Biased random walks in uniform wireless networks

Biased Random Walks inUniform Wireless Networks

Roberto Beraldi

Abstract—A recurrent problem when designing distributed applications is to search for a node with known property. File searching in

peer-to-peer (P2P) applications, resource discovery in service-oriented architectures (SOAs), and path discovery in routing can all be

cast as a search problem. Random walk-based search algorithms are often suggested for tackling the search problem, especially in

very dynamic systems-like mobile wireless networks. The cost and the effectiveness of a random walk-based search algorithm are

measured by the excepted number of transmissions required before hitting the target. Hence, to have a low hitting time is a critical

goal. This paper studies the effect of biasing random walk toward the target on the hitting time. For a walk running over a network with

uniform node distribution, a simple upper bound that connects the hitting time to the bias level is obtained. The key result is that even a

modest bias level is able to reduce the hitting time significantly. This paper also proposes a search protocol for mobile wireless

networks, whose results are interpreted in the light of the theoretical study. The proposed solution is for unstructured wireless mobile

networks.

Index Terms—Algorithm/protocol design and analysis, random walks, mobile ad hoc networks, search algorithms.

Ç

1 INTRODUCTION

CONTEXT of this study. To search for a node with knownproperty is a basic recurrent problem arising in many

distributed applications. For example, in routing protocolsfor mobile wireless networks, e.g., [12] and [24], the searchednode is identified by its IP address, while in peer-to-peer(P2P) architectures by a key, associated to the object the nodestores [19], [16]. Searching is also a central functionality in aservice-oriented architecture (SOA), e.g., see [13]. With theirshift toward wireless communication support, distributedsystems are becoming more dynamic, and the search problemis consequently becoming even more challenging. This paperfocuses on the search problem in the context of mobilewireless networks, i.e., autonomous self-organizing net-works composed of a set of wireless devices.

Broadly speaking, there are two approaches to face asearch problem: structured and unstructured. The formerexploits a logical structure for guiding searches, e.g.,routing tables stored at nodes, Distributed Hash Tables(DHTs), or centralized/distributed directories, while thelatter does not leverage any logical organization in thesearch space. To maintain the structure used to support asearch may become challenging in mobile networks sincethe mobility of nodes makes the topology of the networkalso variable. For this reason, the unstructured approach isregarded as an attractive alternative, as more deeplydiscussed, for example, in [20].

An unstructured search has to potentially explore thewhole network; as such, it is generally carried out byflooding. Alternatively, random walks can be used.

Compared to flooding, a random walk search has a morefine-grained control of the search space, a higher adap-tiveness to termination conditions, and can naturally copewith failures or voluntary disconnections of nodes [22].Examples of concrete exploitations of random walks inwireless networks are found in the context of routingprotocols for MANET—e.g., ANT [23], Hint-Based routing[6], and most recently, in P2P over MANET, e.g., ROAN[14]. Graph theoretical studies on random walks that arerelevant for wireless networks have also recently appearedin the literature, e.g., [5] and [2].

Biased random walks are random walks in which nodeshave statistical preference to forward the walker toward thetarget. The clear advantage of a biased random walk is thatit reduces the excepted number of steps before the target isreached, called the hitting time, significatively. However,the bias level achievable in a real setting is limited, whilethe implementation of any biasing mechanism comes atsome additional cost. Thus, to understand the effect of biason the hitting time is an important preliminary step fordeciding the practical benefit of a random walk-basedsearch algorithm. The effect of bias on the hitting time whenthe random walk is executed over a wireless network is thesubject of this paper.

Contributions of this work. The random walk weconsider exploits look-ahead one. The walker (a packet) isforwarded from one node to a randomly chosen neighboruntil a neighbor of the target is found. To study the effect ofbias analytically, we exploit a network model composed ofinfinitely many nodes located at uniformly random posi-tions inside a circle. This model is useful to study the effectof bias on a real network because during the lifetime of awalk the variation in the network topology can, at firstglance, be neglected.

The contribution of this paper is twofold. First, we foundthat for the uniform model the hitting time is very sensible

500 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 4, APRIL 2009

. The author is with the Dipartimento di Informatica e Sistemistica,Universita di Roma “La Sapienza,” Via Ariosto 25, 00100 Rome, Italy.E-mail: [email protected].

Manuscript received 17 Oct. 2007; revised 30 May 2008; accepted 29 Sept.2008; published online 16 Oct. 2008.For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference IEEECS Log Number TMC-2007-10-0315.Digital Object Identifier no. 10.1109/TMC.2008.151.

1536-1233/09/$25.00 � 2009 IEEE Published by the IEEE CS, CASS, ComSoc, IES, & SPS

to bias. In particular, we show that the hitting time is

conveniently upper bonded by N1=�, where N is propor-

tional to the radius of the circle and �, 0:5 � � � 1, is a

number expressing the bias level of the walk.Second, we present a random walk-based search

protocol designed for mobile networks that runs atopthe data link layer. The random walk is biased via asimple distance estimation mechanism, which generates abias � � 0:6 when the search is for a single target. We giveexperimental evidence of the validity of our theoreticalstudy by simulating nodes moving inside a circle. Then,we present a simulation study for a more general setting,whose results can be interpreted in the light of thetheoretical expectations.

Structure of this paper. The reminder of this paper is

organized as follows: Section 2 studies the bias effect in one

dimension, while Section 3 considers a 2D circular deploy-

ment; Section 4 discusses the possible options for imple-

menting a biased random walk; the proposed protocol is

presented in Section 5 and the simulation results are given

in Section 6. Section 7 discusses the related work; conclu-

sions are finally given in Section 8.

2 EFFECT OF BIAS ON A RANDOM WALK,SIMPLE CASE

A summary of the main symbols used throughout this

paper is given in Fig. 1, while the definitions of the key

terms are now given.

Definition 1. Bias level. A random walk has bias level � with

respect to a given target, and it is called �-biased random walk,

if after a step the walker gets closer to the target with

probability �.

Definition 2. Unbiased random walk. A random walk with

bias level � ¼ 0:5 is said to be unbiased.

Definition 3. Hitting time. The hitting timeh�ði; jÞ of a�-biased

random walk is the expected number of steps before node j is

visited for the first time starting from node i.

Definition 4. Gain. The gain g�ði; jÞ of a �-biased random

walk is the percentage reduction of the hitting time of j

starting from i with respect to the same hitting time for the

unbiased random walk, namely, g�ði; jÞ ¼ h0:5ði;jÞ�h�ði;jÞh0:5ði;jÞ .

Definition 5. Return time. The return time is the expectednumber of steps in a random walk starting at a given node,before the same node is reached again.

Let us start by considering a �-biased random over the finiteline ½0::N �. Without loss of generality, let N be the targetpoint. At each time step, the walker moves closer to thetarget with probability � and away from the target withprobability 1� �. The extreme points are reflective barriers,see Fig. 2. It is well known, e.g., [17], that for the unbiasedcase we have h0:5ð0; NÞ ¼ N2 while for � ¼ 1 it is trivial tosee that h1ð0; NÞ ¼ N . Some obvious question then arises.According to which law the hitting time passes from thequadratic to the linear behavior? Is there some simple ruleof thumb that can be useful to characterize the maximumgain one can achieve under a given bias �? We now willgive an answer to these questions.

