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Probabilistic flooding for efficient information dissemination inrandom graph topologies q

Konstantinos Oikonomou a,*, Dimitrios Kogias b, Ioannis Stavrakakis b

a Ionian University, Department of Informatics, Tsirigoti Square 7, 49100 Corfu, Greeceb National and Kapodistrian University of Athens, Department of Informatics and Telecommunications, Athens, Greece

a r t i c l e i n f o

Article history:Received 24 October 2009Received in revised form 5 January 2010Accepted 13 January 2010Available online 21 January 2010Responsible Editor: M.C. Vuran

Keywords:Probabilistic floodingInformation disseminationRandom graphs

a b s t r a c t

Probabilistic flooding has been frequently considered as a suitable dissemination informa-tion approach for limiting the large message overhead associated with traditional (full)flooding approaches that are used to disseminate globally information in unstructuredpeer-to-peer and other networks. A key challenge in using probabilistic flooding is thedetermination of the forwarding probability so that global network outreach is achievedwhile keeping the message overhead as low as possible. In this paper, by showing that aprobabilistic flooding network, generated by applying probabilistic flooding to a connectedrandom graph network, can be (asymptotically) ‘‘bounded” by properly parameterized ran-dom graph networks and by invoking random graph theory results, asymptotic values ofthe forwarding probability are derived guaranteeing (probabilistically) successful coverage,while significantly reducing the message overhead with respect to traditional flooding.Asymptotic expressions with respect to the average number of messages and the averagetime required to complete network coverage are also derived, illustrating the benefits ofthe properly parameterized probabilistic flooding scheme. Simulation results support theclaims and expectations of the analytical results and reveal certain aspects of probabilisticflooding not covered by the analysis.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

A plethora of network protocols requiring informationdissemination (such as routing broadcasting, content dis-covery etc) are based on traditional flooding since its earlyintroduction, [1]. Traditional flooding is capable of sendingmessages (one message for each neighbor node apart fromthe node that the message arrived from) and traversing allnetwork links and all network nodes in any connected net-

work topology in a small number of (time) steps and it issimple and easy to implement in a distributed fashion.The inherent simplicity of traditional flooding as well asits guaranteed success in disseminating information every-where, have been two of the main reasons for its wide-spread protocol instantiations. However, by traversing allnetwork links, the number of messages forwarded in thenetwork is (asymptotically) of the order of the number ofthe network links (actually, more than the number of net-work links and less than twice this number). Therefore,even though termination time (that is, the time requiredto outreach all network nodes) is small (equal to the eccen-tricity of each node and upper bounded by the networkdiameter), the aforementioned increased number of mes-sages is prohibitive in modern network environments thatare typically large-scale with respect to the number ofnodes and network links.

1389-1286/$ - see front matter � 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.comnet.2010.01.007

q This work has been supported in part by the project ANA (autonomicnetwork architecture) (IST-27489), SOCIALNETS (IST-217141) and thePENED 2003 program of the General Secretariat for Research andTechnology (GSRT) co-financed by the European Social Funds (75%) andby national sources (25%).

* Corresponding author. Tel.: +30 26610 87708; fax: +30 26610 48491.E-mail addresses: okon@ionio.gr (K. Oikonomou), dimkog@di.uoa.gr

(D. Kogias), ioannis@di.uoa.gr (I. Stavrakakis).

Computer Networks 54 (2010) 1615–1629

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Typical examples of a large-scale modern network envi-ronment are peer-to-peer (P2P) networks, [2], which, fol-lowing the Napster legacy, [3], offer to network usershuge amounts of content of, basically, delay tolerant ser-vices (e.g., files). While, the underlying structure presentin structured P2P networks facilitates reaching the nodeswith relatively low delay and message overhead, [2,4–6],the problem of disseminating information is a major chal-lenge in unstructured P2P networks, (e.g., Gnutella, [7]),as there is no structure to take advantage of and supportthe design of an effective scheme. As a result, a brute-forceapproach is followed, typically implemented through re-source wasteful approaches such as traditional floodingand some of its message reducing variations, [2,7–11].For example, in Gnutella, [7], controlled flooding is em-ployed for searching purposes restricting message floodingto a small number of L-hops around the node that has ini-tiated the search. Such an approach – referred to hereafteras L-flooding – is scalable in terms of the number of mes-sages for small values of L but, at the same time, it reducesthe covered network area; the use of this approach wouldreduce the probability of success for the searching processin [7], particularly in large-scale networks.

Message reduction during an information dissemina-tion process is a key goal in many protocol designs,[12,13]. The idea of reducing the messages of traditionalflooding by forwarding a message to neighbors probabilisti-cally (with some forwarding probability) and not determinis-tically lays behind probabilistic flooding, [14–20]. There isclearly a trade-off between the induced total number ofmessages and the network coverage (i.e., the fraction ofnetwork nodes reached by such messages): the smallerthe forwarding probability, the smaller the message over-head and the smaller the network coverage.

The work presented here investigates probabilisticflooding when the underlying network is a random graph,[21], and aims at designing such a scheme in a way thatthe aforementioned trade-off is effectively managed. Thatis, achieve high network coverage with a relatively smallnumber of messages. Analytical tools and results, bor-rowed from random graph theory, [21–26], are consideredfor analyzing probabilistic flooding and comparing its per-formance against traditional flooding. Even though it hasbeen shown that some networks do not follow the randomgraph model but rather power-law models, e.g., [23,24], itis still valid that many useful results may be extracted afterstudying random graph topologies, e.g., [17,27,28].

One of the main contributions of this work is establish-ing a connection between random graphs and the probabi-listic flooding network; the latter is defined to be thenetwork consisted of the (sub)set of links and nodes ofthe underlying random graph network that are traversedby the messages under the probabilistic flooding. This al-lows for the subsequent study of probabilistic flooding inrandom graph topologies based on analytical tools bor-rowed from random graph theory.

Another contribution of this work is the derivation of(asymptotic) analytical expressions on the appropriate va-lue of the forwarding probability, defined to be the valuefor which the probabilistic flooding network consists (withhigh probability) of a certain number of network nodes

(e.g., all network nodes) using the smallest possible num-ber of messages. The case of reaching all network nodes– which is equivalent to full network coverage or globalnetwork outreach – is extensively studied here.

The (asymptotic) analysis has shown a significant de-crease with respect to the number of messages, reducedfrom the order of HðN2Þ (i.e., traditional flooding) to the or-der of HðN lnðNÞÞ (i.e., probabilistic flooding) in randomgraph topologies of N nodes, for the case of global networkoutreach. Furthermore, as it is analytically shown in thispaper, the price paid for this reduction is (a) an increaseof the time required to cover a certain network area; (b)any network coverage (e.g., global node outreach) isachieved with high probability under probabilistic floodingas opposed to certainty under traditional flooding. Theseanalytical findings are supported by simulation results.

The analytical study has been extended to the case of L-coverage, achieved when no node is away from a reachednode by more than L-hops. This notion of coverage (yield-ing overhead savings) is applicable to various practicalcases such as when a supplementing L-hop flooding (forsmall L) allows every node to obtain the disseminatinginformation, as it is the case under Gnutella mentioned be-fore. The cases of L ¼ 0; L ¼ 1 and L ¼ 2 are studied here(the case of L ¼ 0 is equivalent to global network outreachmentioned before) and it is analytically shown that there isa significant reduction with respect to the number of mes-sages when probabilistic flooding is employed and an L-coverage is targeted. Given the small-world phenomenonin random graphs (i.e., significantly small diameters evenfor large network sizes), [22,26], larger values of L wouldhave been significantly close to the network diameter fornetwork sizes of 10,000 nodes (as it will be mentioned inSection 6, the simulation program had memory and perfor-mance limitations for network sizes greater than 10,000nodes). For example, when L is equal to the network diam-eter, information dissemination is obsolete since all nodesare by definition less than or equal to L-hops away fromany node, and thus from the source node as well. Such acase is investigated at the simulations section.

