Efficient Collection of Sensor Data via a New AcceleratedRandom Walk ∗

C.M. Angelopoulos S. Nikoletseas D. Patroumpa C. Raptopoulos{aggeloko, patroump, raptopox}@ceid.upatras.gr, nikole@cti.gr

University of Patras andResearch Academic Computer Technology Institute (RACTI),

Greece

ABSTRACTMotivated by the problem of efficiently collecting data fromwireless sensor networks via a mobile sink, we present an ac-celerated random walk on Random Geometric Graphs. Ran-dom walks in wireless sensor networks can serve as fully lo-cal, lightweight strategies for sink motion that significantlyreduce energy dissipation but introduce higher latency inthe data collection process. In most cases random walksare studied on graphs like Gn,p and Grid. Instead, we herechoose the Random Geometric Graphs model (RGG), whichabstracts more accurately spatial proximity in a wireless sen-sor network. We first evaluate an adaptive walk (the Ran-dom Walk with Inertia) on the RGG model; its performanceproved to be poor and led us to define and experimentallyevaluate a novel random walk which we call γ-stretched ran-dom walk. Its basic idea is to favour visiting distant neigh-bours of the current node towards reducing node overlap andaccelerate the cover time. We also define a new performancemetric called Proximity Cover Time which, along with othermetrics such as visit overlap statistics and proximity vari-ation, we use to evaluate the performance properties andfeatures of the various walks.

Categories and Subject DescriptorsC.2.1 [Network Architecture and Design]: Distributednetworks, Network topology, Wireless communication

KeywordsWireless Sensor Networks, Random Walks, Data Collection,Sink Mobility

1. INTRODUCTIONWireless Sensor Networks are envisioned as large ad-hoc col-lections of very small autonomous devices, that can sense

∗This work was partially supported by EU/FIRE HOBNETproject - STREP ICT-257466

environmental conditions in their immediate surroundingswhile having limited processing, communication capabilitiesand energy reserves (see e.g. [3] for a comprehensive cover-age of the fundamental algorithms and protocols for wirelesssensor networks). The collected sensory data is usually dis-seminated to a static control center (called data sink) in thenetwork, using node to node multi-hop data propagation.Such settings have increased implementation complexity andsensor devices consume significant amounts of energy. Thisis due to the fact that multi-hop data propagation proto-cols in sensor networks, where the sink is static, leads thesensor devices to consume significant amounts of energy inoperations other than sensing (e.g. inter-node communi-cation for synchronisation purposes, exploratory messages,etc). Moreover, the data exchanged are used as input to sev-eral complex distributed algorithms and protocols (such aslocalization) run by the sensor devices. Even though thosealgorithms are designed so as to consume as less resourcesas possible (e.g. [5][16]) the problem still exists. Further-more, in the area around the control center, nodes need toheavily relay the data from the entire network, thus a bottle-neck of increased energy consumption emerges and failuresdue to strained energy resources of these nodes leads to anearly disconnected and dysfunctional network (see e.g. [11]).One of the approaches for more efficient data-centric rout-ing in wireless sensor networks is provided in [4] where anenergy aware distributed heuristic builds a special rootedbroadcast tree with many leaves that facilitates the routing.An Energy Balancing Protocol is presented in [7] where thepresented algorithm decides in each step of the transmissionwhether to propagate data one-hop towards the sink, or tosend data directly to the sink.

Sink mobility can be used as a simple and energy efficientalternative for data collection. The sink is lifting the bur-den of inter-node coordination by traversing the networkitself. A mobile sink may also substitute connectivity as itis capable of bypassing obstacles and reaching disconnectedcomponents of the network. Sensor motes maintain a ratherpassive role, in terms of data propagation, by simply wait-ing to contact the sink. When contact is established theydeliver data via cheap (energy wise) one-hop transmissions.

Apparently, in such data collection schemes, new critical is-sues emerge regarding the pattern of movement to be adoptedby the sink. The network has to be traversed in a timely

and efficient manner. Also, the mobility pattern of the sinkshould guarantee that the entire network area, or at leastthe vast part of it, will eventually be visited.

