On the Value of a Social Network

 1 On the V alue of a Social Network Sande ep Chalas ani Abstract Different laws have been proposed for the value of a social network. According to Metcalfe’s law, the value of a network is proportional to n 2 where n is number of users of the network, whereas Odlyzko et al propose on heuristic grounds that th e value is proportional to n log n, which is the Zipf’s law. In th is paper we have examined scale free, small world and random social networks to determine their value. We have found that the Zipf’s law describes the value for scale free and small world networks although for small world networks the proportionality constant is a function of the pr obability of rewiring. We have estimated the function associated with different values of rewiring to be described well by a quadratic equation. We have also shown experimentally that the value of random networks lies between Zipf’s law and Metcalfe’s law. Introduction  Social networks are structur es consisting of individuals or organizations that create powerful ways of communicating and sharing information. Mill ions of people use social networking websites like MySpace, Facebook, Bebo, Or kut and Hi5. The question of their value related to size is an importan t problem in computer science [1, 2] , both from the point of view of connectivity and that of business investment. Social networks connect people and the cost involved in connecting is low, which benefits businesses and institution s. These netw orks are important in custom er relationship management, and they serve as online meeting places for professionals. Virtual communities allow individuals to be easily accessible. People esta b lish their real identity in a verifiable place, these individuals then interact with each other or within gr oups that share common business interests and goals. False and exaggerated estimates of the valu e of a social networ k can have significant implications for technology investo rs. Until the IT bubble burst in 2001, it had been common to estimate the market value of a social network bas e d on Metcalfe’s law which says that a value of a network is proportional to the square of the size of network [1]. Recently Odlyzko and his collaborators have argu ed against Metcalfe’s law saying that it is sign ificantly overestimates value and they have suggested that the valu e of a general communica tion network of size n grows according to n log n, which is Zipf’s law [2, 27]. An important claim has been made by anthropologist Robin Dunbar [3, 4] on the extent of connectivity in effective so cial organi zations. He argued that th e size of the brain is correlated with the complexity of function and developed an equation, which works for most primates, that relates the neocortex ratio of a part icular species - the size of the neocortex relative  2 to the size of the brain – to the largest size of the social group. For humans, the m ax group size is 147.8, or about 150. This represents Dunbar’s es timate of the maximum number of people who can be part of a close so cial relationship [4]. Support for Dunbar’s ideas come from the community of Hutterites, followers of the sixteenth century Jakob Hutter of Austria, who ar e pacifists and believe in community property and live in a shared community called colony. Se veral thousand Hutterites relocated to North America in the late 19 th century and their colonies are mostly rural [3,4]. A colony consists of about 10 to 20 families, with a population of around 60 to 150. When the colony' s population approaches the upper figure, a daughter colony is established. Dunbar’s ideas can be taken to be an indication of the idea that most social networks are “small world” networks [3, 4, 5, and 9]. Small world networks exhibit clustering and sm all characteristic path lengths that seem to capture m any features of social computing networks. We are interested in re lating valu e to size in such networks.  In this paper we propose to i nvestigate the value of a social network with respect to the probability mechanism underly ing its structure. Specifically we compute the value for small world networks and scale free networks. We provid e evidence in support of the value to be given by Zipf’s law. Zipf’s Law Zipf’s law is an empirical law originally proposed for words in a large text and it states that given some corpus of natural language utte rances, the frequency of any word is inversely proportional to its rank in the fre quency table. The most frequent word will occur approxim ately twice as often as the second most frequent word, which occurs twi ce as of ten as the fourth most frequent word etc. In the networ k context, if the valu e of the most important m e mber to user A is taken to be proportional to 1; that of the sec ond most important member is proportional to ½, and so on. For a network that has n me mbers, this value to the user A will be proportion al to 1 + 1/2 + 1/3 +…+ 1 / (n-1), which approximates to log n. Given that the num ber of users is n , the total value of the network is proportional to n log n. Metcalfe’s law took the value of the network to be proportional to its connectivity, since the total number of connections in a network