.
s latterrealisationexplainswhytheupperboundofk-connectivity
c istighterforstochasticwirelesschannels.Weobtainclosedform
[ analyticresultsandcomparetoextensivenumericalsimulations.

1
Index Terms—Connectivity, outage, channel randomness,
v
stochastic geometry.
8
8
1
I. INTRODUCTION
7
Fig.1. Randomrealizationsofadhocnetworksinasquaredomainofside
.
1 Recent advancements in micro and nano-scale electronics L=10,usingβ=1andη=2,4,6,∞,asdefinedinthepairconnectedness
0 alongwiththedevelopmentofefficientroutingprotocolshave functionH(r).Differentconnectedcomponentsareshownindifferentcolors.
4
rendered current wireless technologies ideal for ad hoc and
1
sensingapplications[1].Makinguseoflowcomplexitymulti-
:
v hop relaying techniques and signal processing capabilities, canalsoimprovetheoverallfunctionality,reliabilityandfault-
i
X sensor networks can often achieve very good coverage and tolerance of the network.
connectivity over large areas, “on the fly” in a decentralized The analysis and resolution of network connectivity is
r
a anddistributedmannerbyself-organisingintoameshnetwork, typically addressed from a physical layer’s point of view
assignedwithsomedatacollectionanddisseminationtask[2]. through the theory of stochastic geometry [5], random geo-
Thespontaneousself-organizationtraitoflargescalesensor metric graphs [6] and complex networks [7], equipped with a
networkshasattractedmuchresearchattentioninrecentyears, plethoraofmethodsandmetricse.g.clusteringandmodularity
withparticularinterestinenhancingtheconnectivityproperties statistics, node importance, correlations between degrees of
of the underlying communication graph [3]. Of practical neighbouring nodes, etcetera. From a communications per-
interestforexampleistheabilitytopredicttheoptimalnumber spective, a popular and well studied observable of random
of nodes necessary to maintain good connectivity [4], or networks is that of full connectivity [8]. This characterizes the
conversely, to predict the optimal average transmission range probability P with which a random realization of an ad hoc
fc
for a given number of nodes. These predictions are essential networkwillconsistofasingleconnectedcomponent(cluster).
network design recommendations which can in turn act as Consequently, every node in the network can communicate
inputsforcognitiveschedulingandroutingprotocolselection. with any other node in a multi-hop fashion.
Furthermore, improvement in the connectivity properties of In the theory of random geometric networks, two nodes
the network can have a significant impact on the operational are said to connect and form a pair if they are within a
lifetime of ad hoc sensor networks by conserving energy, and certain distance r from each other. This is referred to as the
0
1
geometric disk model and is only sufficient when modelling V ⊂ Rd with volume V. The node position coordinates
deterministic, distance-dependent wireless channels. One ex- are given by r ∈ V for i = 1...N. We say that a
i
tensionofthisconnectivitymodel,istoaccountforthechannel communication link between a pair of nodes i and j exists
randomness due to shadow fading and multipath effects in with probability H(r ), where r = |r −r | is the relative
ij ij i j
the spatial domain by ‘softening’ the position dependence of distance between the pair. One physical interpretation of
thepairconnectednessfunctionsuchthatlinksareestablished a communication link is given by the complement of the
in probability space [9]. In static wireless networks, this information outage probability between two nodes for a given
softening can be understood as modelling the randomness in rate x in bits per complex dimension, which can be written as
the received signal power [10]. Thus, in the absence of inter- Pr(cid:0)log (1+SNR×|h|2)>x(cid:1) [11], where h is the channel
2
node interference, two nodes connect with a probability H(r) transfer coefficient, and SNR ∝ r−η is the average received
ij
whichdecaysasafunctionofthedistancer betweenthepair. signaltonoiseratioandη isthepathlossexponent.Typically
Whatthiseffectivelymeansisthatnodeswhicharecloserthan η = 2 corresponds to propagation in free space but for
a distance r from each other may no longer be connected, cluttered environments it is observed to be η ≥2.