To compute the hitting h�ð0; NÞ, we follow the methodillustrated in [17]. The hitting time can be written as

h�ð0; NÞ ¼XNk¼1

h�ðk; k� 1Þ; ð1Þ

while the hitting timeh�ðk; k� 1Þ is one less than the expectedreturn time at position k of a �-biased random walk over thesegment ½0::k�. The return time is the inverse of the stationaryprobability of observing the walker at position k, see [17];thus, if �i;k is the stationary probability that a walker over½0; k� is observed at position i, we have

h�ðk; k� 1Þ ¼ ��1k;k � 1

and

ht�ð0; NÞ ¼ 1þXNk¼2

��1k;k � 1

h i:

The probability �k;k is readily obtained by solving anelementary Discrete Time Markov Chain (DTMC) with statespace ½0; k� and transition probabilities

pi;iþ1 ¼ �; pi;i�1 ¼ 1� � 1 � i � k� 1;

p0;1 ¼ pk;k�1 ¼ 1:

From the balance equation of such a DMTC, we can write

��1k;k ¼ ��1

0;k

1� ��

� �k�1

BERALDI: BIASED RANDOM WALKS IN UNIFORM WIRELESS NETWORKS 501

Fig. 1. List of the main parameters and symbols used throughout this

paper.

Fig. 2. A random walk with bias � over [0, 6].

and

��10;k ¼ 1þ �k�1

ð1� �Þk�1þ

1� �k�1

ð1��Þk�1

2�� 1;

hence,

��1k;k ¼ 1þ ð1� �Þ

k�1

�k�1þ

�k�1

ð1��Þk�1 � 1

2�� 1:

By substituting the above probability into (2), we obtain

ht�ð0; NÞ ¼ 1þXN�1

k¼1

ð1� �Þk

�kþ

�k

ð1��Þk � 1

2�� 1

that after some manipulation can be rewritten as

ht�ð0; NÞ ¼ 1þ N � 1

2�� 1þ 2

�� 1

2�� 1

� �2 1� ��

� �N�1

�1

" #: ð2Þ

2.1 Discussion

Let us now summarize the properties of the biasedrandom walk over the segment in terms of hitting timeand gain.

Property 1 (Character of the hitting time). ht�ð0; NÞ ¼�ð N

2��1Þ, for � such that 0:5 < � � 1.1

Proof. In fact, for 0:5 < � � 1, the fraction 1�� < 1; thus, for

N !1, the third term of (2) tends to be a constant. tuUnder a formal point of view, the hitting time ht�ð0; NÞ is

lower than the approximating values N=ð2�� 1Þ as well asthe unbiased value, N2. The last term in (2) is in factnegative. Now,

N=ð2�� 1Þ > N2

until N reaches the value N 0 ¼ 1=ð2�0 � 1Þ. This is acrossover point. For N < N 0,

N=ð2�� 1Þ �HT > N2 � ht�ð0; NÞ

and then ht�ð0; NÞ is closer to N2, while for N > N 0,ht�ð0; NÞ is closer to N=ð2�� 1Þ. For example, for �0 ¼ 0:505,N 0 ¼ 100. The value N 0 is always finite; thus, eventuallyhitting time behaves like N=ð2�� 1Þ. This last aspect has thefollowing physical interpretation.

Consider an infinite line and a particle (walker)starting at point 0 and moving along the line, steppingof one unit to the right with probability � � 0:5 and withprobability 1� � to the left, i.e., the particle is subject toan external field that pushes it right. On the average, theparticle arrives at point N after the hitting time, ht�ð0; NÞ.The ratio N=ht�ð0; NÞ can then be interpreted as theaverage drift speed in the direction of the field.

Now, using the function N=ð2�� 1Þ of Property 1, we cansee that the speed becomes 2�� 1. This value, 2�� 1, isinterpreted as the characteristic speed, which is acquired bythe particle when it is observed at a distance sufficiently farfrom the origin (for N sufficiently high). When � becomescloser and closer to 0.5, in order to observe the “character” ofthe walker we need to go farther and farther from the origin.

With this interpretation, the speed of an unbiased particleis 1=N . That is, the speed goes to 0 as N goes to infinity. Inother terms, for � ¼ 0:5, the random walk is unbiased andthere is no drift in the direction of the external field.

Property 2 (Upper bound on the hitting time).

ht�ð0; NÞ � N1=�, for � such that 0:5 � � � 1 and N > N0,where N0 is a constant.

Proof. For 0:5 < � < 1, N1=� increases with N more rapidly

than N2��1 does; thus, N1=� > N

2��1 , provided that N > N 0,

for some N 0 > 0. But, from Property 1 the hitting time,

for 0:5 < � � 1, indeed grows as N2��1 , while for � ¼ 0:5

the bounding function and the correct hitting time are

the same [17]. Since N0 can be derived from N 0, the

property holds. tuThe constant value N0 is derived numerically and is

N0 ¼ 5. As an example, Fig. 3a reports ht�ð0; 9Þ and thebounding function as � is varied.

Property 3 (Lower bound on the gain). g�ð0; NÞ � 1�N1�2�� ,

for � such that 0:5 � � � 1 and N > N0, where N0 is aconstant.


1. Recall that f ¼ OðgÞ if and only if lim supt!1jfðtÞj=gðtÞ ¼ k, for somek � 0, and f ¼ �ðgÞ if f ¼ OðgÞ and g ¼ OðfÞ.

Fig. 3. Hitting time of a biased random walk over N ¼ 9, exact value and upper bound: (a) hitting time and (b) gain.

Proof. The property comes from the definition of the gainand Property 2. tu

The results for N ¼ 9 are reported in Fig. 3b. A bias assmall as � ¼ 0:6 reduces the hitting time by approximativelya half.

Property 4 (The minimum gain is obtained when starting

from the farthest point). g�ð0; NÞ < g�ð0; iÞ, for i such that

0 < i < N .

Proof. We have that h�ði;NÞ ¼ h�ð0; NÞ � h�ð0; iÞ, e.g., see

[17]. Since h0:5ði;NÞ ¼ N2 � i2, we have to show that 1�h�ð0;NÞN2 < 1� h�ð0;NÞ�h�ð0;iÞ

N2�i2 .

The inequality can be rewritten as 1� i2

N2 > 1� h�ði;0Þh�ðN;0Þ ,

which holds if and only ifh�ði;0Þi2

>h�ðN;0ÞN2 . But, h�ð0; NÞ is

given by (2), which varies linearly with N ; thus, the

function fðNÞ ¼ h�ð0;NÞN2 decreases with N and then the

property holds. tuFrom Properties 3 and 4, we can derive the followinglemma:

Lemma. The gain of random walk with bias �, which starts from

an arbitrary point of a segment ½0; N�, is at least 1�N1�2�� ,

provided that N > N0 ¼ 5.

