A Growing Network Model with High Hub Connectivity and Tunable Clustering Coefficient

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1 Manuscrpt receved Aug. 23, 2007; revsed ov. 29, 2007 A Growng etwork Model wth Hgh Hub Connectvty and Tunable Clusterng Coeffcent YIHJIA TSAI, PIG-A HSIAO Department of Computer Scence and Informaton Engneerng Tamkang Unversty 151 Yng-chuan Road, Tamsu, Tape TAIWA tsa@cs.tku.edu.tw, danel@ssp.snca.edu.tw Yha Tsa, Png-an Hsao Abstract: - The paper nvestgates a model for socal network based on a growng network. The model we propose can ft the followng two basc propertes: (1) scale-free property whch has power law degree dstrbuton, and (2) small-world property whch has hgh clusterng coeffcent. In addton, the model s modfed for hgh hub connectvty whch s a new concept n scale-free network. Ths study also provded an analyss of emprcal data. Key-Words: - Socal network model, growng network, scale-free, small-world, complex network. 1 Introducton The sgnatures of complex network appear n many felds, such as bology [ 1 ], economy, computer scence, and socal networks [2]. In these felds, the socal network models [3] [4] [5] [6] [7] have been used for the smulaton of epdemologcal transmsson [8], rumor spreadng [9] and the spread of computer vruses n emal network [10]. Snce socal networks can be found n many aspects of our lves, t s worth to nvestgate how they work precsely. In ths paper, we propose a scale-free property wth tunable clusterng coeffcent socal network model based on a growng network. Then we modfy the network model accordng to s max property. In addton, ths study ncludes the analyss of real socal network data. 1.1 Scale-Free A scale-free network s nspred by emprcal studes such as the Internet backbone network [11], web documents lnk network [12], and emal network [ 13 ]. The characterstc property of scale-free network s that t conssts of a few large connected nodes and many nodes wth small connectvty. Barabás and Albert proposed an algorthm to construct a scale-free network [ 14 ].The network grows by addng one new node at a tme, denoted by t, that connects to m nodes already n the network by a probablty functon. The connectng probablty depends on the node degree t was connected to. For node wth node degree k, the connectng probablty s denoted by (k ) = k k. (1) The connectvty dstrbuton pr(k) of the resultng network, whch s defned by the probablty of a node havng k edges. The connectvty dstrbuton for scale-free networks s a decayng functon of k s power followng γ pr ( k) = k. (2) Ths phenomenon s sometmes called power law dstrbuton and γ s the tal ndex. Power law degree dstrbuton s consdered as a defnng characterstc between real communcaton network and random network. The BA model for scale-free network constructon wth a preferental attachment strategy creates a scale-free network wth tal ndex γ = Small-World The small-world network s known to have both small average network dstance and large clusterng coeffcent. The average dstance between network nodes, denoted by L, δ (, ) = 1, > L =, (3) ( 1) / 2 ISS: Issue 12, Volume 4, December 2007

2 Yha Tsa, Png-an Hsao where δ (, ) s the dstance of shortest path between node and node, and s the number of nodes n the network. The L ndcates the average dstance to communcate from one node to the other. The clusterng coeffcent descrbes how close among the neghbors of a node. The formula of calculatng clusterng coeffcent of node s the rato of actual number of edges connectng these nodes n ts k neghbors to the number of edges n a fully connected network of the k nodes, denoted by C, 2E C =, (4) k ( k 1) where E stands for the number of edges n the subnetwork formed by node and ts E neghbors. The clusterng coeffcent of the entre network s defned as the average of all nodes C C = = 1. (5) In 1998, Watts and Strogatz [15] proposed a small-world model that formally defnes an algorthm to construct and analyze such network. The WS network model s based on a regular lattce network and a re-wrng probablty p. Each edge n the base regular network s modfed wth probablty p to re-connect to a dfferent destnaton node. For p=0 the orgnal lattce network s unchanged. As p ncreases the network becomes ncreasngly dsordered untl for p=1 all edges are rewred randomly. Increasng p ncreases randomness. 