0
and nodes that are more than r apart may be connected. We consider the case where the individual channel fading
0
The overall effect of channel randomness is typically en- distributionsfollowaRayleighfadingmodel,andallchannels
codedintothepathlossexponentη(definedlater)[11]andhas arestatisticallyindependent.Itfollowsthat|h|2 hasastandard
beenarguedtohaveanegativeeffectonlocalconnectivity,but exponentialdistributionforasingle-inputsingle-output(SISO)
a positive effect on overall connectivity [12]. The latter con- antenna system. Hence, the connection probability H(r) be-
clusionhasbeenreachedthroughacombinationoftheoretical tween two nodes a distance r apart can be expressed as
andnumericalresults,theformermainlyrelyingonthebound
H(r)=e−βrη, (1)
P ≤ P , where P characterizes the probability with
fc md md
whicharandomrealizationofanadhocnetworkhasminimum where β sets the characteristic connection length r =β−1/η.
0
network degree equal to one i.e. each node is connected to at
Therefore, our system model has two sources of randomness:
least one other node. The bound was proven to be tight by
random node positions, and random link formation according
Penrose in 1999 for the deterministic geometric disk model
to the ‘softness’ of the channel fading model controlled here
in the limit of number of nodes N → ∞ [13]. Since then, a
by η. It is important to note that in the limit of η → ∞, the
number of papers have followed suit extending these results
connection between nodes is no longer probabilistic and con-
in many directions to include for, inter alia, boundary effects,
vergestothegeometricdiskmodel,withanon/off connection
channel randomness, and anisotropic radiation patterns.
rangeatthelimitingr .Wewilllatermakeuseofthislimitin
0
In this paper, we challenge the negative effect of channel
order to compare the connectivity of random networks using
randomness on local connectivity and also investigate the
deterministic or stochastic point-to-point link models.
tightness of the bound Pfc ≤ Pmd for finite yet sufficiently Fig. 1 shows how N = 150 nodes scattered randomly in
large N and examine the role of channel randomness to this a square domain of side L = 10, connect to form different
respect. We show analytically that Rayleigh fading improves networksfordifferentvaluesofη =2,4,6,and∞,usingβ =
local connectivity when η is less than the effective spatial 1. The corresponding H(r) functions are also plotted below
dimension d of the network. We also show that the pair
eachpanelinordertogivethereaderafeelingofhowprobable
isolationprobability Π is whatdistinguishes Pmd from Pfc at shorterandlongerthanr linksareinthepresenceofRayleigh
0
high node densities,and calculate closed formexpressions for
fading. We stress that in practice, an efficient medium access
itassumingaRayleighfadingchannel.Bothofourresultsare
control (MAC) layer protocol is typically required e.g. using
validated through extensive numerical simulations, the latter
aTimeDivisionMultipleAccess(TDMA)schemeinorderto
suggesting that two nodes are more likely to form an isolated
render inter-node interference negligible. Alternatively, a low
pair when η is large i.e. in heavily cluttered environments.
traffic network may be assumed so that the communication
Finally, we discuss the engineering insight provided by our
network can be modelled as seen in Fig. 1.
analysis towards facilitating the design of wireless ad hoc
sensor and mesh networks [14]. III. LOCALCONNECTIVITYANDMEANDEGREE
The paper is structured as follows: Sec. II describes the The network mean degree µ is a local observable of
system model and relevant assumptions. Secs. III and IV network connectivity characterising the average number of
discuss local and global network observables respectively and one-hop neighbours of a typical node. For random geometric
their sensitivity to stochastic/deterministic wireless channels. networks as described in Sec. II, this can be expressed as
Sec. V investigates analytically the pair isolation probability, µ = N−1(cid:82) H(r )dr dr which follows from multiplying
V2 V2 ij i j
and Sec. VI numerically confirms our theoretical predictions N−1bytheprobabilityoftworandomlyselectednodes(iand
which are then summarised and discussed in Sec. VII. j)connectingtoformapair.AssumingthatV islargeandthe
typical length scale of the domain is greater than the effective
II. SYSTEMMODEL connection range r , a typical node is most likely to be found
0
We consider a network of N nodes distributed randomly away from the borders of the domain. Moreover, since H(r)
and uniformly in a d = 2,3, dimensional convex domain is decaying exponentially, it is reasonable to expect that the
2
a wireless link to at least one access point, in decentralized
ad hoc networks, efficient routing protocols can utilize mu-
tually independent paths to communicate information through
the network e.g. for sensing, monitoring, alerting or storage
purposes [1]. Therefore, if a multihop path exists between all
pairs of nodes, then the network is fully connected and in a
sense is both delay and disruption tolerant [15].