3 EFFECT OF THE BIAS ON A UNIFORM WIRELESS

NETWORK

In this section, we study the effect of bias on two dimensions.To study this problem analytically, we examine a circular areaof radiusR�, which contains infinitely many nodes deployedat random, i.e., the probability that a node occupies theinfinitesimal squared area centered at ðx; yÞwith respect to apair of Cartesian axis is 1

�R�2dxdy. The target node occupies the

center of the circle. The transmission range of each node isR.The neighbors of a node are the nodes located at distance atmostR from itself. Nodes have no physical extension; thus, inthe following the terms point and node are used inter-changeably.

The random walk we study exploits look-ahead one, i.e.,the neighbors of the target will send the walker directly tothe target rather than making a random choice. Lookaheadis simple to implement in wireless transmissions (asdetailed in our proposal) while it makes easier to computethe hitting time mathematically.

Due to the circular symmetry, the random walk over thecircle induces a geometric random walk on the finitesegment ½0; R��. This new random walk describes how thedistance of the walker from the target varies. The walkermakes variable steps in the range ½�R;R�.2 For example,Fig. 4 shows a walker started at distance R�. Steps arenumbered progressively starting from zero; the distancefrom the target at step i, ri, is reported on a vertical radius.The walker hits the target when r becomes zero. The circleof radius R is the look-ahead area. The hitting time in theexample is then nine.

The random walk studied in the previous section is a

particular case of our geometric random walk; in fact, it is

obtained for R� ¼ NR and discrete steps of length R. Thus,

we can hope that the hitting time is also subject to a behavior

similar to the one summarized by the properties given in the

previous section. To see if such is the case, we now compute

the exact hitting time numerically and will then try to find a

connection between the two random walks.Let ri be a random variable representing the distance of

the walker at the ith step and fiðrÞ its probability density

function (pdf). The target is reached when ri becomes zero,

i.e., the target is point 0. After hitting the target, the walk

remains alive but its distance does not change. This means

that fið0Þ increases monotonically with i and f1ð0Þ ¼ 1 (the

walker eventually hits the target).The probability that the distance becomes zero the first

time at the ith step is given by fiþ1ð0Þ � fið0Þ; thus, the

hitting time of point 0 starting from r0 is

hðr0; 0Þ ¼Xi�0

i fiþ1ð0Þ � fið0Þ½ �:

The functions fiðÞ can be computed through the

following recursive relationship:

fiðrÞ ¼ZrþRr�R

fi�1ðr0Þpðrjr0Þdr0;

where f0ðrÞ ¼ �ðr� r0Þ is the pdf associated to the initial

position, r0, and pðrjr0Þ ¼ Prfri ¼ rjri�1 ¼ r0g is the transi-

tion probability from point r0 to r. Such a transition pdf is

only defined for jr0 � rj � R. For the natural random walk,

it can be expressed as (bias is introduced later in this

section)

pðrjr0Þ ¼�ðrÞ; 0 � r0 < R;fðrjr0Þ; R � r0 < R� �R;fðrjr0Þ þ fð2R� � rjr0Þ þ; R� �R � r0 � R�:

8<: ð3Þ


2. A geometric random walk starts at some point in Rn and, at each step,moves to a neighboring point chosen according to some distribution thatdepends only on the current point, see [26].

Fig. 4. An example of a random walk on the symmetric setting we are

studying.

The function fðrjr0Þ will be called the progress pdf. Itexpresses the probability that, after a retransmission, thedistance of the walker from the target varies from r0 to r.The value r� r0 is called the progress.

The last expression in (3) takes the border effect intoaccount, i.e., the fact that the region is limited. To deal withsuch a limitation, we allow to select virtual points beyondR� and map point R� þ x to R� � x, i.e., a virtual transitionto R� þ ðR� � rÞ ¼ 2R� � r generates the transition to r.

Let us now compute the progress pdf. Consider the circlec of radius R centered at the selecting node and let C1 ðC2Þbe the circle of radius r0 ðrÞ centered at the target node (seeFig. 5). When the selected node belongs to the archdelimited by the intersection of C2 with c (bold in thefigure), the distance of the packet varies from r0 to r. Let�ðr0; rÞ be the length of such an arch. Since nodes areuniformly deployed we can write

fðrjr0Þ ¼ �ðr0; rÞ

�R2¼ 2�cr;

where �c is the angle under which the arch is seen from thetarget. Therefore, combining

ðr0 þRcos�cÞ2 þ ðRsin�cÞ2 ¼ r2

with

rcos�c ¼ r0 þRcos�c;

we obtain

�c ¼ arccos1

rr0 þ r

2 � r02 �R2

2r0

� �� ¼ arccos r02 þ r2 �R2

2r0r

� �ð4Þ

so that

fðrjr0Þ ¼ 2r

�R2arccos

r02 þ r2 �R2

2r0r

� �: ð5Þ

Model validation. Fig. 6 compares the hitting timecalculated by the model against the hitting time estimated

by simulations. The number of nodes is varied. Theparameters are r0 ¼ R�, R� ¼ 1, and R ¼ 0:3. The analysisprovides accurate results when the total number of nodesis higher than 300, which corresponds to approximatively24 neighbors per node. Note that, due to lookahead, inorder to hit a target it is sufficient to reach any of itsneighbors. Thus, the hitting time decreases as the networkbecomes more dense.

Introducing bias. To include bias in our model, it isenough to modify the transition probability by a scalingfactor. The new transition probability is p0ðrjr0Þ ¼ k1pðrjr0Þ ifr < r0 and p0ðrjr0Þ ¼ k2pðrjr0Þ otherwise, where

k1 ¼�R r0

r0�R pðrjr0Þdr; k2 ¼

1� �R r0þRr0 pðrjr0Þdr

: ð6Þ

3.1 Discussion

The random walk studied in the previous section is aparticular random walk on ½0; R��, which makes onlydiscrete steps of fixed size �R. On the other hand, due tothe circular symmetry, a random walk in 2D also induces arandom on the segment ½0; R�� but with variable steps in therange ½�R;R�. Is there some connection between theperformance of the two random walks? To answer to thisquestion, we follow a fairly pragmatic approach. The valueN in the bounding function for the discrete case is thehitting time for � ¼ 1. We will compute the “equivalent” Nfor the continuous random walk and then, a posteriori, testif the function is still useful.

Let then Ri ¼RRr0�R rfðrjr0Þdr be the average positive

progress at the ith step of the walk, i.e., the expectedprogress when the selected point is closer than the selectingone. The equivalent N is N ¼ K þ 1, where K is such thatPK

i¼1 Ri > R� �R andPK�1

i¼1 Ri < R� �R; in other words,K is the minimum number of steps after which the distancebecomes less than R (recall that due to lookahead thewalker hits the target after its distance becomes less than R).To simplify the computation, we consider Ri ¼ R, where Ris the average progress observed when the packet is very farfrom the target. Hence, K ¼ dR��R

Re. For r!1, the


Fig. 5. Variation of the euclidian distance after retransmitting a packet.