1.3 S-metrc In 2005, L et al. [16] defned an "s-metrc", that measures the extent to whch hgh-degree nodes connect to other hgh-degree nodes. Let g be an undrected graph wth edge-set ε, and let degree of node be d, then defne the metrc s ( g) = d d, (6) (, ) ε and the s max s the maxmum value of s(g). The value of s(g)/s max, denoted by S, descrbes the level of a large hgh-degree nodes connected to another large hgh-degree nodes, and t s more close to 1 f more hgh-degree nodes are connected n the network. In ths paper, we use the algorthm, shown n Table 1, to mplement the s max n smple graph. The tme complexty of the algorthm s O(n 2 lgn), snce there s a sortng algorthm for square edge set. 2 Emprcal etworks Many real world socal networks such as the collaboraton network of move actors [17], emal communcaton of students [7], and the coauthorshp network of neuroscentsts [3], show both the property of the small-world and that of the scalefree. We collected a real data from the People Fsher Webste, denoted by PFW, at September 2006 [18], whch s a web ste of a blog style. Everyone can create a blog, share photos, and make new frends n the web ste. We analyzed the user data wth ther settng of frends. If user A consders user B as hs/her frend whle user B also sets user A as hs/her frend n ther web pages, then one edge between node A and B s added. The total number of users on the web ste s 2789, and the largest connected network conssts of 2738 nodes and 9333 edges. Fg. 1 shows the degree dstrbuton of the PFW network, and t shows the scale-free property obvously. Table 2 s the resultng average dstance L, clusterng coeffcent C, tal ndex γ and s-metrc S. In the PFW network, the average dstance L s very small, snce the maxmum node degree, the creator of the webste, s The creator sets about 96% of the total users to be hs frends. Therefore, n terms of s-metrc, ths causes the results n the maxmum node greater than the second node by a large amount. Table 1 The algorthm of s max n smple graph. Step 1. Let edge set ε ' = U = 1, > d d Step 2. Sort edge set ε ' n decreasng order. Step 3. Construct a new graph g wth ε ' n order wthout mult-lnk, and keep ts degree sequence constant. Step 4. Calculate s(g ) for the new graph, whch s s max. Table 2 The emprcal data of the PFW network. ode 2738 Average degree k Tal ndex γ Clusterng coeffcent C Average dstance L S-metrc S ISS: Issue 12, Volume 4, December 2007

3 Yha Tsa, Png-an Hsao P(k) degree k Fg. 1 The log-log degree dstrbuton plot of the PFW network, the slope of the lne s ew Model We proposed a model based on the BA model to descrbe socal network topology. The BA model constructs a scale-free network wth decreasng clusterng coeffcent when network sze ncreases. We add a parameter f to ncrease the clusterng coeffcent by ncreasng the probablty of frendshp n the network. For example, when you know a nce person, you may want to know hs/her frend. At last, when the network s constructed, we then add extra edges to satsfy the average degree of the real network that s what we need. The average degree K n BA model s 2mt K =, (7) t + m0 when t s large enough, the average degree K wll nearly equal to 2m whch s almost even number. In the real world, the degree of most networks wll not be even, so we add extra edges to ft the average degree k of the real network. The number of extra edges should be added to the network s 2mt ( k K ) / 2 = ( k ) / 2. (8) t + When addng the extra edges(, ) whch follows the weght of s max probablty: d d P (, ) =, (, ) ε, and. (9) d d It wll make the large hub-nodes have large probablty to connect to large ones rather than small degree nodes connect to other small nodes. The m 0 reason s that each person has only lmted tme, so he/she would rather choose to meet a famous person than others. Furthermore, f we hope the negotaton wll be more successful n a socal network, the famous person s lkely to know more other famous persons. For example, n a company, there are usually some departments. If the leaders of the departments not drect communcate wth other leaders, the negotaton of the socal network wll not smooth. Thus, the hgh-degree nodes of a socal network should have more large probablty to connect to other ones. To avod the wnners take all phenomenon [19] n addng extra edges phase, we add a parameter x to adust the weght of s max. The probablty x s for preferental attachment of s max and 1-x s for unform attachment. So (9) wll be modfed to d d 1 P(, ) = x + (1 x), d d (, ) (, ) ε, and. (10) In addng one new node and ts edges, we defne ts preferental attachment as PA, and the attachment of makng frend as FA. Table 3 shows the comparson wth our model, the growng model proposed by Holme and Km [4], and the dynamc model proposed by Tsa, Ln and Hsao [7] when addng one new node. The man dfference of three models s whom s the FA connectng to. The step by step dagram of our model when addng one new node shows n Fg. 2. In fact, the tunable clusterng coeffcent methods of the three models are not too dfferent, and we do not compare them n detal. The only thng that we want to do s to show that we do not need use the same way to control the clusterng coeffcent. Table 3 The comparson wth three models when addng one new node. The means FA connects to the neghbor node of PA or FA. The steps of PA and FA Our model PA 1, FA 1, PA 2, PA 3, FA 2, FA 3, FA 4,... Growng PA 1, FA 1, PA 2, PA 3, FA 2, FA 3, FA 4,... model [4] Dynamc PA 1, FA 1, PA 2, PA 3, FA 2, FA 3, FA 4,... model [7] ISS: Issue 12, Volume 4, December 2007

