Biased Random Walks in Uniform Wireless Networks by linzhengnd

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




                                         Biased Random Walks in
                                        Uniform 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

C     ONTEXT of this study. To search for a node with known
     property is a basic recurrent problem arising in many
distributed applications. For example, in routing protocols
                                                                                                 Compared to flooding, a random walk search has a more
                                                                                                 fine-grained control of the search space, a higher adap-
                                                                                                 tiveness to termination conditions, and can naturally cope
for mobile wireless networks, e.g., [12] and [24], the searched                                  with failures or voluntary disconnections of nodes [22].
node is identified by its IP address, while in peer-to-peer                                      Examples of concrete exploitations of random walks in
(P2P) architectures by a key, associated to the object the node                                  wireless networks are found in the context of routing
stores [19], [16]. Searching is also a central functionality in a                                protocols for MANET—e.g., ANT [23], Hint-Based routing
service-oriented architecture (SOA), e.g., see [13]. With their
                                                                                                 [6], and most recently, in P2P over MANET, e.g., ROAN
shift toward wireless communication support, distributed
                                                                                                 [14]. Graph theoretical studies on random walks that are
systems are becoming more dynamic, and the search problem
is consequently becoming even more challenging. This paper                                       relevant for wireless networks have also recently appeared
focuses on the search problem in the context of mobile                                           in the literature, e.g., [5] and [2].
wireless networks, i.e., autonomous self-organizing net-                                             Biased random walks are random walks in which nodes
works composed of a set of wireless devices.                                                     have statistical preference to forward the walker toward the
    Broadly speaking, there are two approaches to face a                                         target. The clear advantage of a biased random walk is that
search problem: structured and unstructured. The former                                          it reduces the excepted number of steps before the target is
exploits a logical structure for guiding searches, e.g.,                                         reached, called the hitting time, significatively. However,
routing tables stored at nodes, Distributed Hash Tables                                          the bias level achievable in a real setting is limited, while
(DHTs), or centralized/distributed directories, while the                                        the implementation of any biasing mechanism comes at
latter does not leverage any logical organization in the                                         some additional cost. Thus, to understand the effect of bias
search space. To maintain the structure used to support a                                        on the hitting time is an important preliminary step for
search may become challenging in mobile networks since                                           deciding the practical benefit of a random walk-based
the mobility of nodes makes the topology of the network                                          search algorithm. The effect of bias on the hitting time when
also variable. For this reason, the unstructured approach is                                     the random walk is executed over a wireless network is the
regarded as an attractive alternative, as more deeply                                            subject of this paper.
discussed, for example, in [20].                                                                     Contributions of this work. The random walk we
    An unstructured search has to potentially explore the                                        consider exploits look-ahead one. The walker (a packet) is
whole network; as such, it is generally carried out by                                           forwarded from one node to a randomly chosen neighbor
flooding. Alternatively, random walks can be used.                                               until a neighbor of the target is found. To study the effect of
                                                                                                 bias analytically, we exploit a network model composed of
. The author is with the Dipartimento di Informatica e Sistemistica,                             infinitely many nodes located at uniformly random posi-
            `
  Universita di Roma “La Sapienza,” Via Ariosto 25, 00100 Rome, Italy.                           tions inside a circle. This model is useful to study the effect
  E-mail: beraldi@dis.uniroma1.it.                                                               of bias on a real network because during the lifetime of a
Manuscript received 17 Oct. 2007; revised 30 May 2008; accepted 29 Sept.                         walk the variation in the network topology can, at first
2008; published online 16 Oct. 2008.                                                             glance, be neglected.
For information on obtaining reprints of this article, please send e-mail to:
tmc@computer.org, and reference IEEECS Log Number TMC-2007-10-0315.                                  The contribution of this paper is twofold. First, we found
Digital Object Identifier no. 10.1109/TMC.2008.151.                                              that for the uniform model the hitting time is very sensible
                                                          1536-1233/09/$25.00 ß 2009 IEEE        Published by the IEEE CS, CASS, ComSoc, IES, & SPS
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BERALDI: BIASED RANDOM WALKS IN UNIFORM WIRELESS NETWORKS                                                                                                                     501




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

                                                                                               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
                                                                                                                                            h ði;jÞÀh ði;jÞ
                                                                                                 unbiased random walk, namely, g ði; jÞ ¼ 0:5 h0:5 ði;jÞ .
                                                                                               Definition 5. Return time. The return time is the expected
                                                                                                 number of steps in a random walk starting at a given node,
                                                                                                 before the same node is reached again.
Fig. 1. List of the main parameters and symbols used throughout this
                                                                                               Let us start by considering a -biased random over the finite
paper.
                                                                                               line ½0::NŠ. Without loss of generality, let N be the target
                                                                                               point. At each time step, the walker moves closer to the
to bias. In particular, we show that the hitting time is
                                                                                               target with probability  and away from the target with
conveniently upper bonded by N 1= , where N is propor-                                        probability 1 À . The extreme points are reflective barriers,
tional to the radius of the circle and , 0:5  1, is a                                        see Fig. 2. It is well known, e.g., [17], that for the unbiased
number expressing the bias level of the walk.                                                  case we have h0:5 ð0; NÞ ¼ N 2 while for  ¼ 1 it is trivial to
   Second, we present a random walk-based search                                               see that h1 ð0; NÞ ¼ N. Some obvious question then arises.
protocol designed for mobile networks that runs atop                                           According to which law the hitting time passes from the
the data link layer. The random walk is biased via a                                           quadratic to the linear behavior? Is there some simple rule
simple distance estimation mechanism, which generates a                                        of thumb that can be useful to characterize the maximum
bias  % 0:6 when the search is for a single target. We give                                   gain one can achieve under a given bias ? We now will
experimental evidence of the validity of our theoretical                                       give an answer to these questions.
study by simulating nodes moving inside a circle. Then,                                            To compute the hitting h ð0; NÞ, we follow the method
we present a simulation study for a more general setting,                                      illustrated in [17]. The hitting time can be written as
whose results can be interpreted in the light of the
theoretical expectations.                                                                                                                 X
                                                                                                                                          N

   Structure of this paper. The reminder of this paper is                                                               h ð0; NÞ ¼             h ðk; k À 1Þ;                ð1Þ
                                                                                                                                          k¼1
organized as follows: Section 2 studies the bias effect in one
dimension, while Section 3 considers a 2D circular deploy-                                     while the hitting time h ðk; k À 1Þ is one less than the expected
ment; Section 4 discusses the possible options for imple-                                      return time at position k of a -biased random walk over the
                                                                                               segment ½0::kŠ. The return time is the inverse of the stationary
menting a biased random walk; the proposed protocol is
                                                                                               probability of observing the walker at position k, see [17];
presented in Section 5 and the simulation results are given
                                                                                               thus, if i;k is the stationary probability that a walker over
in Section 6. Section 7 discusses the related work; conclu-                                    ½0; kŠ is observed at position i, we have
sions are finally given in Section 8.
                                                                                                                             h ðk; k À 1Þ ¼ À1 À 1
                                                                                                                                              k;k