Simulation results confirm the analytical findings of thispaper and at the same time explore various aspects notcovered by the analysis. In particular, it is shown that theprobabilistic flooding network is (asymptotically)‘‘bounded” by two random graphs and that savings underprobabilistic flooding are significant when compared totraditional flooding (or controlled flooding in the case ofL ¼ 1;2 for fairness issues as it is explained in more detailin the simulations section). On the other hand, it is shownthat termination time does increase (even though at asmaller rate compared to message savings) as it is also ex-pected by the analysis. However, due to the ‘‘small-worldphenomenon” in random graphs (i.e., small network diam-eters even for large network sizes), when L is comparableto the network diameter, simulation results deviate fromthe analytical ones. For the simulation scenarios of 104

nodes considered in this work, L ¼ 2 is comparable to thenetwork diameter. However, larger network sizes (e.g.,108 network nodes) could not be considered due to compu-tational and memory limitations. This issue is further ex-plained in the simulations section.

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In the following section, past related works are pre-sented and particularly those mostly related to this paper.Section 3 summarizes important results from randomgraph theory that will be used throughout this work. Sec-tion 4 presents the probabilistic flooding scheme and dis-cusses its relevance to random graphs. Analytical(asymptotic) results are presented in Section 5, and simu-lation results in Section 6. The conclusions are drawn inSection 7.

2. Past related work

The effectiveness of most of the works (including theone presented in this paper) attempting to reduce thenumber of messages sent in a network, is frequently estab-lished through comparison with traditional flooding. Ap-proaches, like random walkers, e.g., [8,10], may be seenas an entirely different class which – while typicallyachieving low message overhead- results in large networkcover times. Hybrid probabilistic schemes (e.g., a localflooding process initiated after a random walk) have alsobeen proposed and analyzed, [10], as well as other schemesthat adapt the employed values of L for L-flooding in aprobabilistic manner, [11]. Another modification, [9], al-lows for network nodes to selectively forward messagesto their neighbors, thus significantly reducing the numberof messages in the network. Other approaches for fillingthe gap between random walkers and traditional floodinghave also been proposed. For example, in Kogias et al.,[29] replicated random walker strategies have been stud-ied for various network topologies. Gossip strategies, underwhich messages are forwarded based on local information,e.g., [12,30–35], have also been on the edge of this researcharea for many years, e.g., for the design of networkprotocols.

Probabilistic flooding has attracted considerable atten-tion since its introduction, basically for providing scalabledissemination information approaches in modern large-scale environments like P2P networks, e.g., [14–16]. Eventhough – as already mentioned – its main disadvantagewith respect to traditional flooding is its probabilistic nat-ure, it may be employed in cases that probabilistic guaran-tees are affordable given the subsequent savings withrespect to the number of messages. These trade-offs havebeen the focus of recent studies, e.g., [18–20].

The most relevant works to the work presented here arethose in [19,20]. In [19], Stauffer and Barbosa compareprobabilistic flooding against a proposed heuristic floodingthat instead of employing a fixed forwarding probabilityfor all network nodes – as it is the case under probabilisticflooding – it adapts accordingly based on a heuristic rule.In [20], Crisostomo, Schilcher, Bettstetter and Barros, con-sider random graph topologies and – based on dominatingset properties – provide for suitable values of global net-work outreach based on both analytical and simulationresults.

Both [19,20] mention the relevance between randomgraphs and probabilistic flooding in random graph topolo-gies. However, none of these approaches either investigatethis further or exploit this in their work, as it is the case in

this paper. Furthermore, the work in this paper focuses inmore general cases than global network outreach with re-spect to covering (notion of L-coverage), that allow for fur-ther reduction of the messages forwarded in the network.

3. The random graph model

A network topology is typically represented by a graphG defined in terms of a set of nodes VðGÞ and a set of (bidi-rectional) links EðGÞ connecting pairs of nodes in VðGÞ. Letthe number of nodes in the network be denoted by N(N ¼ jVðGÞj, where jXj corresponds to the number of ele-ments of a certain set X). Random graphs, mainly intro-duced by the pioneering work of P. Erd}os and A. Rényi,[21], have certain properties that are useful in sheddinglight on the particulars of various networks like the Inter-net, social contact networks, biological networks, web-page networks etc., [23,24]. For most of the cases, randomgraphs have been considered for studying topological as-pects of these networks (e.g., the average number of neigh-bors, the diameter of the network, the emergence of thegiant component etc.). In this work, random graphs andtheir properties are used for understanding and analyzingthe information dissemination network created after per-forming probabilistic flooding in random graph networktopologies.

In the random graph literature there are two basic mod-els for representing a random graph of N nodes: the bino-mial and the uniform. The first model – denoted asGðN; pðNÞÞ – assumes a certain probability pðNÞ for select-

ing each link out of the N2

� �possible links to become part

of the particular random graph GðN; pðNÞÞ (in order to sim-plify the notation, in the sequel GðN; pÞ will be used in-stead of GðN; pðNÞÞ). Under the second model, a graph ofN nodes and M links is uniformly chosen out of the

N2

� �

M

0@

1A possible different graphs of N nodes and M links

and is denoted as GðN;MÞ. Both models are asymptotically

equivalent provided that pðNÞ N2

� �is close to M, [26,22].

For the rest of this paper, the binomial model GðN; pÞ isconsidered. For this model, the expected number of links is

equal to pðNÞ N2

� �and therefore, in order to ensure the

equivalence of both previous models, it is assumed that

pðNÞ N2

� �� M. Furthermore, it is also assumed that N is

significantly large (e.g., such that pðNÞðN � 1Þ � pðNÞN issatisfied), which is a basic assumption in the literature ofrandom graphs, e.g., [22,26].

Based on the previously mentioned model description,it is easily derived that for pðNÞ ¼ 0, there are no links inthe resulting graph (M ¼ 0) and GðN; pÞ consists only of Nnodes with no links among them (Fig. 1a), whereas forpðNÞ ¼ 1, the resulting graph is the complete graph

M ¼ N2

� �� �. For values of pðNÞ slightly higher than 0 (al-

ways under the assumption that N is fairly large), some

K. Oikonomou et al. / Computer Networks 54 (2010) 1615–1629 1617

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links start to be created among some nodes, creating smallcomponents of a few nodes (Fig. 1b). Note that Fig. 1 illus-trates the various stages of a random graph of N nodes – asprobability pðNÞ (or the number of links M) increases –using two-dimension drawings for better visualization. Ingeneral, a random graph is not expected to be a two-dimensional graph, [22].

For many properties of random graphs (and particularlyfor those useful for the subsequent analysis of probabilisticflooding), it has been shown in the literature that there ex-ists a critical probability pQ ðNÞ – for some property Q – suchthat if pðNÞ grows faster than pQ ðNÞ as N increases, thenGðN; pÞ has property Q, otherwise it does not. More for-mally, [23,25],

limN!1

PrfGðN;pÞ has property Qg ¼0 if pðNÞ

pQ ðNÞ! 0;

1 if pðNÞpQ ðNÞ

! þ1:

8<:

ð1Þ

In the sequel and based on this definition, any argumentthat a random graph has property Q is made with highprobability (w.h.p.). For the rest of this paper, the followingwell-known asymptotic notations will be used:

a. pðNÞ ¼ OðpQ ðNÞÞ, if pðNÞ 6 CpQ ðNÞ for some constantC > 0 (equivalently, limN!1

pðNÞpQ ðNÞ

! 0() OðpðNÞÞ< OðpQ ðNÞÞ, [37]);

b. pðNÞ ¼ XðpQ ðNÞÞ, if pðNÞP cpQ ðNÞ for some constantc > 0 (equivalently, limN!1

pðNÞpQ ðNÞ!þ1()OðpðNÞÞ>

OðpQ ðNÞÞ, [37]);c. pðNÞ ¼ HðpQ ðNÞÞ, if CpQ ðNÞP pðNÞP cpQ ðNÞ for

some constants C; c > 0.

Note that for some property Q and pðNÞ such thatpðNÞ ¼ HðpQ ðNÞÞ; limN!1

pðNÞpQ ðNÞ

¼ c0, [37], where c0 is a posi-

tive constant (0 < c0 < þ1). Therefore, the case ofpðNÞ ¼ HðpQ ðNÞÞ is not covered by the definition given byEq. (1). The basic reason is that for pðNÞ ¼ HðpQ ðNÞÞ, thegraph passes a phase transition period, [22,26,23], duringwhich it ‘‘moves” from a state in which it does not haveproperty Q to a state in which it does have property Qand therefore, it is not known for certain (not even withhigh probability) whether it has or not property Q. Forthe rest of this paper, notation pðNÞ ¼ HðpQ ðNÞÞ is used toindicate such a phase transition with respect to propertyQ (e.g., the emergence of the giant component that indi-cates that the network becomes connected).

