Significance of Eigenvector Centrality for Routing in a Delay Tolerant Network

Transcription

1 Journal of Computatons & Modellng, vol.1, no.1, 2011, ISSN: (prnt), (onlne) Internatonal Scentfc Press, 2011 Sgnfcance of Egenvector Centralty for Routng n a Delay Tolerant Network R. J. D Souza 1 and Johny Jose 2 Abstract Centralty measure s an mportant concept n networks. It ndcates the relatve mportance of nodes n a network. Varous centralty measures have been proposed n the lterature, such as degree centralty, closeness centralty etc. Practcally all these measures are some values based on the propertes of the node concerned. Egenvector centralty takes nto account the centralty value of the neghbours of a node to assgn a centralty value to t. In ths paper, we show how ths value can be utlzed to select relay nodes n a delay tolerant network and mprove the delvery delay. Keywords: Egenvector, Routng Protocols, Delay Tolerant Networks C.2.2 [Computer Communcaton Networks]: Network Protocols Routng Protocols. 1 Department of Mathematcal and Computatonal Scences, Natonal Insttute of Technology Karnataka Surathkal, Karnataka, , Inda, e-mal: rjd@ntk.ac.n 2 Department of Mathematcal and Computatonal Scences, Natonal Insttute of Technology Karnataka Surathkal, Karnataka, , Inda, e-mal: johnysdb@gmal.com Artcle Info: Revsed : August 3, Publshed onlne : August 31, 2011.

2 92 Sgnfcance of Egenvector Centralty for Routng 1 Introducton A Delay Tolerant Network (DTN) s a sparse and moble ad hoc network. In a typcal DTN, some node pars never meet each other. The moblty of the nodes causes connectvty to be ntermttent and may lead to even more unpredctablty of routng nformaton. A routng protocol for a DTN should address all these ssues. Several methods have been proposed to acheve effectve routng n a DTN envronment, startng wth the floodng approach, whch blndly sends the message to all the nodes that t meets [12]. Other methods ntellgently choose the relay nodes based on metrcs lke hstory of contacts, context nformaton etc. [2, 3]. Ths work proposes a method that chooses the relay nodes based on human moblty characterstcs. Fundamental to the development of ths concept s the fact that human moblty pattern s 93% determnstc [4] and the dstrbuton of the moblty parameters lke contact duraton, nter-contact duraton and locaton vstng preference follows heavy taled power law, up to a characterstc tme [4]. The objectve of ths work s to dentfy sutable relay nodes that wll mprove the delvery latency n a DTN, usng centralty measure. The rest of the paper s organzed as follows. Secton 2 explans the human moblty characterstcs and how t can be utlzed n the desgn of a routng protocol for DTN. Secton 3 brngs out the related work n ths area. Secton 4 explans the theory behnd egenvector centralty and secton 5 develops an algorthm that utlzes the same. Secton 6 studes the smulaton result. The concludng remarks are made n secton 7.

3 R.J. D Souza and J. Jose 93 2 Characterstcs of Human Moblty For long, human moblty pattern was consdered to be fundamentally stochastc and random moblty models were used n the studes that based tself on human moblty. Falure of these models to relate to real lfe scenaro, gave rse to other moblty models that captured some aspects of human moblty. Chantreau et al. [4] showed that human moblty s sem determnstc and followed power law over a large perod of tme. Power law dstrbuton s characterzed by the equaton p( t) t a (1) where p() t s the nter contact tme (ICT) dstrbuton and s the power law exponent. It was notced that the value of a was typcally 1, n all the avalable traces. Ths means that the mean delay for any routng algorthm based on ICT dstrbuton s nfnte. Karaganns et al. [8] further showed that ICT followed power law up to a characterstc tme and after that, t followed exponental decay. The authors found that the characterstcs tme was of the order of about 12 hours. They also notced that the mean ICT s of the same order as the characterstc tme. The mplcaton of ths dchotomy s that the expected forwardng delay becomes fnte, unlke the case where a 1 and wthout the exponental decay. Hence the node contacts can be utlzed to transfer message n a DTN, wth a fnte delay. 3 Related Work The recent research on routng messages n DTNs utlzng human moblty characterstcs has shown that ncorporatng socal nformaton n routng decson mproves the performance. The HBOp algorthm proposed by Boldrn et al. [1], work of Mklas et al. [9], and SmBet routng protocol proposed by Daly and Haahr [5] are a few examples.

