Complex Networks: Small-World, Scale-Free and Beyond 黃崇源
Complex Network <ul><li>Nodes    Objects; Edges    Relations among objects </li></ul><ul><ul><li>Internet    a network ...
Types of Complex Networks <ul><li>Social Network </li></ul><ul><ul><li>Patterns of friendships between individuals </li></...
Aims of Complex Network Theory <ul><li>Find global features that characterize the structure and behavior of networked syst...
Properties of Complex Networks <ul><li>Small-world effect </li></ul><ul><ul><li>Random, small-world, scale-free networks <...
Small-World Effect <ul><li>Definition </li></ul><ul><ul><li>The distance  d ij  between two nodes </li></ul></ul><ul><ul><...
Local Clustering <ul><li>Your friend’s friend is also your direct friend; or two of Your friends are quite possibly friend...
Degree Distribution <ul><li>Simplest and most important characteristic of node </li></ul><ul><ul><li>The node degree  k i ...
Complex Network Models <ul><li>Regular networks (e.g., Cellular Automata) </li></ul><ul><ul><li>Local clustering property ...
Random Networks <ul><li>RN model Algorithm </li></ul><ul><ul><li>Starts with  N  nodes. </li></ul></ul><ul><ul><li>Connect...
Small-World Networks
Scale-Free Networks
SFN Examples
Robustness vs. Fragility of Internet
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Scott Complex Networks

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Scott Complex Networks

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

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