Connectivity Based Node Clustering in Decentralized Peer-to-Peer Networks

Lakshmish Ramaswamy Bug̃ra Gedik Ling Liu

College of ComputingGeorgia Institute of Technology

Altanta, GA 30324Email: �laks, bgedik, lingliu�@cc.gatech.edu

Abstract

Connectivity based node clustering has wide ranging applica-tions in decentralized Peer-to-Peer (P2P) networks such as P2P filesharing systems, mobile ad-hoc networks, P2P sensor networks andso forth. This paper describes a Connectivity-based DistributedNode Clustering scheme (CDC). This scheme presents a scalableand an efficient solution for discovering connectivity based clustersin peer networks. In contrast to centralized graph clustering algo-rithms, the CDC scheme is completely decentralized and it only as-sumes the knowledge of neighbor nodes, instead of requiring a globalknowledge of the network (graph) to be available. An important fea-ture of the CDC scheme is its ability to cluster the entire networkautomatically or to discover clusters around a given set of nodes. Weprovide experimental evaluations of the CDC scheme, addressing itseffectiveness in discovering good quality clusters. Our experimentsshow that utilizing message-based connectivity structure can consid-erably reduce the messaging cost, and provide better utilization ofresources, which in turn improves the quality of service of the appli-cations executing over decentralized peer-to-peer networks.

1 Introduction

In recent years the field of distributed data managementsystems has witnessed a paradigm shift from the traditionalClient-Server model to the Peer-to-Peer (P2P) computingmodel. While file sharing applications like Gnutella [4] andKazaa [6] were the harbingers of this change, various othersystems like mobile ad-hoc networks, P2P sensor networksetc. have adopted this model of computation and communica-tion.

Although these systems appear to be diverse and disparate,all of them share some distinct characteristics: (1) Networkand data management are completely decentralized. (2) In-dividual nodes have limited knowledge about the structure ofthe network. (3) Networks are dynamic with frequent entryand exit of nodes.

In this paper we use the term Peer-to-Peer systems in

a generic sense to refer to any system that possesses thesecharacteristics. Although, the P2P distributed computingparadigm alleviates the scalability problem that has doggedclient-server systems and enables a lot of interesting and use-ful applications, it also raises key research challenges thatneed to be addressed by the community including:

1. Scalable techniques for data discovery and peer look-up

2. Efficient mechanisms for communication among nodesin the network

The strategies adopted by most of the present systems toaddress these challenges are costly in terms of the number ofmessages required.

It is our contention that the absence of any knowledge ofthe network structure is proving to be a stumbling block inutilizing the full capabilities of P2P networks. We believe thatevery such network exhibits some unique structural propertiesand discovering these network structures is crucial to efficientdata discovery, node look-up and communication.

Connectivity-based node clustering is one such interestingand important network structure that can be utilized in variousways to improve the quality of service of applications run-ning on these networks. Informally, a connectivity-based nodeclustering (hereafter referred to as node clustering) can be de-fined as a partition of network nodes into one or more groupsbased on their connectivity. We provide a formal definition ofa node-cluster in Section 2. For now we shall assume that twonodes that are highly connected are placed in the same cluster.

To illustrate the utility of node clustering, consider a P2Pfile sharing system like Gnutella [4] or Freenet [3]. The pre-dominant file discovery mechanism in such systems has beenbroadcast-based breadth first search. Though, this is a simplesolution its major drawback immediately comes to the fore:The number of messages in the network is very high. To over-come this drawback some researchers have proposed tech-niques to limit the breadth first search by sending the querymessage to a few neighbors that are either random or selected

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based on some criteria. The trade-off here is between the num-ber of messages and the quality of search results.

Let us see how we can utilize the cluster structure for lim-iting the number of messages, while ensuring minimal degra-dation of search results. Suppose the nodes in the networkare clustered and each node in the network knows which ofits neighbors belong to its own cluster. Now a node �� in thesystem, on receiving a query message forwards it to all itsneighbors that do not belong the same cluster as ��. From theneighbors that belong to its cluster, �� selects a few nodes andforwards the message to them. The logic behind this schemeis that nodes in same clusters are highly connected and hencethe query message will eventually reach them.