In this paper the interest is on three properties (Q0;Q1

and Q 2) of random graphs: (a) Q0: the graph is connected;(b) Q 1: the graph consists of a giant component and someisolated nodes; and (c) Q 2: the giant component has justemerged. The corresponding critical probabilities

lnðNÞN ; lnðNÞ

2N ; and 1N ; respectively

� �as a function of N for each

property are shown in Table 1, [22,26]. Note that,lnðNÞ

N > lnðNÞ2N > 1

N, for large values of N. The analysis of proba-bilistic flooding in random graph topologies presented inthe following sections is based on the emergence of thesethree properties.

Fig. 1. An example of a random graph GðN; pÞ (depicted in two-dimensions).

Table 1Critical probability functions for properties Q0 ;Q 1, and Q2.

Criticalprobabilityfunction

Property description

pQ2ðNÞ ¼ 1

NQ2: emergence of giant component. Somecomponents are also present along with someisolated nodes

pQ1ðNÞ ¼ lnðNÞ

2NQ1: giant component and some isolated nodes

pQ0ðNÞ ¼ lnðNÞ

NQ0: connected network

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Figs. 1c and 1d depict a two-dimensional illustration ofthese properties. Note that in general random graphs arenot two-dimensional as depicted here for illustration pur-poses. In Fig. 1c, where OðpðNÞÞ < OðpQ2

ðNÞÞ, the graph con-sists of numerous isolated components. These isolatedcomponents become larger as pðNÞ increases and at somepoint, the largest components are so large that it is likelya new link to merge two of them resulting in a new, larger(merged) component which, in turn, is likely to be furtherincreased by a new link, etc. Eventually, for pðNÞ ¼HðpQ2

ðNÞÞ the giant component emerges, as depicted inFig. 1d. As pðNÞ increases further such thatOðpðNÞÞ > OðpQ2

ðNÞÞ and OðpðNÞÞ < OðpQ1ðNÞÞ, the giant

component increases further by connecting to the smallercomponents (i.e., graph has property Q2 and not propertyQ 1 w.h.p.). For pðNÞ ¼ HðpQ1

ðNÞÞ, there exist only the giantcomponent and some isolated nodes, as depicted in Fig. 1e.Further increase of pðNÞ such that OðpðNÞÞ > OðpQ1

ðNÞÞ andOðpðNÞÞ < OðpQ0

ðNÞÞ connects the giant component to theisolated nodes (i.e., graph has property Q 1 and not propertyQ 0 w.h.p.). Finally, the network becomes connected w.h.p.for pðNÞ ¼ HðpQ0

ðNÞÞ, as shown in Fig. 1f, and remains con-nected thereafter, for OðpðNÞÞ > OðpQ0

ðNÞÞ.The diameter (i.e., the maximum shortest path of all

pairs of nodes in the same component) of a random graphGðN; pÞ, denoted as DðGðN; pÞÞ, is also important for thesubsequent analysis of probabilistic flooding. ForOðpðNÞÞ > OðpQ0

ðNÞÞ is known, [23,36], to be proportionalto:

DðGðN;pÞÞ ¼ lnðNÞlnðpðNÞNÞ : ð2Þ

For the rest of this paper, it is assumed that

DðGðN; pÞÞ ¼ H lnðNÞlnðpðNÞNÞ

� �, when OðpðNÞÞ > OðpQ 0

ðNÞÞ. For

the case of OðpðNÞÞ < OðpQ0ðNÞÞ and OðpðNÞÞ > OðpQ2

ðNÞÞ,the network diameter is equal to the diameter of the giantcomponent, (the network is not connected and there existsa giant component w.h.p.). Since the number of nodes inthe giant component are of the order of OðNÞ, [26,23], avery rough idea about the diameter of the giant componentis given by OðlnðNÞÞ (for this case the giant componentlooks mostly like a tree even though some cycles do exist).

4. Probabilistic flooding and random graphs

4.1. On probabilistic flooding

Under traditional flooding, messages arrive to all nodeshaving traversed all network links, by sending one messageto each neighbor node each time a node receives the for-warding message for the first time (excluding the particu-lar neighbor node that the message has arrived from). Thetotal number of messages under traditional flooding can begreater than the number of links of the network topologysince either one or – at most – two messages is possibleto be forwarded over any link. Therefore, the number ofmessages under traditional flooding over a network topol-ogy G is of the order of HðjEjÞ (greater than jEj and lessthan 2jEj). This number turns out to be fairly large for largeN ¼ jVðGÞj resulting in the scalability problems of tradi-

tional flooding. For instance, for a network topology mod-elled as a random graph, GðN; pÞ, the number of messagesunder traditional flooding is of the order of

H pðNÞ N2

� �� �¼ H pðNÞN N�1

2

� �which becomes very large

for large networks. This large message overhead leads, onthe other hand, to relatively short termination time (definedas the number of time steps until the message dissemina-tion is completed) bounded by the network diameterDðGðN; pÞÞ (i.e., OðDðGðN; pÞÞÞ).

Under probabilistic flooding, [14], the initiator node (i.e.,the source of the information to be disseminating) sends amessage to each of its neighbor nodes with an (indepen-dent) constant forwarding probability pf ðNÞ. Any nodereceiving such a message forwards it to its neighbor nodes(apart from the one the message has arrived from) withprobability pf ðNÞ. Clearly, for pf ðNÞ ¼ 0, no messages aresent in the network, while for pf ðNÞ ¼ 1, probabilisticflooding reduces to traditional flooding. As a result of prob-abilistic flooding, a network can be defined that consists:(a) of the set of nodes that have been reached by themessages plus the initiator node; and (b) of the set of linksover which these messages have been forwarded. This par-ticular network will be referred to hereafter as the probabi-listic flooding network; an example of such a network isdepicted in Fig. 2. It is trivial to show (based on the defini-tion of probabilistic flooding) that a probabilistic floodingnetwork is actually a connected network each link ofwhich is created as a result of message forwarding overthe corresponding network link. Let PðGðN; pÞ; pf Þ denotethe probabilistic flooding network generated by applyingprobabilistic flooding with probability pf ðNÞ over networkGðN; pÞ.

An observation, that will be utilized later, is that someof the links of GðN; pÞ become part of PðGðN; pÞ; pf Þ withprobability pf ðNÞ if only one of the end nodes receivesthe message from an other link and take a forwarding deci-sion, some with probability ~pðNÞ if both end nodes of thelink receive the message from a different than their com-mon link and, thus, both make an (assumed independent)decision that each fails with probability 1� pf ðNÞ andsome with probability 0 if none of end nodes receives themessage.

From the above it can be proved that ~pðNÞ equals:

~pðNÞ ¼ 1� ð1� pf ðNÞÞð1� pf ðNÞÞ ¼ 2pf ðNÞ � p2f ðNÞ: ð3Þ

4.2. L-coverage of a network

The network coverage is typically defined as the frac-tion of network nodes that are forwarded a message, or‘‘covered ” by an information dissemination process. In thispaper (as also in [29]) the notion of coverage is extended tothat of L-coverage (L an integer and L P 0) referring to theportion of network nodes which are at most L-hops awayfrom a node that has received the message and denotedby CðLÞ. Clearly, Cð0Þ corresponds to the portion of nodesthat have actually received the message and if Cð0Þ ¼ 1 aglobal outreach has been achieved. Notice that CðLÞ is

K. Oikonomou et al. / Computer Networks 54 (2010) 1615–1629 1619

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non-decreasing with L. Notice, also, that for traditionalflooding Cð0Þ ¼ 1 and, thus, CðLÞ ¼ 1 for any L P 0 as well.