2    EFFECT OF BIAS                 ON A      RANDOM WALK,                                     and
     SIMPLE CASE
                                                                                                                                                Xh
                                                                                                                                                N            i
A summary of the main symbols used throughout this                                                                     ht ð0; NÞ ¼ 1 þ               À1 À 1 :
                                                                                                                                                       k;k
paper is given in Fig. 1, while the definitions of the key                                                                                      k¼2
terms are now given.                                                                           The probability k;k is readily obtained by solving an
Definition 1. Bias level. A random walk has bias level  with                                  elementary Discrete Time Markov Chain (DTMC) with state
  respect to a given target, and it is called -biased random walk,                            space ½0; kŠ and transition probabilities
  if after a step the walker gets closer to the target with
                                                                                                               pi;iþ1 ¼ ; pi;iÀ1 ¼ 1 À  1                 i        k À 1;
  probability .
                                                                                                                 p0;1 ¼ pk;kÀ1 ¼ 1:
Definition 2. Unbiased random walk. A random walk with
  bias level  ¼ 0:5 is said to be unbiased.                                                   From the balance equation of such a DMTC, we can write
Definition 3. Hitting time. The hitting time h ði; jÞ of a -biased                                                                                
                                                                                                                                                1 À  kÀ1
  random walk is the expected number of steps before node j is                                                               À1
                                                                                                                              k;k   ¼   À1
                                                                                                                                         0;k
                                                                                                                                                  
  visited for the first time starting from node i.

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

and                                                                                              and then ht ð0; NÞ is closer to N 2 , while for N > N 0 ,
                                                                 kÀ1
                                                                                                 ht ð0; NÞ is closer to N=ð2 À 1Þ. For example, for 0 ¼ 0:505,
                                                              
                                         kÀ1           1 À ð1ÀÞkÀ1                             N 0 ¼ 100. The value N 0 is always finite; thus, eventually
                    À1
                     0;k    ¼1þ                     þ                  ;                         hitting time behaves like N=ð2 À 1Þ. This last aspect has the
                                    ð1 À ÞkÀ1            2 À 1
                                                                                                 following physical interpretation.
hence,
                                                                                                     Consider an infinite line and a particle (walker)
                                                          kÀ1                                   starting at point 0 and moving along the line, stepping
                       ð1 À ÞkÀ1 ð1ÀÞkÀ1 À 1                                                   of one unit to the right with probability  ! 0:5 and with
                 ¼1þ À1
                      k;k        þ             :
                          kÀ1       2 À 1                                                      probability 1 À  to the left, i.e., the particle is subject to
By substituting the above probability into (2), we obtain                                        an external field that pushes it right. On the average, the
                                                                                                 particle arrives at point N after the hitting time, ht ð0; NÞ.
                                                                  k
                                                                                                The ratio N=ht ð0; NÞ can then be interpreted as the
                                         X
                                         N À1
                                                ð1 À Þk ð1ÀÞk À 1
                 ht ð0; NÞ ¼ 1 þ                       þ                                        average drift speed in the direction of the field.
                                         k¼1
                                                   k      2 À 1                                    Now, using the function N=ð2 À 1Þ of Property 1, we can
that after some manipulation can be rewritten as                                                 see that the speed becomes 2 À 1. This value, 2 À 1, is
                                                                                                 interpreted as the characteristic speed, which is acquired by
                                     "         #
                   N À1         À 1 2 1 À  NÀ1                                                 the particle when it is observed at a distance sufficiently far
  ht ð0; NÞ ¼ 1 þ        þ2                     À1 : ð2Þ                                        from the origin (for N sufficiently high). When  becomes
                   2 À 1     2 À 1      
                                                                                                 closer and closer to 0.5, in order to observe the “character” of
                                                                                                 the walker we need to go farther and farther from the origin.
2.1 Discussion                                                                                       With this interpretation, the speed of an unbiased particle
Let us now summarize the properties of the biased                                                is 1=N. That is, the speed goes to 0 as N goes to infinity. In
random walk over the segment in terms of hitting time                                            other terms, for  ¼ 0:5, the random walk is unbiased and
and gain.                                                                                        there is no drift in the direction of the external field.
Property 1 (Character of the hitting time). ht ð0; NÞ ¼                                         Property 2 (Upper bound on the hitting time).
  Âð2À1Þ, for  such that 0:5 <  1.1
      N                                                                                            ht ð0; NÞ N 1= , for  such that 0:5  1 and N > N0 ,
Proof. In fact, for 0:5 <  1, the fraction 1À < 1; thus, for                                     where N0 is a constant.
                                             
   N ! 1, the third term of (2) tends to be a constant. t    u                                   Proof. For 0:5 <  < 1, N 1= increases with N more rapidly
   Under a formal point of view, the hitting time ht ð0; NÞ is                                    than 2À1 does; thus, N 1= > 2À1 , provided that N > N 0 ,
                                                                                                          N                        N

lower than the approximating values N=ð2 À 1Þ as well as                                          for some N 0 > 0. But, from Property 1 the hitting time,
the unbiased value, N 2 . The last term in (2) is in fact                                                                                N
                                                                                                   for 0:5 <  1, indeed grows as 2À1 , while for  ¼ 0:5
negative. Now,                                                                                     the bounding function and the correct hitting time are
                                  N=ð2 À 1Þ > N 2                                                 the same [17]. Since N0 can be derived from N 0 , the
                                                                                                   property holds.                                         u
                                                                                                                                                           t
until N reaches the value N 0 ¼ 1=ð20 À 1Þ. This is a
                                                                                                   The constant value N0 is derived numerically and is
crossover point. For N < N 0 ,
                                                                                                 N0 ¼ 5. As an example, Fig. 3a reports ht ð0; 9Þ and the
                    N=ð2 À 1Þ À HT > N 2 À ht ð0; NÞ                                           bounding function as  is varied.
                                                                                                                                                                                      1À2
                                                                                                 Property 3 (Lower bound on the gain). g ð0; NÞ ! 1 À N  ,
   1. Recall that f ¼ OðgÞ if and only if lim supt!1 jfðtÞj=gðtÞ ¼ k, for some                     for  such that 0:5  1 and N > N0 , where N0 is a
k ! 0, and f ¼ ÂðgÞ if f ¼ OðgÞ and g ¼ OðfÞ.                                                      constant.