A generalization of the concept of full connectivity is
that of k−connectivity [16]. A fully connected network is
Fig.2. Plotofthemeandegreeµasafunctionofη,obtainedfromnumerical said to be k-connected if the removal of any k − 1 nodes
simulationsofadhocnetworksinasquareofsideL=10(blue),andacube leavestheremainingnetworkfullyconnected.Theremovalof
ofsideL=7(purple).Inbothcasesβ=ρ=1.Theanalyticalpredictions
nodes may model technical failures (e.g. a hardware/software
of(2)areshownassolidcurvesandthelimitµ∞ asdashedlines.
malfunction)orattackswhichcandisruptthefunctionalityand
operation of the network, in some cases leading to cascades
of catastrophic failures [17]. Equivalently, k-connectivity also
degree of a typical node is very much insensitive to boundary
guarantees that for each pair of nodes there exist at least k
effects, thus justifying the following approximation
mutually independent paths connecting them [6]. Therefore,
(cid:90) (cid:90) ∞ ΩΓ(d) k-connectivityisanimportantmeasureofnetworkrobustness,
µ ≈ρ H(r)dr=ρΩ rd−1e−βrηdr =ρ η (2)
η d resilience but also of routing diversity.
Rd 0 ηβη
It is clear that a k-connected network has minimum degree
d
where ρ≈N−1 and Ω= 2π2 is the solid angle in d dimen- k, i.e. each node has at least k neighbouring nodes. The
V Γ(d)
sions. In (2) we have effectiv2ely assumed that the network is opposite statement is not true however and hence the former
homogeneous i.e. it is translation and rotation invariant, such set is a subset of the latter and so Pfc(k) ≤ Pmd(k), where
that we can set ri = 0 and extend the radial integral of rj Pfc(k) denotes the probability that a random realization of
to infinity taking on exponentially small errors. Substituting an ad hoc network is k-connected, and Pmd(k) denotes the
β =r−η andtakingthelimitη →∞weobtainanexpression probability that it has minimum degree k. These two ob-
0
for the mean degree for the deterministic geometric disk servables are however strongly related through a fundamental
model µ∞ = ρΩrd0d. Comparing this to the result of (2), we cthoantce“piftNorigisinbalilgyepnroouvgenh,inth[e1n3]w(iTthhehoigrehmpr1o.1b)abwilhitiyc,hisftaotnees
conclude that µ = µ when η = d and that µ < µ
∞ η η ∞
starts with isolated points and then adds edges connecting
for η > d. In other words, the stochastic and deterministic
the points in order of increasing separation length, then the
connectivity models are equivalent at η = d when viewed
resulting graph becomes k-connected at the instant when it
locally. More importantly however, Rayleigh fading reduces
achieves a minimum degree of k”. Ever since this realization,
the local network connectivity when η > d, but improves it
P (k)hasbeenapproximatedbyP (k)[18]asitiseasierto
whenη <d.Thelatterconditiondescribesaveryspecialcase fc md
expressmathematically,andevaluatenumerically.Specifically,
when the network resides in an effectively 2D plane, but may
we have that [16]
well be significant in 3D networks deployed in multi-storey
buildings for example where the path loss exponent may be N
(cid:89)
in the range 2 ≤ η < 3. Consequently, one may significantly Pmd(k)=(cid:104) P(degree(ri)≥k)(cid:105)
overorunderspecifynetworkdesignfeaturesanddeployment i=1
(3)
methods if the network is not correctly modelled. (cid:34) k(cid:88)−1ρm 1 (cid:90) (cid:35)N
Numerical verification of the above result is presented in ≈ 1− m!V MHm(ri)e−ρMH(ri)dri ,
Fig. 2 showing computer Monte Carlo simulations of ad m=0 V
(cid:82)
hoc networks in two and three dimensions. The analytical where M (r )= H(r )dr and V is assumed to be much
H i V ij j
predicationof(2)andthelimitcaseofµ∞ arealsoshownfor larger than πr0d. The angled brackets in (3) represent a spatial
comparison.Analmostperfectagreementisobservedbetween average of a network observable O over all possible node
theory and simulations with the theoretical result typically configurations and is defined as
being smaller than the numerical one. This is because µη 1 (cid:90)
wasapproximatedassumingnoboundaryeffects,whichhinder (cid:104)O(cid:105)= VN O(r1,r2,...,rN)dr1dr2...drN. (4)
connectivity for nodes near the borders of V. The minimum VN
We will use this notation in order to calculate expectation
value of µ is obtained numerically at η ≈4.33 and η ≈6.50
η
values of different random network observables.