Since random graph networks have small diameters(known as the small-world phenomenon), [23,26], theinterest is basically for coverage of L ¼ 0; L ¼ 1 and L ¼ 2and the main focus is to reduce as much as possible thenumber of messages required to achieve CðLÞ ¼ 1, for suchvalues of L. It should be mentioned that probabilistic flood-ing employs a probabilistic mechanism and therefore anyclaim that ‘‘a certain coverage CðLÞ ¼ 1” is always madewith high probability.

4.3. Probabilistic flooding in random graphs

The objective when constructing a probabilistic flood-ing network is to keep the number of its links (i.e., numberof forwarded messages) as low as possible, while strivingto achieve a certain coverage CðLÞ ! 1. Clearly (and assum-ing always that GðN; pÞ is connected), for pf ðNÞ ¼ 1 theprobabilistic flooding network contains all network nodesand all network links of GðN; pÞ. That is,PðGðN; pÞ;1Þ ¼ GðN; pÞ. The interest in the sequel of thissection is on determining a value of pf ðNÞ (to keep thenumber of messages under probabilistic flooding low) suchthat CðLÞ ¼ 1 w.h.p., for L ¼ 0;1;2.

Given that each node of a connected network is associ-ated with at least one link and most likely with several,removing a link from a network does not necessarily dis-connect (or ‘‘remove”) an associated node as well. In otherwords, the decrease in the number of nodes in a network asa result of a decrease in the number of links is expected tobe lower than the decrease in the number of links. Conse-quently, it is conceivable that all network nodes continueto be included in a network (i.e., be connected) despitethe removal of a number of links w.h.p. This observationsuggests that a probabilistic flooding network with suffi-ciently high forwarding probability may still keep all thenodes ‘‘connected” and in the network, despite a poten-tially significant removal of links due to a decision not toforward a message over such links.

In view of the above discussion it is evident that aspf ðNÞ decreases, the number of links in PðGðN; pÞ; pf Þ de-creases as well, while the number of nodes inPðGðN; pÞ; pf Þ decreases at a lower rate. Consequently, for

a small reduction in pf ðNÞ below the value of 1, it is ex-pected that all network nodes still be included inPðGðN; pÞ; pf Þ, or Cð0Þ ¼ 1 w.h.p. Further reduction in thevalue of pf ðNÞ will result in a further decrease in the num-ber of network links, until a small number of nodes ofGðN; pÞ do not belong in PðGðN; pÞ; pf Þ (see also Section3). It is thus expected that there is a certain value for theforwarding probability, denoted by pf ;0ðNÞ, such that: (a)if Oðpf ðNÞÞ < Oðpf ;0ðNÞÞ, then the probabilistic flooding net-work does not include all network nodes w.h.p.; (b) ifOðpf ðNÞÞ > Oðpf ;0ðNÞÞ, then the probabilistic flooding net-work does include all network nodes w.h.p.

The goal in the sequel is to derive an asymptotic expres-sion for pf ;0ðNÞ by utilizing properties of random graphs(those already described in Section 3), taking into consid-eration that since Cð0Þ ¼ 1 w.h.p. all network nodes are ex-pected to be part of the probabilistic flooding network and,in view of the latter and as discussed in 4.1, each of thelinks of GðN; pÞ will be included in PðGðN; pÞ; pf Þ eitherwith probability pf ðNÞ or ~pðNÞ. Thus, it is evident thatPðGðN; pÞ; pf Þ contains on average more links thanGðN; p� pf Þ. Consequently, when GðN; p� pf Þ is connectedw.h.p., then PðGðN; pÞ; pf Þ is also connected w.h.p. and,thus, includes all network nodes w.h.p.

Note also that since pf ðNÞ 6 2pf ðNÞ � p2f ðNÞ ¼ ~pðNÞ (the

equality holds for pf ðNÞ ¼ 1),GðN; p� ~pÞ contains (on aver-age) more links than GðN; p� pf Þ and when the latter net-work is connected the former is also connected w.h.p.

Based on the previous two observations, and assuminga certain value for pðNÞ, as pf ðNÞ increases, it is expectedthat there will be some probability value for pf ðNÞ forwhich GðN; p� ~pÞ becomes connected w.h.p. As pf ðNÞ in-creases further, PðGðN; pÞ; pf Þ becomes connected (equiva-lently, Cð0Þ ¼ 1) w.h.p. For further increment, GðN; p� pf Þalso becomes connected. Consequently, the particular va-lue of pf ðNÞ (i.e., pf ;0ðNÞ for Cð0Þ ¼ 1) for which probabilis-tic flooding disseminates information to all network nodes(global network outreach) is ‘‘between” the values of pf ðNÞfor which GðN; p� ~pÞ and GðN; p� pf Þ become connected.This is demonstrated later in Section 6 using simulationresults.

From the discussion in Section 3 it follows that randomgraph GðN; p� pf Þ becomes connected (or has property Q0)

w.h.p. when pðNÞpf ðNÞ ¼ HðpQ0ðNÞÞ, or pf ðNÞ ¼ H

pQ0ðNÞ

pðNÞ

� �.

Fig. 2. An example of nodes (marked with ellipses) reached by probabilistic flooding messages and the corresponding links (marked with thick lines) thatthese messages have been forwarded over.

1620 K. Oikonomou et al. / Computer Networks 54 (2010) 1615–1629

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On the other hand, random graph GðN; p� ~pÞ becomesconnected when pðNÞ~pðNÞ ¼ HðpQ0

ðNÞÞ, or

~pðNÞ ¼ HpQ0ðNÞ

pðNÞ

� �, or pf ðNÞ ¼ H 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� pQ0

ðNÞpðNÞ

q� �(since

~pðNÞ ¼ 2pf ðNÞ � p2f ðNÞ and OðpðNÞÞ > OðpQ0

ðNÞÞ in order toensure connectivity of GðN; pÞ), [18,37].

An interesting result is that even though GðN; p� ~pÞ be-comes connected for smaller values of pf ðNÞ when com-pared to GðN; p� pf Þ (as already mentioned ~p > pf for0 < pf < 1), these values have the same asymptotical orderand eventually, or pf ;0ðNÞ ¼ H

pQ0ðNÞ

pðNÞ

� �as it is shown in Lem-

ma 1. This asymptotic result suffices for the analysis pur-poses in the following section and the subsequentsimulation results.

Lemma 1. Random graph GðN; p� ~pÞ and random graphGðN; p� pf Þ become connected for values of pf ðNÞ of the sameasymptotic order, provided that random graph GðN; pÞ isconnected.

Proof. Given that random graph GðN; pÞ is connected,based on Eq. (1), this is equivalent to pðNÞ

pQ0ðNÞ ! þ1, or

pQ0ðNÞ

pðNÞ < 1. It is easy to show thatpQ0ðNÞ

pðNÞ P 1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� pQ0

ðNÞpðNÞ

qis

satisfied. For this, it is enough to show thatffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� pQ0

ðNÞpðNÞ

qP 1� pQ0

ðNÞpðNÞ is satisfied. The latter expression

is valid sinceffiffiffiffiffiffiffiffiffiffiffi1� xp

P 1� x for 0 6 x 6 1, which in this

case for x ¼ pQ0ðNÞ

pðNÞ .

The next step is to show that there exists some constant

0 < c 6 1, such that 1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� pQ0

ðNÞpðNÞ

rP c

pQ0ðNÞ

pðNÞ is satisfied.

Let x ¼ pQ0ðNÞ

pðNÞ . The case of x ¼ 0 and x ¼ 1 is trivial so the

focus next is on 0 < x < 1. It is sufficient to show that thereexists a constant 0 < c 6 1, such that 1�

ffiffiffiffiffiffiffiffiffiffiffi1� xp

P cx, or

1� cx Pffiffiffiffiffiffiffiffiffiffiffi1� xp

, or, c 6 1�ffiffiffiffiffiffiffi1�xp

x . Let f ðxÞ ¼ 1�ffiffiffiffiffiffiffi1�xp

x . The first

derivative f 0ðxÞ ¼ 2�2ffiffiffiffiffiffiffi1�xp

�x2ffiffiffiffiffiffiffi1�xp ¼ 0, has no solution in

0 < x < 1. That means that for 0 < x < 1; f ðxÞ either mono-tonically increases or monotonically decreases. For

x! 1; limx!1f ðxÞ ¼ limx!11�ffiffiffiffiffiffiffi1�xp

x ¼ 1. For x! 0; limx!0

f ðxÞ ¼ limx!01�ffiffiffiffiffiffiffi1�xp

x ¼ limx!0ð1�

ffiffiffiffiffiffiffi1�xp

Þ0ðxÞ0 ¼ limx!0

12ffiffiffiffiffi1�xp

1 ¼ limx!0

12ffiffiffiffiffiffiffi1�xp ¼ 0:5. Since limx!0f ðxÞ< limx!1f ðxÞ -and in view of

the monotonicity of f ðxÞ argued above- it is implied thatf ðxÞ monotonically increases and therefore, f ðxÞP 0:5, for0< x< 1.