In addition to the above example, researchers in the pasthave applied clustering information to address certain keyproblems in various decentralized P2P networks [1, 5, 7, 8].

Although node cluster information has been utilized to im-prove performance and scalability in P2P networks, very fewresearchers have studied the problem of discovering and main-taining node clusters in P2P systems. There has been con-siderable research in the algorithm community addressing theproblem of clustering nodes in directed and undirected graphs.Although the P2P systems are essentially undirected graphs,most existing graph clustering algorithms assume that the en-tire graph information is available in one central location. Un-fortunately, none of the P2P networks maintain their completeand up-to-date connectivity information. Therefore the needis to design schemes that can cluster the nodes of a network ina completely distributed and decentralized manner.

With these problems in mind, this paper presents a Connec-tivity based Decentralized node Clustering scheme (CDC), ascalable and an efficient solution for discovering connectiv-ity based clusters in peer networks. In contrast to centralizedgraph clustering algorithms, the CDC scheme only requiresthe local knowledge about neighboring nodes. An importantstrength of the CDC scheme is its ability to cluster the entirenetwork automatically or to discover clusters around a givenset of nodes in the network. Our experimental results indicatethat these schemes yield high quality clusters. An initial ex-perimental evaluation of the CDC scheme is provided, show-ing the effectiveness of the CDC scheme in discovering goodquality clusters.

2 Definitions and Terminologies

The connectivity structure of every P2P network can berepresented by an undirected graph with nodes of the P2P net-work forming the vertices of the graph and connections be-tween nodes being the edges of the graph. Henceforth we usethe terms graph and network interchangeably. Similarly, theterms node and vertex are used equivalently, and so are theterms edges and connections.

Let � � ����� be an undirected graph, where� � ���� ��� ��� ���� ��� is the set of nodes and � ����� ��� ��� ���� ��� is the set of edges in the graph G. Two

nodes �� and �� in the graph are said to be Connected if thereexists a series of consecutive edges ���� ��� ��� ���� �� � suchthat �� is incident upon the vertex �� and �� is incident uponthe vertex �� . The series of edges leading from �� to �� iscalled a Path from vertex �� to �� . The length of a path � isthe number of edges in the path.

A Similarity function � on the vertices of a graph � is asymmetric function mapping � �� to ��� , where ��� is theset of positive real numbers and ���� �� � ���� ��. Furtherthe function satisfies the condition that ���� �� �� iff � � �.The similarity and dissimilarity function may be appropriatelydefined according to the semantics of the particular graph un-der consideration and the particular application at hand.Twoof the most popular similarity functions have been the num-ber of K-Paths between the vertices and the Reach Probabilityfrom one vertex to another in the graph.

A Clustering of a graph � � ����� is collection ofthe sets ��� �� ���� ��, satisfying the following threeconditions: (1) each � is a non-empty subset of vertices(��� � � � ) and

������� � � � . (2) Any two nodes

in �, � � � �, are similar. (3) Any two nodes belongingto two different sets, say � and�, are not similar. Each ofthe � in is termed as a cluster. A clustering � is termedas a Disjoint Clustering if the clusters are pair-wise disjoint.

For graph clustering problem, the similarity can be definedin host of meaningful ways. Each of these definitions leadto different natural clusters and have different applications.However there are some properties that most useful clusteringschemes share.

� In a graph �, for any two nodes �� and �� which belongto the same cluster �, there exists at least one path suchthat all intermediate vertices along that path lie in �.

� For a graph �, any two vertices lying in the same clustertend to have large number of paths connecting them.

� A random walk on the graph tends to visit most nodes ina cluster multiple times before it leaves the cluster.

Based on these properties of good clustering, a numberof graph clustering algorithms have been proposed. The twomost popular schemes are: the K-Path Clustering Algorithmand the MCL Algorithm [10]. Like most existing graph clus-tering algorithms, both of them assume that the global infor-mation about the entire graph (i.e., the number of vertices, thenumber of edges, and their connectivity) is available in onecentral location.

3 CDC Scheme for Graph ClusteringIn contrast to centralized graph clustering algorithms, the

problem of distributed clustering assumes that each node haslimited view of the entire network. In this section we presentour scheme for distributed node clustering, termed as the Con-nectivity based Decentralized node Clustering scheme (CDC).First, we first formalize the distributed node clustering prob-lem and then discuss the CDC approach.

