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Small World Networks

Small World Networks. Jean Vaucher Ift6802 - Avril 2005. Contents. Pertinence of topic Characterization of networks Regular, Random or Natural Properties of networks Diameter, clustering coefficient Watt’s network models ( alpha & beta ) Power Law networks

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Small World Networks

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  1. Small World Networks Jean Vaucher Ift6802 - Avril 2005

  2. Contents • Pertinence of topic • Characterization of networks • Regular, Random or Natural • Properties of networks • Diameter, clustering coefficient • Watt’s network models (alpha & beta) • Power Law networks • Clustered networks with short paths • Can these short paths be found ? ift6802

  3. Duncan J. Watts • Six degrees - the science of a connected age, 2003, W.W. Norton. I read somewhere that everybody on this planet is separated by only six other people. Six degrees of separation between us and everybody on this planet. Six degrees of separation by John Guare ift6802

  4. Networks • Networks are everywhere • Internet • Neurons is brains • Social networks • Transportation • Networks have been studied long time • Euler (1736): Bridges of Königsberg  theory of graphs, which is now a major (and difficult! – or almost obvious) branch in mathematics ift6802

  5. So what is new? • Global interconnections • Internet • Power grids • Mass travel, mass culture • FAILURES • Computer Viruses • Power Blackouts • Epidemics • Modeling & analysis ift6802

  6. Found short chains of acquaintances linking pairs of people in USA who didn’t know each other; Source person in Nebraska Target person in Massachusetts. Sends message by forwarding to people they knew personally (who should be closer to target) Average length of the chains that were completed was between 5 and 6 steps “Six degrees of separation” principle Milgram’s Experiment ift6802

  7. Correct question • WHY are there short chains of acquaintances linking together arbitrary pairs of strangers??? • Or • Why is this surprising ift6802

  8. Random networks • In a random network, if everybody has 100 friends distributed randomly in the world population, this isn’t strange • In 6 hops, you can reach 1006 people - a million million > 6,000 million (world pop.) • BUT: our social networks tend to be clustered. ift6802

  9. Social networks • Not random • But Clustered • Most of our friends come from our geographical or professional neighbourhood. • Our friends tend to have the same friends BUT • In spite of having clustered social networks, there seem to exist short paths between any random nodes. ift6802

  10. Social network research • Devise various classes of networks • Study their properties ift6802

  11. Network parameters • Network type • Regular • Random • Natural • Size: # of nodes • Number of connexions: • average & distribution • Selection of neighbours ift6802

  12. REGULAR Network Topologies STAR TREE GRID BUS RING ift6802

  13. Connectivity in Random graphs • Nodes connected by links in a purely random fashion • How large is the largest connected component? (as a fraction of all nodes) • Depends on the number of links per node • (Erdös, Rényi 1959) ift6802

  14. Connecting Nodes ift6802

  15. Random Network (1) • add random paths ift6802

  16. Random Network (2) • paths • trees ift6802

  17. Random Network (3) • paths • trees • networks ift6802

  18. Random Network (3+) • paths • trees • networks • ….. ift6802

  19. Network Connectivity (4) • paths • trees • networks • fully connected ift6802

  20. Connectivity of a random graph 1 Disconnected phase Fraction of all nodes in largest component Conected phase 0 1 Average number of links per node ift6802

  21. Regular or Ordered Network ift6802

  22. Network measures • Connectivity is not main measure. • Characteristic Path Length (L) : • the average length of the shortest path connecting each pair of agents (nodes). • Clustering Coefficient (C) is a measure of local interconnection • if agent i has ki immediate neighbors, Ci, is the fraction of the total possible ki*(ki-1) / 2 connections that are realized between i's neighbors. C, is just the average of the Ci's. • Diameter: maximum value of path length ift6802

  23. Regular vs Random Networks Regular Random Average number of connections/node few, clustered fewer, spread Number of connections needed to fully connect many fewer (<2/3) Diameter large moderate ift6802

  24. Natural networks • Between regular grids and totally random graphs • Need for parametrized models: • Regular -> natural -> random • Watts • Alpha model ( not intuitive) • Beta rewiring model  ift6802

  25. Clustering • Clustering measures the fraction of neighbors of a node that are connected themselves • Regular Graphs have a high clustering coefficient • but also a high diameter • Random Graphs have a low clustering coefficient • but a low diameter • Both models do match the properties expected from real networks! Random Graph (k=4) Short path length • L~logkN Almost no clustering • C~k/n Regular Graph (k=4) Long paths • L ~ n/(2k) Highly clustered • C~3/4 Base metwork is circle ift6802

  26. Small-World Networks • Random rewiring of regular graph (by Watts and Strogatz) • With probability p (or )rewire each link in a regular graph to a randomly selected node • Resulting graph has properties, both of regular and random graphs • High clustering and short path length • FreeNet has been shown to result in small world graphs ift6802