in two and three dimensional networks respectively. We now
Fig. 3 shows Monte Carlo computer simulations for η =2
turn to investigate global network observables.
(left panels) and η =∞ (right panels). 105 random networks
IV. k-CONNECTIVITYANDMINIMUMDEGREE weresimulatedinasquaredomainofsideL=10forarange
In the absence of a fixed infrastructure (e.g. cellular, or of node density values ρ ∈ (1,8) using β = 1, in order to
WLANs), where it is sufficient that each network node has obtain curves for P (k) and P (k) for k = 1,2,3, and 4.
fc md
3
Using the spatial average defined in (4), and letting H =
ij
H(r ) in order to save space, we write
ij
(cid:88) (cid:89)
Π(1)=(cid:104) H (1−H )(1−H )(cid:105)
ij ik jk
i<j k(cid:54)=j(cid:54)=i
N(N −1) (cid:89)
= (cid:104)H (1−H )(1−H )(cid:105)
2 ij ik jk
k(cid:54)=j(cid:54)=i
ρ2(cid:90) (cid:20)1(cid:90) (cid:21)N−2
≈ H (1−H )(1−H )dr dr dr
2 ij V ik jk k i j
V2 V
≈ ρ22 (cid:90) Hije−ρ(cid:82)VHik+Hjk−HikHjkdrkdridrj,
V2
(6)
Fig.3. ComputersimulationofPmd(k)(filledmarkers)andPfc(k)(hollow where we have assumed that N (cid:29)1 such that (N −1)/V ≈
markers)fork∈[1,4]usingβ=1andη=2(left)andη=∞(right)in (N−2)/V ≈ρ,and(1−x)N ≈e−Nx.Inthethirdlineof(6),
a square-shaped domain of side length L = 10. Bottom: 1−Pmd(k) and weusedthefactthatN−2oftheN integralsdefinedin(4)are
1−Pfc(k)onalog-linearscale.
separable since the possible connections of the pair (i,j) to
theremainingN−2nodesarestatisticallyindependentevents.
Whilst equation (6) appears to be very complicated, there
Indeed, it can be seen from Fig. 3 that the two observables
are several important observations to be highlighted which
areingoodagreementwitheachother,withP (k)following
fc
can simplify it. Firstly, in the high density limit we expect
P (k) closely from below. What is also evident however,
md
equation(6)tobedominatedbycontributionswheretheinner
is that the gap between the two ∆(k) = P (k)−P (k)
md fc
integral in the exponent is small. The integral in the exponent
is significantly larger and persistent even at high densities
represents the probability that a randomly selected node k
for deterministic (η = ∞) rather than probabilistic (η < ∞)
connectswithnodeiornodej orboth,aneventleastprobable
wirelesschannels.Thevisibledifferencecanbeseentopersist
if both i and j are near the boundary of the domain which
even when plotted on a log-linear scale (bottom panels).
physically corresponds to the most hard to connect to region
This observation has been investigated numerically in several
of V. Secondly, the dominant contribution of (6) also requires
articles [9], [10], [12], [18] but has not been fully understood.