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Let � � ����� be an un-weighted, undirected graph.Each node �� in this graph is mapped to an autonomous andindependent computing element ���. Further each of thesecomputing elements knows only its neighbors. In other words,the node ��� has the knowledge of the existence of anothernode ��� iff ��� and ��� are neighbors in the graph �.This condition has important connotations. First, it impliesthat we do not have a centralized global view of the graph �.Second, it also means that each node can communicate onlywith its immediate neighbors. If it ever wants to reach a nodethat is not its neighbor, then it would have to route the messagethrough one of its neighbors. The problem is how to discoverreasonable node clusters in the graph � in a completely dis-tributed fashion, i.e. without ever constructing a global viewof the graph.

The problem setting described above reflects the scenarioin real-world systems like P2P file sharing systems, P2P sen-sor networks and ad-hoc mobile networks. As an example letus consider the Gnutella network [4]. Each peer in the net-work maintains a live TCP connection with a few other peers,which are called its neighbors. The knowledge of each peerabout the network is limited only to its immediate neighbors.

The central idea in the CDC scheme is to simulate flow inthe network in a distributed and a scalable fashion. Cluster-ing a graph through flow simulation is based on the followingintuition. Let us think of the graph as a network of mutu-ally intersecting roads. The roads are the edges of the graphand the intersection of two or more roads are the nodes of thegraph. Suppose a large number of people who do not knowthe structure of the roads start out from a node �� in the roadgraph, which we call the Originator node. Let each personcarry a weight �� along with him. As these persons are notaware of the structure of the roads, they choose any of theroads starting out from the node �� and travel along the roadto reach another intersection. Whenever they reach an inter-section, they drop some of the weight they are carrying at theintersection. Then they choose another road at random andcontinue to travel along that road to execute the same cycletill they are tired of walking or the weight they carry becomesnegligible. Now if one were to aerially observe the roads andthe intersections, he would observe two facts:

� If the graph structure has a densely connected graphstructure around the originator node, then a high percent-age of people can be observed in the nodes (intersections)and edges (roads) that lie in the dense region (i.e. cluster)around the originator.

� Nodes that lie inside the cluster would have accumulateda higher weight from all the people who passed throughthe node and dropped part of their weight. In compari-son, the nodes that are remotely approachable from theoriginator would have accumulated very small weight.

These observations lead us to the central idea of the algo-rithm. If there are a few originators in the graph from where

people would start their random walk, the nodes would ac-quire weight from different originators. The idea then is thateach node would join a cluster from whose originator, it re-ceived the maximum weight. If some node did not receiveenough weight from any node, then it decides to be an outliernode (a node that does not belong to any cluster).

In a distributed P2P network, peer nodes are analogous tointersections, and connections between peers represent roads.People moving about are simulated by messages that are cir-culated in the network. Each message has a predefined Timeto Live(��� for short). Each message executes only ���

hops, after which it expires and is discarded.

3.1 CDC Algorithms

The CDC algorithm starts out by initiating messages froma set of nodes that are the originators of the clustering al-gorithm. The set of originators is represented by � ����� ��� � ���, which initiate the process of message cir-culation by sending out messages to all its neighbor nodes.

Each cluster message is a tuple consisting of the followingfive fields:

� Originator ID (OID): A field uniquely identifying theOriginator node

� Message ID (MID): A field distinguishing each messagefrom all other messages from the same originator.

� Message Weight (MWeight): The weight carried by themessage

� Source ID (SourceID): A field indicating the most re-cent node the message visited

� Time to Live (TTL): The maximum number of hops thismessage can be re-circulated

The algorithm for the originator selection itself is com-pletely distributed and is explained in detail in Section 3.2.For now we assume that we have been provided with a set oforiginators.