In the light of the above discussion, random walks in wire-less sensor networks can serve as fully local, very simple, dis-tributed strategies for sink motion that significantly reduceenergy dissipation and also probabilistically guarantee thateventually the entire network area will be covered. Severalreal life applications exploit sink mobility for more efficientdata collection, e.g. robots that move inside a building thatneeds to be evacuated or foresters moving through a forestwhere sensors have been deployed.

In this study, we choose to model wireless sensor networksvia Random Geometric Graphs (RGG). We choose this modelinstead of other graph models (e.g. Gn,p and Grid), that arewell established in random walk studies, as RGG better cap-ture certain relevant characteristics of real WSN’s such aslink existence dependencies of neighbouring nodes due togeometric proximity.

Our goal is mainly to accelerate the random walk coverageof the network via reducing the node overlaps. Also, toimprove other (network related) features of the walk, suchas how fast it gets close to the network nodes.

A preliminary version of this research has appeared in [1].

2. RELATED WORK AND COMPARISONRandom Walks have been extensively studied in the pastdecades in the context of several disciplines. However, de-spite their numerous applications in ad hoc, vanets (seee.g. [6][12]) and overlay networks, not much research hasbeen done on how they can be particularly applied in sen-sor networks with mobile entities, in a way that addressesthe peculiarities of such networks (such as severe computingand communication constraints, the small memory and con-strained battery, as well as the time-criticality of importantapplications).

A comparison of different random walk strategies for ad hocnetworks is performed in [9]. The authors investigate theeffectiveness of each strategy in terms of the expected hopcount and the occurrence of deadlocks. In our work, we pro-pose a new random walk strategy and compare two varia-tions of it with two known walks. We study the performanceof the strategies in terms of cover time, approximate covertime, proximity variation and a new metric that we definehere, proximity cover time.

In [13] the problem of data gathering in a large-scale wire-less sensor network with static nodes and a mobile patrolnode is formulated as a classical random walk on a randomgeometric graph. The authors derive analytical bounds forthe performance of the random walk in terms of node cover-age. In order to improve this performance they propose analgorithm to constrain the random walk using the availableside information, such as the awareness of previously visitedsites. We also use some information (just the information ofgeometric distance) in order to speed up the network traver-sal and our new walks are different.

In [18] authors investigate the Random Waypoint Model,which is widely used in the simulation studies of mobile adhoc networks and show that it fails to provide a steady statein that the average nodal speed consistently decreases overtime; therefore it should not be directly used for simulation.They also propose a simple fix of the problem and discussa few alternatives. In our study, the moving element of thenetwork (the sink) is visiting the nodes in a sequential way.Therefore we abstract its motion by considering that it ismoving on the edges of a Random Geometric Graph, thatmodels a wireless sensor network.

Nice research has also been conducted for deterministic sinkmobility, such as in [14, 15] where authors address the prob-lem of maximising the network lifetime. In [14] authors sug-gest that the base station can be mobile and conclude thatthe best mobility strategy consists in following the periph-ery of the network (assuming that the sensors are deployedwithin a circle). In [15] authors investigate the approachthat makes use of a mobile sink for balancing the trafficload and in turn improving network lifetime. They engineera routing protocol that effectively supports sink mobilityand through intensive simulations they prove the feasibilityof their mobile sink approach.

3. THE NEW RANDOM WALK AND PER-FORMANCE METRICS

3.1 The network modelSensor networks comprise of a vast number of ultra-small ho-mogeneous sensor devices (which we also refer to as sensors),whose purpose is to monitor local environmental conditions.Each sensor is a fully-autonomous computing and commu-nication device, characterized mainly by its available powersupply (battery), its transmission range r, the energy cost ofdata transmission and the (limited) processing and memorycapabilities. Sensors (in our model here) do not move. Thepositions of sensors within the network area are random andfollow a uniform distribution. We focus on data collectionmethods, so we assume that initially all sensors have somedata to deliver to the sink. For clarity, we also assume thatno data is generated during the network traversal. That isif a node is called “visited”, then it has no data to send tothe sink.

