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Basic Models of Complex Networks

Basic Models of Complex Networks . Peter Sloot: Computational Science, University of Amsterdam, The Netherlands. Agenda. Elementary Statistical Properties of Networks The Small-World Property Clustering Scale-Free Degree Distributions And their Basic Models

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Basic Models of Complex Networks

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  1. Basic Models of Complex Networks Peter Sloot: Computational Science, University of Amsterdam, The Netherlands.

  2. Agenda • Elementary Statistical Properties of Networks • The Small-World Property • Clustering • Scale-Free Degree Distributions • And their Basic Models • Erdős-Rényi (Exponential) Random Networks • Watts-Strogatz Networks • The Preferential Attachment Model • Some further models

  3. Reminder

  4. Complex Systems, Definitions • Systems composed of interacting components • Simple entities yield complicated dynamics • Nonlinearity, self-organization (pattern development) • „The whole is more than the sum of its parts” • Recursive effects from interactions; path dependence; dynamically emergent properties • Typically not amenable to analytic solutions • Size and computational complexity, explosion • Nonexistence of „solution”: infititely long lived transients, nonequilibrium cascades, sensitive dependencies, etc.

  5. Complex Systems (Examples) • N-Particle Systems • Protein Interactions • Metabolic Pathways • Financial Markets • Market Research • Supply Chain Optimization • Ecological and Population Dynamics • Stellar Systems

  6. Friendship and acquaintance nets Co-authorship nets Network of sexual relationships World-wide web Foodwebs Metabolic pathways Gene regulation networks Protein interactions E G B D F H C A Basic Models of Complex Networks »All the world's a network,And all the men and women merely nodes; «

  7. Basic Statistical Properties of Complex Networks • „Small World” (low average path length).

  8. Frigyes Karinthy: Chains (Láncszemek, 1929) „ (…) A fascinating game grew out of this discussion. One of us suggested performing the following experiment to prove that the population of the Earth is closer together now than they have ever been before. We should select any person from the 1.5 billion inhabitants of the Earth—anyone, anywhere at all. He bet us that, using no more than five individuals, one of whom is a personal acquaintance, he could contact the selected individual using nothing except the network of personal acquaintances. (…)”

  9. Frigyes Karinthy: Chains (Láncszemek, 1929) (…) A nehezebb feladatot: egy szögecselő munkást a Ford-művek műhelyéből, ezekután magam vállaltam és négy láncszemmel szerencsésen meg is oldottam. A munkás ismeri műhelyfőnökét, műhelyfőnöke magát Fordot, Ford jóban van a Hearst-lapok vezérigazgatójával, a Hearst-lapok vezérigazgatójával tavaly alaposan összeismerkedett Pásztor Árpád úr, aki nekem nemcsak ismerősöm, de tudtommal kitűnő barátom - csak egy szavamba kerül, hogy sürgönyözzön a vezérigazgatónak, hogy szóljon Fordnak, hogy Ford szóljon a műhelyfőnöknek, hogy a szögecselő munkás sürgősen szögecseljen nekem össze egy autót, éppen szükségem lenne rá. Így folyt a játék és barátunknak igaza lett - soha nem kellett ötnél több láncszem ahhoz, hogy a Földkerekség bármelyik lakosával, csupa személyes ismeretség révén, összeköttetésbe kerüljön a társaság bármelyik tagja."

  10. „It’s a small world!” • „Six degrees of separation” • Karinthy • Drama of John Guare and film of Fred Schepisi • Stanley Milgram’s experiment (and that of Duncan Watts) • Erdős Number • The „Kevin Bacon Game”

  11. Recent Experiments • In 2001, Duncan Watts, a professor at Columbia University, attempted to recreate Milgram's experiment on the internet, using an e-mail message as the "package" that needed to be delivered, with 48,000 senders and 19 targets (in 157 countries). Watts found that the average (though not maximum) number of intermediaries was around six. • A 2007 study by Jure Leskovec and Eric Horvitz examined a data set of instant messages composed of 30 billion conversations among 240 million people. They found the average path length among Microsoft Messenger users to be 6.6

  12. Number of nodes: N Notions: „Path” between two nodes Shortest path b/w two nodes l– avg of shortest paths The small-world property: l~log(N) The „small-world property”

  13. The Erdős-Rényi Network(Random Network, 1959.) • Nnodes, all edges with probability p. • Small world, if connected. • Almost always connected: • „Giant Component” promptly and suddenly. • 1/N+ • Small world within the components • Almost always connected Original term: almost surely, meaning: p(statement)  1, if N. • Exponentially growing number of reachable nodes.

  14. The Erdős-Rényi Network(Random Network, 1959.)

  15. Basic Statistical Properties of Complex Networks • „Small World” (low average path length). • Clustering (a friend of my friend is my friend).

  16. D. Watts applied mathematician (turned sociologist). Real networks are small worlds, but they are also clustered – in contrast to Erdős-Rényi networks. C – Clustering (between 0and 1). Number of closed triplets/number of triplets The Story by Duncan Watts

  17. Illustration

  18. Local clustering Watts-Strogatz clustering (close to C, equal if Ci is constant through the network). Alternative measures

  19. Real networks are small worlds, but they are also clustered – in contrast to Erdős-Rényi networks. C – Clustering (between 0and 1). The Story by Duncan Watts

  20. Complex Networks, BIOINF The Watts-Strogatz model • Also misnamed to „the small world model”: • In arbitrary Ddimensions (D=1,2) • Ordered lattice with k-neighborhood • With probability w: rewiring or shortcuts w w

  21. Properties of the Watts-Strogatz model • The average shortest path (l)drops earlier than clustering (C). • Thus for relatively loww’s: • It is both a small world andclustered.

