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Understanding Complex Networks: Applications and Analysis

Explore the world of complex systems made of interconnected elements and diverse interactions in business, biotech industry, and social networks. Learn about network structures, models, and analysis techniques for robust network growth.

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Understanding Complex Networks: Applications and Analysis

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  1. Complex Networks - Anurag Singh 1

  2. Complex systems • Made of many non-identical elements connected by diverse interactions. • NETWORK

  3. Business ties in US biotech-industry • Nodes: companies: investment • pharma • research labs • public • biotechnology • Links: financial • R&D collaborations • http://ecclectic.ss.uci.edu/~drwhite/Movie

  4. Business ties in US biotech-industry • Nodes: companies: investment • pharma • research labs • public • biotechnology • Links: financial • R&D collaborations • http://ecclectic.ss.uci.edu/~drwhite/Movie

  5. Structure of an organization • Red, blue, or green: departments • Yellow: consultants • Grey: external experts • www.orgnet.com

  6. Internet

  7. Friendship Network

  8. Network Collaboration Network

  9. 9-11 Terrorist (?) Network • Social Network Analysis is a mathematical methodology for connecting the dots -- using science to fight terrorism. Connecting multiple pairs of dots soon reveals an emergent network of organization.

  10. Some interesting Problems • Consonants (Language) Networks • Marriage Networks • Collaboration Networks • Build Networks which are robust as well as efficient • Actors Network

  11. Outline • Techniques to analyze networks • Special types of networks – random networks, power law networks, small world networks • Models of network growth • Processes taking place on network – search,

  12. Traditional vs. Complex Systems Approaches to Networks • Traditional Questions: • Social Networks: • Who is the most important person in the network? • Graph Theory: • Does there exist a cycle through the network that uses each edge exactly once? • Complex Systems Questions: • What fraction of edges have to be removed to disconnect the graph? • What kinds of structures emerge from simple growth rules?

  13. Introduction to Complex Networks • Complex network is a network (graph) with non-trivial topological features (heavy tail in the degree distribution, a high clustering coefficient, assortavity among vertices, and community structure) • Features that do not occur in simple networks • Lattices or random graphs, does not have these features.. degreedist. clustering assortativity comunity Random Lattice

  14. Introduction to Complex Networks (contd..) • Many systems in nature can be described by models of complex networks • Structures consisting of nodes or vertices connected by links or edges.

  15. Introduction to Complex Networks (contd..) • Examples : • The Internet is a network of routers or domains. • The World Wide Web (WWW) is a network of websites • The brain is a network of neurons. • Social network • Citation networks • Diseases are transmitted through social networks • Man-made infrastructures, and in many physical systems such as the power grids.

  16. Network structures of the Internet and the WWW.

  17. Social network Citation network

  18. Evolution of Complex Network research • Erdös and Rényi (ER) described a network with complex topology by a random graph • Many real-life complex networks are neither completely regular nor completely random, • Two significant recent discoveries are small-world effect and the scale-free nature of most complex networks.

  19. Introduction to Complex Networks (Contd..) • Watts and Strogatz (WS) introduced the concept of small-world phenomenon • A prominent common feature of the ER random graph and the WS small-world model is • The connectivity distribution of a network peaks at an average value and decays exponentially. • Each node has about the same number of link connections. • Such networks are called “exponential networks” or “homogeneous networks,”

  20. Introduction to Complex Networks (Contd..) • A significant recent discovery in the complex networks is the observation that many large-scale complex networks are scale-free, • That is, their connectivity distributions are in a power-law form that is independent of the network scale . • Differs from an exponential network, • A scale-free network is inhomogeneous in nature • Most nodes have very few link connections and yet a few nodes have many connections.

  21. Decision parameters and its definitions • Average path length • Clustering coefficient • Degree distribution • Degree exponent

  22. Decision parameters and its definitions (contd..) Average Path Length • In a network, the distance dij between two nodes, labeled i and j respectively, is defined as the number of edges along the shortest path connecting them. The diameter D: of a network, therefore, is defined to be the maximal distance among all distances between any pair of nodes in the network. • The average path length L of the network, then, is defined as the mean distance between two nodes, averaged over all pairs of nodes.

  23. Decision parameters and its definitions (contd..) • L determines the effective “size” of a network, • The average path length of most real complex networks is relatively small.] D = max l (A,B)

  24. Decision parameters and its definitions (contd..) Clustering Coefficient • Two of our friends are quite possibly friends of each other. • This property refers to the clustering of the network. • A clustering coefficient C as the average fraction of pairs of neighbors of a node that are also neighbors of each other. • a node i in the network has kiedges • they connect this node to ki other nodes (neighbors). • at most ki (ki − 1)/2 edges can exist among them

  25. Decision parameters and its definitions (contd..) • The clustering coefficient : Ei -- edges that actually exist among ki nodes • Cluster coefficient C of whole network: • 0 ≤ C ≤ 1 • In most large-scale real networks clustering coefficients are much greater than completely random network

  26. Decision parameters and its definitions (contd..) Degree Distribution • The degree ki of a node i is the total number of its neighbors. • The larger the degree, the “more important” the node is in a network. • The average of kiover all i is called the average degree of the network ( < k >). • The spread of node degrees over a network is characterized by a distribution function P(k) • P(k) is the probability that a randomly selected node has exactly k edges.

  27. A regular lattice has a simple degree sequence because all the nodes have the same number of edges • so a plot of the degree distribution contains a single sharp spike • Any randomness in the network will broaden the shape of this peak. In the limiting case of a completely random network • the degree sequence obeys the familiar Poisson distribution • the shape of the Poisson distribution falls off exponentially away from the peak value <k>

  28. Small World Networks 28

  29. 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 29

  30. Correct question • WHY are there short chains of acquaintances linking together arbitrary pairs of strangers??? • Or • Why is this surprising 31

  31. 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. 32

  32. 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. 33

  33. Social network research • Devise various classes of networks • Study their properties 34

  34. Network parameters • Network type • Regular • Random • Natural • Size: # of nodes • Number of connexions: • average & distribution • Selection of neighbours 35

  35. REGULAR Network Topologies STAR TREE GRID BUS RING 36

  36. 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) 37

  37. Connecting Nodes 38

  38. Random Network (1) • add random • paths 39

  39. Random Network (2) • paths • trees 40

  40. Random Network (3) • paths • trees • networks 41

  41. Random Network (3+) • paths • trees • networks • ….. 42

  42. Network Connectivity (4) • paths • trees • networks • fully connected 43

  43. 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 44

  44. Regular or Ordered Network 45

  45. 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 46

  46. 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 47

  47. Classes of Complex Networks • 1.Random Graphs Model • First studied by Erdos and Renyi • Some properties of E-R networks: • if nodes in in graph = N • Average number of edges (= size of graph): • Let n ,vertices and connect each pair (or not) with probability p (or 1-p). • E = pN (N - 1) / 2 • Average degree: • 〈k〉 = 2 E/N = p (N - 1) ~ p N 48

  48. Classes of Complex Networks (contd..) • Erdos and Renyi proposed the following model of a network : • the model called Gn,p , is the ensemble of all such graphs in which a graph having m edges appears with pm(1-p)M-m, where , ,is the maximum possible number of edges. 49

  49. Classes of Complex Networks (contd..) • another model, called Gn,m, which is the ensemble of all graphs having n vertices and exactly m edges, each possible graph appearing with equal probability. • presence or absence of edges is independent, and hence the probability of a vertex having degree k is: • (for large n and fixed k) • Where, z =expected degree = p(n-1) 50

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