There is a special node within the network region, whichwe call the sink S, that represents a control center wheredata should be collected. Here, we assume that the sinkis mobile. The sink is not resource constrained i.e. it isassumed to be powerful in terms of computing, memory andenergy supplies.

We consider that the random uniform placement of the sen-sors inside the network area is abstracted by a Random Ge-ometric Graph. Random Geometric Graphs are formed byn vertices that are placed uniformly at random in the [0,1]2

square. An edge (u, v) exists iff the Euclidean distance ofvertices u and v is at most r, where r corresponds to thewireless communication radius r of the sensors. This holdsassuming a disc radio model; two sensors can communicatewith each other iff each one lies inside the communicationrange of the other. Random Geometric Graphs also have animportant nice property: unlike other random graphs, like

Gn,p, edges are not statistically independent of each other.That is, the existence of an edge (u, v) is not independentof the existence of edges (u,w) and (w, v). This propertymakes RGG a quite realistic model for wireless sensor net-works that captures to a great extent the communicationstructure of real WSNs (at least their spatial aspects).

More strictly, consider an area A ⊂ R2 in two dimensionalspace. An instance of the random geometric graphs modelG(Xn; r) is constructed as follows: select n points Xn uni-formly at random in A. The set V = Xn is the set of verticesof the graph and we connect two vertices if their euclideandistance is at most r. For any vertex v ∈ V we will denote byN(v) the set of neighbours of v and by deg(v) = |N(v)| itsdegree. Furthermore, we will denote by ‖u−v‖ the euclideandistance between the points corresponding to vertices v, u.

In [10, 17] it is shown that the connectivity threshold for

G(Xn; r) is rc =√

lnnπn

. In this paper we will consider ran-

dom instances of G(Xn; r) of varying density, by selecting

r =√

c lnnπn

, for different values of c > 1, which guarantees

that the produced random instance is connected with highprobability.

Besides the information about the set of neighbours of eachvertex v ∈ V , an instance G(Xn; r) of the random geometricgraphs model also contains extra information about the ex-act euclidean distance between v and any of its neighbours.We used this information in order to define a new randomwalk, namely the the γ-stretched random walk on G(Xn; r),which is described below. This random walk aims to accel-erate the data collection process while keeping the memoryrequirements restricted. The basic idea is that “more dis-tant” neighbours of the current vertex (state of the walk)are favoured in a probabilistic manner. Therefore, our newwalk is in fact a biased transitions random walk (see e.g. [2]for a nice discussion of several types of random walks basedon the assumptions they made).

3.2 The γ-Stretched Random WalkLet G(Xn; r) be a random instance of the random geometricgraphs model with vertex set V = Xn, where |V | = n. Con-sider a particle moving on the vertices of the graph G(Xn; r).Given that it occupies a specific vertex v ∈ V at time t ≥ 0,it decides where to move at time t+ 1 by choosing a vertexu ∈ V with probability

pu,v =

{‖u−v‖γ∑

w∈N(v) ‖w−v‖γif u ∈ N(v)

0 otherwise.(1)

The Markov chain describing the above process will be calledγ-stretched random walk on G(Xn; r) and will be denoted by

W(γ)

G(Xn;r). Furthermore, we will denote byWγv (t) the state of

the walk that begins at v at time t. More formally, the statespace of the γ-stretched random walk is the set of verticesV = Xn of the graph and its transition probability matrixis given by P = [pu,v]u,v∈V .

This new random walk has minimum memory requirements,

since every step is decided only by using information of thecurrent state. Notice also that the larger γ is, the moredistant neighbours are favoured over neighbours that areclose by. In the simulation, we will often set γ = r

rc, so that

the bias to visit distant neighbours is stronger as the densityof G(Xn; r) increases. The special case γ = 1 will be referredto simply as stretched random walk.