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BERALDI: BIASED RANDOM WALKS IN UNIFORM WIRELESS NETWORKS                                                                                                             503


Proof. The property comes from the definition of the gain
  and Property 2.                                       u
                                                        t

   The results for N ¼ 9 are reported in Fig. 3b. A bias as
small as  ¼ 0:6 reduces the hitting time by approximatively
a 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 À h ð0; iÞ, e.g., see
   [17]. Since h0:5 ði; NÞ ¼ N 2 À i2 , we have to show that 1 À
   h ð0;NÞ      h ð0;NÞÀh ð0;iÞ
      N2    < 1 À  N 2 Ài2     .
                                                     i2   h ði;0Þ
        The inequality can be rewritten as 1 À N 2 > 1 À hðN;0Þ ,
                                   h ði;0Þ h ðN;0Þ
   which holds if and only if i2 > N 2 . But, h ð0; NÞ is
   given by (2), which varies linearly with N; thus, the
                        h ð0;NÞ
   function fðNÞ ¼  N 2 decreases with N and then the
   property holds.                                               u
                                                                 t                             Fig. 4. An example of a random walk on the symmetric setting we are
From Properties 3 and 4, we can derive the following                                           studying.
lemma:
                                                                                                   The random walk studied in the previous section is a
Lemma. The gain of random walk with bias , which starts from
                                                           1À2                                particular case of our geometric random walk; in fact, it is
  an arbitrary point of a segment ½0; NŠ, is at least 1 À N  ,
                                                                                               obtained for RÃ ¼ NR and discrete steps of length R. Thus,
  provided that N > N0 ¼ 5.
                                                                                               we can hope that the hitting time is also subject to a behavior
                                                                                               similar to the one summarized by the properties given in the
3    EFFECT OF            THE     BIAS      ON A      UNIFORM WIRELESS                         previous section. To see if such is the case, we now compute
     NETWORK                                                                                   the exact hitting time numerically and will then try to find a
In this section, we study the effect of bias on two dimensions.                                connection between the two random walks.
To study this problem analytically, we examine a circular area                                     Let ri be a random variable representing the distance of
of radius RÃ , which contains infinitely many nodes deployed                                   the walker at the ith step and fi ðrÞ its probability density
at random, i.e., the probability that a node occupies the                                      function (pdf). The target is reached when ri becomes zero,
infinitesimal squared area centered at ðx; yÞ with respect to a                                i.e., the target is point 0. After hitting the target, the walk
                           1
pair of Cartesian axis is RÃ2 dxdy. The target node occupies the                              remains alive but its distance does not change. This means
center of the circle. The transmission range of each node is R.                                that fi ð0Þ increases monotonically with i and f1 ð0Þ ¼ 1 (the
The neighbors of a node are the nodes located at distance at                                   walker eventually hits the target).
most R from itself. Nodes have no physical extension; thus, in                                     The probability that the distance becomes zero the first
the following the terms point and node are used inter-                                         time at the ith step is given by fiþ1 ð0Þ À fi ð0Þ; thus, the
changeably.                                                                                    hitting time of point 0 starting from r0 is
   The random walk we study exploits look-ahead one, i.e.,                                                                  X
the neighbors of the target will send the walker directly to                                                    hðr0 ; 0Þ ¼   i½fiþ1 ð0Þ À fi ð0ފ:
the target rather than making a random choice. Lookahead                                                                              i!0

is simple to implement in wireless transmissions (as                                              The functions fi ðÞ can be computed through the
detailed in our proposal) while it makes easier to compute                                     following recursive relationship:
the hitting time mathematically.
   Due to the circular symmetry, the random walk over the                                                                           ZþR
                                                                                                                                    r

circle induces a geometric random walk on the finite                                                                   fi ðrÞ ¼             fiÀ1 ðr0 Þpðrjr0 Þdr0 ;
segment ½0; Rà Š. This new random walk describes how the                                                                           rÀR
distance of the walker from the target varies. The walker
makes variable steps in the range ½ÀR; RŠ.2 For example,                                       where f0 ðrÞ ¼ ðr À r0 Þ is the pdf associated to the initial
Fig. 4 shows a walker started at distance RÃ . Steps are                                       position, r0 , and pðrjr0 Þ ¼ P rfri ¼ rjriÀ1 ¼ r0 g is the transi-
numbered progressively starting from zero; the distance                                        tion probability from point r0 to r. Such a transition pdf is
from the target at step i, ri , is reported on a vertical radius.                              only defined for jr0 À rj R. For the natural random walk,
The walker hits the target when r becomes zero. The circle                                     it can be expressed as (bias is introduced later in this
of radius R is the look-ahead area. The hitting time in the                                    section)
example is then nine.                                                                                   8
                                                                                                        < ðrÞ;                          0 r0 < R;
                                                                                                    0
  2. A geometric random walk starts at some point in Rn and, at each step,                     pðrjr Þ ¼ fðrjr0 Þ;                       R r0 < RÃ À R; ð3Þ
                                                                                                        :
moves to a neighboring point chosen according to some distribution that                                     fðrjr0 Þ þ fð2RÃ À rjr0 Þ þ; RÃ À R r0 RÃ :
depends only on the current point, see [26].


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                                                                                                 Fig. 6. Analysis versus simulation, a variable number of nodes deployed
                                                                                                 in a circular-shaped area of radius RÃ ¼ 1; the target node occupies the
                                                                                                 center, the random walk starts at distance RÃ .
Fig. 5. Variation of the euclidian distance after retransmitting a packet.
                                                                                                 by simulations. The number of nodes is varied. The
The function fðrjr0 Þ will be called the progress pdf. It                                        parameters are r0 ¼ RÃ , RÃ ¼ 1, and R ¼ 0:3. The analysis
expresses the probability that, after a retransmission, the                                      provides accurate results when the total number of nodes
distance of the walker from the target varies from r0 to r.                                      is higher than 300, which corresponds to approximatively
The value r À r0 is called the progress.                                                         24 neighbors per node. Note that, due to lookahead, in
   The last expression in (3) takes the border effect into                                       order to hit a target it is sufficient to reach any of its
account, i.e., the fact that the region is limited. To deal with                                 neighbors. Thus, the hitting time decreases as the network
such a limitation, we allow to select virtual points beyond                                      becomes more dense.
RÃ and map point RÃ þ x to RÃ À x, i.e., a virtual transition                                       Introducing bias. To include bias in our model, it is
to RÃ þ ðRÃ À rÞ ¼ 2RÃ À r generates the transition to r.                                        enough to modify the transition probability by a scaling
   Let us now compute the progress pdf. Consider the circle                                      factor. The new transition probability is p0 ðrjr0 Þ ¼ k1 pðrjr0 Þ if
c of radius R centered at the selecting node and let C1 ðC2 Þ                                    r < r0 and p0 ðrjr0 Þ ¼ k2 pðrjr0 Þ otherwise, where
be the circle of radius r0 ðrÞ centered at the target node (see
Fig. 5). When the selected node belongs to the arch                                                                                                        1À
                                                                                                               k1 ¼ R r0                   dr;   k2 ¼ R r0 þR            :             ð6Þ
delimited by the intersection of C2 with c (bold in the                                                                 r0 ÀR   pðrjr0 Þ               r0     pðrjr0 Þdr
figure), the distance of the packet varies from r0 to r. Let

ðr0 ; rÞ be the length of such an arch. Since nodes are                                         3.1 Discussion
uniformly deployed we can write                                                                  The random walk studied in the previous section is a
                                          