4 94 Sgnfcance of Egenvector Centralty for Routng Ths motvates us to nvestgate how the delvery latency can be mproved n a DTN, by ntellgent selecton of the relay nodes. A relay node s an ntermedary node that meets the source n slot p and the destnaton n slot q, such that p < q< r, where r s the slot where the nodes drectly meet. 4 Centralty Measures It s well known that n a network some nodes are more popular than others. Such nodes are known as central nodes and can functon as effectve relay nodes. Several metrcs have been proposed as a measure of ths centralty. Degree, closeness, betweenness etc. are a few of them. Ths work proposes another method to dentfy these popular nodes and demonstrates that t mproves the delvery rato and delvery latency. 4.1 Egenvector centralty Degree centralty s one of the smplest centralty measures. It s the count of the connectons that a node has. Havng a large number of connectons has ts sgnfcance. However, a node wth a few hgh-qualty contacts may outrank another one wth a large number of medocre contacts. We wll now derve a method that contans ths concept. Let the centralty of a node be x and A be the adjacency matrx of node. Snce the connecton to nodes that are themselves mportant, makes a node more central, we can say that x s proportonal to the average of the centraltes of the neghbours of node,.e., x n Ax j j j= 1 1 n Ax j j λ j = 1 x =, (2)

5 R.J. D Souza and J. Jose 95 where λ s a constant. Let x be the vector of centraltes. Then, Now (2) can be rewrtten n matrx form as x= ( x, x, ) Ax 1 2 = λx (3) Hence we see that x s an egenvector of the adjacency matrx A of the network wth egenvalue λ. We want all the components x, for all, of the egenvector to be postve, as they are centralty values. Perron-Frobenus theorem states that all the elements of the egenvector correspondng to the largest egenvalue wll be postve. Hence λ must be the largest egenvalue of the adjacency matrx of the network. Ths makes x to be the prncpal egenvector of the adjacency matrx defnng the network. Each element of ths vector represents the centralty of the correspondng node n the network. The hghest element s the largest centralty value. Hence the correspondng node s the most central node. It can be seen that, a node that has a hgh egenvector score s adjacent to nodes that are themselves of hgh scorers. Thus egenvector centralty s an nfluence measure, that depends both on the number and qualty of ts connectons. We apply ths approach to dentfy the central nodes n a network and use them as relay nodes. 5 Proposed Algorthm R. J. D Souza and Johny Jose [10] have proposed an algorthm that uses a sngle relay, utlzng human moblty characterstcs, to mprove the delvery rato and delvery latency n a DTN. Ths algorthm s based on the sem-determnstc nature of human moblty, as proved by Song et al [4]. It dvdes the tme nto several slots and keeps track of the neghbours n each slot, as a contact graph.

6 96 Sgnfcance of Egenvector Centralty for Routng The hgh degree of sem-determnsm n human moblty results n the contact pattern to be repeated daly. Ths knowledge s used to fnd routes by R. J. D Souza and Johny Jose [10] and the results are found to be better than those of the exstng DTN routng algorthms. In the present work, we dentfy relay nodes by computng the egenvector of the contact graph n each tme slot. The result s compared wth that of [10]. 5.1 Algorthm to Fnd the Optmal Path Assume that the smulaton tme s dvded nto S slots of duraton T. Check f the source meets the destnaton n any slot. If the meetng takes place n the 1 st or 2 nd slot, Handover the message drectly. Endf. If the meetng takes place n a later slot, say slot r, Generate the adjacency matrx of the node contacts n slot r. Fnd the egenvalues of ths matrx. Fnd the largest egenvalue. Fnd the correspondng egenvector. Fnd the hghest value n ths egenvector. Fnd the node correspondng to ths value. Assgn ths node as the relay node. Endf 5.2 Calculaton of Delvery Latency The above algorthm gves the earlest slot for handng over the message. In the best case, the message generaton and the message handover takes place n the