3.3 Random Walk with InertiaIn the heuristic Random Walk with Inertia the walk triesto maintain the same direction while traversing the graphG(Xn; r). Let v denote the current state of the walk, i.e.the current vertex. Let Sv denote the set of vertices thatare 1-hop neighbours of v. Finally, let vprevious denote theprevious state of the walk. At the beginning of the walkgiven its current state, the next hop is chosen uniformly atrandom among the neighbouring nodes and vprevious is notdefined. From the second hop and so forth the next state ofthe walk is chosen among the vertices of the set Sv/vpreviousaccording to the following probability distribution function:

P{next state == i} =

{φvi∑

j∈Sv φvj

if i ∈ Sv

0 otherwise

where φvj is the angle defined by vertices v, vprevious and jand 0 ≤ φvj ≤ π.

Intuitively, this walk tries to change the subregions of thegraph it visits frequently by favouring the vertices lying to-wards the same direction of its motion. The choice of thenext vertex is probabilistic, in an effort to reduce overlapsby avoiding to move on the same path.

We note that this walk has light-weight requirements. Itassumes zero knowledge of the network and is relativelysimple with low computational complexity. Furthermore,it requires a small, constant sized memory since the nextstep of the walk depends solely on the previous one. Byfavouring vertices lying towards the same direction, the sinkmakes long paths and traverses many different sub-regions ofthe network area very quickly, thus avoiding early overlaps.However, after most of the network area has been covered,there exist small unvisited sub-regions that are hard for thewalk to discover. The fact that many different sub-regionsare visited very soon makes this walk suitable for time crit-ical applications, such as reactive event detection.

3.4 Known Random WalksIn this section we present two known random walks.

3.4.1 Blind Random WalkThis is the usual random walk model. The blind randomwalk on a random instanceG(Xn; r) of the random geometricgraphs model is the simplest of all possible sink mobilitypatterns, since the next move of the sink is stochasticallyindependent to the previous ones. Furthermore, given thatthe current vertex is v ∈ V , the probability of moving toany neighbouring vertex u ∈ N(v) is pv,u = 1

deg(v). This

method is very robust, since it probabilistically guaranteesthat eventually all network regions and nodes will be visitedand thus all data will be collected given that the network

is connected. However, in some network structures it maybecome inefficient, mostly with respect to high latency, sincethe sink uses no memory of the past movements in order toselect the next one and thus overlaps (i.e. visits to alreadyvisited vertices) occur.

3.4.2 Random Walk with MemoryThe performance of the blind random walk can be improvedusing some memory of past visits. In random walk withmemory K, the sink maintains a first-in-first-out (FIFO)list M which contains the last K nodes visited during therandom walk, i.e M = {c1, c2, ..., cK}. The next hop is cho-sen uniformly at random among the neighbours of the nodethat are not in the memory list M. The use of memoryeliminates the possibility of short loops in random walks.Setting K = 0, we simply get the blind random walk. Inthis study the random walk with memory 1 will be usedfor comparison purposes. Note that this walk has strongermemory requirements and overhead than both the blind andthe γ-stretched random walk.

3.5 MetricsIn this section we define and discuss the metrics that willbe used in the evaluation of our new random walk, whencompared to the blind random walk and the random walkwith memory 1. In the following, let W denote any randomwalk model defined on a random instance G(Xn; r) of therandom geometric graphs model.

3.5.1 Cover time and approximate cover timeConsider the random walk Wv that begins on vertex v ofa random instance of the random geometric graphs model

G(Xn; r). Define Tvdef= inf{t|∀u ∈ V, ∃t′ ≤ t : Wv(t′) = u}

to be the time needed until Wv has visited all the verticesin the graph. The cover time C of G(Xn; r) is defined asC = maxv∈V E[Tv], where E denotes the expected value ofthe random variable Tv. It was shown in [8] that when r =√

c lnnπn

, for c > 1, the cover time of G(Xn; r) for the blind

random walk is asymptotically c ln(

cc−1

)n lnn.

For ε ∈ (0, 1) define also the ε-approximate cover time asthe mean number of steps that W needs in order to visit a

fraction 1 − ε of the vertices. More formally, define T(ε)v to

be the time needed until Wv has visited (1 − ε)n vertices.The ε-approximate cover time C of G(Xn; r) is defined as

C(ε) = maxv∈V E[T(ε)v ].