ðr0 ; rÞ                                              particular random walk on ½0; Rà Š, which makes only
                            fðrjr0 Þ ¼              ¼ 2c r;                                     discrete steps of fixed size ÆR. On the other hand, due to
                                           R2
                                                                                                 the circular symmetry, a random walk in 2D also induces a
where c is the angle under which the arch is seen from the                                      random on the segment ½0; Rà Š but with variable steps in the
target. Therefore, combining                                                                     range ½ÀR; RŠ. Is there some connection between the
                                                                                                 performance of the two random walks? To answer to this
                       ðr0 þ Rcosc Þ2 þ ðRsinc Þ2 ¼ r2
                                                                                                 question, we follow a fairly pragmatic approach. The value
with                                                                                             N in the bounding function for the discrete case is the
                                                                                                 hitting time for  ¼ 1. We will compute the “equivalent” N
                              rcosc ¼ r0 þ Rcosc ;                                             for the continuous random walk and then, a posteriori, test
we obtain                                                                                        if the function is still useful.
                                                                                                                     RR
                                               02                                              Let then Ri ¼ r0 ÀR rfðrjr0 Þdr be the average positive
                    1 0 r2 À r02 À R2              r þ r2 À R2                                   progress at the ith step of the walk, i.e., the expected
c ¼ arccos            r þ               ¼ arccos
                    r        2r0                       2r0 r                                     progress when the selected point is closer than the selecting
                                                                                        ð4Þ                                is N ¼
                                                                                                 PK The equivalent N PKÀ1 K þ 1, where K is such that
                                                                                                 one.
                                                                                                              Ã                           Ã
                                                                                                    i¼1 Ri > R À R and        i¼1 Ri < R À R; in other words,
so that                                                                                          K is the minimum number of steps after which the distance
                                              02                                               becomes less than R (recall that due to lookahead the
                                   2r         r þ r2 À R2
                    fðrjr0 Þ ¼        arccos                :                           ð5Þ      walker hits the target after its distance becomes less than R).
                                  R2             2r0 r
                                                                                                 To simplify the computation, we consider Ri ¼ R, where R
   Model validation. Fig. 6 compares the hitting time                                            is the average progress observed when the packet is very far
                                                                                                                                        Ã
                                                                                                                                          ÀR
calculated by the model against the hitting time estimated                                       from the target. Hence, K ¼ dR R e. For r ! 1, the

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BERALDI: BIASED RANDOM WALKS IN UNIFORM WIRELESS NETWORKS                                                                                                            505




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


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

probability that the progress is x is given by the length of                                   Fig. 10. Gain versus starting point (distance from the target); N ¼ 400,
theffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi at distance x from the selecting node, that is,                            R ¼ 0:3.
 p chordffi
2 R2 À x2 , divided half of the area of the circle, see
Fig. 7; thus,                                                                                  our model. Second, the function upper bounds the hitting
                                                                                               time correctly.
                                   ZR pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
                                                      ffi                                           Fig. 9 shows the gain as a function of the bias level for
                        4                               4R
                    R¼               x R2 À x2 dx ¼        :                                   RÃ ¼ 1 and R ¼ 0:3, 0.25, 0.2. The average progress is
                       R2                              3                                     R ¼ 0:127, 0.106, 0.084, which corresponds to K ¼ 6, 8, 10.
                                   0
                                                                                               Since simulation results are equal to the analytical one, they
  Then, we conjecture the hitting time and the gain are                                        are not reported to simplify the plot. Again, the bounding
bounded, respectively, by                                                                      function provides a useful approximation.
                                                   1                                          Finally, Fig. 10 reports the gain estimated by simulation
                                              RÃ À R                                          as a function of the starting point for 400 nodes and R ¼ 0:3.
                       h ðr; 0Þ           1þ            ;                            ð7Þ
                                                R                                              When the starting point gets closer to the target, the gain
                                                                                               increases, as predicted by Property 4 of the previous
                                       Ã   1À2                                            section.
                                        R ÀR     
                   g ðr; 0Þ ! 1 À 1 þ              :                                 ð8Þ
                                           R
                                                                                               4      HOW       TO IMPLEMENT A                BIASED RANDOM WALK
These bounds are the equivalent of Property 2 and
Property 3. Fig. 8 shows the hitting time as a function of                                     The previous sections provide us with a theoretical flavor
                                                                                               about the goodness of biased random walks for searching.
the bias level for RÃ ¼ 1 and R ¼ 0:3. The figure reports: the
                                                                                               As our analysis has shown, the hitting time is very sensible
upper bound function—with these values, K ¼ 6; the hitting
                                                                                               to the bias level. Motivated by this encouraging result, we
time computed with the biased transition probabilities—see                                     will now drill down to a practical implementation of a
(6); and the hitting time estimated via simulations for                                        search algorithm. Before that, we discuss the possible
n ¼ 400 nodes. Confidence intervals are not reported. This                                     options for achieving bias and present general implementa-
figure shows two things. First, the simulation results validate                                tion frameworks that adhere to these frameworks.