Fig. 6. Analysis versus simulation, a variable number of nodes deployed

in a circular-shaped area of radius R� ¼ 1; the target node occupies the

center, the random walk starts at distance R�.

probability that the progress is x is given by the length of

the chord at distance x from the selecting node, that is,

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � x2p

, divided half of the area of the circle, see

Fig. 7; thus,

R ¼ 4

�R2

ZR0

xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � x2p

dx ¼ 4R

3�:

Then, we conjecture the hitting time and the gain are

bounded, respectively, by

h�ðr; 0Þ � 1þ R� �RR

� �� 1�

; ð7Þ

g�ðr; 0Þ � 1� 1þ R� �RR

� �� 1�2��

: ð8Þ

These bounds are the equivalent of Property 2 and

Property 3. Fig. 8 shows the hitting time as a function of

the bias level for R� ¼ 1 and R ¼ 0:3. The figure reports: the

upper bound function—with these values,K ¼ 6; the hitting

time computed with the biased transition probabilities—see

(6); and the hitting time estimated via simulations for

n ¼ 400 nodes. Confidence intervals are not reported. This

figure shows two things. First, the simulation results validate

our model. Second, the function upper bounds the hittingtime correctly.

Fig. 9 shows the gain as a function of the bias level forR� ¼ 1 and R ¼ 0:3, 0.25, 0.2. The average progress isR ¼ 0:127, 0.106, 0.084, which corresponds to K ¼ 6, 8, 10.Since simulation results are equal to the analytical one, theyare not reported to simplify the plot. Again, the boundingfunction provides a useful approximation.

Finally, Fig. 10 reports the gain estimated by simulationas a function of the starting point for 400 nodes and R ¼ 0:3.When the starting point gets closer to the target, the gainincreases, as predicted by Property 4 of the previoussection.

4 HOW TO IMPLEMENT A BIASED RANDOM WALK

The previous sections provide us with a theoretical flavorabout the goodness of biased random walks for searching.As our analysis has shown, the hitting time is very sensibleto the bias level. Motivated by this encouraging result, wewill now drill down to a practical implementation of asearch algorithm. Before that, we discuss the possibleoptions for achieving bias and present general implementa-tion frameworks that adhere to these frameworks.


Fig. 7. Computation of the average progress for a far point.

Fig. 8. Validity of the bounding function for the hitting time. Hitting timeversus bias, n ¼ 400 nodes, R ¼ 0:3.

Fig. 9. Validity of the bounding function for the gain. Gain versus bias,n ¼ 400 nodes, transmission range R given as a parameter.

Fig. 10. Gain versus starting point (distance from the target); N ¼ 400,R ¼ 0:3.

In a random walk, the walker moves by making blindand memoryless random selections. The decision of the nextnode to visit is blind because the walker does not use anyexternal information to decide, and it is memoryless since ifthe walker visits two times the same node, it behaves thesame. The bias arises when such a fairly simple decisionmechanism is somehow altered in a way that the walker isstatistically “pushed” toward the target. In this paper, weexplore two orthogonal ways to bias the walk, dubbed asbias-by-information and bias-by-memory.

In Bias-by-information, the walker exploits informationavailable at the currently visited node, which indicates themost appropriate decision to take for reaching the target.The information used for deciding is maintained by someprotocol, which basically corresponds to the oracle used inour model.

In the literature, many examples fall in this category.In P2P architectures, where this technique is widely used,the information can consist of some topological knowl-edge, e.g., the connectivity degree of the neighbors of theselecting node; another option is leveraging on a learningprotocol that estimates the goodness of the candidatenodes, according to previous searches, see [10]. A routingprotocol can also be considered as a special case of thisclass. The routing tables represent the information storedat nodes while the routing protocol, which is in charge ofmaintaining the table up to date, corresponds to theoracle. In this particular case, the “search” becomesdeterministic.

In Bias-by-memory, the walker maintains memory of itsprevious selections, so that bias merely consists of forcing tovisit new portions of the network. No sense of direction isexploited. For example, in [3], visited nodes are hidden witha given probability.

Bias-by-information is potentially more effective thanbias-by-memory, especially if the number of target is low.For example, the best one can expect from bias-by-memoryis that at each step a new node is visited. If we assume that anewly visited node has the same probability of being thetarget, then the lowest average hitting is nþ1

2 , where n is totalthe number of candidate nodes (see Appendices A and B).The hitting time may be much shorter under bias-by-information, the lowest value being the average networkdistance of a requesting node from the target.

We now discuss two implementation frameworks thatallow one to design random walks of these two classes,along with their performance assessment.

4.1 Bias-by-Information ImplementationFrameworks

The first implementation strategy for an informed randomwalk is called ALLð�Þ. In this case, all nodes are equippedwith an oracle functionality, which tags all neighbors eitheras near or far. A near (far) neighbor is a node whosedistance from the target is less (higher) than the taggingone. The oracle is required to provide correct discrimina-tions with probability �; namely, a near (far) node iscorrectly tagged as near (far) with probability � andwrongly as far (near) with probability 1� �. The next nodeto visit is selected at random among the estimated nearnodes. If no nodes are tagged as near, then the selection is at

random among all nodes. A given bias level is obtained byregulating the estimation correctness, �.

The expected bias level of this framework can becomputed as follows: Let C be the average number of nearnodes of a randomly observed selecting node and F theaverage number of far nodes; then, the expected number ofnear nodes that the oracle correctly tags as such is �C, whilethe expected number of far nodes, wrongly tagged as nearnodes, is ð1� �ÞF . Hence, the probability that the selectednode is actually closer than the selecting one to the target,i.e., the bias level, is

� ¼ C�

ðC � F Þ� þ F :

Note that if the neighbors of a node are equally likely closeror farther to the target, i.e., F ¼ C, we obtain � ¼ �. Thismeans that the bias level corresponds to the estimationcorrectness of the oracle.

The second framework is called PARTIALðI; kÞ. Thisimplementation strategy requires only a subset I of nodes tobe equipped with the oracle, which correctly tagsminfk;NNeighg near nodes—NNeigh being the total numberof near neighbors. When the walker arrives at an informednode, i.e., one equipped with the oracle, it makes aninformed step by selecting one of the recognized nearnodes, at random. In this case, the bias level � is

� ¼ iþ C

C þ F ð1� iÞ;

where i is the probability that the selecting node isinformed. Assuming C ¼ F , we obtain

� ¼ 1þ i2

: ð9Þ

4.1.1 Evaluation

To assess the performance of these frameworks, we havesimulated a random walk over 400 nodes, placed atrandom inside a unit circle. The target occupies the centerof the area while the source node lies on the circumfer-ence. In order to also study the effect of node density, thetransmission range R is varied and takes the values of 0.2,0.25, or 0.3, which correspond to roughly 15, 23, and32 average neighbors per node, respectively; the numberof independent trials was 10,000 times. All the protocolsuse lookahead. The oracle is simulated according to theframework studied.

The results report the bias level, namely the number oftimes that, after a step, the walker reduced its distance fromthe target and total number of steps done before the target isreached as well as the correlation between bias level andhitting time/gain.