These cover times are related to latency, as they capturethe time the sink needs to collect the sensory data from theentire network. The approximate cover time metric is ofgreat interest as the majority of overlaps occur while thesink tries to locate the last few unvisited sub-regions thatare scattered in the network area; however, in most sensornetwork applications it is sufficient to collect a vast percent-age of the total sensory data, so this metric is relevant andinformative.

3.5.2 A new metric: Proximity Cover TimeIn real-life sensor networks, the sink is capable of collect-ing data not only from the currently visited node, but also

from all nodes that are inside the communication range r(via a single-hop data transmission). Therefore, there isstrong motivation from real-life WSN to consider as visitednot only the vertex of the current state of the walk, but alsoall neighbouring vertices. Following is the formal definitionof the corresponding metric, Proximity Cover Time.

Denote by D(v, ρ) the disc of radius ρ > 0, centered at v.Consider the random walk Wv that begins on vertex v ofa random instance of the random geometric graphs modelG(Xn; r). We will say that Wv is within distance ρ to ver-tex u at time t if Wv(t) ∈ D(u, ρ), i.e. the random walkstarted at v occupies a vertex that lies in D(v, ρ) at time t.

Let Tv(ρ)def= inf{t|∀u ∈ V, ∃t′ ≤ t : Wv(t′) ∈ D(u, ρ)} be

the time needed until Wv has come within distance ρ to allvertices of the graph. We define the ρ-proximity cover timeof G(Xn; r) as C(ρ) = maxv∈V E[Tv(ρ)].

A similar but different metric is presented in [2] where eachtime the random walk visits an already visited vertex, thenu.a.r. picks an unvisited neighbour (if one exists), marksit as visited and then continues the random walk from itscurrent vertex (i.e. it does not make a transition to themarked neighbour).

3.5.3 Proximity VariationFor a vertex v ∈ V and time t0 ≥ 0 let

dist(v, t0) = min0≤t≤t0

‖W(t)− v‖

be the function that returns the minimum distance betweenthe random walk W up to time t0 and node v. Then theproximity variation PV (t0) at time t0, is given by

PV (t0) =

∑v∈V dist(v, t0)

n

where n is the total number of nodes.

The rationale behind this metric is the following: In a realnetwork, if the PV metric converges to zero quickly witht, this means that the sink gets close to all sensors quitesoon and data collection progresses fast; this is especiallyrelevant in case when the role of sensors is not completelypassive but includes some limited multi-hop propagation ofdata to accelerate data propagation at a reasonable energycost. On the contrary, when the PV converges slowly tozero, it means that the network traversal is performed in away that some areas may stay unvisited for long time.

3.5.4 Visit overlap statisticsOther metrics that will be used in order to further charac-terize each walk’s evolution include the number of visits to aspecific vertex v, as well as the distribution of the number ofvisits for every vertex in the graph. These overlap statisticswill provide an insight on how exactly each walk traversesthe graph, how different components of the graph are visitedand in what rate.

4. PERFORMANCE EVALUATION4.1 Simulation Set-up

We conducted our experimental evaluation using MatlabR2009a as our simulation environment. We evaluate thewalks on several instances of the Random Geometric Graphsmodel in order to address its random nature. More partic-ularly, we construct Random Geometric Graphs consistingof n = 2000 nodes placed in a 100 × 100 square area. Forsuch RGG graphs, the connectivity threshold is computed to

be of radius r = 100√

ln 2000π2000

= 3.47. For our performance

evaluation we use three characteristic values for r: r1 = 5,r2 = 7, r3 = 10, corresponding to sparse, average and densergraphs respectively. For every value of r, we construct 50RGG instances on which we evaluate all walks for 100 itera-tions per instance. We consider these randomly constructedRGG instances to be representative of the RGG space.