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    In a random walk, the walker moves by making blind                                           random among all nodes. A given bias level is obtained by
and memoryless random selections. The decision of the next                                       regulating the estimation correctness, .
node to visit is blind because the walker does not use any                                           The expected bias level of this framework can be
external information to decide, and it is memoryless since if                                    computed as follows: Let C be the average number of near
the walker visits two times the same node, it behaves the                                        nodes of a randomly observed selecting node and F the
same. The bias arises when such a fairly simple decision                                         average number of far nodes; then, the expected number of
mechanism is somehow altered in a way that the walker is                                         near nodes that the oracle correctly tags as such is C, while
statistically “pushed” toward the target. In this paper, we                                      the expected number of far nodes, wrongly tagged as near
explore two orthogonal ways to bias the walk, dubbed as                                          nodes, is ð1 À ÞF . Hence, the probability that the selected
bias-by-information and bias-by-memory.                                                          node is actually closer than the selecting one to the target,
    In Bias-by-information, the walker exploits information                                      i.e., the bias level, is
available at the currently visited node, which indicates the
most appropriate decision to take for reaching the target.                                                                                  C
                                                                                                                                 ¼                  :
The information used for deciding is maintained by some                                                                                ðC À F Þ þ F
protocol, which basically corresponds to the oracle used in
                                                                                                 Note that if the neighbors of a node are equally likely closer
our model.
                                                                                                 or farther to the target, i.e., F ¼ C, we obtain  ¼ . This
    In the literature, many examples fall in this category.
                                                                                                 means that the bias level corresponds to the estimation
In P2P architectures, where this technique is widely used,
                                                                                                 correctness of the oracle.
the information can consist of some topological knowl-
                                                                                                    The second framework is called PARTIALðI; kÞ. This
edge, e.g., the connectivity degree of the neighbors of the
                                                                                                 implementation strategy requires only a subset I of nodes to
selecting node; another option is leveraging on a learning
                                                                                                 be equipped with the oracle, which correctly tags
protocol that estimates the goodness of the candidate
nodes, according to previous searches, see [10]. A routing                                       minfk; NNeigh g near nodes—NNeigh being the total number
protocol can also be considered as a special case of this                                        of near neighbors. When the walker arrives at an informed
class. The routing tables represent the information stored                                       node, i.e., one equipped with the oracle, it makes an
at nodes while the routing protocol, which is in charge of                                       informed step by selecting one of the recognized near
maintaining the table up to date, corresponds to the                                             nodes, at random. In this case, the bias level  is
oracle. In this particular case, the “search” becomes                                                                                      C
deterministic.                                                                                                                ¼iþ            ð1 À iÞ;
                                                                                                                                          CþF
    In Bias-by-memory, the walker maintains memory of its
previous selections, so that bias merely consists of forcing to                                  where i is the probability that the selecting node is
visit new portions of the network. No sense of direction is                                      informed. Assuming C ¼ F , we obtain
exploited. For example, in [3], visited nodes are hidden with
                                                                                                                                              1þi
a given probability.                                                                                                                   ¼         :                                    ð9Þ
                                                                                                                                               2
    Bias-by-information is potentially more effective than
bias-by-memory, especially if the number of target is low.
For example, the best one can expect from bias-by-memory                                         4.1.1 Evaluation
is that at each step a new node is visited. If we assume that a                                  To assess the performance of these frameworks, we have
newly visited node has the same probability of being the                                         simulated a random walk over 400 nodes, placed at
target, then the lowest average hitting is nþ1 , where n is total
                                             2
                                                                                                 random inside a unit circle. The target occupies the center
the number of candidate nodes (see Appendices A and B).                                          of the area while the source node lies on the circumfer-
The hitting time may be much shorter under bias-by-                                              ence. In order to also study the effect of node density, the
information, the lowest value being the average network                                          transmission range R is varied and takes the values of 0.2,
distance of a requesting node from the target.                                                   0.25, or 0.3, which correspond to roughly 15, 23, and
    We now discuss two implementation frameworks that                                            32 average neighbors per node, respectively; the number
allow one to design random walks of these two classes,                                           of independent trials was 10,000 times. All the protocols
along with their performance assessment.                                                         use lookahead. The oracle is simulated according to the
                                                                                                 framework studied.
4.1   Bias-by-Information Implementation
                                                                                                    The results report the bias level, namely the number of
      Frameworks
                                                                                                 times that, after a step, the walker reduced its distance from
The first implementation strategy for an informed random                                         the target and total number of steps done before the target is
walk is called ALLðÞ. In this case, all nodes are equipped                                      reached as well as the correlation between bias level and
with an oracle functionality, which tags all neighbors either                                    hitting time/gain.
as near or far. A near (far) neighbor is a node whose                                               ALLðÞ framework. Fig. 11 shows the bias level as a
distance from the target is less (higher) than the tagging
                                                                                                 function of . The relationship is almost linear. The
one. The oracle is required to provide correct discrimina-
                                                                                                 discrepancy from a line is due to the border effect and to
tions with probability ; namely, a near (far) node is
                                                                                                 the fact that F > C.3 Note that for R ¼ 0:2 and  ¼ 1 we
correctly tagged as near (far) with probability  and
                                                                                                 obtain  < 1, since some nodes have no near neighbors at all.
wrongly as far (near) with probability 1 À . The next node
to visit is selected at random among the estimated near                                             3. Far nodes fall into an area higher than the area where near nodes
nodes. If no nodes are tagged as near, then the selection is at                                  should fall, see Fig. 5.


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BERALDI: BIASED RANDOM WALKS IN UNIFORM WIRELESS NETWORKS                                                                                                           507




Fig. 11. Bias level versus estimation correctness.                                            Fig. 13. Bias level versus number of informed nodes.




Fig. 12. Hitting time and gain versus bias.                                                   Fig. 14. Hitting time and gain versus bias.


   By approximating the relationship with a line, we obtain                                   recognizes some near nodes, or by equipping some node
a useful rule of thumb. The bias level  directly maps to the                                 with a precise oracle that recognizes all its near nodes. As an
correctness in the estimations performed by the oracle.                                       example, to obtain  ¼ 0:6 either all nodes should have an
   Fig. 12 shows the correlation among the hitting time/                                      oracle that correctly tags near nodes with probability
gain and the bias level. We can see that for all the                                           ¼ 0:6 or 80 nodes must have a precise oracle, which
transmission range, the bias level greatly affects the search                                 detects all near neighbors.
performance. The upper bounds predicted by the analysis                                          PARTIALðI; kÞ framework. This implementation strat-
are also reported as solid lines in the plot. Since the                                       egy requires a less “powerful” oracle, in the sense only a
analytical model assumes infinitely many nodes, the bound                                     subset of near nodes must be known, not all. In the
derived from the model becomes more and more correct as                                       extreme case, we may ask the oracle to recognize just one
the node density increases, i.e., R increases.                                                near node, i.e., k ¼ 1.
   PARTIALðI; 1Þ framework. Fig. 13 reports the bias as a                                        Fig. 15 shows the bias as a function of the number of
function of I. The bias level varies almost linearly with the                                 informed nodes for such a framework. The bias first
number of informed nodes, meaning that, as a rule of                                          increases rapidly with I and then more slowly, meaning
                              I
thumb, we can assume i ¼ N and use (9) to calculate the                                       that the usefulness of each new informed node depends on
number of informed nodes required to achieve a given bias                                     how many informed nodes are already in the system. As far
level . More specifically, to achieve a bias level  the                                     as the hitting time is concerned, see Fig. 16, the same
percentage of informed nodes must be 2 À 1. For example,                                     relationship previously found holds.
to obtain  ¼ 0:6, the fraction of informed nodes must be                                        Reducing k ¼ 1 provides somehow a counterintuitive
20 percent. Fig. 14 shows how the hitting time and the gain                                   result. For a fixed I, having just one option is better than
are correlated to the bias level.                                                             having more chances. Having no options among which to
   Discussion. Figs. 12 and 14 put in evidence that the bias                                  choose has a positive effect on the bias level. By inspecting
level has the same effect on the search performance,                                          the simulation results, we saw that, on the average, the
regardless the implementation strategy used to obtain it.                                     informed nodes are encountered more frequently than
Thus, the same search performance are achieved either by                                      when all near nodes are detected. And, this increases the
equipping all nodes with an imprecise oracle, which                                           bias level, according to (9). Roughly speaking, the fact that