4 Yha Tsa, Png-an Hsao FA 2 PA 2 FA 2 FA 1 PA 1 new node PA 3 (a) (b) (c) (d) Fg. 2 The double crcle nodes are the neghbor nodes of PA or FA. (a)fa 1 connects to the neghbor node of PA 1 ; (b)fa 2 connects to the neghbor node of PA 3 ; (c)fa 3 connects to the neghbor node of FA 2 ; (d)fa 4 connects to the neghbor node of FA 3. 4 Smulatons and Dscusson We use our model to smulate the proposed socal network model and compare wth the PFW expermental results. Table 4 summarzed the comparson. In the smulaton, we use (9), that s x=1 n (10), n the proposed model snce the S of the PFW network s too large, and the sutable f n our model for the PFW network s Fg. 3 shows the degree dstrbuton of our model for the PFW network wth f=0.35, and t shows the scale-free property. Fg. 4 demonstrates the S n two condtons: constructed network before extra edges added, and extra edges added by (9) for s max and by (10) for random when x=0 wth dfferent f. We fnd that our model can rase the S sgnfcantly. Fg. 5 shows the clusterng coeffcent relatonshp wth the same parameters of Fg. 4, and ths s absolutely for large f made large C. Fg. 6 shows the tal ndex relatonshp wth the same parameters of Fg. 4, and the more f makes, the less γ s. At the same tme, we use other parameters to smulate the socal network. Fg. 7 shows the results of S wth =2000, m=3 and f=0 for varous addtonal edges. We fnd that more extra edges added by (10) wth larger x, the hgher the S wll become than by random when x=0. In the rght porton of ths fgure, the breadth of ncreasng s lower than left. The reason s that the more addtonal edges added by s max property cause the most large hgh-degree nodes to be connected, and ths s less helpful for the S by connectng nodes wth small degree. Fg. 8 shows the clusterng coeffcent C wth the same parameters of Fg. 7. When we add addtonal edges by (10) wth larger x, the C wll ncrease slowly; but f we add by (10) wth x below 0.3, the result s decreasng C. The reason s that the more large hub-nodes connected make the neghbor nodes of the large-hub nodes wll be large ones. Fg. 9 demonstrates the tal ndex γ wth the same parameters of Fg. 8. We fnd that the more addtonal edges added by (10) wth larger x, the lower γ wll become than by (10) wth small x. The reason s that (10) wll make large hub-nodes become larger when x=1, and the poston of these nodes wll be on the rght porton n log-log degree dstrbuton plot, and that causes the slope of lne, that s γ, to appear flatter than before, thus resultng n smaller γ. P(k) degree k Fg. 3 The log-log degree dstrbuton plot of our model wth =2738, k=6.817, f=0.35. The slope of the lne s Table 4 The comparson between the PFW and our model. PFW The dfferent parameter f n our model wth parameter x set 1 for all the cases. f=0.05 f=0 f=5 f=0.20 f=0.25 f=0.30 f=0.35 f=0.40 f=0.45 C γ L S FA 3 FA 3 FA 4 ISS: Issue 12, Volume 4, December 2007

5 Yha Tsa, Png-an Hsao s(g)/smax S Extra edges added by smax Extra edges added by random Parameter f Fg. 4 The value of S wth =2738, f=0.05~0.55. The average degree K by (7) n the constructed network before extra edges added s 5.991, and the k of the others after edges added by (8) s clusterng coeffcent C Added extra edges by smax Added extra edges by random Parameter f Fg. 5 The clusterng coeffcent C wth =2738, f=0.05~0.55. The average degree K by (7) n the constructed network before extra edges added s 5.991, and the k of the others after edges added by (8) s Tal ndex r Added extra edges by smax Added extra edges by random Parameter f Fg. 6 The tal ndex γ wth =2738, f=0.05~0.55. The average degree K by (7) n the constructed network before extra edges added s 5.991, and the k of the others after edges added by (8) s s(g)/smax S Added extra edges, x=1.0 Added extra edges, x=0.7 Added extra edges, x=0.5 Added extra edges, x=0.3 Added extra edges, x= Added extra edges Fg. 7 The S wth =2000, m=3, f=0, and the extra edges added are from 100 to The parameter x s from 0 to 1. Clusterng coeffcent C Added extra edges, x=1.0 Added extra edges, x=0.7 Added extra edges, x=0.5 Added extra edges, x=0.3 Added extra edges, x= Added extra edges Fg. 8 The clusterng coeffcent C wth =2000, m=3, f=0, and the extra edges added are from 100 to Tal ndex r Added extra edges, x=1.0 Added extra edges, x=0.7 Added extra edges, x=0.5 Added extra edges, x=0.3 Added extra edges, x= Added extra edges Fg. 9 The tal ndex γ wth =2000, m=3, f=0, and the extra edges added are from 100 to Conclusons and Future Work In ths paper, we present a socal network model that combnes the degree dstrbuton property of a scalefree network model and the clusterng coeffcent ISS: Issue 12, Volume 4, December 2007