Eventually, for any c 6 0:5 it is satisfied that

c 6 f ðxÞ ¼ 1�ffiffiffiffiffiffiffi1�xp

x , for 0 < x < 1, or, 1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� pQ0

ðNÞpðNÞ

rP

cpQ0ðNÞ

pðNÞ is satisfied. SincepQ0ðNÞ

pðNÞ P 1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� pQ0

ðNÞpðNÞ

rP c

pQ0ðNÞ

pðNÞ

is satisfied, then 1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� pQ0

ðNÞpðNÞ

r¼ H

pQ0ðNÞ

pðNÞ

� �. Consequently,

pf ;0ðNÞ ¼ HpQ0ðNÞ

pðNÞ

� �, and the lemma is proved. h

So far, the case of global network outreach (i.e., CðLÞ ¼ 1for L ¼ 0) w.h.p. has been studied. The study for L ¼ 1 andL ¼ 2 follows the previously described steps with respect

to properties Q 1 and Q2. For instance, if GðN; p� pf Þ hasproperty Q1 w.h.p., it is expected PðGðN; pÞ; pf Þ to includethe majority of the network nodes (as it is the case forthe giant component of GðN; p� pf Þ that consists of themajority of the network nodes) and those nodes that arenot part of PðGðN; pÞ; pf Þ, to be at most 1 hop away fromat least one node of PðGðN; pÞ; pf Þ (since an increase inpf ðNÞ could result in the creation of a new link betweenthese nodes and a node of the giant component). To sup-port this claim, in [26] it is also mentioned that when theproperty Q1 exists (w.h.p.), then apart from the giant com-ponent there exist isolated nodes in the network for whichany new link will connect them to the giant componentw.h.p. Assuming pf ;1ðNÞ to be the particular probabilitysuch that Cð1Þ ¼ 1 is satisfied w.h.p., it is derived thatpf ;1ðNÞ ¼ H

pQ1ðNÞ

pðNÞ

� �. The same applies for L ¼ 2 and prop-

erty Q 2. As already mentioned, in [26] when property Q 2

exists, the giant component is present in the network alongwith small components of one, two or three nodes (w.h.p.)for which any new link will connect them to the giant com-ponent (thus some of these being at distance 2 hops awayfrom nodes already within the giant component) w.h.p.Therefore, assuming pf ;2ðNÞ to be the particular probabilitysuch that Cð2Þ ¼ 1 is satisfied w.h.p., it is derived thatpf ;2ðNÞ ¼ H

pQ2ðNÞ

pðNÞ

� �. Fig. 3 depicts asymptotic numerical

values of pf ;LðNÞ as a function of N for L ¼ 0;1;2, usingthe results depicted in Table 1. For this particular case,the underlying network topology is considered fixed, andtherefore pðNÞ is a positive constant.

5. Asymptotic analysis

The asymptotic expressions derived in the previous sec-tion are considered next to asymptotically analyze proba-bilistic flooding and compare it against traditionalflooding, with respect to the number of messages requiredand termination time (upper bounds) until a certain cover-age CðLÞ ¼ 1 is achieved.

For the case of global network outreach or Cð0Þ ¼ 1, it isknown that the number of messages under traditionalflooding (Mtrad) is of the order of:

Mtrad ¼ H pðNÞN � 12

N� �

: ð4Þ

Under probabilistic flooding, a probabilistic floodingnetwork PðGðN; pÞ; pf Þ is created, and therefore, the num-ber of messages is of the order of the number of links of

0

0.05

0.1

0.15

0.2

0.25

100 200 300 400 500 600 700 800 900 1000N

)(, Np Lf

0L1L

2L

Fig. 3. pf ;LðNÞ as a function of N for L ¼ 0;1;2.

K. Oikonomou et al. / Computer Networks 54 (2010) 1615–1629 1621

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this network, or HðEðPðGðN; pÞ; pf ÞÞÞ. Based on the studypresented in the previous section, the number of messagesunder probabilistic flooding ðMprobÞ for global network out-reach are of the order of:

Mprob ¼ H pQ0ðNÞNðN � 1Þ

2

� �¼ H

N � 12

lnðNÞ� �

¼ HðN lnðNÞÞ: ð5Þ

Assuming pðNÞ to be constant note that under tradi-tional flooding, the number of messages is of the order ofHðN2Þ while under probabilistic flooding are of the orderof HðN lnðNÞÞ, which is a significant reduction comparedto traditional flooding that becomes more significant as Nincreases, as illustrated in Fig. 4a. Let RM;LðNÞ denote the(asymptotic) fraction of messages under probabilisticflooding over those under traditional flooding for some L,or, RM;LðNÞ ¼

EðPðGðN;pÞ;pf ÞÞEðGðN;pÞÞ . For the case of L ¼ 0,

RM;0ðNÞ ¼lnðNÞpðNÞN : ð6Þ

Obviously, RM;0ðNÞ ! 0, when N ! þ1OðpðNÞÞ > OðpQ 0

ðNÞÞ ¼ O lnðNÞN

� �� �. Note that in strict terms,

RM;0ðNÞ ¼ H lnðNÞpðNÞN

� �. However, in order to simplify the nota-

tions in the sequel Hð�Þwill not appear for both RM;LðNÞ andRT;LðNÞ (to be defined later).

As already mentioned, since GðN; pÞ is a connected net-

work w.h.p., then OðpðNÞÞ > OðpQ0ðNÞÞ ¼ O lnðNÞ

N

� �. For

pðNÞ ¼ H lnðNÞN

� �(which means that GðN; pÞ has just become

connected), it is interesting to see that RM;0ðNÞ ¼ Hð1Þ,which apparently demonstrates the fact that there is noadvantage under probabilistic flooding when comparedto traditional flooding for this case (the number of mes-sages is the same under both probabilistic flooding and tra-ditional flooding). Actually, this particular case is -asymptotically- equivalent to pf ðNÞ ¼ 1, for which probabi-listic flooding reduces to traditional flooding. In order toexplain further this observation, note that for the case ofpðNÞ ¼ H lnðNÞ

N

� �;GðN; pÞ has just become connected w.h.p.

which apparently means that the number or ‘‘redundant”links (links over which traditional flooding forwards mes-sages and probabilistic flooding ‘‘saves” by probabilistically

‘‘avoiding” to do so) is significantly reduced. The shape ofthe network – even though it contains cycles – looksmostly like a tree, [22,26], and therefore, the ability ofprobabilistic flooding to ‘‘avoid” forwarding messages over‘‘redundant” links is reduced.

As pðNÞ increases asymptotically, the potential advan-tage of probabilistic flooding increases, as it is shown inFig. 4b for various functions of pðNÞ of different asymptotic

order i:e:;ffiffiffiNp

N ;ffiffiffiNp

lnðNÞN ;

ffiffiffiNp

ln2ðNÞN

� �, [37], in order to illustrate in

more depth how pðNÞ affects RM;0ðNÞ. Note that as pðNÞ in-

creasesffiffiffiNp

N <ffiffiffiNp

lnðNÞN <

ffiffiffiNp

ln2ðNÞN

� �;RM;0ðNÞ decreases which

means that the savings in messages under probabilisticflooding increase when compared to traditional flooding.However, assuming large values of N but not that largesuch that N ! þ1, and pðNÞ ¼ 1;RM;0ðNÞ ! lnðNÞ

N > 0, whichis the asymptotic lower bound with respect to N. Still, asN ! þ1;RM;0ðNÞ ! 0.