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

an informed node always selects the same neighbor has the                                        number of nodes. In this case, each node maintains a flag
effect of “hiding” to the walker the portion of the network                                      associated to the walker that indicates if the node was
behind the not selected neighbors. Although some informed                                        visited or not. At selection time, a node probes the status of
node is also hidden, the overall effect is that the walker                                       all neighbors and sends the walker to one unvisited node,
meets less uninformed nodes.                                                                     selected uniformly at random. The overhead due to probing
    To better understand how the number of detected nodes                                        can easily be avoided by leveraging the broadcast nature of
affects the search performance, in Fig. 17, we report the bias                                   transmissions (see Section 4.2.1).
Fig. 17a and the hitting time Fig. 17b as a function of the
detected near nodes, k, when R ¼ 0:3 (the same results are                                       4.2.1 Evaluation
obtained for other transmission ranges). We can see how the                                      Fig. 18 shows the bias induced by the first solution as a
search performance becomes worst as k increases, especially                                      function of H. The distributed memory option corresponds
if the percentage of informed nodes is high.                                                     to H ¼ 400. Since we saw that the bias and the hitting time
                                                                                                 for H ! 100 are the same, the plot shows only results until
4.2 Bias-by-Memory Implementation Frameworks                                                     H ¼ 100. The bias level increases rapidly as a memory
To implement bias-by-memory, the identifiers of the visited                                      capacity is added, and then it stabilizes to a value.
nodes must be available at the walker. The first implemen-                                          Consider that in this case the bias cannot be regulated.
tation is based on a Memory List, carried by the walker.                                         Rather, it is just a way to model the side effect of the
The list contains at most H different identifiers of the most                                    memory. Nevertheless, see Fig. 19, a strong correlation
recently visited nodes.                                                                          between the hitting time and the bias exists.
   The next node to visit is selected at random among the
neighbors that do not appear in the list. If all neighbors are
in the list, the selection is at random among all neighbors.                                     5      PROPOSED PROTOCOL
   The other implementation option is called Distribu-                                           After analyzing the characteristics of the frameworks, we are
ted Memory and corresponds to H that is equal to the                                             now ready to describe our protocol. The protocol is designed




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


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BERALDI: BIASED RANDOM WALKS IN UNIFORM WIRELESS NETWORKS                                                                                                           509




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

for mobile settings; it combines bias-by-information with                                     tLBj . When i misses K consecutive beacons, i.e.,
bias-by-memory and uses one hop lookahead.                                                    t À tLBj > KÁT , where t is the time when the Kth beacon
   The information is computed by each node, i.e., no                                         should have been received, i sets a local variable tij to t. tij
central authority is required, and consists of the estimated                                  represents the (estimated) time when j exited from the i’s
euclidian distance of a node from the target. Estimations are                                 transmission range. At time t tLBj þ KÁT , dij ¼ 0, while
carried out through a simple distributed algorithm, as                                        for t > tLBj þ KÁT , it is
detailed later in this section.
   The bias is embedded in the next hop’s selection logic in
the form of an effective distributed algorithm, which also                                                                                   t À tij
                                                                                                                                 dij ðtÞ ¼           :
encompasses lookahead. The protocol can be adapted to                                                                                        kij ÁT
multitarget search straightforwardly. We will first present
how the biasing information is achieved—both for single
target and multitarget searches. Then, we present the                                            The denominator kij ÁT represents the dwell time of j with
forwarding protocol, which implements the steps of the                                        respect to i, namely how long j remained in i’s transmission
random walk.                                                                                  range. The above formula assumes that the dwell time is
                                                                                              inversely proportional, in expectation, to the relative speed
5.1 The Biasing Information Source                                                            between the two nodes, see [12] and [6] for a deeper
Basic idea. The information exploited for biasing is the                                      discussion. Appendix A reports an experimental evidence
estimated distance of the neighbor of the selecting node                                      about the bias level the estimated distances may provide.
from the target. Such a form of information is well suited for                                   Bias in multitarget search. A multitarget search arises
mobile networks, exactly due to mobility. During its                                          when the target node is any node among a subset G of the
lifetime, the target node comes frequently in contact with                                    network nodes. The targets share a common unique ID,
other nodes, where a contact of i with j occurs and i                                         which is included into their beacons. For example, in SOA,
receives a packet, e.g., a beacon from j. The time elapsed                                    the ID could be the description of the same service. diG ðtÞ
since the last contact, namely the contact time, provides a                                   denotes the distance of node i from the set G. It is the
node with a rough indication about how far the target could                                   minimum distance of i from a member of G, formally
currently be. To exemplify, suppose that a node needs to                                      diG ðtÞ ¼ minfdij ðtÞjj 2 Gg.
decide to which of two neighbors, say a and b, to send the
                                                                                              5.2 Packet Forwarding Algorithm with Lookahead
packet; assume further that the target node sends a periodic
beacon every second and that a received the last beacon                                       The forwarding algorithm aims at selecting the neighbor
10 seconds ago while b received the last beacon 20 seconds                                    closest to G, which is not yet visited. It assumes that the
ago. It is clear that, unless movements are highly irregular, a                               walking packet, say m, is uniquely identified and carries the
is more likely to be closer than b to the target; thus, bias may                              G’s ID. Moreover, each node i is required to manage a local
be introduced by selecting a. The above idea of contact time                                  list, F W Di , of the last packets it has forwarded. m 2 F W Di
was originally proposed in [9] and exploited in the Last                                      means that node i has forwarded packet m.
Encounter Routing (LER) protocol, described in [25]. In this                                      The idea of the protocol is simple. The selecting node
paper, we adopt a slightly different estimation mechanism,                                    probes the neighbors with Request To Send (RT S) control
which also takes an estimation of the relative speed into                                     packets and then sends m to the first neighbor that replies
account, see [6].                                                                             back with a Clear To Send (CT S) packet. The CT S packet is
    Implementation details. To estimate the distance be-                                      emitted by a node after a suitable delay, which takes bias and
tween a target node, say j, and another node, say i, the                                      lookahead into account. When a node hears that the selecting
target node is required to send a beacon periodically, every                                  node is sending m to another node, it aborts its own reply.
ÁT s. The estimated distance from node i to j will be                                             Implementation details. Let k be the selecting node. k
denoted as dij . Initially, dij is set to unknown. As soon as i                               broadcasts an RT S control packet containing the G’s ID and
receives a beacon from j, i sets dij ¼ 0 and starts counting                                  m’s ID. On receiving such a control packet, a node i
the number kij of consecutive beacons it is receiving.                                        schedules the transmission of a CT S control packet after a
Moreover, it stores the time of the last beacon received into                                 delay t. The following four cases are considered:


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Fig. 20. Parameters of the experimental evaluation.