H ≈1, and thus nodes i and j must be close to each other
What is the impact of channel randomness? Is fading good or ij
in order to form the isolated pair. Both these conditions are
bad? These are but a few related questions we aim to revisit
met at a corner of V.
in the next section by attempting to understand where ∆(k)
We therefore consider each of the four right angled corners
stems from and the significance of the path loss exponent η.
of a square1 domain V = [0,L]2 independently and attempt
V. THEORETICALANALYSIS to calculate Π(1) from (6). We assume that nodes i and j are
It was recently shown in [8], that Pfc(1) at high node both near this corner and hence Taylor expand Hik and Hjk
densities is given by the complement of the probability of in the integrand of in the exponential using polar coordinates
an isolated node up to linear order in ri and rj respectively to obtain
Pfc(1)=1−ρ(cid:90) e−ρ(cid:82)VH(rij)drjdri+..., (5) Kˆ(ri,rj)=(cid:90) Hik+Hjk−HikHjkdrk
V V
where the (...) indicate higher order terms, possibly ex- ≈2−η+η2β−η2(cid:104)π(2η+η2 −1)Γ(cid:18)η+2(cid:19)
ponentially smaller than the leading order term. Physically, 2 η
(7)
(5) makes sense since a dense network which is not fully (cid:18)η+1(cid:19)
1 η+1
connected will most probably involve a single isolated node. +(2β)η(2 η −1)Γ
η
It is thus reasonable to extend this argument and expect (cid:105)
that at high node densities a network which has minimum ×(ri(cosθi+sinθi)+rj(cosθj +sinθj)) ,
degree 1 but is not fully connected will most likely consist
where we have ignored correction terms of order O(r r ).
of a large N −2 cluster and an isolated pair of nodes. We i j
The integration limits in (7) were taken to be r ∈ [0,∞)
therefore define Π(1) as the probability that two randomly k
and θ ∈ [0,π/2]. The semi-infinite integration is allowed
selected nodes are connected to each other, but are isolated k
here since H(r) decays exponentially fast with r and so if
from the remaining nodes of the network, and argue that
βL(cid:29)1 any added errors in (7) will be exponentially small.
∆(1) = P (1) − P (1) ≈ Π(1) at high node densities.
md fc Substituting back into (6) and assuming that the two nodes
In order to confirm our hypothesis, we will analytically
aresufficientlyclosesuchthatH ≈1,wemaynowperform
calculatethedominantcontributiontoΠ(1)andthencompare ij
against numerical simulations of ∆(1) for different values of
1Note that the current analysis is not restricted to the simple case of a
η obtained. squaredomain,northeSISOpoint-to-pointRayleighfadingmodel(1).
4
with diffuse power (TWDP) model [19]. We postpone further
discussiononthephysicalinterpretationofthisobservationto
Sec. VII and turn to numerical simulations in order to verify
that ∆(1)≈Π(1) at high node densities.
VI. NUMERICALSIMULATIONS
In this section we will numerically investigate ∆(1) =
P (1)−P (1) obtained from computer Monte Carlo sim-
md fc
ulations of ad hoc networks (as in Fig. 3) for different values
of η = 2,4,6,∞, in a square domain of side L = 10. Our
Fig.4. Computersimulations(filledmarkers)inasquaredomainshowing
∆(1)=Pmd(1)−Pfc(1)asafunctionofthedensityρfordifferentvalues results are plotted in Fig. 4 and are contrasted against the
ofη=2,4,6,∞andβ =1.TheanalyticalapproximationofΠ(1)isalso theoretical prediction of equations (8) and (9). We observe
plotted(solidcurves).
that ∆(1) is a unimodal function of ρ, which peaks around
ρ ∈ (2,3) with a value of approximately 0.1. Therefore, at
such densities, approximately 1 out of 10 random realisations
the remaining integrals to arrive at our main result of ad hoc networks have minimum degree one, but are not
2η4βη4 exp(cid:104)−ρπ(1−2−η+η2)β−η2Γ(cid:16)η+2(cid:17)(cid:105) fullyconnected.Atlowerdensitiesρ<2,thedifference∆(1)
2 η is small because the set space of possible graphs is small,
Π(1)≈32 , (8)
ρ2(cid:16)(2−η+η1 −1)Γ(cid:16)η+1(cid:17)(cid:17)4 essentially enforcinga large overlap betweenfull connectivity
η and minimum degree. At higher node densities, where the
where we have also multiplied the result by 4 due to the four set space grows exponentially, the difference between the two
identicalrightanglecornersofV.Theresultingapproximation observablesgrowstoamaximum,onlytodecayexponentially
of Π(1) is a monotonically decreasing (increasing) function as the two distributions converge to 1. Fig. 4 confirms with
with respect to the density ρ (path loss exponent η), which a surprisingly good accuracy that ∆(1) ≈ Π(1) at high node
diverges like ρ−2 near the origin (see Fig. 4). Therefore, in densities,andthereforeanetworkwhichhasminimumdegree
order for (8) to be interpreted as a probability, one needs to 1 but is not fully connected will most likely consist of a large
carefully investigate regions of convergence and bound any N −2 cluster and an isolated pair of nodes. Moreover, the
approximating errors. We will refrain from doing so for the probabilitythattwonearbynodesformanisolatedpairismuch
sakeofbrevityandinsteadaimatobtainingusefulengineering less for stochastic H(r) than for deterministic ones.