of n users is n (n-1) or about n 2 . In practice many users will be connected socially only to a fraction of all the us ers though the networks provide a full connectivity of n 2 . Reed’s law [7] is based on the insi ght that in a communication network as flexible as internet, in addition to linking pairs of m embers. With n participants, there are 2 n possible groups, and if they are all equally valuable, the value of the network grows like 2 n . Probabilistic Random Networks We consider probabilistically generated social networks. These networks are based on the variable binomial distribution in which sets of nodes are connect ed to other nodes with different probability distributions. A sample random networ k with 12 nodes and 62 connectio ns is shown in Fig. 1. Fig.2 Graph showing the values of example networks compared with n 2 and n log n Figure 2 shows the values of the network in co mparsion with values n 2 and n log n . The number of nodes in the n etwork is the X-axis an d the associated value for each node is in the Y- axis. From this observation we clearly understand that the actual value of the network lies somewhere in between the values n 2 and n log n .  3 Small World Networks We have simulated small world networks a nd and the value associated to coressponding graphs are observed on an average case. We generated a Watts-Strogatz sm all world network consisting of N nodes [17, 18, 21]. Each node is directly connected to k immediate neighbours that are located symm etrically in th e ring lattice on two sides of the node. Fig. 3 Watts Strogatz ring lattice for a small world ne tworks with 15 Nodes and 4 local contacts for each node A small world network is generated by “rewir ing” the basic n etwork i.e. ring lattice. Rewiring at each node consists of redirecting one of the outgoing ar cs at the node to som e other destination node. The extent of re wiring is controlled by probability p. We generate a random number which is uniformly distributed and check whether the generated random number is less than or greater than the given probability. If the random number gene rated is less than the assumed probability we rewire an arc, otherwise the arc is left unchanged. Fig.4 Small world network with the p robability of rewiring is p =0.08 As we increase the value of the p from 0 to 1.0 we see a randomly rewired graph almost all the nodes connected differently  4 Fig. 5 Small world network with probability of rewiring p =1.0. Value of Small World Networks Several small world networks with differe nt number of nodes are generated using different binomial dist ribution for random number generati on and with variable probability values for the rewiring. In every network for ev ery node we count the number of other nodes to which it is connected and the total v alue of the network is estimated. We considered s everal repetitions of the generations and the av erage case value is considered. No. of Nodes Calculated Value Metcalfe’s Odlyzko 100 747 10000 200 90 689 8100 176 80 634 6400 152 70 518 4900 129 60 490 3600 107 50 345 2500 86 40 256 1600 64 Table. 1. Comparison between the calculated value, n 2 and n log n values of small world networks with different sizes with p =0.18 The graph showing the different value curves as observed for a small world network with a probability of rewiring p as 0.18 and 0.32 in Figures 6 and 7.  5 Fig. 6 Graph comparing the Values of small world network with a p =0.18 No. of Nodes Calculated value Metcalfe Zipf’s 100 1296 10000 200 90 1221 8100 176 80 1025 6400 152 70 830 4900 129 60 730 3600 107 50 522 2500 85 40 384 1600 64 Table.2. Comparison between the calculated Value, n 2 and n log n values of small world networks with different sizes with probability of rewiring p =0.32  6 Fig. 7 Graph comparing the values of small world network with a p =0.32 We observe that in Table1 and Table 2 that the calculated value is 4 and 6 times that of n log n respectively. W e established a relation between probability and number of tim es the calculated value is more than that of n log n as and where the functional relationship is given by the following quadratic relationship Y=12.045x 2 + 6.59x+2.5533 The regression value for this quadra tic function is quite close to 1. Fig. 8 Graph showing the relation between probabil ity and the no. of times calculated value is more than n log n.  7 Scale Free Networks A scale free network is network whose degree distribution follows a power law. We have simulated Barabasi and Albert (B-A) [10, 11] model of scale free networks. W e generated a network of small size, and then used that netw ork as a seed to build a greater sized network, continuing this process until the actual desired n e twork size is reached. The initial seed used need not have scale free properties, while the la ter seeds may happen to have these properties. Fig. 9 B-A Scale Free graph with 30 nodes We can draw a best fit line to the frequenc y of degrees distribution of the nodes. Degree is the number of links that connect to and fro a single node. For s cale free networks, the frequency of degrees distribution forms a power- la w curve, with a expone nt usually between -2 and -3. Fig.10 Power-law curve for the sm all world netw ork in Fig.9.  