The �� ����, the ��� and the ��� fields in the mes-sage tuple are self explanatory and straightforward to initial-ize. The weight function that we use estimates the probabil-ity of reaching any node from originator nodes. An origina-tor node �� initializes message weight as ���������� �

�����������

.Each node �� maintains a set of values, represented as

��������������� ���. This value indicates the sum of theweights from all the messages that originated at �� andreached ��. On receiving a message ���, the recipient ��updates the ����������� function corresponding to the mes-sage originator. Then �� checks whether the ��� of the mes-sage is greater than �. If so, �� forwards the message to all itsneighbors. Before the re-circulation, the recipient updates the������� and the ���. The message weight is divided bythe degree of �� and the ��� is decremented by 1.

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Figure 1: CDC Illustration

Each node may receive multiple messages from severalnodes. It calculates the ���������� function for each ofthe originators from which it received messages. Each nodejoins the cluster led by the Originator, for which the value of���������� is the maximum. If all ���������� valueslie below a predefined threshold, then the node remains anoutlier. The psuedo-code for the CDC scheme is provided inAlgorithm 1 and 2.

Now we provide an example to illustrate the functioningof the CDC algorithm. In Figure 1 we have a graph of 14nodes, labeled from � to ��. We illustrate how the algorithmworks when the process is initiated by three Originator nodes�� � ���. The figure illustrates ���������� acquired byeach node at different steps of the algorithm execution. Thearrows in the diagram denote the messages flowing throughthe system. The weight carried by each message is indicatednext to the arrow. The last diagram shows the ����������

each node has gained due to the three originators after 4 hops.Each node joins the cluster from which it has gained the maxi-mum ���������� leading to three clusters marked in the di-agram. However, for node �, the ���������� received fromall Originator nodes are less than the threshold and hence itbecomes an outlier.

Algorithm 1 Algorithm Executed by Message Originator ��

Create a New Message ���

������� � �� , ��������� � ���������� �

���� ������ � ��, ������� � ��������������� � Current System Time �A unique value�for Each node �� � ������� do

Send ��� to ��end for

The algorithms described above are self-explanatory. How-ever, we want to discuss an important issue regarding theweight function we are using in the algorithm. If a node �in the graph receives messages whose ��� � from orig-inator ��, then the quantity,

����������������������

has special significance. This quantity indicates the probabil-ity of being in node �, if one were to start from node �� and

Algorithm 2 Algorithm Executed by Node � on Receiving���

�Check whether I have received messages from ��������if I have seen messages from ������� before then�Check if the ������������� �� ��������if ������������� ������������� then������������ � ���� ������������ � ��� ����������

else������������ � ���� ���������

�������������� �����������end if

else�This is the first message from ��������������� � ���� ������������ � ��� ����������

�������������� �������end ifif ������������ � ��� � ������� then������� � ������������ � ������������ � �������

end ifif ������� � � and ��������

��������� �� ������ then

Create a New Message�������������� � �������, ������ ������ � ��

����������� � ��������

��������� �

��������� � �������� � ��, ��������� � �������

for Each node �� � ������� doSend ��� to ��

end forend ifWait for ���� � in anticipation of other messagesif ������� � ������������ then

Join the cluster led by ���������

elseRemain an outlier

end if

perform a random walk of exactly ��������������� steps.This section described algorithms to discover clusters

around a given set of Originators. In the next section we ad-dress the question of how to select Originator nodes so thatthe scheme yields good clusters.

3.2 THP Originator Determination Scheme

Choice of Originators is critical to the performance of theCDC scheme discussed in the previous section. We discuss anexample that elucidates the significance of selecting “good”originator nodes. The diagrams in Figure 2 shows four differ-ent scenarios, indicating the clusters we obtain when we start

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out with four different sets of Originators. In each scenario,the originators are indicated by the shaded nodes.

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Scenario 1 Scenario 3

Scenario 2 Scenario 4

Figure 2: Importance of Good Originators

Scenario 1 is the best clustering we can obtain for thegraph. In this case we have three clusters and a single out-lier. The clustering in scenario 2, though not ideal is again agood clustering. The clusters we obtain in other two scenariosare unintuitive and are in no way close to ideal clustering inscenario one. Though we have used the same ���, the sameweight function and the same number of Originators, we ob-tain different clusters that not only vary in number but alsoin their quality. This example demonstrates the importance ofselecting good Originators.

Now we briefly discuss the properties a good originator setshould possess.

� First, the set of originators should be spread out in allregions of the graph.