6 Yha Tsa, Png-an Hsao property of a small-world network. In the connectvty of hgh-degree nodes, our model can provde very hgh value when parameter x=1 than random when x=0. In addton, the smulaton result of the proposed model closely matches the emprcal network degree dstrbuton and the clusterng coeffcent. Although the shortest path and s-metrc can not ft perfectly, snce the maxmum node of emprcal network s too large, our model stll provde a good way to tune the level of hgh hub connectvty. The socal network analyss can be used n busness for mprovng knowledge creaton and Reference: [1] R. Dobrescu, Modelng complex bologcal systems usng Scale Free etworks, Proceedngs of the 5th WSEAS Internatonal Conference on on-lnear Analyss, on- Lnear Systems and Chaos, 2006, pp [2] S. H. Strogatz, Explorng complex networks, ature 410, 2001, pp [3] A.-L. Barabás, H. Jeong, Z. eda, E. Ravasz, A. Schubert, T. Vcsek, Evoluton of the socal network of scentfc collaboratons, Physca A 311, 2002, pp [4] P. Holme, B. J. Km, Growng scale-free networks wth tunable clusterng, Physcal Revew E 65, 2002, [5] K. Klemm, V. M. Eguluz, Growng scale-free networks wth small-world behavor, Physcal Revew E 65, 2002, [6] R. Tovonen, J.-P. Onnela, J. Saramäk, J. Hyvönen, K. Kask, A Model for socal networks, Physca A 371, 2006, pp [7] Y. Tsa, C.-C. Ln, P.-. Hsao, Modelng Emal Communcatons, IEICE Transactons on Informaton and Systems, Vol.E87-D o.6, 2004, pp [8] L. A. Meyers, B. P. Pourbohloul, M. E. J. ewman, D. M. Skowronsk, R. C. Brunham, etwork theory and SARS: predctng outbreak dversty, Journal of Theoretcal Bology 232, 2005, pp [9] Y. Moreno, M. ekovee, A.F. Pacheco, Dynamcs of rumor spreadng n complex networks, Physcal Revew E 69, 2004, [10] M. E. J. ewman, S. Forrest, J. Balthrop, Emal networks and the spread of computer vruses, Physcal Revew E 66, 2002, [11] M. E. Crovella, M. S. Taqqu, A. Bestavros, Heavy-taled probablty dstrbutons n the World Wde Web, In A Practcal Gude To Heavy Tals, Chap.1, 1998, pp sharng, and cty plannng for desgnng spaces to support human nteracton [20]. Besdes, t can also be used for the smulaton of epdemologcal transmsson, computer vruses dssemnaton [21], the rumor spreadng and opton formaton. In recent years, we were warned of the great probablty of brd flu spreadng [22], snce f brd flu turns out to be person-to-person nfectous dseases, the ablty of transmsson wll rse substantally such as SARS durng 2002 to Usng socal network model to smulate t wll allow us to predct ts damage under dfferent measures. That s mportant why we study socal network. [12] R. Albert, H. Jeong, A.-L. Barabás, Internet: Dameter of the World-Wde Web, ature 401, 1999, pp [13] H. Ebel, L.-I. Melsch, S. Bornholdt, Scale-free topology of e-mal networks, Physcal Revew E 66, 2002, [14] R. Albert, A.-L. Barabás, Statstcal mechancs of complex networks, Revews of Modern Physcs, Vol.74, 2002, pp [15] D. J. Watts, S. H. Strogatz, Collectve dynamcs of small-world networks, ature 393, 1998, pp [16] L. L, D. Alderson, J. C. Doyle, W. Wllnger, Towards a theory of scale-free graphs: defnton, propertes, and mplcatons, Internet Mathematcs, 2(4), 2005, pp [17] A.-L. Barabás, R. Albert, Emergence of Scalng n Random etworks, Scence 286, 1999, pp [18] [19] D. M. Pennock, G. W. Flake, S. Lawrence, E. J. Glover, and C. L. Gles, Wnners don t take all: Characterzng the competton for lnks on the web, Proceedngs of the atonal Academy of Scences, 99(8), 2002, pp [20] M. Huysman, V. Wulf, IT to Support Knowledge Sharng n Communtes, Towards a Socal Captal Analyss, Journal of Informaton Technology, Vol. 21, 2006, pp [21] S. Mocanu, S. Taralunga, Immunzaton strateges for networks wth scale-free topology, Proceedngs of the 5th WSEAS Internatonal Conference on on-lnear Analyss, on- Lnear Systems and Chaos, 2006, pp [22] A. Abbott, H. Pearson, Fear of human pandemc grows as brd flu sweeps through Asa, ature 427, 2004, pp ISS: Issue 12, Volume 4, December 2007