1、1,The Small World Phenomenon: An Algorithmic Perspective,Jon Kleinberg,Review of Literature,Reviewed by: Siddharth Srinivasan,2,Oh, its such a small world!,Milgram (1967, 69) performed an empirical validation of the small world concept in sociology. Previous work- Pool and Kochen model 2 people at r
2、andom connected with k intermediaries. Assumes synthetic, homogenous structure. Rapaport and Horvath empirical study on school friendships. Asymmetric nets and Universe is small. Packet sent by a randomly chosen source to a random target. Mean chain length = 5.2 Variables of geographic proximity, pr
3、ofession and sex Funneling of chains by certain individuals,3,Small world! Small world!,White (1970) tries fitting a simple model to Milgrams work. Gives hints to future work Killworth few contacts for non-local targets. The size of geographical area that a single contact is responsible for decrease
4、s as a function of the distance of the target from starter. Most choices based on cues of occupation and geographic location.,4,Small Worlds Everywhere,Watts and Strogatz (1998) Very small number of long range contacts needed to decrease path lengths without much reduction in cliquishness. Long rang
5、e contact picked uniformly at random (u.a.r) Small world networks in 3 different areas esp. spread of infectious disease. Probabilistic reach. No specific destinations. Doesnt require knowledge of paths and no active path selection. Barabasi et al.(1999) diameter of the WWW Power-law distribution; L
6、ogarithmic diameter. Need for search engines to intelligently pick links,5,Two Important Properties of Small World Networks,Low average hop count High clustering coefficientAdditionally, may be searchable on the basis of local information,6,Enter Kleinberg,Two issues of concern in small-world networ
7、ks: Presence of short paths in a small world network how do you find the short chains? Gives an infinite family of small world network models on a grid n/w with power-law distributed random long-range links. K(n,k,p,q,r) p radius of neighbours to which short, local links q no. of random long range l
8、inks k - dimension of mesh (k=2 in this paper) r - clustering exponent of inverse power-law distribution. Prob.(x,y) dist(x,y)-r. Decentralized greedy routing algorithm Decisions based on local information only.,7,Bounds on Kleinbergs Model,Expected Delivery time = O(log n)2), for r = 2. (n(2-r)/3),
9、 for 0 r 2. (n(r-2)/(r-1), for 2 r Disproves usefulness of Watts & Strogatz model (r=0). Only for special case of r = k, possible to find short chains always of length O(log n)2) and dia = O(log n) (dia bound not proved by Kleinberg in this paper). Cues used in small world networks propounded to be
10、provided through a correlation between structure and distribution of long-range connections.,8,For r=2, p=1, q=1. Event Eu(v) - u chooses v as its random long range contact ProbEu(v) = ProbEu(v) 4 ln(6n) d(u,v)2-1. In phase j, 2j d(u,t) 2j+1. For log(log n) j log n, No. of nodes in Bj each within la
11、ttice distance 2j + 2j+1 2j+2 of u ProbEnters Bj Steps in j = Xj; ,Proof of the upper bound,9,Proof of lower bound 1,As in the previous proof, where, assumed that n2-r 23-r. Let = (2-r)/3 and U be the set of nodes witihin radius pn of t.where, assumed that pn2. Let be the event that the msg reaches
12、a node in Ut in n steps. Let i be the event that this happens in the ith step.where,10,Proof of lower bound 1 contd.,Let events F (s and t separated by n/4). PrF ; Pr!F ; and so PrF ! . Let - event that msg reaches t from s in n steps. cannot occur if (F !) occurs. Pr | (F !) = 0 and EX|(F !) n step
13、s. EX EX|(F !) . PrF ! n steps,where, X is the random variable denoting the no. of steps. Thus, lower bound on expected no. of steps is (n(2-r)/3), for 0 r 2.,11,Proof of lower bound 2,Similar to the previous proof,where, = r-2. Let = /1+, = 1/1+, and = min(1,)/8q. Assumed that n p. Let i be the eve
14、nt that in the ith step, msg reaches u w/ a long range contact v such that d(u,v)n.Let be the event that this happens in n steps. Similar to the previous proof, max dist. Covered w/o occuring is and hence,Thus, lower bound on expected no. of steps is (n(r-2)/(r-1), for 2 r,12,Major Ideas Contributed
15、,Gives a model of a small world network where local routing is possible using small paths. Shows the more generalized results for k dimensions in a subsequent publication. Correlation between local structure and long range links provides fundamental cues for finding paths. When rk, long range links
16、do not provide sufficiently long jumps and path becomes long.,13,Questions Raised,Can the expected delivery time be reduced to the bounds of the diameter? Is the model extendable to more general networks? Can less regular base graphs also produce navigable small worlds?,14,Work Done post-papyri,Furt
17、her analysis and generalization of Kleinbergs models and other small world models Conversion of general networks to small world networks Applications of the small world idea to real networks,15,Further Analysis and Generalizations 1,Barriere et al.(2001) proves (log n)2) bound on routing complexity.