It should be mentioned that the tightest lower boundwith respect to the number of messages for global networkoutreach is N � 1 for any (connected) network of N nodes,assuming that there is global knowledge available regard-ing the topology of the network in order to derive theappropriate paths for the forwarded messages in the net-work (e.g., creating a spanning tree). Such global knowl-edge cannot be assumed for the large-scale and dynamicnetwork environments considered in this work. What isinteresting is that the number of messages under probabi-listic flooding – that assumes no global knowledge for thenetwork topology GðN; pÞ – is (asymptotically) close tothe optimal one – that requires global knowledge – inthe order of HðlnðNÞÞðHðN lnðNÞÞÞ and N � 1 messages,respectively.

The reduced number of messages under probabilisticflooding is achieved at the expense of larger terminationdelays. This is shown by comparing the network diameter

of GðN; pÞ and PðGðN; pÞ; pf Þ, for pf ðNÞ ¼ H lnðNÞN

� �(the

upper bound of termination time corresponds to thenetwork diameter). Let RT;LðNÞ denote the (asymptotic)fraction of the network diameter of the probabilistic flood-ing network over the network diameter minus L (for fair-ness issues as it will be explained in the sequel in Section

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

10000 20000 30000 40000 50000 60000 70000 80000 90000 100000N

10000 20000 30000 40000 50000 60000 70000 80000 90000 100000N

)(0 NRM , )(0 NRM ,

)(Np)(Np

)(Np

0

0.05

0.1

0.15

0.2

N

NNp )(

N

NNNp

ln)(

N

NNNp

2ln)(

0.40.6

0.8

a b

Fig. 4. RM;0ðNÞ as a function of N for various values and asymptotic orders of pðNÞ.

1622 K. Oikonomou et al. / Computer Networks 54 (2010) 1615–1629

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6), for some L ¼ 0;1;2, or RT;LðNÞ ¼DðPðGðN;pÞ;pf ÞÞDðGðN;pÞÞ�L . Given that

for

L ¼ 0;DðGðN; pÞÞ ¼ H lnðNÞlnðpðNÞNÞ

� �;DðPðGðN; pÞ; pf ÞÞ ¼ HðDðG

ðN; p� pf ÞÞÞ ¼ H lnðNÞlnðpðNÞpf ðNÞNÞ

� �according to Lemma 1, and

pðNÞpf ðNÞ ¼ H lnðNÞN

� �, it follows that,

RT;0ðNÞ ¼lnðpðNÞNÞln lnðNÞð Þ : ð7Þ

Fig. 5 depicts RT;0ðNÞ for various values and asymptoticorders of pðNÞ. It is observed that the upper bound of ter-mination time does not increase as rapidly as the numberof messages decreases. For instance, for N ¼ 10;000 andpðNÞ ¼ 0:8, as it is depicted in Fig. 4a, RM;0ðNÞ � 0:1, andas depicted in Fig. 5a, RT;0ðNÞ � 4:03. So, in asymptoticterms and the particular depicted case, probabilistic flood-ing employs 0.1% of the messages required under tradi-tional flooding, at the expense of increasing the (average)upper bound of termination time 4.03 times. Given thatdiameters in random graph topologies are basically small(as already mentioned it is known as the ‘‘small-world”phenomenon, [22,23]) a time delay upper bound of the or-der of 4:03�DðGð10;000;0:8ÞÞ may be viewed as smalland thus not be prohibitive for employing probabilisticflooding.

Apart from the aforementioned increment of the (aver-age) upper bound of termination time, another drawbackof probabilistic flooding is the (unavoidable) employmentof a probabilistic process instead of a deterministic oneas it is the case under traditional flooding. For example,reduction of the number of messages under probabilisticflooding takes place at the cost of being achieved only withhigh probability, whereas under traditional flooding it isguaranteed.

So far in this section the global network outreach case(i.e., L ¼ 0 or Cð0Þ ¼ 1) has been studied. The cases corre-sponding to L ¼ 1 and L ¼ 2 are naturally expected to yieldsmaller number of messages under probabilistic flooding –compared to traditional flooding – since the particular val-ues of pf ðNÞ are expected to be (on average) smaller thanthose ensuring global network outreach (i.e., Cð0Þ ¼ 1)

w.h.p. Given that the case of L ¼ 1, corresponds to propertyQ1, as it can be seen from Table 1, pQ1

ðNÞ ¼ lnðNÞ2N ¼ H lnðNÞ

N

� �.

It is evident that the asymptotic analysis that has been pre-viously followed for L ¼ 0 applies to this particular case aswell. Note, that the resemblance is only asymptotic andsavings with respect to the number of messages are greaterfor the case of L ¼ 1 than for the case of L ¼ 0 under prob-abilistic flooding, as it will be demonstrated by simulationsin the following section.

The case of L ¼ 2, is different than the case of L ¼ 1 andL ¼ 0, since pQ2

ðNÞ ¼ 1N. Following the same steps as for the

case of L ¼ 0, it is concluded that,

RM;2ðNÞ ¼1

pðNÞN : ð8Þ

RM;2ðNÞ for various values of pðNÞ is depicted in Fig. 6.Note that for N ¼ 10;000 and pðNÞ ¼ 0:8 as depicted inFig. 6a, the (asymptotic) savings with respect to the num-ber of messages is about 0:01% as opposed to 0:1% de-picted in Fig. 4a.

6. Simulation results and evaluation

The purpose of the simulation results presented in thissection is two-fold: (a) to support the claims and expecta-tions of the analysis; and (b) to provide for further insightinto various aspects of probabilistic flooding in randomgraphs. A simulation program was developed in C pro-gramming language and random graph topologies up to10,000 nodes were created, while traditional and probabi-listic flooding was considered over these topologies. Thepresented results are averaged values of 100 independentruns. For each run the initiator node was randomly se-lected and different seeds with respect to pseudorandomsequences (governing the selection of initiator nodes, thecreation of topologies and message forwarding underprobabilistic flooding) were used. In many casespðNÞ ¼ 0:0008, that is a value large enough for GðN; pÞ tobe connected and at the same time small enough in orderfor the resulting topology to be of reduced number of linkswhich corresponds to a worst-case scenario under probabi-listic flooding when compared against traditional flooding.

3

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

10000 20000 30000 40000 50000 60000 70000 80000 90000 100000

)(0 NRT , )(0 NRT ,

)(Np

)(Np

)(Np

N N10000 20000 30000 40000 50000 60000 70000 80000 90000 100000

N

NNp )(

N

NNNp

ln)(

N

NNNp

2ln)(

1.5

2

2.5

3

3.5

4

a b0.8

0.6

0.4

Fig. 5. RT;0ðNÞ as a function of N for various values and asymptotic orders of pðNÞ.

K. Oikonomou et al. / Computer Networks 54 (2010) 1615–1629 1623

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Let CðLÞ denote the averaged (coverage) fraction of nodesthat belong to either the probabilistic flooding networkor the giant component of the under consideration randomgraph, depending on the case. Let RM;LðNÞðRT;LðNÞÞ denotethe averaged fraction of the number of messages (termina-tion time) under probabilistic flooding over that under tra-ditional flooding for L ¼ 0;1;2.

6.1. Bounded probabilistic flooding

Fig. 7a depicts simulation results with respect to cover-age for L ¼ 0. It is observed that probabilistic flooding’sperformance lays between the performance of GðN; p� ~pÞand GðN; p� pf Þ, as was expected from the analysis. In par-ticular, as pf ðNÞ increases, PðGðN; pÞ; pf Þ includes all net-work nodes after GðN; p� ~pÞ has achieved global networkoutreach (pf ðNÞ � 0:4 in Fig. 7) and slightly beforeGðN; p� pf Þ (pf ðNÞ � 0:5 in Fig. 7). Global network out-reach within the simulation section is considered to beachieved when network coverage is equal to 98%.

It is also interesting to note that Cð0Þ for the probabilis-tic flooding network is closer to the particular value of Cð0Þfor GðN; p� pf Þ than to Cð0Þ for GðN; p� ~pÞ. This is moreclearly depicted in Fig. 7b that is a zoom version ofFig. 7a in the range of values pf ðNÞ 2 ½0:25;0:5�. However,

by the time GðN; p� pf Þ achieves global network outreach(i.e., pf ðNÞ ¼ 0:5 according to the simulation results) allthree curves are close (Cð0Þ ¼ 99:8%, 98.2% and 98.14%,for GðN; p� ~pÞ;PðGðN; pÞ; pf Þ and GðN; p� pf Þ,respectively).