  22. Basic Statistical Properties of Complex Networks • Power Law degree distribution • Important consequences(see WWW, epidemics). • Not true for all networks • „Small World” (low average path length). • Clustering (a friend of my friend is my friend). • Not all nodes are created equal.

  23. Degree DistributionPower Law, Poisson, etc. • Hatványfüggvény • y ~ axb • log(y) = log(a) + blog(x) • Scale-free and fat tail distributions • The importance of scales (i.e., average sizes)

  24. Barabási and Albert’s„Preferential Attachment” model • Minitial ‘core’nodesarbitrarily (e.g.,fully) connected. • In each step, one new node arrives with Enew links. • Random links, butprobability proportional to degree (c.f., preferential linking).

  25. Properties of Barabási-Albert networks • Degree distribution is asymptotically power law (scale-free). • γ~3. • Trivially small world, • BUT trivially not clustered. • „The rich get richer” or the Matthew principle "for whomsoever hath, to him shall be given“ [Matthew 13:12] • Pareto-distribution, investments, etc. • Its robustness and fragility. • C.f. Internet, WWW, etc.

  26. Barabási and Albert’s„Preferential Attachment” model „Hubs” in the centre…

  27. Further (basic) models of networks

  28. Scale-Free Networks • Is the Barabási-Albert model the only possible way to yield scale free networks? Or • Are scale-free networks necessarily generated by preferential attachment?

  29. The copying model of Kumar et al. (linear case) • (Original version with directed graphs.) • In each step, F new node with E new link. • New links are added: • Random „prototype node” V is selected: • The ith link • Connects to a random node with probability q, • Copies the next link from V with probability (1-q).

  30. Is there a model that yields networks with all three basic properties? • Is there a model that is • A small world, • Clustered, and • Has a Power Law degree distribution the same time?

  31. Is there a model that yields networks with all three basic properties? • Yes, several. • For example, • The hierarchical model of Ravasz and Barabási

  32. Generating Ravasz-Barabási Hierarchical Networks

  33. The iterative construction leading to a hierarchical network • Start from a fully connected cluster of five nodes • Note that the diagonal nodes are also connected — links not visible! • Create four identical replicas, connecting the peripheral nodes of each cluster to the central node of the original cluster. • Thus obtaining a network of N=25 nodes • Create four replicas of the obtained cluster, and connect the peripheral nodes again to the central node of the original module. • Thus obtaining a N=125-node network. • This process can be continued indefinitely.

  34. Properties of the hierarchical model of Ravasz and Barabási I. • Deterministic. • Trivially small world: • Why? • Its degree distribution is scale-free. • Why? • Clustering coefficient: • High • Degree dependent • Independent of system size!

  35. Properties of the hierarchical model of Ravasz and Barabási II.

  36. Reality Check • Real Networks that are Similar: • Movie actors • English synonimes • WWW • Internet domains • Real Networks that are Dissimilar: • Internet at the level of routers • Power grid

  37. More Importantly I. Vverage clustering coefficient, C(N), for 43 organisms, as a function of the numberof substrates N present in each of them. Species belonging to Archae, Bacteria and Eukaryotes are shown. The dashed line indicates the dependence of C on the network size for a module-free scale-free network, while the diamonds denoteC for a scale-free network with the same parameters (N and number of links)as observed in the43 organisms.

  38. More Importantly II. • Three organisms: • Aquidex Aeolicus (archaea) (c), • Escherichia coli (bacterium) (d) • S. cerevisiae(eukaryote) (e). • In (f) The C(k) curves averaged over all 43 organisms is shown, while the insetdisplays all 43 species together. • The dashedlines correspond to C(k) ~ k-1 • The diamonds represent C(k)expected for a scale-freenetwork.

  39. Is there a model that yields networks with all three basic properties? • Ravasz and Barabási. • But a little „too regular”…

  40. Step 1: Initial “five core”. Step 2: 4 copies A p fraction of the new nodes are randomly connected to the nodes of the central core. With preferential attachment. Step 3: 4 copies of the current 25 nodes network. A p2fraction of the new nodes are randomly connected to the nodes of the central module. With preferential attachment. Step 4: … The Stochastic version of the Ravasz-Barabási model

  41. Properties of the stochastic Ravasz-Barabási model Exponent of the degree distribution p-dependent slopes! Exponent of Clustering

  42. Summary • Basic properties of Complex Networks • 3 of them • Basic models of Complex Networks • 3+ of them

  43. Exercices • Write a program • To generate Erdős-Rényi networks. • To generate Watts-Strogatz networks. • To generate Barabási-Albert networks and analyze the generated networks’ basic properties in Pajek. • Calculate • The average shortest pathlength on a regular D-dimensional grid. • The clustering of a regular 2-dimensional grid with given k. • Try to prove**** • That connected Erdős-Rényi networks have the small-world property (almost surely).

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