For the performance evaluation of the protocols we used twonetwork topologies. At the first topology sensor nodes aredeployed uniformly at random over the planar area. Thistopology constitutes the base of our performance evaluationstudy by providing a first insight on the performance of eachprotocol. At the second network topology a portion of thesensors (in our experiments 60%) is deployed uniformly atrandom in order to establish connectivity. The rest of thesensors are deployed over the network area following thenormal distribution with mean value µ = 50 (i.e. half thedimension of the network area) and standard deviation σ =7 for the x and y co-ordinates (independently of each other)of each sensor. This set-up leads to a heterogeneous networkdeployment in which a high concentration of sensor nodes islocated around the center of the network area.

4.2 Simulation FindingsWe first experimentally evaluate the performance of the Ran-dom Walk with Inertia on the Random Geometric Graphand compare it with the Blind Random Walk (only in theuniform topology). Then we compare four random walks;the well known Blind Random Walk; two versions of ournewly proposed walk: the Stretched Walk where γ = 1 andthe γ-Stretched Walk, where γ = r

rc, in order to investigate

the impact of γ; the Random Walk with Memory 1, thatsomehow represents an upper bound in terms of memoryusage and therefore is considered to be more powerful thanthe rest of the walks.

4.2.1 Uniform TopologyFigure 1 presents the comparison between the Random Walkwith Inertia and the Blind Random Walk. Although theRandom Walk with Inertia on the Grid performs better thanthe Blind, however the RGG model causes the Inertia topresent a really weak performance. This result led us to thedesign of the newly proposed random walk, the so calledγ-Streched Random Walk.

Figure 2 depicts the mean number of hops each walk needsin order to visit all the nodes of the network for three repre-sentative values of the communication range r. For all therandom walks we observe that as the density of the graph in-creases, the total cover time is reduced even by nearly 50%.This is due to the fact that in a dense network the walkhas more edges to traverse on, and therefore there is higherprobability to visit any given vertex. In other words, a ran-dom walk reaches its stationary distribution much faster.

25000

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40000

45000

50000

55000

60000

Blind Random Walk

Random Walk with Inertia

Number of Hops

Type of Walk

Cover Time

r = 5r = 7r = 10

Figure 1: Cover Time of the Blind Random Walkand the Inertia Random Walk

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45000

50000

55000

Blind Random Walk

Stretched Walk

g-Streched Walk

Walk with Memory 1

Number of Hops

Type of Walk

Cover Time

r = 5r = 7r = 10

Figure 2: Cover Time of the Blind Random Walk,Stretched Walk, γ−Stretched Walk and Walk withMemory 1

We also observe that for all graph densities, the Blind Ran-dom Walk has the biggest cover time, while the impact ofthe fixed-sized memory of the Walk with Memory declines asthe density of the graph increases. In fact, in sparse graphs,where the available options are limited, avoiding 1-hop over-laps by not visiting the last position significantly reduces thecover time. In our Stretched Walks, on the contrary, the im-pact of the γ factor is proportional to the number of neigh-bouring nodes as the graph density increases. This is whyfor small values of radius r the performance of the StretchedWalks is very similar. However, in more dense graphs, bias-ing the walk towards the more distant neighbours seems tohave a well noticeable effect on the speed the random walktraverses the graph.

Figures 3 and 4 depict the rate at which each walk discov-ers unvisited nodes inside the network area for range r = 5and r = 10 respectively. For sparse networks (r = 5) wenote that the Stretched Walk outperforms even Random

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Figure 3: Percentage of discovered nodes over num-ber of hops with range r = 5 and uniform distribu-tion

Walk with Memory. The Random Walk with Memory, whileavoiding its previous position, may move towards a neigh-bouring node that geometrically is very approximate; thusallowing a high probability for early overlaps. On the otherhand, the Stretched Walk, by favouring the more distantneighbours, quickly changes the subregions of the networksit traverses. This way it avoids early overlaps, thus dis-covering unvisited nodes much faster. However, in sparsenetworks, if the bias factor towards distant neighbours isvery crude it may have an opposite effect. As seen in theperformance of the γ-Stretched Walk, a crude bias factormay narrow the available options for the next step down tothe point the walk is forced to make long cycles during itstraversal, thus visiting already visited nodes.