The analysis presented in Section 4.3 have shown theasymptotic ‘‘equivalence” among the previously discussedgraphs with respect to L ¼ 0 (i.e., Lemma 1). As it is de-picted in Fig. 8, this is also the case for L ¼ 1 and L ¼ 2 aswell (cases not covered by the analysis in Section 4.3).Obviously, coverage under probabilistic flooding is be-tween coverage of both (i.e., GðN; p� ~pÞ and GðN; p� pf Þ)random graphs for L ¼ 1;2 as well. The sudden increasein the performance of the probabilistic flooding is due tothe ‘‘phase transition phenomenon” that takes place. As itwas mentioned in Section 3, there is a critical probabilityfor larger values of which a property of the graph existswhile for lower values it does not. Therefore, coverageCðLÞ increases suddenly by the time property Q L issatisfied.

Fig. 9 depicts the particular values of pf ;LðNÞ for whichCðLÞ ¼ 0:98, for pðNÞ ¼ 0:0008 and different values of L.CðLÞ ¼ 0:98 is considered instead of CðLÞ ¼ 1 due to theprobabilistic nature of both random graphs and probabilis-tic flooding (e.g., sometimes for networks of 10,000 nodes

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0C

0.75

0.8

0.85

0.9

0.95

1

0.25 0.3 0.35 0.4 0.45 0.5Np fNp f

0C

a b

Fig. 7. Cð0Þ as a function of pf ðNÞ for N ¼ 10; 000 and pðNÞ ¼ 0:0008.

0

0.0005

0.001

0.0015

0.002

10000 20000 30000 40000 50000 60000 70000 80000 90000 100000

)(2 ,NRM

)(Np)(Np

)(Np

N10000 20000 30000 40000 50000 60000 70000 80000 90000 100000

N

0

5e-006

1e-005

1.5e-005

2e-005

2.5e-005

3e-005

)(2, NRM

N

NNp )(

N

NNNp

ln)(

N

NNNp

2ln)(

0.4

0.8

0.6

a b

Fig. 6. RM;2ðNÞ as a function of N for various values and asymptotic orders of pðNÞ.

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it is unreasonably time consuming for CðLÞ ¼ 1 to be satis-fied). As L increases, the particular value of pf ;LðNÞ de-creases as expected, since smaller fractions of thenetwork nodes are needed in order for probabilistic flood-ing to cover the network for the particular value of L. At thesame time, the particular value of pf ;LðNÞ significantly de-creases, as the network size N increases. This is due tothe fact that as N increases and given that pðNÞ is fixed,the number of redundant links significantly increases (forexample, if 10,000 nodes are already present in the net-work, the addition of a new node for pðNÞ ¼ 0:0008 – acomparably small value of pðNÞ – corresponds to on aver-age eight new links), thus allowing for the coverage ofthe network under probabilistic flooding for small valuesof pf ;LðNÞ. It should be also noted that the shape of thecurves in Fig. 9 is similar (in asymptotic terms) to thecurves depicted in Fig. 3 corresponding to the asymptoticvalues of pf ;LðNÞ as derived by the analysis.

6.2. Links and messages

Fig. 10 depicts simulation results for the number oflinks of the probabilistic flooding network and the numberof messages under probabilistic flooding that reveal someinteresting aspects of the probabilistic flooding process

not captured by the analysis. In particular, in Fig. 10a, thenumber of links is depicted for the previously mentionedrandom graphs and the probabilistic flooding network. Itis interesting to observe that for small values of pf ðNÞ(e.g., smaller than 0:18), the number of links of the proba-bilistic network EðPðGðN; pÞ; pf ÞÞ is smaller than the num-ber of links for either random graph. This is not acontradiction to the analytical results, since the analysiswas based on the existence of the giant component. Forsuch small values of pf ðNÞ < 0:18 (and given the small va-lue of pðNÞ ¼ 0:0008), the giant component is not present.This can also be observed by looking at the value of theCð0Þ in Fig. 7a, where the larger existing component thatcontains – at least 50% of the network nodes – is createdfor pf P 0:18 (when 50% of the nodes belong to a certaincomponent, it is safe to assume that a giant componentstarts to emerge, [26]). As soon as the giant componentemerges (for pf ðNÞ > 0:18), the number of links of theprobabilistic flooding network lays between the numberof links of the other two random graph topologies, as it isobserved from Fig. 10a and expected from the analysis.

Fig. 10b presents a comparison between the number ofthe links of the probabilistic flooding network and thenumber of messages sent under probabilistic flooding con-sidering a GðN; pÞ network topology. The vertical dottedlines correspond to the particular values of pf ðNÞ for whichCðLÞ ¼ 0:98, for L ¼ 0;1;2 (e.g., as also observed from Figs.7 and 8). It is interesting to note that for small values ofpf ðNÞ the number of links and the number of messages isabout the same, particularly for those values of pf ðNÞ forwhich Cð2Þ ¼ 0:98 and Cð1Þ ¼ 0:98. However, after the par-ticular value of pf ðNÞ for which Cð1Þ ¼ 0:98 and as pf ðNÞ in-creases – and before Cð0Þ ¼ 0:98 – the number of messagesgradually deviates (increases) from the number of links.The maximum difference takes place for pf ðNÞ ¼ 1, (around20,000 more messages than links in a network of 10,000network nodes) which is the case when probabilistic flood-ing reduces to traditional flooding. Even for pf ðNÞ ¼ 0:7(i.e., Cð0Þ ¼ 0:98) the difference is about 10,000 more mes-sages than links. This means that for a certain link of theprobabilistic flooding network, in many cases two mes-sages were forwarded over it (instead) of one, resultingin a significant number of redundant links particularly forCð0Þ ¼ 0:98 (i.e., global network outreach). Such redun-

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dancy is not observed for L ¼ 1 and L ¼ 2 as it is also dem-onstrated in Fig. 10b. The main reason is that bothCð1Þ ¼ 0:98 and Cð2Þ ¼ 0:98 are achieved for small valuesof pf ðNÞ (thus, redundancy is limited).

6.3. Global network outreach ðL ¼ 0Þ

Fig. 11 depicts simulation results with respect to RM;0ðNÞand RT;0ðNÞ for three different values of pðNÞ(0.004, 0.005and 0.01). As it can be observed from Fig. 11a, the savingson the number of messages increase with the number ofnetwork nodes N, for the same value of pðNÞ, as it is ex-pected by the analysis and particularly Eq. (6). There is aclear resemblance between the simulation results pre-sented in Fig. 11a and the (asymptotic) analytical resultspresented in Fig. 4. The savings on the number of messagesincrease as pðNÞ increases. This is due to the fact that largevalues for pðNÞ provide for an increase on the number ofnetwork links and, therefore, an increase on the numberof redundant links that are used under traditional floodingto forward messages.

Fig. 11b depicts simulation results with respect to ter-mination time ratio RT;0ðNÞ. RT;0ðNÞ increases by the in-crease on the network’s size, as it is already mentionedin the analysis (Section 5). It is observed that the largerthe value for pðNÞ, the larger the increase of RT;0ðNÞ as it

is also expected given Eq. (7). Note that an increase onpðNÞ corresponds to networks of more redundant links,smaller diameters and smaller termination times undertraditional flooding, which leads to an increase in RT;0ðNÞ.As before there is a clear resemblance between the simula-tion results presented in Fig. 11b and the (asymptotic) ana-lytical results presented in Fig. 5a.

6.4. The case of L ¼ 1 and L ¼ 2

Note that the comparison between probabilistic flood-ing and traditional flooding with respect to coverage, num-ber of messages and termination time, for L ¼ 1 and L ¼ 2is not a fair one since traditional flooding deterministicallyachieves CðLÞ ¼ 1 for L ¼ 0;1;2, having ‘‘spent” as many/much messages/time as required for Cð0Þ ¼ 0:98 � 1 (glo-bal network outreach). A fairer comparison – as it is thecase in the sequel – would be one under which controlledflooding is used for flooding information ðrðuÞ � LÞ hopsaway from any initiator node u, where rðuÞ is the maxi-mum number of hops between any node in the networkand node u over a shortest path (rðuÞ is usually referredto as the radius or eccentricity of node u and is upperbounded by the network diameter).