7 R.J. D Souza and J. Jose 97 same slot. If the message handover takes place n a later slot, t entals latency. Ths latency can be calculated as follows. Let the message handover for the th message take place n slot s. Wdth of each slot Delvery duraton = t tme unts. = s t tme unts. Delvery latency = ( s 1) t tme unts. The latency experenced by each packet s dfferent. The average latency experenced by a packet s calculated as the average of the sum of latences, experenced by the packets that were successfully delvered. Average latency = ( delvery tme of the delvered packets) number of packets delvered. 6 Expermental Results and Dscusson The communcaton opportuntes for varous values of communcaton range and node densty were smulated usng Matlab. The synthetc moblty trace avalable from North Carolna State Unversty [11] was used for the same. Most of the modern handheld devces are equpped wth IEEE n wreless port, whch provdes an ndoor range of 70 m [12]. Keepng ths n mnd, the communcaton range was set as 70 m. Scenaros wth varous node denstes were created. The smulaton tme was dvded nto slots of 10 mnutes each. For each source-destnaton par, the relay node s found out, usng the egenvalue approach. The average tme taken by a sngle packet to be delvered was noted for these scenaros. They were compared aganst the algorthm n [10]. The result s plotted n Fgure 1.

8 98 Sgnfcance of Egenvector Centralty for Routng Fgure 1: Comparson of delvery latency When the relay nodes are selected based on the egenvector centralty, there s a reducton n the delvery tme. However, the dfference n delvery tme s less when the node densty s hgh. We also notced that at hgh node densty, the number of relay nodes avalable s less. But these nodes were able to delver the message wth shorter delvery latency. Ths s the reason for the mproved performance. Ths shows that egenvector centralty s able to select more approprate relay nodes, than the contact based algorthm. 7 Concluson Identfyng sutable relays s mportant whle desgnng a routng protocol for a DTN. Utlzng human moblty characterstcs s a novel concept n ths drecton. Ths work proposes to apply the concept of egenvector, to dentfy sutable relay nodes for routng messages. Smulaton results show mprovement n delvery latency, compared to smlar works.

9 R.J. D Souza and J. Jose 99 References [1] C. Boldrn, M. Cont, I. Iacopn and A. Passarella, HBOp: a Hstory Based Routng Protocol for Opportunstc Networks, Proceedngs of IEEE WoWMoM, (2007). [2] J. Burgess, B. Gallagher, D. Jensen and B. Levne, Maxprop: Routng for vehcle-based dsrupton-tolerant networks, Proceedngs of INFOCOM, (2006). [3] B. Burns, O. Brock and B. N. Levne, MV routng and capacty buldng n dsrupton tolerant networks, Proceedngs of IEEE INFOCOM, (2005). [4] Chantreau, P. Hu, J. Crowcroft, C. Dot, R. Gass and J. Scott, Impact of human moblty on opportunstc forwardng algorthms, IEEE Transacton on Moble Computng, 6(6), (2007), [5] E. Daly and M. Haahr, Socal network analyss for routng n dsconnected delay-tolerant MANETs, Proceedngs of MobHoc, (2007). [6] Human Moblty Model and DTN Group, North Carolna State Unversty, avalable onlne at downloaded on [7] IEEE Standards Assocaton, avalable onlne at downloaded on [8] T. Karaganns, J. L. Boudec and M. Vojnovc, Power Law and Exponental Decay of Inter Contact Tmes between Moble Devces, n MobCom 07, (2007), [9] A.G. Mklas, K.K. Gollu, K.K.W. Chan, Stefan Sarou, K. P. Gummad and E. Lara, Explotng Socal Interactons n Moble Systems, UbComp 2007, (2007),

10 100 Sgnfcance of Egenvector Centralty for Routng [10] C. Song, Z. Qu, N. Blumm and A. Barabas, Lmts of Predctablty n Human Moblty, Scence, 327(5968), (2010), [11] R.J D Souza and Johny Jose, Effectve utlzaton of n wreless standard for message transfer n delay tolerant networks: the sngle relay approach, Proceedngs 2 nd IEEE Internatonal Conference on Computer and Communcaton Technology, (September 2011). [12] A. Vahdat and D. Becker, Epdemc routng for partally connected ad hoc networks, Techncal Report CS , Duke Unversty, 2000.