� Second, a node �� is considered to be a good origi-nator if it acquires more weight due to messages initi-ated by it than the weight acquired by messages initiatedby any other originator. i.e. ������������� ��� �

������������� ������� � � .

In any graph, it is not desirable to have originators that ac-cumulate more weight from messages that originated at someother node than the messages initiated by it. If such were to bethe case, then the originator itself would “defect” to a clusterinitiated by some other originator. This again would result inthe formation of bad clusters.

The originators Scenario 1 and Scenario 2 in Figure 2 sat-isfy both these conditions and yield good clusters. In contrastthe originators in Scenario 3 are concentrated in one single re-gion. The originators in Scenario 4 do not satisfy the secondcondition. Both of these yield bad clusters.

Although the second property is logical, the crucial ques-tion is how do we determine whether a node satisfies this cri-terion? This property demands that we know ���������

for each pairs of vertices, which cannot be determined till weactually execute the CDC scheme.

We adopt an approximation technique to solve thisproblem, which we call the Two Hop Return Prob-ability technique. In this technique we determinethe probability of returning to a node �� in the graphin two hops, if we were to perform a random walkon the graph starting at ��. This is calculated as� ������������ �

�����������

� ����������������������

�.A higher � ������������ indicates that the node�� has a higher chance of satisfying the condition������������� ��� � ������������� ������� � � �,and hence has a lesser chance of “defecting” into anothercluster. As this scheme relies upon the two hop returnprobability, we term it as the Two-Hop-Probability scheme orTHP scheme for short.

The THP scheme performs two tests. First, it checkswhether the node has already received any clustering mes-sages from other nodes in its vicinity. If so, it means thatthere are other nodes in its vicinity that have already cho-sen to be Originators. Hence the node opts not to becomean originator and does not initialize messages. If the nodediscovers that there are no Originators in its vicinity, thenit obtains the degree of each of its neighbors and computesthe Two Hop Return Probability (� ��������). If the TwoHop Return Probability is higher than a pre-defined threshold(� �������������) then the node chooses to be an Origi-nator. Otherwise, the node will not become an Originator.

The two configurable parameters for this algorithm are the� ���� factor and the � ������������� factor. If thesefactors are set to high values, the number of originators woulddrop and vice-versa.

The THP algorithm is completely distributed. The numberof messages circulated in this phase is also very small. Eachnode, which has not received a message from an originator inits vicinity, has to just get the degree of its neighbors. Hencethis scheme is very efficient in terms of the messaging cost.

4 Handling Node Dynamics

Most P2P networks exhibit some degree of dynamism intheir structures, although the nature and the scale of dynamismare specific to each network. For example in P2P file sharingsystems, the peer nodes enter and exit the system at arbitrarypoints in time. This dynamics in the structure affects the ex-isting clusters of the nodes in the network. Re-clustering theentire network on each node-entry or exit is impractical fortwo reasons: 1. Re-clustering on each node-entry and exitcauses the network to be loaded with clustering messages. 2.If the entry and exit of the nodes are frequent then the clustersnever stabilize. Even before the clusters are discovered an-other node might enter or exit the system, causing the wholeprocess to restart. Therefore efficient and effective schemesare needed to handle the node entry and exit.

We have designed algorithms to handle the entry and exitof the nodes in P2P networks. These schemes are based on thefact that node entry and exit are localized phenomena that af-

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fect the network structure in the close vicinity of the enteringor the exiting node. The key idea of the node entry mecha-nism is to calculate the “attraction” between the existing clus-ters and the entering node. The node then joins the cluster towhich it is most attracted to. Similarly when a node exits thesystem, its immediate neighbors recalculate their “attraction”to various clusters and “defect ” to the cluster to which theyare most attracted to. These algorithms are completely dis-tributed and involves messaging between the entering/exitingnode and its immediate neighbors. Due to space limitations,we do not discuss these schemes in this paper. Interested read-ers may refer to our technical report [8], where we discuss thealgorithms in detail and experimentally evaluate the schemes.

5 Experiments and ResultsThis section reports the experiments we performed to eval-

uate the proposed schemes and the results we obtained. Webegin by discussing the metric used for measuring graph clus-tering accuracy.