       i 2 G, i.e., i is a target. In this case, t ¼ 0.
      1.                                                                                            The metrics of interests are: 1) the hitting time, measured
       i 62 G, diG ¼ 0, m 62 F W Di , i.e., i is not a target, but
      2.                                                                                         as the ratio of the number of times the packet is
       the target is likely a neighbor of i; moreover, i is                                      retransmitted before it hits the target with the total number
       going to forward the packet for the first time. In this                                   of successful random walks; 2) the gain with respect to the
       case, t ¼ randomð0; 1 Þ, i.e.,t is a uniform random                                    unbiased walk. They are estimated using five independent
       value in the range Š0; 1 ½:                                                              replications and 95 percent confidence interval.
   3. i 62 G, 0 < dij H, m 62 F W Di , i.e., i is not a target,
       but it has a valid estimation of its distance from the                                    6.1 Baseline Assessment on the Circular Area
       closest target; moreover, i is going to forward the                                       In this first set of experiments, the target node is static and
       packet for the first time. In this case, t ¼ 1 þ                                        positioned at the center of the area, while the other nodes
                  dij
       ð2 À 1 Þ H , i.e., t is proportional to a value in the                                 move. A search is initiated either by the Farthest node
       range ½1 ; 2 Š.                                                                         (Biased-F), or by Any node (Biased-A).
   4. None of the previous conditions are met.                                                      For this topology, the bounds given in (7) and (8) should
       t ¼ randomð2 ; 3 Þ.                                                                                                  Ã
                                                                                                                                 ÀR
                                                                                                 apply. We have 1 þ K ¼ dR R e ¼ 9; thus, the hitting time
                                                                                                                         1
   Node k sends the packet to the one from which the first                                       should at most be 9 , where  is the bias level. Fig. 21
RT S is received. Thus, node k tries to select a neighbor                                        shows such a predicted hitting time along with the hitting
according to the following order:                                                                time measured via simulations. The bias  used in the
       a target node (lookahead);
      1.                                                                                         expression was also estimated during the simulations. It
       a new node (bias-by-memory) whose neighbor is
      2.                                                                                         was found that  % 0:54; 0:65, 0:58, and 0:59 for vmax ¼ 0,
       very likely the target (bias-by-information);                                             10, 30, and 50 m/s, respectively. When the network is
   3. a new node (bias-by-memory) with the highest                                               static, the random walk is biased on the basis of the
       chance of being the closest one to the target (bias-                                      memory, as discussed in Section 4.2.
       by-information);                                                                             When mobility is added, the information starts also to
   4. a node at random.                                                                          bias the walk. The hitting time is then reduced until a
   All nodes that overhear such transmissions delete their                                       minimum value. Increasing the speed further has two
scheduled CT S transmissions. Also, k ignores any subse-                                         effects. From one hand, nodes come more often in contact
quent CT S packet it would receive.


6      AN EXPERIMENTAL EVALUATION
To assess the suitability of our proposal, we have conducted
a simulation study by exploiting a custom discrete event
simulator, already used in [6]. The simulator has the
following main characteristics. The transmission of a packet
starts after the channel is sensed free for a Random
Assessment Delay (RAD) randomly chosen in the range
[0..500] ms; the packet reception is notified to a sender’s
neighbor provided that it remained for the whole duration
of the transmission within the transmission range and such
that no collisions with other transmissions occurred; a FIFO
buffer of 20 packets in size is used at each node. Nodes
move according to the round trip mobility model with
waypoints (RWP); we adapted the public code available
                                                                                                 Fig. 21. Hitting time versus speed. Biased-F is the biased random walk
in [15]; the speed varies in the range ½1::vmax Š m/s, there is                                  when the farthest node starts the walk; Biased-A is the biased random
no pause time. The main simulation and protocol para-                                            walk when any node can start the walk. The upper bound predicted by
meters are reported in Fig. 20.                                                                  our model is also reported.


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BERALDI: BIASED RANDOM WALKS IN UNIFORM WIRELESS NETWORKS                                                                                                           511




Fig. 22. Gain time versus speed. Biased-F is the biased random walk                           Fig. 24. Impact of speed on the gain.
when the farthest node starts the walk; Biased-A is the biased random
walk when any node can start the walk. The lower bound predicted by
our model is also reported.                                                                   decreasing again. This can be explained considering that a
                                                                                              higher speed helps the target node to come more frequently
with the target while from the other hand estimations                                         in contact with other nodes, which can thus refresh their
become more evanescing. The net effect is that information                                    estimations and make the bias stronger again.
becomes less effective. The figure shows how the hitting                                          For jGj > 1, a low correctness is less critical; rather, it turns
time is upper bounded by the theoretical value.                                               out that mobility helps in reducing the hitting time because
   Fig. 22 reports the gain for the same setting. The                                         targets have more chances of becoming neighbors of more
theoretical gain of (8) is now a valid lower bound.                                           nodes. Since increasing the speed does not have a significant
                                                                                              impact on the hitting time, we can deduce that the hitting time
6.2 Results for the Square Area                                                               is now determined by the bias-by-memory effect.
Fig. 23 reports the average hitting time as a function of the                                     Compared to the natural random walk, the hitting is
maximum speed for the square area. In these experiments,                                      reduced considerably, e.g., roughly the hitting time passes
the target node is also mobile. For a single target search                                    from 90 to 30 for 10 m/s and one target only. The
ðjGj ¼ 1Þ, the bias level is highly affected by the informa-                                  improvement over the natural walk is shown systematically
tion. When the network is static, bias is obtained by                                         in Fig. 24, in the form of gain. We can see how for all the
remembering the previous choices. Memory alone is able                                        mobility conditions the hitting time reduced by at least a half.
to decrease the hitting time by a half. When mobility is
added, it decreases the other half. Thus, mobility indeed
allows one to gather information useful to bias the search.                                   7      RELATED WORK
   By increasing the maximum speed from 10 to 30 m/s, the                                     Random walks are used in several algorithms proposed for
hitting time for the singleton case increases; this is a                                      wireless networks. In [4], the RAndom Walk-based Member-
consequence of a reduced correctness of estimations.                                          ship Service (RaWMS) for ad hoc networks is described. The
Directing the packet away from the region where the target                                    service provides each node with a partial uniformly chosen
is currently located has a strong negative effect on the                                      view of network nodes. The algorithm uses random walk as a
hitting time; roughly speaking, the packet has to return                                      sampling technique, whereas the aim of our protocol is to
back. By further increasing the speed, however, we can                                        locate a target. Dolev et al. [8] propose a randomized self-
observe that the hitting time for the singleton case starts                                   stabilizing full group membership service for ad hoc net-
                                                                                              works. The group membership list is collected by a single
                                                                                              random walk agent traversing the network. They apply a
                                                                                              single random walk that covers the whole network, not for
                                                                                              searching. An efficient token passing algorithm is exploited in
                                                                                              NASCENT to provide a network layer service dedicated to
                                                                                              group communication in ad hoc networks [18]. Again, the
                                                                                              goal 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 visited
                                                                                              node is hidden to subsequent selections with a given
                                                                                              probability, which is called the bias of the walk. The work
                                                                                              exploits only one form of bias. Differently from our case, no
                                                                                              information is used.
                                                                                                 In [1], nodes are allowed to choose the next hop among a
                                                                                              small subset selected at random. The authors discuss the
                                                                                              power of such a strategy for improving the performance of
Fig. 23. Impact of speed on the hitting time.                                                 a random walk considerably. Their results are consistent