8 Fig. 11 B-A small world network with 150 nodes Fig. 12 Graph showing the power law distribu tion for the small world network in Fig.11 Value of Scale Free Networks Several of these scale free networ ks are generated and the average case for the value calculation is taken into account. These scale free networks follow the Power law and therefore the values associated with them correspond to n log n.  9 No. of nodes Calculated value Odlyzko n log n 30 60 44.31 40 80 64.082 50 100 84.94 60 120 106.689 70 140 129.156 80 160 152.247 90 180 175.88 100 200 200 Table 3. Showing the Values Scale Free networks with different nodes Fig. 13 Graph comparing the value of scale free network and n log n This is also seen in Figure 13. We conclude that the property of be ing scale free captures the underlying foundation of the Zipf’s law.  10  11 Conclusion We have demonstrated that the Zipf’s law, originally proposed on heuristic grounds, is valid for scale free and small world networks. We have shown em pirically that the expression of value for a Watts- Strogatz small world network of n nodes is f(p) n log n f (p) = 12.054p 2 +6.59p+2.5533 where p is the probability of rewiring. We have computed the value of f ( p) for various p and found that the quadratic function prov ides an excellent fit. W e belie ve that this is the first study broadly validating the heur istic claim of Odlyzko et al on the value of social networks. Although no specific relationship between si ze and value can be fixed for random networks, our simulation shows that this value lies between Zipf’s law and Metcalfe’s law. As future study one would like to determine if non-Watts-Strogatz sm all world networks also follow the Zipf’s law. References 1. Metcalfe, B, There oughta be a law, New York Times, July 15 1996. 2. Briscoe, B, Odlyzko, A, and Tilly B, Metcalfe's Law is Wrong . IEEE Spectrum, July 2006. 3. Dunbar, R . Neocortex size as a constraint on group size in primates, Journal of Human evolution, Vols. 20, 469-493, 1992. 4. Kak, S. The Future of Social Computing Networks . p. 7, 2008. 5. Kleinberg, J. The Small-World Phenomenon:An Algorithmic Perspective , Proc. 32nd ACM Symposium on Theory of Computing, 2000. 6. Freeman, L. The Development of Social Netw ork Analysis . Vancouver: Empirical Pres, 2006. 7. Reed, D.P. Weapon of math destruction: A si mple formula explains why the Internet is wreaking havoc on business models, Context Magazine . Spring 1999. 8. Zipf, G.K. Human Behavior ad the Prin ciple of Least Effort, Addison Westey, 1949. 9. Admaic, L. The Small W orld Web, Proceedings of th e European Conf. on DigitalLibraries, 1999. 10. Albert, R. Jeong, H, Barabasi, A-L. The di ameter of the World Wide W eb, Nature 401,130, 1999. 11. Barabasi, A-L. Linked, Th e New Science of Networks . Cambridge,Perseus, 2002. 12. Cohen, D and Prusak, L. In good company: Ho w social capital makes organizations work, Boston : Harvard Business School Press, 2001.  12 13. Cross, R, and Parker, A The hidden power of social networks :Understanding how work really gets done in organizations . Boston : Harvard Business School, 2004. 14. Kak, S. the golden mean and the physics of aesthetics . Foarm Magazine,. Issue 5,73-81, 2006; arXiv:physics/0411195  15. Odlyzko, A. The History of Communications and its implications for the Internet. Florham Park, New Jersey : AT&T Labs-Research, 2000. 16. White, H. Identity and Control: How Social Formations Em erge . Princeton University Press, 2008. 17. Watts, D. Six Degrees: The science of a connected age , New York : W.W.Norton & Company, 2003. 18. Watts, D. The new Science of Networks . Annual Review of sociology 30, 243-270, 2004.  19. Erdos, P and Renyi, A. Random Graphs, Math’s Review, pp.290-297, 1959. 20. Barabasi, A-L and Albert, R. Emergence of scaling in random ne tworks, Science, pp.509- 512, 1999. 21. Watts, D and Strogatz, S. Collective Dynami cs of Small World Networks, Nature, pp. 440- 442, 1998. 22. Sandberg, O. Distributed Routing in Small-World Networks, ALENEX, 2006. 23. Kak, A. Small World Peer-to-Peer Networks and Their Security Issues. Lecture Notes on Computer and Network Security. Purdue University, http://cobweb.ecn.purdue.edu/~kak/c ompsec/NewLectures/Lecture25.pdf, 2008. 24. Reed, M “Simulating Small W orl d Networks” Dr. Dobb’s Portal http://www.ddj.com/184405611, April 2004. 25. Amaral, LAN, Scala, A, Barthelemy, M., Stan ley, HE, Classes of behavior of small-world networks. Proceedings of the National Academ y of Sciences (USA) 97:11149-11152, 2000. 26. Jeffrey, T & Milgram, S. An experimental study of the sm all world problem. Sociometry, Vol. 32, No. 4, pp. 425-443, 1969. 27. Odlyzko, A, Tilly, B, A refutation of Metcalfe ’s law and a better estimate for the value of networks and networ k interconnections. www.dtc.umn.edu/~odlyzko/doc/metcalfe.pdf. 28 . Borgatti, S.P. Everett, M.G. and Freeman , L.C. Ucinet for Window s: Software for Social Network Analysis. Harvard, MA: Analytic Technologies, 2002. 29 . Erd ő s, P, Rényi, A. The Evolution of Random Gr aphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 : 17–61, 1960. 30. Erd ő s, P.; Rényi, A. On Random Graphs. I .. Publicationes Math em aticae 6: 290–297,1959.