5.1 Accuracy Measure for Graph Clustering

Measuring the accuracy of a given clustering on graph isin general a tricky task. This is primarily because, unlike datapoints in Euclidean spaces, the distance measure for graphscan be defined in many different meaningful ways. Hence itis possible to obtain different accuracy measures based on thedifferent similarity measures. In this paper we use an intuitiveperformance measure proposed by [11] termed as the ScaledCoverage Measure.

Let � � ����� be a graph and let �� ������ ���� ���� ���� be a given clustering on the graph. Letus consider any node ��.

� Let ������ denote the set of neighbors of ��.

� Let �������� denote the set of all nodes that belong tothe same cluster as that of vertex ��, i.e. �������� ���� if �� � ���.

� False Positive Set is defined as the set of all nodes, whichare not neighbors of ��, but are included in the samecluster as ��, i.e. ���������������� ��� � ��� ��� ��������� � �� �� �������.

� False Negative Set is the set of all neighborsof ���� but are excluded from ��������, i.e. ���������������� ��� � ������ � ������ � �� �����������.

Then define Scaled Coverage Measure of the �� in � withrespect to the clustering � as:

��������� ��� � ��� ������������ � ����� � �������������� � ����

�������� � ����������(1)

Some of the salient features of the Scaled Coverage Mea-sure are:

� It assumes a value of ��� at best and ��� at worst.

� It “punishes” a clustering for both false positives andfalse negatives.

� A node in a sparse region of the graph is penalized morefor false positives and false negatives than a node in adense region.

The accuracy of a clustering � over a graph � is definedas the average of the Scaled Performance Measure of all of itsnodes.

������������������� �

�����

��������� ���

�� �(2)

The highest clustering accuracy achievable for any graph � iscalled its optimal clustering accuracy. The optimal clusteringaccuracy for any graph depends upon its structure. It evalu-ates to 1 for graphs which contain only fully connected com-ponents, with no edges across the components. For all othergraphs the optimal clustering accuracy is strictly less than �.

5.2 Experimental Data Sets

For our experimental evaluation, we have used two kindsof datasets:

Power-Law Topology: We use power-law topology togenerate graphs that resemble P2P data sharing networks intheir topological structure. Studies, in past few years, ontopology of the Internet [2] and more recently on P2P datasharing networks [9], have revealed that topology of such net-works closely follow what is well known as a Power-Law Dis-tribution. For our experiments we have used power law topol-ogy graphs with 100, 200, 500, 1000, 2500 and 5000 nodes.

Range Topology: Range topology models the connectivityrelationship in wireless/sensor networks. In wireless networkstwo computing units have the knowledge of the existence ofthe other, only if they fall in each others radio range. Rangetopology graphs model this phenomenon. In our experimentswe have use range topology graphs with 100, 200, 500, 1000,2500 and 5000 nodes.

Table 1 lists some of the important properties of bothdatasets.

5.3 Experimental Results

In this section we provide a brief description of eachexperiment and the results obtained. We begin by comparingthe accuracy of the CDC algorithm.

Cluster Accuracy of CDC SchemeOur first experiment is aimed towards demonstrating the

effectiveness of the CDC scheme we have proposed. In or-der to do so, we compare the accuracy of the CDC schemewith the Centralized MCL Clustering and the Distributed K-Path Clustering schemes. We use the public domain softwaredeveloped by the author of [10] for obtaining clusters in thecentralized MCL scheme.

The MCL graph clustering software requires us to set aconfigurable parameter, which controls the clustering granu-larity. This parameter can take on values from ��� to �. When

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Parameter Power RangeGraphs Graphs

Total Nodes 5000 5000Total Edges 11446 27083Average Degree 4.57 10.83Maximum Degree 623 25Minimum Degree 1 1Variance in Degree 212.53 11.08

Table 1: Parameter Values for Power andRange Topology Graphs 100 200 500 1000 2500 5000

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Figure 3: Accuracy of CDC Scheme onPower Graphs

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Figure 4: Accuracy of CDC Scheme onRange Graphs

the parameter is set to lower values, the algorithm yields fewernumber of large clusters and vice versa. We have obtainedclusters by setting it to 1.2, 2, 3, 4 and 5. We measure the ac-curacy of the clusters obtained and use the highest value as ourbenchmark. We would like to make it clear that a value, whichis in between these values might yield cluster with slightlyhigher accuracy.