ALLð�Þ framework. Fig. 11 shows the bias level as afunction of �. The relationship is almost linear. Thediscrepancy from a line is due to the border effect and tothe fact that F > C.3 Note that for R ¼ 0:2 and � ¼ 1 weobtain � < 1, since some nodes have no near neighbors at all.


3. Far nodes fall into an area higher than the area where near nodesshould fall, see Fig. 5.

By approximating the relationship with a line, we obtaina useful rule of thumb. The bias level � directly maps to thecorrectness in the estimations performed by the oracle.

Fig. 12 shows the correlation among the hitting time/gain and the bias level. We can see that for all thetransmission range, the bias level greatly affects the searchperformance. The upper bounds predicted by the analysisare also reported as solid lines in the plot. Since theanalytical model assumes infinitely many nodes, the boundderived from the model becomes more and more correct asthe node density increases, i.e., R increases.

PARTIALðI;1Þ framework. Fig. 13 reports the bias as afunction of I. The bias level varies almost linearly with thenumber of informed nodes, meaning that, as a rule ofthumb, we can assume i ¼ I

N and use (9) to calculate thenumber of informed nodes required to achieve a given biaslevel �. More specifically, to achieve a bias level � thepercentage of informed nodes must be 2�� 1. For example,to obtain � ¼ 0:6, the fraction of informed nodes must be20 percent. Fig. 14 shows how the hitting time and the gainare correlated to the bias level.

Discussion. Figs. 12 and 14 put in evidence that the biaslevel has the same effect on the search performance,regardless the implementation strategy used to obtain it.Thus, the same search performance are achieved either byequipping all nodes with an imprecise oracle, which

recognizes some near nodes, or by equipping some nodewith a precise oracle that recognizes all its near nodes. As anexample, to obtain � ¼ 0:6 either all nodes should have anoracle that correctly tags near nodes with probability� ¼ 0:6 or 80 nodes must have a precise oracle, whichdetects all near neighbors.

PARTIALðI; kÞ framework. This implementation strat-egy requires a less “powerful” oracle, in the sense only asubset of near nodes must be known, not all. In theextreme case, we may ask the oracle to recognize just onenear node, i.e., k ¼ 1.

Fig. 15 shows the bias as a function of the number ofinformed nodes for such a framework. The bias firstincreases rapidly with I and then more slowly, meaningthat the usefulness of each new informed node depends onhow many informed nodes are already in the system. As faras the hitting time is concerned, see Fig. 16, the samerelationship previously found holds.

Reducing k ¼ 1 provides somehow a counterintuitiveresult. For a fixed I, having just one option is better thanhaving more chances. Having no options among which tochoose has a positive effect on the bias level. By inspectingthe simulation results, we saw that, on the average, theinformed nodes are encountered more frequently thanwhen all near nodes are detected. And, this increases thebias level, according to (9). Roughly speaking, the fact that


Fig. 11. Bias level versus estimation correctness.

Fig. 12. Hitting time and gain versus bias.

Fig. 13. Bias level versus number of informed nodes.

Fig. 14. Hitting time and gain versus bias.

an informed node always selects the same neighbor has theeffect of “hiding” to the walker the portion of the networkbehind the not selected neighbors. Although some informednode is also hidden, the overall effect is that the walkermeets less uninformed nodes.

To better understand how the number of detected nodesaffects the search performance, in Fig. 17, we report the biasFig. 17a and the hitting time Fig. 17b as a function of thedetected near nodes, k, when R ¼ 0:3 (the same results areobtained for other transmission ranges). We can see how thesearch performance becomes worst as k increases, especiallyif the percentage of informed nodes is high.

4.2 Bias-by-Memory Implementation Frameworks

To implement bias-by-memory, the identifiers of the visitednodes must be available at the walker. The first implemen-tation is based on a Memory List, carried by the walker.The list contains at most H different identifiers of the mostrecently visited nodes.

The next node to visit is selected at random among theneighbors that do not appear in the list. If all neighbors arein the list, the selection is at random among all neighbors.

The other implementation option is called Distribu-

ted Memory and corresponds to H that is equal to the

number of nodes. In this case, each node maintains a flagassociated to the walker that indicates if the node wasvisited or not. At selection time, a node probes the status ofall neighbors and sends the walker to one unvisited node,selected uniformly at random. The overhead due to probingcan easily be avoided by leveraging the broadcast nature oftransmissions (see Section 4.2.1).

4.2.1 Evaluation

Fig. 18 shows the bias induced by the first solution as afunction of H. The distributed memory option correspondsto H ¼ 400. Since we saw that the bias and the hitting timefor H � 100 are the same, the plot shows only results untilH ¼ 100. The bias level increases rapidly as a memorycapacity is added, and then it stabilizes to a value.

Consider that in this case the bias cannot be regulated.Rather, it is just a way to model the side effect of thememory. Nevertheless, see Fig. 19, a strong correlationbetween the hitting time and the bias exists.

5 PROPOSED PROTOCOL

After analyzing the characteristics of the frameworks, we arenow ready to describe our protocol. The protocol is designed


Fig. 15. Bias level versus informed nodes, k ¼ 1. Fig. 16. Hitting time and gain versus bias.

Fig. 17. (a) Bias level and (b) hitting time versus number of near nodes detected; transmission range R ¼ 0:3.

for mobile settings; it combines bias-by-information withbias-by-memory and uses one hop lookahead.

The information is computed by each node, i.e., nocentral authority is required, and consists of the estimatedeuclidian distance of a node from the target. Estimations arecarried out through a simple distributed algorithm, asdetailed later in this section.

The bias is embedded in the next hop’s selection logic inthe form of an effective distributed algorithm, which alsoencompasses lookahead. The protocol can be adapted tomultitarget search straightforwardly. We will first presenthow the biasing information is achieved—both for singletarget and multitarget searches. Then, we present theforwarding protocol, which implements the steps of therandom walk.

5.1 The Biasing Information Source

Basic idea. The information exploited for biasing is theestimated distance of the neighbor of the selecting nodefrom the target. Such a form of information is well suited formobile networks, exactly due to mobility. During itslifetime, the target node comes frequently in contact withother nodes, where a contact of i with j occurs and ireceives a packet, e.g., a beacon from j. The time elapsedsince the last contact, namely the contact time, provides anode with a rough indication about how far the target couldcurrently be. To exemplify, suppose that a node needs todecide to which of two neighbors, say a and b, to send thepacket; assume further that the target node sends a periodicbeacon every second and that a received the last beacon10 seconds ago while b received the last beacon 20 secondsago. It is clear that, unless movements are highly irregular, ais more likely to be closer than b to the target; thus, bias maybe introduced by selecting a. The above idea of contact timewas originally proposed in [9] and exploited in the LastEncounter Routing (LER) protocol, described in [25]. In thispaper, we adopt a slightly different estimation mechanism,which also takes an estimation of the relative speed intoaccount, see [6].