  27. Example: 4096 node ring K=4 Regular graph: n nodes, k nearest neighbors  path length ~ n/2k 4096/16 = 256 Rewired graph (1% of nodes): path length ~ random graph clustering ~ regular graph  Small World Graph Random graph: path length ~ log (n)/log(k) ~ 4 ift6802

  28. Beta network 1 C Small- world networks 0 L 0 1 Rewiring probability  ift6802

  29. More exactly …. (p = ) C L Small world behaviour ift6802

  30. Effect of short-cuts • Huge effect of just a few short-cuts. • First 5 rewirings reduces the path length by half, regardless of size of network • Further 50% gain requires 50 more short-cuts ift6802

  31. The strength of weak ties • Granovetter (1973): effective social coordination does not arise from densely interlocking strong ties, but derives from the occasional weak ties • this is because valuable information comes from these relations (it is valuable if/because it is not available to other individuals in your immediate network) ift6802

  32. Two ways of constructing ift6802

  33. Alpha model • Watts’ first Model (1999) • Inspired by Asimov’s “I, Robot” novels • R. Daneel Olivaw • Elijah Baley • Caves of Steel (Earth) • Solaria ift6802

  34. Two extreme types of social networks • Caveman’s world • people live in isolated communities • probability meeting a random person is high if you have mutual friends and very low if you don’t • Solaria • people live isolated from each other but with supreme communication capabilities • your social history is irrelevant to your future ift6802

  35. Alpha network • Alpha () distance parameter • =0 : if A and B have a friend in common, they know each other (Caveman world) • =∞ : A & B don’t know each other, no matter how many common friends they have (Solarian world) ift6802

  36. =0 =1 = Caveman world Likelihood that A meets B Solaria world Number of mutual friends shared by A and B ift6802

  37. Clustering coefficient C Small- world net- works Fragmented networks Alpha network Path length L critical  L drops because we only count nodes that are connected ift6802

  38. How about real networks • All nodes in alpha and beta networks are equal in the sense that the number of connections each nodes has is not very far from the average • Watts and Strogatz had used normal distribution • Real world is not like that • Sizes of cities, Wealth of individuals in USA, Hubs in transportation systems • Barabási and Albert (1999) • Scale-free networks, whose connectivity is defined by a power-law distribution ift6802

  39. Random Networks Each node is connected to a few other nodes. The number of connections per node forms a Poisson distribution, with a small average of number of connections per node. This & three following graphics from: Linked: The New Science of Networksby Albert-Laszlo Barabasi; 2002 ift6802

  40. Scale-Free Networks Each node is connected to at least one other; most are connected to only one, while a few are connected to many. The number of connections per node forms a hyperbolic distribution, with no meaningful average number of connections per node. ift6802

  41. Random Scale-Free Scale-free networks are associated with networks that grow by “natural” processes in which the number of nodes increases with time not just the number of connections. ift6802

  42. Power law phenomena • Average & median are far apart • Whales and minnows • Average from a few large nodes • Median governed by majority of small nodes ift6802

  43. Performance • Real power law networks also have short distances • Existence of central backbone of highly connected HUBS nodes • Similar phenomena noted in linguistics and economics • Zipf • Pareto ift6802

  44. Zipf's law - linguistics • Zipf, a Harvard linguistics professor, sought to determine the frequency of use of the 3rd or 8th or 100th most common words in English text. • Zipf's law states that the frequency y is inversely proportional to it's rank r: • Y~r -b, with b close to unity. •  Zipf Presentations ift6802

  45. The Pareto Income Distribution • The Pareto distribution gives the probability that a person's income is greater than or equal to x and is expressed as ift6802

  46. Vilfredo Pareto, 1848-1923 • Italian economist • Born in Paris • Polytechnic Institute in Turin in 1869, • Worked for the railroads. • Pareto did not study economics seriously until he was 42. • In 1893 he succeeded his mentor, Walras, as chair of economics at the University of Lausanne. ift6802

  47. Pareto’s contributions • Pareto optimality. • A Pareto-optimal allocation of resources is achieved when it is not possible to make anyone better off without making someone else worse off. • Pareto's law of income distribution. • In 1906, Italian economist Vilfredo Pareto created a mathematical formula to describe the unequal distribution of wealth in his country, observing that 20% of the people owned 80% of the wealth. ift6802

  48. log-log plot Pareto distribution, m=10000, k=1 Pareto distribution is said to be scale-free because it lacks a characteristic length scale ift6802

  49. Building Power-law networks • It is easy to create PL networks • Build network node by node • Connect new node to an existing node • Probability of connection proportional to its number of links • The rich get richer • The poor get poorer ift6802

  50. Structure and dynamics • The case of centrality • centers are in networks • by design (central control, dictatorship) • by non-design (unnoticed critical resources, informal groups) • or they emerge as a consequence of certain events • ”he was at the right place at a right time” • clapping in unison ift6802

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