In this experiment we want to test the accuracy of the clus-ters yielded by the bare CDC algorithm. Hence we turn off theTHP originator determination mechanism. Instead, we ran-domly select originators. In the experiments we report werandomly select ��� of the total nodes in the graph. Thesenodes act as the originators, initiating the messages. As theoriginator selection is random, we have performed each ex-periment ��� times and we report the average of the accuracyvalues obtained.

Figure 3 and 4 indicate the clustering accuracy of CDCscheme, the centralized clustering scheme and the DistributedK-path clustering scheme on power-law topology and rangetopology graphs respectively. The graphs show that the CDCscheme performs better than the K-path clustering scheme forall graph sizes of both power and range topology. The ac-curacy value of the CDC scheme is around ��� and ���

higher than the accuracy of the distributed K-path scheme fora power-law graph and range topology graph of 5000 nodesrespectively.

However the centralized MCL scheme performs better thanthe CDC scheme for both power-law and range graphs. Thecentralized MCL scheme beats the CDC scheme by almost��� and ��� for power graph and range graph with 5000nodes respectively.

These results show that the CDC scheme can yieldclustering results comparable to the centralized scheme, thusdemonstrating the reasonableness of the CDC approach.

THP Originator Mechanism AccuracyHaving demonstrated that CDC scheme is a reasonable

approach for the distributed graph clustering problem, wenow demonstrate the effectiveness of THP originator selec-tion mechanism.

Figure 5 and 6 indicates the clustering accuracy of the Two-

Hop Probability originator determination scheme and com-pares it with the clustering accuracy of the CDC scheme withrandom originator selection and centralized MCL scheme onpower-law and range topology graphs respectively

The results show that THP originator determinationscheme improves the clustering accuracy of the CDC schemeconsiderably for graphs of both topologies. For example, ona power-law graph of 5000 nodes, the THP originator mech-anism yields a mean accuracy value 0.381 as against 0.267given by the CDC scheme with random originators, whichamounts to an improvement of over ���. Similarly for rangegraph of 5000 nodes, the improvement is over ���.

Surprisingly, the CDC scheme coupled with the THPoriginator determination mechanism yields better accuracyvalues than the centralized MCL scheme for both power andrange graphs with 200, 500, 1000, 2500 and 5000 nodes.We did not expect our scheme to perform better than thecentralized scheme. We think that this phenomenon is dueto the structural properties of these two topologies. We feelthat for graphs of other topologies, the centralized schemewould perform slightly better than the CDC scheme with THPoriginator mechanism. Hence, though we do not claim thatour algorithm performs better than the centralized clustering,we certainly feel that the accuracy of the clusters given by ourscheme is very close to the accuracy of the clustering yieldedby centralized MCL scheme.

Does CDC Scheme Need High TTL?One concern which we had regarding the CDC scheme was

whether we need to initialize messages with high TTL valuesin order to obtain good clusters? Using messages with highTTL values has two problems which might be detrimental tothe practicality of the scheme. First, it increases the messageload on the network. Second, it increases the time required forthe clusters to emerge.

In order to figure out the effect of Initial-TTL on the clus-tering accuracy, we obtained various clusters by setting theInitial-TTL values from 1 to 5. Due to space constraints welimit our discussion to power-topology graphs. The Figure 7indicate the accuracy values of power-law topology graphswith 200, 500 and 1000 nodes.

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100 200 500 1000 2500 50000.0

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Figure 5: Accuracy of THP Scheme onPower Graphs

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Figure 6: Accuracy of THP Scheme onRange Graphs

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Figure 7: Accuracy at Various TTL forPower Law Graphs

The results indicate that the algorithm yields good clusterseven when the TTL is set to 2. The accuracy values for TTL of5 are almost equal to the accuracy yielded by the scheme whenthe TTL is 2. This demonstrates that the scheme stabilizesfast, which is a necessity for any distributed algorithm.