In dense networks (r = 10) the rate at which unvisitednodes are discovered is accelerated for all walks. However,γ-Stretched Walk performs slightly better as the bias factorstrongly favours distant nodes over geometrically approxi-mate ones. We note that for all walks while they all discoverrelatively quick the 95% of the total number of vertices inthe graph, they spend nearly half of the cover time in or-der to visit the last 5% of the vertices, thus creating a longconvergence tail at the end of the traversal.

As discussed in subsection 3.5.2, there is strong motivationfrom real-life wireless sensor networks to consider as visited

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Figure 5: Proximity Cover Time

not only the vertex of the current state of the walk, butalso all neighbouring vertices in the RGG network. Figure 5depicts the performance evaluation of all walks in RGG forthe Proximity Cover Time metric. Intuitively, the dominantfactor for this metric is the dislocation attribute of the mo-tion (i.e. the mean geometric distance the sink covers per

hop). In sparse networks the use of memory by the RandomWalk with Memory slightly improves the Proximity CoverTime of the Blind Random Walk. The Stretched Walk withγ = 1 shows the best performance from all other walks. Theγ-Stretched Walk is outperformed due to the crude way itis biased towards the few distant neighbours. However, forhigher network densities this factor aids the walk to outper-form even the Walk with Memory. In a network consisting of2000 nodes, in less than 2000 hops the sink has been insidethe communication radius of every node in the network.

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Figure 6: Overlaps distribution over the nodes ofthe graph for the random uniform topology.

Figure 6 depicts the overlaps distribution over the nodes ofthe graph. That is what percentage of the total number ofthe nodes has been visited i times by the sink during the en-tire network traversal process. In sparse graphs the RandomWalk with Memory outperforms the two Stretched Walks,that is more nodes of the network have been visited fewertimes (i.e. the curve corresponding to Walk with Memoryis above and to the left than the rest). At first glance thiscontradicts the fact that the Stretched Walk is discoveringunvisited nodes in sparse networks at a higher rate than theWalk with Memory (Fig. 2). We note however that the

Stretched Walk avoids early overlaps, while it needs moretime to cover the entire network (Fig. 2). This leads to theconclusion that most of the overlaps occur towards the endof the network traversal process, thus the converging tails inFig. 2.

For more dense networks the impact of the bias factor ofγ-Stretched Walk is increasing along with the value of r,thus eventually demonstrating the best performance for thismetric. Also, as the network graph density increases, thenumber of revisits for the majority of the nodes is more orless the same (10-15 revisits). However, the total numberof revisits is increased significantly, as the last few unvisitednodes are hard to find in a dense network.

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Figure 7: The Proximity Variation Metric for r = 5and uniform topology

There lies a different intuition behind each walk on how thenetwork area is traversed, regardless of the total number ofhops needed. For example the strategy of avoiding previouspositions is getting less effective when the density of thenetwork increases, while the γ factor in Stretched walks isallowing the sink to visit many different network subregionsvery early in the traversal process. The proximity variancemetric captures these differences. In Fig. 6 we note thatthe two walks which perform better in sparse graphs are theStretched Walk (where γ = 1) and the Random Walk withMemory. As there are relatively few neighbouring nodesslightly favouring distant nodes or just avoiding previousposition seems to suffice. On the other hand, the γ-StretchedWalk (where γ = r

rc) favours distant nodes in a more crude

manner (for r = 5 and rc ≈ 2.3, γ ≈ 2). This way, the walkis left with very few available options, forced to make severalcycles during its traversal process.

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Figure 8: The Proximity Variation Metric for r = 10and uniform topology

In dense networks, where there are many available optionsfor the next step of the walk, this crude selection of thedistant nodes allows the walk not to get attracted by ap-proximate neighbours. Therefore, in Fig 8 the γ-StretchedWalk is the one converging to zero at a higher rate.

