Some random graphs for network models - Birgit Plötzeneder


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Some random graphs for network models - Birgit Plötzeneder

  1. 1. Some random graphs for network models Birgit Plötzeneder
  2. 2. Bell-shaped node degree distributions
  3. 3. Random model Erdös,Renyi (1960s) On random graphs I; On the evolution of random graphs; On the strength of connectedness of a random grap h - start with N disconnected nodes - connect nodes with probability p to each other
  4. 4. Watts and Strogatz Watts, Strogatz (1998), Collective dynamics of "small-world" networks - one-dimensional ring lattice of N nodes connected to its 2 K nearest neighbors - goes through each of the edges in turn and, independently with probability p "rewire" it to a randomly selected (different) node
  5. 5. Watts and Strogatz - average distance grows like O(log(N) and not O(N). - support high levels of clustering „ The small-world effect (short average distance between nodes and high levelsof clustering) has been detected in networks that include a network of actors in Hollywood, the power generator network in the western US...“ Gerardo Chowell and Carlos Castillo-Chavez, Worst-Case Scenarios and Epidemics
  6. 6. Newman and Watts Newmann, Watts (1999): Renormalization group analysis of the small-world network model , <ul><li>Ring like with Watts and Strogatz's
  7. 7. Don't replace edges, instead create shortcuts </li></ul>
  8. 8. Power-law degree distributions = Pareto distributions
  9. 9. Pareto distributions - small number of highly connected nodes, most nodes have a small number of connections - Barabasi and Albert called them scale-free networks
  10. 10. Barabási and Albert Barabàsi, Albert (1999) Emergence of scaling in random networks - starts with a small number of nodes - a new node connects with higher probability to nodes that have already accumulated a higher number of connections
  11. 11. Klemm and Eguíluz <ul>&quot; When a node is created it is linked to nodes that are popular at the time. It then receives links from nodes created subsequently, he said. &quot;This continues until eventually the node under consideration loses its popularity.&quot; <li>From:
  12. 12. Klemm, Eguíluz (2002) Growing scale-free networks with small-world behavior </li></ul>
  13. 13. fat-tailed degree distribution
  14. 14. Dorogovtsev, Mendes, Samukhin Dorogovtsev, Mendes, Samukhin : How to generate a random growing network - with each step, the edges of a growing network are transformed into configurations of edges and new vertices according to some probability function