6 Related Work and ConclusionIn this paper we have proposed the Connectivity based Dis-

tributed Node Clustering (CDC) scheme for clustering nodesof a decentralized peer-to-peer network. The scheme can ei-ther cluster the entire network automatically or detect clustersaround a given set of nodes. To the best of our knowledge,this paper is the first one to report a completely distributedand decentralized scheme for node clustering. This work hasbeen primarily inspired by the MCL algorithm proposed byDongen [10]. Researchers in algorithmic graph theory haveproposed a couple of algorithms for general graph clustering.Out of these algorithms the two most significant ones from apractical view point have been the K-path and the MCL clus-tering algorithms [10].

The K-path clustering algorithm is based on the observa-tion that in a graph �, any two nodes in the same cluster(if such clusters indeed exist in �) would be connected bylarge number of paths of lengths �, � � �. The MCL algo-rithm introduced by [10] views the graph as a Markov Chainand operates on the corresponding Markov matrix, which con-tains one-step transition probabilities between all pairs of ver-tices. The algorithm defines a non-linear operator termedas the Inflation operator. Alternate application of the self-multiplication operator and the inflation operator, reveals theclusters in the graph.

The key difference between our work and the ones dis-cussed in these papers is that these are centralized graph algo-rithms working on the global view of the graph, whereas ourscheme is completely distributed and does not need a com-plete connectivity structure.

The work in the distributed computing community address-ing the problem of electing cluster-heads bears some resem-blance to our work [1]. These algorithms although distributed,do not attempt to cluster the network based on its connectivitystructure. Hence the clusters discovered are not necessarily

“good” clusters from a connectivity stand-point. In contrastour scheme is entirely based on connectivity structure of thenetwork and hence leads to high quality clusters.

Several researchers have proposed applying node cluster-ing information to various systems for improving the effi-ciency and quality of service [1, 5, 7, 8]. However very few ofthem address the question of discovering good quality clus-ters. In short, the work reported in this paper is unique andvery few researchers have addressed the connectivity baseddistributed node clustering problem in such detail as we havedone in this paper.

We plan to extend this work in a variety of directions. First,we want to experiment with graphs of various other topologi-cal structures to study how our scheme performs. The secondline of research we want to pursue is to design variants of ourscheme to suit specific needs of different networks like P2Pnetworks with low bandwidth or networks with devices whichhave low battery power etc. We think it is not only feasible butalso important to design variants of our scheme to suit specificconstraints. Finally, we also plan to study in detail, the ap-plication of cluster information in various decentralized P2Psystems such as P2P file sharing systems, sensor networks,mobile ad-hoc networks etc.

References[1] G. Chen, F. G. Nocetti, J. S. Gonzalez, and I. Stojmenovic. Connectivity Based

k-hop Clustering in Wireless Networks. In 35th Hawaii International Conferenceon System Sciences, 2002.

[2] M. Faloutsos, P. Faloutsos, and C. Faloutsos. On power-law relationships of theinternet topology. In SIGCOMM, pages 251–262, 1999.

[3] Freenet Home Page. http://www.freenet.sourceforge.com.[4] Gnutella development page. http://gnutella.wego.com.[5] W. Heinzelman, A. Chandrakasan, and H. Balakrishnan. Energy-Efficient Coomu-

nication Protocols for Wireless Microsensor Networks. In Hawaii InternationalConference on System Sciences, 2000.

[6] Kazaa home page. http://www.kazaa.com.[7] P. Krishna, N. Vaidya, M. Chatterjee, and D. Pradhan. A Cluster-based Approach

for Routing in Dynamic Networks, 1997. ACM SIGCOMM Computer Communi-cation Review, April 1997.

[8] L. Ramaswamy, B. Gedik, and L. Liu. Connectivity Based Node Clustering inDecentralized Peer-to-Peer Networks. Technical report, College of Computing,Georgia Tech, 2003.

[9] M. Ripeanu. Peer-to-peer architecture case study: Gnutella network. In Proceed-ings of International Conference on Peer-to-peer Computing, August 2001.

[10] S. van Dongen. A New Cluster Algorithm for Graphs. In 281, page 42. Centrumvoor Wiskunde en Informatica (CWI), ISSN 1386-3681, 31 1998.

[11] S. van Dongen. Performance Criteria for Graph Clustering and Markov ClusterExperiments. Technical report, National Research Institute for Mathematics andComputer Science in the Netherlands, 2000.

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