insight on the impact of stochastic wireless channel models. Given the numerical verification of our hypothesis, we may
Wethereforesubstituteβ =r−η into(8)andexpandforη (cid:29)1 speculate for the more general case k ≥ 1, and conjecture
0
exp(cid:104)−ρπr02 +f(η)(cid:105) tchoantsitdheerindgomthineanmtocsotntlrikibeulytiosncetnoar∆io(kin)vcoalvninbgeade(rkive−d1b)y-
4 (9)
Π(1)≈32 +O(1/η),
r4ρ2 connected network of minimum degree k in the high density
0
limit. For example, for k =2 this would correspond to a pair
where f(η) is always negative and decays monotonically to
of nodes which connect to the main cluster via a single node.
zerolike|f(η)|∼η−1.Notethattheπr2/4termappearingin
0 Similarly, for k =3 this would correspond to a pair of nodes
the exponential indicates the available connection region for
which connect to the main cluster via 2 nodes. See Fig. 5 for
the pair to establish a link with the rest of the network. We
moreillustrativeexamplesfork =2,3,4,5.Wemaytherefore
can thus conclude that the probability that two nearby nodes
attempt to write down a general expression for Π(k)≈∆(k)
formanisolatedpairismuchlessforstochasticH(r)thanfor
deterministicones.Moreover,comparing(9)totheprobability (cid:88) k(cid:89)−1 (cid:89)
Π(k)=(cid:104) H H H (1−H )(1−H )(cid:105),
of an isolated node in a square domain given by [8] ij itn jtn im jm
exp(cid:104)−ρπ4r02(cid:105) (10) i<j tnn(cid:54)==i1,j m(cid:54)=i,j,tn (11)
4 ,
r2ρ
0 where the nodes tn with n < k act as relays or “bridging”
in the high density limit, we observe that for a stochastic nodes, which if removed, would cut off the pair (i,j).
connectivity function (i.e. η < ∞), Π(1) is exponentially We will not however investigate (11) any further as it is
smallerthan(10)whilstforadeterministicconnectivitymodel beyond the scope of this paper. Instead, we discuss briefly the
(i.e. η =∞) it differs only by an algebraic function of ρ. concept of k-edge-connectivity related to the removal/failure
We have shown that the probability that two nearby nodes of k − 1 edges (rather than nodes). Clearly, a k-connected
form an isolated pair is much less for stochastic H(r) than network is k-edge-connected and so P(n)(k) ≤ P(e)(k),
fc fc
for deterministic ones in the dense regime. Furthermore, where we have used superscript (n) and (e) to denote node-
there is good reason to expect that this observation is not connectivity and edge-connectivity respectively. This is be-
only restricted to Rayleigh fading channels but also to other cause when you remove a node from a connected network,
small-scale fading models such as for example the two-wave you always remove at least one edge. Further, it is also clear
5
subsequent theoretical analysis successfully captures the main
physical picture and therefore has significant implications on
real-world network emulation and deployment. Namely, the
factthat networkbehaviour ishighlydependent uponwhether
a stochastic or deterministic connection model is employed
must be taken into account by network design engineers,
for one can envisage the potential for significantly over or
under specifying network parameters if the incorrect model
is chosen. Such an error could lead to costly deployments,
or worse yet, networks that frequently experience faults or
failures.
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heavilyclutteredorlossyenvironments,twonearbynodeswill
be more correlated with regards to their network topology i.e.
they will typically connect to the same nodes, rather than in
less cluttered environments. While hardware behaviour may
be affected by other unconsidered factors, our model and
6