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    Scott Complex Networks Scott Complex Networks Presentation Transcript

    • Complex Networks: Small-World, Scale-Free and Beyond 黃崇源
    • Complex Network
      • Nodes  Objects; Edges  Relations among objects
        • Internet  a network of routers or domains
        • WWW  a network of websites
        • Brain  a network of neurons
        • Social Network, Sexual Network, Food Webs, Market, …
      • Research Problems
        • Diseases are transmitted through social networks.
        • Computer viruses spread through the Internet.
        • Energy is distributed through transportation networks
    • Types of Complex Networks
      • Social Network
        • Patterns of friendships between individuals
        • Business relationships between companies
      • Information Network
        • WWW, citation network
      • Technological Network
        • Power grid, network of airline routes, roads and railways
      • Biological Network
        • Food webs, neural networks
    • Aims of Complex Network Theory
      • Find global features that characterize the structure and behavior of networked systems.
        • Local clustering, small-world, power-law properties, …
      • Create network models to understand these properties.
        • Random, small-world, scale-free networks, …
      • Predict the behavior of networked systems.
        • Network Resilience and robustness (WWW, sexual network)
        • Epidemic Transmission Dynamics (SARS, Flu, HIV, …)
        • Synchronization in Complex Dynamical Networks
    • Properties of Complex Networks
      • Small-world effect
        • Random, small-world, scale-free networks
      • Local clustering
        • Small-world network
      • Degree distribution
        • Normal distribution  random and small-world networks
        • Power-Law distribution  scale-free network
    • Small-World Effect
      • Definition
        • The distance d ij between two nodes
          • the number of edges along the shortest path connecting them.
        • The network diameter, D
          • The maximal distance among all distances d ij in the network.
        • The average path length, L
          • The mean distance averaged over all pairs of nodes.
      • The average path length in real complex networks is relatively small.
        • Logarithmic increase in L with the size of the network.
        • E.g., “six degree of separation” in social network
    • Local Clustering
      • Your friend’s friend is also your direct friend; or two of Your friends are quite possibly friends of each other.
        • Node clustering coefficient c i = 2  E i / ( k i  ( k i – 1))
          • The average fraction of pairs of neighbors of a node that are also neighbors of each other.
        • Network clustering coefficient C
          • The average of ci over all node i. (0  C  1)
      • C of random networks consisting of N nodes are very small as compared to most real networks. ( C ~ 1/ N )
        • C of real networks are much greater than  (1/ N ).
    • Degree Distribution
      • Simplest and most important characteristic of node
        • The node degree k i
          • The total number of its connections.
        • The node degrees over a network is characterized by a distribution function P(k) .
    • Complex Network Models
      • Regular networks (e.g., Cellular Automata)
        • Local clustering property
      • Random networks (RNs)
        • Small-world property
      • Small-world networks (Watts and Strogatz’ SWNs)
        • Local clustering and small-world properties
      • Scale-free networks (SFNs)
        • Small-world and power-law properties
    • Random Networks
      • RN model Algorithm
        • Starts with N nodes.
        • Connects each pair of nodes with probability p.
        • Creates a random networks with approximately pN ( N – 1) / 2 randomly placed links.
      • Poisson distribution.
      • Cclustering coefficient C ~ 1/ N
      • Average path legnth L ~ log N.
    • Small-World Networks
    • Scale-Free Networks
    • SFN Examples
    • Robustness vs. Fragility of Internet