18、 Simplified analysis using a ring instead of a grid. Oblivious greedy routing. Basic concept used in analysis (f, c)-long range contact graph if for any pair (u,t) at distance at most d, we have PruBd/c(t) 1/f(d). If graph (G, p) is an (f, c)-long range contact graph then greedy routing in O(i=1logc
19、D f(D/ci) expected steps. If p is a non-decreasing fn., then PruBd/c(t) Pr(c+1)d/c . |Bd/c(t)| extends results to any ring by epimorphisms (embedding) one graph to another.,16,Martel, C. and Nguyen, V. (2004): Shows that Kleinbergs algo is tight (log2 n) expected delivery time and diameter tight at
20、(log n). For k-dimensional grid as well. If additional info, then O(log3/2 n) for k=2 and O(log1+1/k n) for k1. Proof done in a manner that uses some interesting conceptual ideas (used by others previously as well): p(u, v) = d2(u, v)/cu , cu = d2(u, v) = bj(u) j-2 ; bj(u) = (j), so, cu approx. as a
21、 harmonic sum. Inherently uses the concept of gradient, (v) = d(v,t) d(N(v),t), to show the lower bound. Uses the concept of harmonics to get for any integer 1 m d(v, t): Expected delivery time is (log2n) for any s and t w/ probability 0.5 when d(s,t) is O(n).,Further Analysis and Generalizations 2,
22、17,Extended algo Window (no. of neighbouring nodes whose long range contacts are known) = log n. In k dimensions, O(log1+1/k n). Prove only for k=2. Diameter = (log n). Extended to all possible K|K*(k,n,p,q) where k, p, q 1 and even for 02k.,18,Further Analysis and Generalizations 3,Fraigniaud et al
23、. (2004) “Eclecticism shrinks even small worlds” Dimensions need not mean only geographical dimensions but can refer to the various parameters used for routing in social networks geography, occupation, education, socio-economic status etc. Higher dimensions intuitively must give better performance,
24、dimension not considered in routing performance in the greedy algo proposed by Kleinberg since O(log2n) in all dimensions. Giving O(log2n) bits of topological awareness per node decreases the expected number of steps of greedy routing to O(log1+1/k n) in k-dimensional augmented meshes.,19,Called ind
25、irect greedy routing. Completely oblivious routing. Analysis proves that between two nodes in a sequence of long-range nodes, dist(zi, zi+1) log1/kn. And, totally O(log n) such nodes. Augmenting the topological awareness above this optimum of O(log2 n) bits would drastically decrease the performance
26、 of greedy routing. Perhaps a first step towards the formalization of arguments in favor of the sociological evidence stating that eclecticism shrinks the world.,20,Further Analysis and Generalizations 4,Raghavan et al. (2005). “Theoretical Analysis of Geographic Routing in Social Networks.” rank-ba
27、sed friendship - probability that a person v is a friend of a person u is inversely proportional to the number of people w who live closer to u than v does. ranku(v) = no. of people w such that d(u,w) d(u,v). prob(u,v) = ranku(v)-1. more accurately models the behaviour of social networks verified ag
28、ainst LiveJournal data. in a grid setting, prob(u,v) = rank-1 = d-k. Halves distance in expected polylogarithmic steps Starting from s, expected number of steps before reaches a point in Bd(s,t)/2(t) is O(log n log m) = O(log2 n) Finds short paths in all 2-D meshes For any 2-dimensional mesh populat
29、ion network with n people and m locations, expected path length is O(log n log2m) = O(log3 n). Interesting proof methodology using only balls. Plus rank and balls is general over all dimensions.,21,Further Analysis and Generalizations 5,Watts et al. (2002) and Motter et al. (2003). hierarchies of so
30、cial groups with groups having some correlation between them. social ties generated by picking links from social groups according to p.distribution governed by social affinity. Manku et al. (2004). Know thy neighbours neighbour. Shows that if every node is aware of the long-range links of its neighb
31、ours then greedy routing in O(log2n/(clog c) with c long range contacts per node.,22,Conversion to small world networks,Duchon et al. (2006). At INRIA On bounded growth graphs and extended to polylogarithmic expansion rates. Using O(n) rounds and O(polylog n) space. No need for a node to have comple
32、te knowledge of the graph. Any synchronized n-node network of bounded growth, of diameter D, and maximum degree , can be turned into a small world via the addition of one link per node, in O(n) rounds, with an expected number of messages O(nD log n), and requiring O( log n logD) memory size with hig
33、h probability, or, in O(D) rounds with an expected number of messages O(nlog D log n), and requiring O(n) bits of memory in each node with high probability In the augmented network, the greedy routing algorithm computes paths of expected length O(logDlog + log n) between any pair of nodes at mutual
34、distance in the original network. Sampling of leader nodes. Only leader nodes explore a ball Bv(3l), when asked by a node u at a distance l (l=2i), to select a random long range link for it, where i is selected u.a.r.,23,Some Applications Areas,P2P overlay networks Distributed hashing protocols Secu
35、rity systems in mobile ad hoc networks Hybrid sensor networks Referral systems,24,Applications: Distributed Hashing,Manku et al. (2002) Symphony arrange all participants in a ring I 0,1). A node manages that sub-range of I which corresponds to the segment between itself and its two neighbours equip
36、them with long range contacts drawn randomly from a family of harmonic distributions p = 1/(x ln n) where x1/n, 1 drawn u.a.r. advantages low degree, can handle heterogeneity by variable number of long range links and only two mandatory short links, low latency O(log n)/k). for fault tolerance, add
37、f number of backups but only on the short link neighbours.,25,Applications: P2P Overlay Networks,Bonsma (2002) - SWAN (Small World Adaptive Network) each node has 3 types of links bootstrap, local (short-range) and long-range (random). Hui et al. - SWOP (Small World Overlay Protocol) Cluster links a
38、nd long links Head nodes and inner nodes Pdf: ProbX=x = p(x) = 1/(x ln m) where, x1,m and m is no. of clusters To handle flash crowds, demand-driven replication over long links.,26,Applications: Hybrid Sensor Networks,Sharma & Mazumdar (2005) Adding of a few shortcut wires between wireless sensors.