Implementation details. To estimate the distance be-tween a target node, say j, and another node, say i, thetarget node is required to send a beacon periodically, every�T s. The estimated distance from node i to j will bedenoted as dij. Initially, dij is set to unknown. As soon as ireceives a beacon from j, i sets dij ¼ 0 and starts countingthe number kij of consecutive beacons it is receiving.Moreover, it stores the time of the last beacon received into

tLBj. When i misses K consecutive beacons, i.e.,

t� tLBj> K�T , where t is the time when the Kth beacon

should have been received, i sets a local variable tij to t. tijrepresents the (estimated) time when j exited from the i’stransmission range. At time t � tLBj

þK�T , dij ¼ 0, whilefor t > tLBj

þK�T , it is

dijðtÞ ¼t� tijkij�T

:

The denominator kij�T represents the dwell time of jwithrespect to i, namely how long j remained in i’s transmissionrange. The above formula assumes that the dwell time isinversely proportional, in expectation, to the relative speedbetween the two nodes, see [12] and [6] for a deeperdiscussion. Appendix A reports an experimental evidenceabout the bias level the estimated distances may provide.

Bias in multitarget search. A multitarget search ariseswhen the target node is any node among a subset G of thenetwork nodes. The targets share a common unique ID,which is included into their beacons. For example, in SOA,the ID could be the description of the same service. diGðtÞdenotes the distance of node i from the set G. It is theminimum distance of i from a member of G, formallydiGðtÞ ¼ minfdijðtÞjj 2 Gg.

5.2 Packet Forwarding Algorithm with Lookahead

The forwarding algorithm aims at selecting the neighborclosest to G, which is not yet visited. It assumes that thewalking packet, say m, is uniquely identified and carries theG’s ID. Moreover, each node i is required to manage a locallist, FWDi, of the last packets it has forwarded. m 2 FWDi

means that node i has forwarded packet m.The idea of the protocol is simple. The selecting node

probes the neighbors with Request To Send (RTS) controlpackets and then sends m to the first neighbor that repliesback with a Clear To Send (CTS) packet. The CTS packet isemitted by a node after a suitable delay, which takes bias andlookahead into account. When a node hears that the selectingnode is sending m to another node, it aborts its own reply.

Implementation details. Let k be the selecting node. kbroadcasts an RTS control packet containing the G’s ID andm’s ID. On receiving such a control packet, a node ischedules the transmission of a CTS control packet after adelay �t. The following four cases are considered:


Fig. 18. Bias level versus history. Fig. 19. Search performance.

1. i 2 G, i.e., i is a target. In this case, �t ¼ 0.2. i 62 G, diG ¼ 0, m 62 FWDi, i.e., i is not a target, but

the target is likely a neighbor of i; moreover, i isgoing to forward the packet for the first time. In thiscase, �t ¼ randomð0; 1Þ, i.e.,�t is a uniform randomvalue in the range �0; 1½:

3. i 62 G, 0 < dij � H, m 62 FWDi, i.e., i is not a target,but it has a valid estimation of its distance from theclosest target; moreover, i is going to forward thepacket for the first time. In this case, �t ¼ 1 þð2 � 1Þ dijH , i.e., �t is proportional to a value in therange ½1; 2�.

4. None of the previous conditions are met.�t ¼ randomð2; 3Þ.

Node k sends the packet to the one from which the firstRTS is received. Thus, node k tries to select a neighboraccording to the following order:

1. a target node (lookahead);2. a new node (bias-by-memory) whose neighbor is

very likely the target (bias-by-information);3. a new node (bias-by-memory) with the highest

chance of being the closest one to the target (bias-by-information);

4. a node at random.

All nodes that overhear such transmissions delete theirscheduled CTS transmissions. Also, k ignores any subse-quent CTS packet it would receive.

6 AN EXPERIMENTAL EVALUATION

To assess the suitability of our proposal, we have conducteda simulation study by exploiting a custom discrete eventsimulator, already used in [6]. The simulator has thefollowing main characteristics. The transmission of a packetstarts after the channel is sensed free for a RandomAssessment Delay (RAD) randomly chosen in the range[0..500] ms; the packet reception is notified to a sender’sneighbor provided that it remained for the whole durationof the transmission within the transmission range and suchthat no collisions with other transmissions occurred; a FIFObuffer of 20 packets in size is used at each node. Nodesmove according to the round trip mobility model withwaypoints (RWP); we adapted the public code availablein [15]; the speed varies in the range ½1::vmax� m/s, there isno pause time. The main simulation and protocol para-meters are reported in Fig. 20.

The metrics of interests are: 1) the hitting time, measuredas the ratio of the number of times the packet isretransmitted before it hits the target with the total numberof successful random walks; 2) the gain with respect to theunbiased walk. They are estimated using five independentreplications and 95 percent confidence interval.

6.1 Baseline Assessment on the Circular Area

In this first set of experiments, the target node is static andpositioned at the center of the area, while the other nodesmove. A search is initiated either by the Farthest node(Biased-F), or by Any node (Biased-A).

For this topology, the bounds given in (7) and (8) shouldapply. We have 1þK ¼ dR��R

Re ¼ 9; thus, the hitting time

should at most be 91�, where � is the bias level. Fig. 21

shows such a predicted hitting time along with the hittingtime measured via simulations. The bias � used in theexpression was also estimated during the simulations. Itwas found that � � 0:54; 0:65, 0:58, and 0:59 for vmax ¼ 0,10, 30, and 50 m/s, respectively. When the network isstatic, the random walk is biased on the basis of thememory, as discussed in Section 4.2.

When mobility is added, the information starts also tobias the walk. The hitting time is then reduced until aminimum value. Increasing the speed further has twoeffects. From one hand, nodes come more often in contact


Fig. 20. Parameters of the experimental evaluation.

Fig. 21. Hitting time versus speed. Biased-F is the biased random walkwhen the farthest node starts the walk; Biased-A is the biased randomwalk when any node can start the walk. The upper bound predicted byour model is also reported.

with the target while from the other hand estimationsbecome more evanescing. The net effect is that informationbecomes less effective. The figure shows how the hittingtime is upper bounded by the theoretical value.

Fig. 22 reports the gain for the same setting. Thetheoretical gain of (8) is now a valid lower bound.

6.2 Results for the Square Area

Fig. 23 reports the average hitting time as a function of themaximum speed for the square area. In these experiments,the target node is also mobile. For a single target searchðjGj ¼ 1Þ, the bias level is highly affected by the informa-tion. When the network is static, bias is obtained byremembering the previous choices. Memory alone is ableto decrease the hitting time by a half. When mobility isadded, it decreases the other half. Thus, mobility indeedallows one to gather information useful to bias the search.

By increasing the maximum speed from 10 to 30 m/s, thehitting time for the singleton case increases; this is aconsequence of a reduced correctness of estimations.Directing the packet away from the region where the targetis currently located has a strong negative effect on thehitting time; roughly speaking, the packet has to returnback. By further increasing the speed, however, we canobserve that the hitting time for the singleton case starts

decreasing again. This can be explained considering that ahigher speed helps the target node to come more frequentlyin contact with other nodes, which can thus refresh theirestimations and make the bias stronger again.

For jGj > 1, a low correctness is less critical; rather, it turnsout that mobility helps in reducing the hitting time becausetargets have more chances of becoming neighbors of morenodes. Since increasing the speed does not have a significantimpact on the hitting time, we can deduce that the hitting timeis now determined by the bias-by-memory effect.