Fig. 12 depicts simulation results for RM;1ðNÞ and RT;1ðNÞ.The curve corresponding to RM;1ðNÞ (Fig. 12a) is similar to

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the one expected from the analysis (i.e., asymptoticallysimilar to RM;0ðNÞ as also observed from Fig. 11a). In fact,RM;1ðNÞ is slightly better than RM;0ðNÞðRM;1ðNÞ < RM;0ðNÞÞbecause of the smaller values for pf ðNÞ required for cover-age to become equal to 0:98 for either case (L ¼ 0 andL ¼ 1). RT;1ðNÞ is depicted in Fig. 12b and – as it is the casefor RM;1ðNÞ – the corresponding graph is similar to RT;0ðNÞ.

Fig. 13 depicts simulation results for both RM;2ðNÞ andRT;2ðNÞ. It can be observed from Fig. 13a that RM;2ðNÞ isslightly smaller than RM;1ðNÞ, as expected (as already men-tioned smaller values of pf ðNÞ are capable of achievingCð2Þ ¼ 0:98 rather than Cð1Þ ¼ 0:98).

The depicted simulation results with respect to RT;2ðNÞin Fig. 13b allow for an interesting observation. Apart froman increase at the beginning for small values of N, as N in-creases it is clear that RT;2ðNÞ decreases, in contrast to bothRT;0ðNÞ and RT;1ðNÞ or RT;0ðNÞ and RT;1ðNÞ.

In order to explain this observation it is important to getinto more details regarding the case of L ¼ 2. For a certain(fixed) value of pðNÞ as N increases, the number of linksamong nodes increases and eventually the diameter ofthe network decreases, [26]. For the conducted simulationsof network sizes between 1000 and 10,000 nodes (the sim-ulator had performance and memory limitations runningfor topologies larger than 10,000 nodes), the consideredvalues of pðNÞ (selected large enough such that all net-

works to be connected w.h.p.), were large enough to resultin small network diameters (for Gð10;000;0:0005Þ thediameter is 4) and therefore, comparable to 2. For suchcase, it is evident that the performance under the probabi-listic flooding is comparable to that under controlled flood-ing (rðuÞ � L hops away for the initiator node u). As Nincreases further, the network diameter becomes smaller,thus allowing for probabilistic flooding to faster cover thenetwork. Note, however, that this is a limited case sincefor a network diameter comparable to L, information dis-semination is becoming obsolete (e.g., for L equal to thenetwork diameter all nodes are at most L-hops away fromany network node by definition).

7. Conclusions

The problem of limited information dissemination inlarge, unstructured networks employing probabilisticflooding is studied and analyzed here. Global network out-reach is available in a network if the employed informationdissemination scheme is capable of taking a message fromany originating node to any other network node. Suchnetwork outreach is needed in order to support routingprotocols, advertise a new service, search for some infor-mation, etc. Traditional flooding schemes achieve global

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Fig. 13. RM;2ðNÞ and RT;2ðNÞ as a function of N for various values of pðNÞ.

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network outreach in unstructured networks with certainty(deterministically, for a connected network) at a large mes-sage overhead cost. In this paper, probabilistic floodingschemes have been considered in order to reduce theirassociated large overhead, at the price of providing proba-bilistic global network outreach guarantees. It is shownhere that the network created by probabilistic floodingover a random graph network asymptotically lays betweentwo random graph networks – which are determined-facil-itating this way the derivation of asymptotic analyticalexpressions on the value of the forwarding probability thatresults in a fairly decreased (compared to the traditionalflooding) number of messages, while covering the networkwith high probability (w.h.p.). Apart from global networkoutreach (i.e., L ¼ 0 case), the cases of L ¼ 1 and L ¼ 2 havealso been analytically studied.

A comparison of probabilistic flooding under the de-rived forwarding probability with the traditional floodingis carried out. For the case of global network outreach itis shown that the number of messages under probabilisticflooding asymptotically increases as N lnðNÞ as opposed toN2 under traditional flooding, for any fixed value of the for-warding probability. Thus, significant message overheadreduction can be achieved, especially for large networks(large N). The relative message overhead (compared to thatunder traditional flooding) is also derived and shown toyield substantial message overhead savings even for lowto medium values of N. However, as it was mentioned,the global network outreach achieved under traditionalflooding with certainty is now achieved only with highprobability. Termination time is also studied and as ex-pected, it is slightly higher under probabilistic flooding(compared to message reduction) but this is practicallyinsignificant if one considers typical time scales of floodingand usage of network information. Extensive simulationresults support the analytical claims and expectation andin addition reveal certain interesting aspects with respectto probabilistic flooding.

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Konstantinos Oikonomou, IEEE Member,received his M.Eng. in Computer Engineeringand Informatics from University of Patras,Greece, in 1998. In September 1999 hereceived his M.Sc. in Communication andSignal Processing from Imperial College(London) and his Ph.D. degree in 2004 fromthe Department of Informatics and Telecom-munications, National and Kapodistrian Uni-versity of Athens, Greece. His Ph.D. thesisfocuses on Topology-Unaware TDMA MACPolicies for Ad Hoc Networks. Between

December 1999 and January 2005 he was employed at Intracom S.A., as aresearch and development engineer. Since 2007 he holds a faculty posi-tion as Lecturer in Computer Networks with the Department of Infor-matics of the Ionian University, Corfu, Greece. His current researchinterests involve, among others, performance issues for ad hoc and sensornetworks, autonomous network architectures, efficient service discovery(placement, advertisement and searching) in unstructured environments,scalability in large networks.

Dimitrios G. Kogias received the B.S. degreein physics and the M.S. degree in radio-tele-communications from the University of Ath-ens, Greece, in 2001 and 2004, respectively.He is currently working towards the Ph.D.degree in Computer Networks at the Univer-sity of Athens, Greece. His research interestsinclude the study of algorithms for informa-tion dissemination in unstructured environ-ments (P2P and MANETs) and graph theoryproblems.

Ioannis Stavrakakis, IEEE Fellow: Diploma inElectrical Engineering, Aristotelian Universityof Thessaloniki, (Greece), 1983; Ph.D. in EE,University of Virginia (USA), 1988; Assist.Prof. in CSEE, University of Vermont (USA),1988–1994; Assoc. Prof. of ECE, NortheasternUniversity, Boston (USA), 1994–1999; Assoc.Prof. of Informatics and Telecommunications,University of Athens (Greece), 1999–2002 andProf. since 2002. Teaching and researchinterests are focused on resource allocationprotocols and traffic management for com-

munication networks, with recent emphasis on: peer-to-peer, mobile, adhoc, autonomic, delay tolerant and future Internet networking. Hisresearch has been published in over 180 scientific journals and confer-ence proceedings and was funded by NSF, DARPA, GTE, BBN and Motorola(USA) as well as Greek and European Union (IST, FET, FIRE) Fundingagencies. He has served repeatedly in NSF and EU-IST research proposalreview panels and involved in the TPC and organization of numerousconferences sponsored by IEEE, ACM, ITC and IFIP societies, including:organizer of the 1999 IFIP WG6.3 workshop, the COST-NSF NeXtwork-ing’03, the Workshop on Autonomic Communications (WAC2005); co-organizer of the 1996 ITC Mini-Seminar, the IEEE Autonomic Opportu-nistic Communications (AOC’07&’08); technical program co-chair for theIFIP Networking’00, EWC’04, IFIP WiOpt’05, COST-NSF NeXtworking’07;general co-Chair for Networking’2002, IFIP MedHocNet’07. He has servedas the chairman of IFIP WG6.3 and elected officer for the IEEE TechnicalCommittee on Computer Communications (TCCC). He is an associateeditor for the ACM/Kluwer Wireless Networks and Computer Communi-cations journals and has served in the editorial board of the IEEE/ACMtransactions on Networking and the Computer Networks Journals. He iscurrently the head of the Communications and Signal Processing Divisionof his Dept.

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