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with our findings, because making an informed choice is a
way to achieve a strong bias.
   Random walk over wireless networks are also studied
from a graph theoretical point of view in several papers,
e.g., [2] and [5]. However, all these studies focus on
unbiased walks.
   Biased random walks are widely adopted for search in
unstructured P2P architectures, both in the form of bias-by-
memory and bias-by-information. The interested reader
can, for example, refer to [10] for a survey.
   However, several key aspects make search in P2P
different from search in wireless networks. First, the
topology of P2P networks is best modeled as a power-law
graph, whereas wireless networks adhere to the random
geometric graph model. Second, the channel model in the                                          Fig. 25. Experimental evidence of the validity of the estimations.
two networks is quite different. While in P2P nodes are
connected via unicast channels (a TCP connection), in                                            networks. For a network with uniform node distribution
wireless networks a transmission is inherently broadcast.                                        and a circular symmetry, we have presented an analytical
The cost of implementing the same technique, like looka-                                         study, which shows the sensibility of the hitting with
head or next hop selection, is then quite different. Last, the                                   respect to bias; in particular, the hitting time varies as N 1= ,
changes in the topology are strongly correlated; this makes                                      where N is proportional to the radius of the circle and ,
some source of information, like the distance among nodes,                                       0:5  1, is a number expressing the biasing level of the
meaningful only in mobile networks.                                                              walk. We have then suggested a protocol that exploits
   We remark that physicians often use random walks to                                           biased random walk. A simple beacon-based biasing
model numerous dynamical processes that occur in nature,                                         mechanism is used. The simulation study shows that the
the most notorious being the Brownian motion. The work                                           hitting time is reduced by at least a half with respect to
described in [21] is of particular interest for our work. It                                     natural walks. The results are interpreted in the light of the
presents a method for calculating the properties of biased                                       theoretical study.
random walks on complex networks in general, and for a
segment in particular. Specifically, the Mean First passage                                      APPENDIX A
Time (MFT)—which is synonymous to the hitting time—is                                            To give experimental evidence of the bias level we can
computed for a particle (the walker) moving on a segment of                                      achieve by estimating the distance from the target, we have
size N under an external biasing field, discretely in space and
                                                                                                 simulated the following scenario. Four nodes, i, j, k, and g
time (hopping). This paper shows how the behavior of the
                                                                                                 move into a unitary square region, according to the random
walker, when it moves in the direction of the bias, changes
                                                                                                 waypoint mobility model with an average speed of 30 m/s
from a diffusive regime to a drift one as a bias is applied. The
                                                                                                 and no pause time. The transmission range of the nodes is
diffusive regime is characterized by the MFT, which grows
as N 2 and it is observed in the limit of a weak bias (it basically                              R ¼ 0:3. Node j plays the role of target node, g is the
corresponds to a natural random walk). As the bias is                                            selecting node while i and k are test nodes. When these two
increased, the MFT varies linearly with N, i.e., the so-called                                   nodes are both neighbors of node g and they are both out of
drift regime arises. The results discussed in that paper                                         the j’s transmission range, their estimated and actual
provide an interesting physical interpretation of our study.                                     distance from j, dij , dkj and d0ij , d0kj , respectively, are
   Finally, the idea of using a random delay before sending                                      observed. The following measurement are then performed:
a packet has already been used in other protocols, including                                     1) probability P ðHÞ that both dij and dkj are below the
counter-based, distance-based, and position-based broad-                                         threshold H; 2) probability Pdwn ðHÞ that the estimations are
casting schemes [11]. The basic idea is to collect duplicate                                     correct, namely, dij < dkj and d0ij < d0kj , given that they are
packets received from neighbors for a random period of                                           both below H. Pdwn ðHÞ measures the bias level. If g would
time after the first packet is received and to use knowledge                                     select the node with the lowest estimation, i.e., node i, then
from these packets to make a forwarding decision. The                                            the packet will actually get closer to node j with probability
counter-based scheme exploits the total number of received                                       Pdwn ðHÞ.
duplicates, and the packet is forwarded if it is below a                                            The results are reported in Fig. 25. When H is very low,
counter threshold. The distance-based scheme uses the                                            the difference in the two estimations is not high and both
minimum distance from the node to the sender of these                                            nodes are equally close to the target j. This, however,
packets, which is an estimation of the node’s additional                                         happens with a very low probability—see P ðHÞ in the plot.
(broadcast) coverage area, and the packet is forwarded if it                                     As H increases, the potential bias level also increases. For
is over a distance threshold. The location-based scheme                                          very high H, the bias level becomes % 0.6.
leverages the precise location information to provide a more
accurate estimation of the additional coverage area.
                                                                                                 APPENDIX B
                                                                                                 Consider an ideal bias-by-memory search initiated by a
8      CONCLUSION                                                                                node over n other nodes. Assume that the target replication
In this paper, we have studied the effect of bias on the                                         degree is k and that each node can be targeted with the
hitting time for a random walk executed over wireless                                            same probability of 1=n. The hitting time can be computed

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BERALDI: BIASED RANDOM WALKS IN UNIFORM WIRELESS NETWORKS                                                                                                                513

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REFERENCES
[1]    A. Avin and B. Krishnamachari, “The Power of Choice in Random                                                      Roberto Beraldi received the Laurea degree in
       Walks: An Empirical Study,” Proc. Ninth ACM/IEEE Int’l Symp.                                                       computer science from the University of Calabria,
       Modeling, Analysis, and Simulation of Wireless and Mobile Systems                                                  Cosenza, Italy, in 1991 and the PhD degree in
       (MSWiM ’06), 2006.                                                                                                 computer science in 1996. He has been an
[2]    C. Avin and G. Ercal, “On the Cover Time of Random Geometric                                                       assistant professor in the Dipartimento di Infor-
       Graphs,” Proc. 32nd Int’l Colloquium Automata, Languages, and                                                                                               `
                                                                                                                          matica e Sistemistica (DIS), Universita di Roma
       Programming (ICALP ’05), 2005.                                                                                     “La Sapienza,” since 2002. From 1996 to 2002,
[3]    C. Avin and C. Brito, “Efficient and Robust Query Processing in                                                    he was an expert in computer networks at the
       Dynamic Environment Using Random Walk Techniques,” Proc.                                                           Italian’s National Institute of Statistica (ISTAT).
       Third Int’l Symp. Information Processing in Sensor Networks                                                        He has published more than 50 peer-reviewed
       (IPSN ’04), 2004.                                                                         papers in various fields including computer networks, wireless networks,
[4]    Z. Bar-Yossef, R. Friedman, and G. Kliot, “RaWMS—Random                                   and distributed systems. He participates in many research projects and
       Walk Based Lightweight Membership Service for Wireless                                    regularly serves as a reviewer for international conferences and journals
       Ad Hoc Networks,” Proc. ACM MobiHoc, 2006.                                                on the above areas. He was a program cochair of the First International
[5]    S. Boyd, A. Ghosh, B. Prabhakar, and D. Shah, “Gossip and                                 Workshop on Dynamic Distributed Systems held in 2006.
       Mixing Times of Random Walks on Random Graphs,” Proc.
       SIAM Second Workshop Analytic Algorithmics and Combinatorics
       (ANALCO ’05), 2005.


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