39、Reduced energy dissipation per node as well as non-uniformity in expenditure. Deterministic as well as probabilistic placement of wires. Few wires unlike 1 long range contact per node in Kleinbergs model. One in a cell / group of cells of sensors is wired. Very good performance in static sink node c
40、ase with addition of (nl(n)/log n) wires, average hop count reduced to (1/l(n) and EDS to (1/l(n). In dynamic case, with greedy routing, hop count cant be reduced below (1/l(n).,27,Applications: Security Systems in Ad Hoc N/ws,Hubaux et al. (2002). Gray et al. (2003). Trust propagation,28,Bibliograp
41、hy,Albert, Jeong, Barabasi (1999). Diameter of the World Wide Web, Nature. Barriere, Fraigniaud, Kranakis, Krizanc (2001). Efficient routing in networks with long range contacts Bonsma and Hoile (2002). A distributed implementation of the SWAN peer-to-peer look-up system using mobile agents. Duchon,
42、 Hanusse, Lebhar, Schabanel (2006). Fully distributed scheme to turn a network in to a small world. Research report No. 2006-03, INRIA Lyon. Fraigniaud, Gavoille, Paul (2004). Eclecticism shrinks even small worlds. Gray, Seigneur, Chen, Jensen (2003). Trust propagation in small world networks. Helmy
43、, A. (2003). Small Worlds in Wireless Networks. IEEE Commun. Lett., vol.7, no.10, pp. 490-492, Oct. 2003. G/A, 14. Hawick & James (2004). Small-World Effects in Wireless Agent Sensor Networks. Hubaux, J.P., Capkun, S., Buttyan, L., (2002). Small Worlds in Security Systems: an Analysis of the PGP Cer
44、tificate Graph. In: New Security Paradigms Workshop, Norfolk, VA. Hui, Lui, Yau (2006). Small world overlay P2P networks: construction and handling dynamic flash crowds. Accepted in J. of Comp. Networks. Killworth, Bernard (1979). Reverse Small World Experiment, Social Networks. Kleinberg (2000). Na
45、vigation in a small world, Nature.,29,Manku, Bawa, Raghavan (2003). Symphony: Distributed hashing in a small world. USENIX Symposium on Internet Technologies and Systems. G. Manku, M. Naor, and U. Wieder (2004). Know Thy Neighbors Neighbor: The Power of Lookahead in Randomized P2P Networks. In 36th
46、ACM Symp. On Theory of Computing (STOC). Martel, C. and Nguyen, V. (2004). Analyzing Kleinbergs (and other) small world networks. ACM PODC 04. Milgram, Travers (1969). An experimental study of the small world problem, Sociometry. Motter, Nishikawa and Lai (2003). Large scale structural organization
47、of social networks. Physical Review. Raghavan, Kumar, Liben-Nowell, Novak, Andrew Tomkins (2005). Geographic Routing in Social Networks. Raghavan, Kumar, Liben-Nowell, Novak, Andrew Tomkins (2005). Theoretical Analysis of Geographic Routing in Social Networks. Sharma, Mazumdar (2005). Hybrid Sensor
48、Networks: a small world. Watts and Strogatz (1998). Collective dynamics of small world networks, Nature. Watts, D., Dodds, P., Newman, M.: Identity and Search in Social Networks. Science, 296 (2002) 13021305 White (1970). Search parameters for the small world problem, Social Forces. Yu, Singh (2003). Searching social networks.,
