1 Algorithmic Networks & Optimization Maastricht, November 2008 Ronald L. Westra, Department of Mathematics Maastricht University

2 Network Models

3 Properties of Networks

4 Networks consist of: * nodes * connections (directed or undirected) * update rules for the nodes Network properties

5 Node Network properties x Connection (directed arrow) Update rule: x t+1 = some_function_of (x t ) = f(x t )

6 Until now we have encountered a number of interesting models 1 entity that interacts only with itself The meaning of networks x

7 Examples: population growth, exponential growth, Verhulst equation: x t+1 = some_function_of (x t ) = f(x t ) The meaning of networks x

8 2 entities that interact: The meaning of networks Yx

9 Example: predator-prey-relations as the Lotka-Volterra equation: x t+1 = f( x t, y t ) y t+1 = g( x t, y t ) The meaning of networks Yx

10 Multiple entities that interact in a network structure: The meaning of networks This is a general model for multi-agent interaction: … x1x1 x3x3 xnxn

11 Small-World Networks

12 Growth of knowledge semantic networks apple orange pear lemon Newto n Einstein gravitation Average separation should be small Local clustering should be large

13 Semantic net at age 3

14 Semantic net at age 4

15 Semantic net at age 5

16 The growth of semantic networks obeys a logistic law

17 Given the enormous size of our semantic networks, how do we associate two arbitrary concepts?

18 Clustering coefficient and Characteristic Path Length Clustering Coefficient (C) –The fraction of associated neighbors of a concept Characteristic Path Length (L) –The average number of associative links between a pair of concepts Branching Factor (k) –The average number of associative links for one concept

19 Example apple orange pear lemon Newto n Einstein gravitation

20 Four network types a c b d fully connectedrandom regular “small world

21 Network Evaluation Type of networkkCL Fully-connectedN-1LargeSmall Random<<NSmall Regular<<NLarge Small-world<<NLargeSmall

22 Varying the rewiring probability p: from regular to random networks C(p)/C(0) L(p)/L(0) 1 0 p

23 Data set: two examples APPLE PIE(20) PEAR(17) ORANGE(13) TREE( 8) CORE( 7) FRUIT( 4) NEWTON APPLE(22) ISAAC(15) LAW( 8) ABBOT( 6) PHYSICS( 4) SCIENCE( 3)

24 L as a function of age (× 100) = semantic network = random network

25 C as a function of age (× 100) = semantic network = random network

26 Small-worldliness Walsh (1999) Measure of how well small path length is combined with large clustering Small-wordliness = (C/L)/(C rand /L rand )

27 Small-worldliness as a function of age adult

28 Some comparisons Semantic Network Cerebral Cortex Caenorhabditis Elegans Small-Worldliness

29 What causes the small- worldliness in the semantic net? TOP 40 of concepts Ranked according to their k-value (number of associations with other concepts)

30 Semantic top 40

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32 Special Networks

33 Special Networks Small-world networks Scale-free networks

34 Consider a set of nodes Network properties: branching factor k x1x2x3x4x5

35 Now make random connections Network properties: branching factor k x1x2x3x4x5 This is a random network

36 This approach results in an average branching number k av If we plot a histogram of the number of connections we find: Network properties: branching factor k

37 Now consider an structured network Network properties: branching factor k x1x2x3x4x5

38 This approach results in an equal branching number k av for all nodes If we plot a histogram of the number of connections we find: Network properties: branching factor k

39 The same for a fully connected network: this results in an equal branching number k av = n - 1 Network properties: branching factor k

40 Now what for a small-world network? Network properties: branching factor k

41 History Using a Web crawler, physicist Albert-László Barabási at the University of Notre Dame mapped the connectedness of the Web in 1999 (Barabási and Albert, 1999). To their surprise, the Web did not have an even distribution of connectivity (so-called "random connectivity"). Instead, some network nodes had many more connections than the average; seeking a simple categorical label, Barabási and his collaborators called such highly connected nodes "hubs". Scale-Free Networks (Barabasi et al, 1998)

42 Scale-Free Networks (Barabasi et al, 1998)

43 History (Ctd) In physics, such right-skewed or heavy-tailed distributions often have the form of a power law, i.e., the probability P(k) that a node in the network connects with k other nodes was roughly proportional to k −γ, and this function gave a roughly good fit to their observed data. Scale-Free Networks (Barabasi et al, 1998)

44 History (Ctd) After finding that a few other networks, including some social and biological networks, also had heavy-tailed degree distributions, Barabási and collaborators coined the term "scale-free network" to describe the class of networks that exhibit a power-law degree distribution. Soon after, Amaral et al. showed that most of the real-world networks can be classified into two large categories according to the decay of P(k) for large k. Scale-Free Networks (Barabasi et al, 1998)

45 A scale-free network is a noteworthy kind of complex network because many "real-world networks" fall into this category. “Real-world" refers to any of various observable phenomena that exhibit network theoretic characteristics (see e.g., social network, computer network, neural network, epidemiology). Scale-Free Networks (Barabasi et al, 1998)

46 In scale-free networks, some nodes act as "highly connected hubs" (high degree), although most nodes are of low degree. Scale-free networks' structure and dynamics are independent of the system's size N, the number of nodes the system has. In other words, a network that is scale-free will have the same properties no matter what the number of its nodes is. Scale-Free Networks (Barabasi et al, 1998)

47 The defining characteristic of scale-free networks is that their degree distribution follows the Yule-Simon distribution — a power law relationship defined by where the probability P(k) that a node in the network connects with k other nodes was roughly proportional to k−γ, and this function gave a roughly good fit to their observed data. The coefficient γ may vary approximately from 2 to 3 for most real networks. Scale-Free Networks (Barabasi et al, 1998)

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49 In the late 1990s: Analysis of large data sets became possible Finding: the degree distribution often follows a power law: many lowly connected nodes, very few highly connected nodes: Examples – Biological networks: metabolic, protein-protein interaction – Technological networks: Internet, WWW – Social networks: citation, actor collaboration – Other: earthquakes, human language Scale-Free Networks (Barabasi et al, 1998)

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51

52 Random versus scale-free

53 Random (□) and scale free (○) Linear axesLogarithmic axes

54 Nodes: people, links: # of sexual partners

55 Nodes: -addresses, links: s

56 Protein network C.elegans

Internet routers and the physical connections between them

58 c. Web pages Inlinks and outlinks (red and blue) d. Network nodes (green)

59 END LECTURE