4.2.2 Normal DistributionFigure 9 depicts the cover time of each walk for a specificcharacteristic communication range r. As shown in the fig-ure, the γ-Stretched Random Walk performs better thanthe Blind Random Walk. The γ biased factor allows theStretched Walk not to get trapped inside dense areas of thenetwork, contrary to the Blind Walk. On the other hand, thesimple Stretched Random Walk has the worst performance.

A better insight on the performance of our two proposedwalks for the normal distribution can be obtained by analysingtheir performance over the uniform network topology shownin figure 2. We notice that in sparse areas both walks havesimilar performance regarding their cover time and the im-pact of the γ factor is negligible. This is due to the fact thatin sparse areas both walks have few alternatives for theirnext move. On the contrary, in dense areas the γ factor al-lows the γ-Stretched Walk to quickly traverse the network

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area by favouring the more distant neighbours resulting inbetter performance. As a result, in the case of a heteroge-neous topology, the γ-Stretched Walk behaves similarly inthe sparse areas of the network, but much better in the moredense areas, yielding a better overall performance.

Figure 10 presents the rate at which each walk discoversunvisited nodes inside the network area. We note that theγ-Stretched Random Walk outperforms even the RandomWalk with Memory. As in the uniform topology, the StretchedWalk, by favouring the more distant neighbours, quicklychanges the subregions of the networks it traverses, thisway avoiding early overlaps and discovering unvisited nodesmuch faster.

Figure 11 depicts the overlaps distribution over the nodesof the graph. That is what percentage of the total num-ber of the nodes has been visited i times by the sink duringthe entire network traversal process. One can see that theRandom Walk with Memory outperforms the two StretchedWalks, that is more nodes of the network have been visitedfewer times. This does not contradict the fact that the γ-Stretched Walk discovers unvisited nodes at a higher ratethan the Walk with Memory. We note however that theStretched Walk avoids early overlaps, while it needs moretime to cover the entire network. Actually, most of the over-laps occur towards the end of the network traversal process,especially in this heterogeneous topology, since it is moredifficult for the Stretched Walks to discover the nodes of thedense area thus leading to multiple overlaps.

Finally, figure 12 depicts the proximity variation metric foreach walk. As shown in the figure, the Stretched RandomWalk (γ = 1) has the best performance, since it favoursdistant neighbours, but is not that biased as the γ-Stretched,thus succeeding to visit many different network subregions,both in the sparse but also in the dense areas of the network.On the other hand, the γ-Stretched Walk (where γ = r

rc)

favours distant nodes in a more crude manner (for r = 7and rc ≈ 3.5, γ ≈ 2). This way, the walk is left with veryfew available options, forced to make several cycles during

(a) Hops rate for r = 7

(b) Hops rate for r = 7 (zoom-in)

Figure 10: The Hops rate for r = 7 and heteroge-neous topology

its traversal process.

5. CONCLUSIONSIn this work we address the problem of efficient data col-lection in wireless sensor networks via a mobile sink. Wemodel the data collection process by random walks on Ran-dom Geometric Graphs as they better capture the geometricdependencies that characterise inter-node wireless commu-nication in WSN’s. We use random walks as local, simplemotion strategies for the sink that have very low compu-tational and a priori knowledge requirements. We define anew walk called Stretched Walk and evaluate two variationsof it. The basic idea of the new walk is to favour distantneighbours of the current node towards avoiding visit over-laps. Also, motivated by real-life WSN’s, we define a newmetric called Proximity Cover Time; this captures collectingdata from all neighbour nodes of a currently visited node.

Findings show that the use of constant sized memory inorder to accelerate the data collection process has limitedresults, particularly when the network density increases. In-stead, walks that use local (immediate neighbour) informa-tion in a smart manner can demonstrate significantly betterresults. However, their use should be wise as they may leadto opposite results (e.g. strong bias factor in sparse networksleads to bigger cover times).

Figure 11: The overlaps distribution for r = 7 andheterogeneous topology

(a) Proximity Variation for r = 7

(b) Proximity Variation for r = 7 (zoom-in)

Figure 12: The Proximity Variation for r = 7 andheterogeneous topology

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