Compared to the natural random walk, the hitting isreduced considerably, e.g., roughly the hitting time passesfrom 90 to 30 for 10 m/s and one target only. Theimprovement over the natural walk is shown systematicallyin Fig. 24, in the form of gain. We can see how for all themobility conditions the hitting time reduced by at least a half.

7 RELATED WORK

Random walks are used in several algorithms proposed forwireless networks. In [4], the RAndom Walk-based Member-ship Service (RaWMS) for ad hoc networks is described. Theservice provides each node with a partial uniformly chosenview of network nodes. The algorithm uses random walk as asampling technique, whereas the aim of our protocol is tolocate a target. Dolev et al. [8] propose a randomized self-stabilizing full group membership service for ad hoc net-works. The group membership list is collected by a singlerandom walk agent traversing the network. They apply asingle random walk that covers the whole network, not forsearching. An efficient token passing algorithm is exploited inNASCENT to provide a network layer service dedicated togroup communication in ad hoc networks [18]. Again, thegoal of the random walk is not to perform a search.

Avin and Brito [3] apply what we have called bias-by-memory to query in sensor networks. A previously visitednode is hidden to subsequent selections with a givenprobability, which is called the bias of the walk. The workexploits only one form of bias. Differently from our case, noinformation is used.

In [1], nodes are allowed to choose the next hop among asmall subset selected at random. The authors discuss thepower of such a strategy for improving the performance ofa random walk considerably. Their results are consistent


Fig. 22. Gain time versus speed. Biased-F is the biased random walkwhen the farthest node starts the walk; Biased-A is the biased randomwalk when any node can start the walk. The lower bound predicted byour model is also reported.

Fig. 23. Impact of speed on the hitting time.

Fig. 24. Impact of speed on the gain.

with our findings, because making an informed choice is away to achieve a strong bias.

Random walk over wireless networks are also studiedfrom a graph theoretical point of view in several papers,e.g., [2] and [5]. However, all these studies focus onunbiased walks.

Biased random walks are widely adopted for search inunstructured P2P architectures, both in the form of bias-by-memory and bias-by-information. The interested readercan, for example, refer to [10] for a survey.

However, several key aspects make search in P2Pdifferent from search in wireless networks. First, thetopology of P2P networks is best modeled as a power-lawgraph, whereas wireless networks adhere to the randomgeometric graph model. Second, the channel model in thetwo networks is quite different. While in P2P nodes areconnected via unicast channels (a TCP connection), inwireless networks a transmission is inherently broadcast.The cost of implementing the same technique, like looka-head or next hop selection, is then quite different. Last, thechanges in the topology are strongly correlated; this makessome source of information, like the distance among nodes,meaningful only in mobile networks.

We remark that physicians often use random walks tomodel numerous dynamical processes that occur in nature,the most notorious being the Brownian motion. The workdescribed in [21] is of particular interest for our work. Itpresents a method for calculating the properties of biasedrandom walks on complex networks in general, and for asegment in particular. Specifically, the Mean First passageTime (MFT)—which is synonymous to the hitting time—iscomputed for a particle (the walker) moving on a segment ofsizeN under an external biasing field, discretely in space andtime (hopping). This paper shows how the behavior of thewalker, when it moves in the direction of the bias, changesfrom a diffusive regime to a drift one as a bias is applied. Thediffusive regime is characterized by the MFT, which growsasN2 and it is observed in the limit of a weak bias (it basicallycorresponds to a natural random walk). As the bias isincreased, the MFT varies linearly with N , i.e., the so-calleddrift regime arises. The results discussed in that paperprovide an interesting physical interpretation of our study.

Finally, the idea of using a random delay before sendinga packet has already been used in other protocols, includingcounter-based, distance-based, and position-based broad-casting schemes [11]. The basic idea is to collect duplicatepackets received from neighbors for a random period oftime after the first packet is received and to use knowledgefrom these packets to make a forwarding decision. Thecounter-based scheme exploits the total number of receivedduplicates, and the packet is forwarded if it is below acounter threshold. The distance-based scheme uses theminimum distance from the node to the sender of thesepackets, which is an estimation of the node’s additional(broadcast) coverage area, and the packet is forwarded if itis over a distance threshold. The location-based schemeleverages the precise location information to provide a moreaccurate estimation of the additional coverage area.

8 CONCLUSION

In this paper, we have studied the effect of bias on thehitting time for a random walk executed over wireless

networks. For a network with uniform node distributionand a circular symmetry, we have presented an analyticalstudy, which shows the sensibility of the hitting withrespect to bias; in particular, the hitting time varies as N1=�,where N is proportional to the radius of the circle and �,0:5 � � � 1, is a number expressing the biasing level of thewalk. We have then suggested a protocol that exploitsbiased random walk. A simple beacon-based biasingmechanism is used. The simulation study shows that thehitting time is reduced by at least a half with respect tonatural walks. The results are interpreted in the light of thetheoretical study.

APPENDIX ATo give experimental evidence of the bias level we canachieve by estimating the distance from the target, we havesimulated the following scenario. Four nodes, i, j, k, and gmove into a unitary square region, according to the randomwaypoint mobility model with an average speed of 30 m/sand no pause time. The transmission range of the nodes isR ¼ 0:3. Node j plays the role of target node, g is theselecting node while i and k are test nodes. When these twonodes are both neighbors of node g and they are both out ofthe j’s transmission range, their estimated and actualdistance from j, dij, dkj and d0ij, d0kj, respectively, areobserved. The following measurement are then performed:1) probability P ðHÞ that both dij and dkj are below thethreshold H; 2) probability PdwnðHÞ that the estimations arecorrect, namely, dij < dkj and d0ij < d0kj, given that they areboth below H. PdwnðHÞ measures the bias level. If g wouldselect the node with the lowest estimation, i.e., node i, thenthe packet will actually get closer to node j with probabilityPdwnðHÞ.

The results are reported in Fig. 25. When H is very low,the difference in the two estimations is not high and bothnodes are equally close to the target j. This, however,happens with a very low probability—see P ðHÞ in the plot.As H increases, the potential bias level also increases. Forvery high H, the bias level becomes � 0.6.

APPENDIX BConsider an ideal bias-by-memory search initiated by anode over n other nodes. Assume that the target replicationdegree is k and that each node can be targeted with thesame probability of 1=n. The hitting time can be computed


Fig. 25. Experimental evidence of the validity of the estimations.

exploiting the urn model. We have an urn with n balls, k of

which are black and the remaining n� k white. Balls are

extracted from the urn without replacement. The hitting

time corresponds to the average number of extractions we

need to perform before a black ball is extracted from the

urn. After j extractions, no black balls are extracted if all the

k balls are still in the urn. This can happen only if n� j � k.

The first black ball is in the urn with probability n�jn ¼ 1� j

n .

The second black ball is still in the urn with probabilityn�j�1n�1 ¼ 1� j

n ; . . . , the kth black ball is in the urn with

probability n�j�kþ1n�kþ1 ¼ 1� j

n�kþ1 . Thus,

F ðjjn; kÞ ¼Yk�1

i¼0

1� j

n� i

� �

is the probability that after j extractions none of the k black

balls are extracted, given n balls in total. The average

number of extraction before a black ball is picked, i.e., our

hitting time, is

htðn; kÞ ¼Xn�kþ1

j¼1

j F ðj� 1jn; kÞ � F ðijn; kÞ½ �:

Fig. 26 shows htð300; kÞ as a function of k. For k ¼ 1, the

effect of “bias” is quite light; it becomes significative as

k � 10.

REFERENCES

[1] A. Avin and B. Krishnamachari, “The Power of Choice in RandomWalks: An Empirical Study,” Proc. Ninth ACM/IEEE Int’l Symp.Modeling, Analysis, and Simulation of Wireless and Mobile Systems(MSWiM ’06), 2006.

[2] C. Avin and G. Ercal, “On the Cover Time of Random GeometricGraphs,” Proc. 32nd Int’l Colloquium Automata, Languages, andProgramming (ICALP ’05), 2005.

[3] C. Avin and C. Brito, “Efficient and Robust Query Processing inDynamic Environment Using Random Walk Techniques,” Proc.Third Int’l Symp. Information Processing in Sensor Networks(IPSN ’04), 2004.

[4] Z. Bar-Yossef, R. Friedman, and G. Kliot, “RaWMS—RandomWalk Based Lightweight Membership Service for WirelessAd Hoc Networks,” Proc. ACM MobiHoc, 2006.

[5] S. Boyd, A. Ghosh, B. Prabhakar, and D. Shah, “Gossip andMixing Times of Random Walks on Random Graphs,” Proc.SIAM Second Workshop Analytic Algorithmics and Combinatorics(ANALCO ’05), 2005.

[6] R. Beraldi, L. Querzoni, and R. Baldoni, “A Hint-Based Probabil-istic Protocol for Unicast Communications in MANETs,” ElsevierJ. Ad Hoc Networks, 2006.

[7] N. Chang and M. Liu, “Optimal Controlled Flooding Search inLarge Wireless Networks,” Proc. Third Int’l Symp. Modeling andOptimization in Mobile, Ad Hoc, and Wireless Networks (WIOPT ’05),2005.

[8] S. Dolev, E. Schiller, and J. Welch, “Random Walk for Self-Stabilizing Group Communication in Ad Hoc Networks,” Proc.21st ACM Symp. Principles of Distributed Computing (PODC ’02),2002.

[9] H. Dubois-Ferriere, M. Grossglauser, and M. Vetterli, “AgeMatters: Efficient Route Discovery in Mobile Ad Hoc NetworksUsing Encounter Ages,” Proc. ACM MobiHoc, 2004.

[10] C. Gkantsidis, M. Mihail, and A. Saberi, “Random Walks in Peer-to-Peer Networks: Algorithms and Evaluation,” Elsevier Perfor-mance Evaluation J., vol. 63, no. 3, Mar. 2006.

[11] S. Ni, Y. Tseng, Y. Chen, and J. Sheu, “The Broadcast StormProblem in a Mobile Ad Hoc Network,” Proc. ACM MobiCom,Aug. 1999.

[12] D.B. Johnson and D.A. Maltz, “Dynamic Source Routing in AdHoc Wireless Networking,” Mobile Computing, T. Iemielinski andH. Korth, eds., chapter 5, Kluwer Academic, 1996.

[13] J. Kopena et al., “Service-Based Computing on Manets: EnablingDynamic Interoperability of First Responders,” IEEE IntelligentSystems, Sept./Oct. 2005.

[14] G. Lau, M. Jaseemuddin, and G.M. Ravindran, “RAON: A P2PNetwork for MANET,” Proc. Second IFIP Int’l Conf. Wireless andOptical Comm. Networks (WOCN ’05), 2005.

[15] http://lrcwww.epfl.ch/RandomTrip/, 2008.[16] X. Li and J. Wu, “Searching Techniques in Peer-to-Peer Net-

works,” Handbook of Theoretical and Algorithmic Aspects of Sensor,Auerbach Publications, 2006.

[17] L. Lovazs, “Random Walks on Graphs: A Survey,” Combinatorics,Paul Erdos in Eighty, vol. 2, Janos Bolyai Math. Soc. Budapest, 1993.

[18] J. Luo and J.-P. Hubaux, “NASCENT: Network Layer Service forVicinity Ad-Hoc Groups,” Proc. First IEEE Conf. Sensor and Ad HocComm. and Networks (SECON ’04), 2004.

[19] Q. Lv, P. Cao, E. Cohen, K. Li, and S. Shenker, “Search andReplication in Unstructured Peer-to-Peer Networks,” Proc. 16thInt’l Conf. Supercomputing (ICS ’02), 2002.

[20] M. Gerla, C. Lindemann, and A. Rowstron, “Perspectives Work-shop: Peer-to-Peer Mobile Ad Hoc Networks—New ResearchIssues,” Proc. Dagstuhl Seminar, 2005.

[21] I. Goldhirsch and Y. Gefen, “Biased Random Walk on Networks,”Physical Rev. A, vol. 35, no. 3, Feb. 1987.

[22] C. Gkantisidis, M. Michail, and A. Saberi, “Random Walks inPeer-to-Peer Networks,” Proc. IEEE INFOCOM, 2004.

[23] M. Gunes and O. Spaniol, “Ant-Routing-Algorithm for MobileMulti-Hop Ad-Hoc Networks,” Proc. Int’l Workshop Ad HocNetworking (IWAHN ’02), 2002.

[24] C.E. Perkins, E.M. Royer, and S.R. Das, Ad-Hoc on Demand DistanceVector (AODV) Routing, IETF Internet draft, work in progress, July2008.

[25] N. Sarafijanovic-Djukic and M. Grossglauser, “Last EncounterRouting under Random Waypoint Mobility,” Proc. Third Int’lIFIP-TC6 Networking Conf. (NETWORKING ’04), May 2004.

[26] S. Vempala, Geometric Random Walks: A Survey, http://www-math.mit.edu/~vempala/papers/survey.pdf, 2008.

Roberto Beraldi received the Laurea degree incomputer science from the University of Calabria,Cosenza, Italy, in 1991 and the PhD degree incomputer science in 1996. He has been anassistant professor in the Dipartimento di Infor-matica e Sistemistica (DIS), Universita di Roma“La Sapienza,” since 2002. From 1996 to 2002,he was an expert in computer networks at theItalian’s National Institute of Statistica (ISTAT).He has published more than 50 peer-reviewed

papers in various fields including computer networks, wireless networks,and distributed systems. He participates in many research projects andregularly serves as a reviewer for international conferences and journalson the above areas. He was a program cochair of the First InternationalWorkshop on Dynamic Distributed Systems held in 2006.


Fig. 26. Average hitting time as a function of the number of targets, k, for

an ideal Full Memory random walk on n ¼ 300 nodes. This hitting time is

the lowest achievable one using bias-by-memory.

Biased random walks in uniform wireless networks

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