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Network Evolution Marco&Mirco

Network Evolution Marco&Mirco. Dingding Xu. Overview. Characters Marcoscopic Mircoscopic Models. Characters. Evolution characters in networks:. No t Grow Local variation at a timestamp Burst analysis: node variation rate Growing -- Adding New Node Most of networks

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Network Evolution Marco&Mirco

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  1. Network EvolutionMarco&Mirco Dingding Xu

  2. Overview • Characters • Marcoscopic • Mircoscopic • Models

  3. Characters Evolution characters in networks: • Not Grow • Local variation at a timestamp • Burst analysis: node variation rate • Growing -- Adding New Node • Most of networks • Exceptions : some biological networks How many edges a new node produces?

  4. History • 2005~ • Microscopic • Membership (2006) • Community Growth • Community Change • Giant Connector Component • Next Largest Connector Component (2008) • … • Before 2005 • Macroscopic • Small World • Scale-free • Rich club • Self-similarity • Cluster • Densification Power Law(2005) • Shrinking Diameters • …

  5. Properties • Macroscopic Properties • What networks look like in their evolution? • Small world, Scale-free, Cluster… • More macroscopic properties? Who knows! • Microscopic Properties • How and why networks look like what they • Try to explain macro from multiple perspective • But know less about why

  6. Macroscopic Properties • What looks like • Small world phenomenon • Scale-free distribution • Rich club phenomenon • Clusters • Densification power law • Shrinking diameters

  7. Densification • What is the relation between the number of nodes and the edges over time? • That is α=? • Common sense: linear(α=1) or E=log(N) • But Networks are denser over time • That is in most networks 1<α<2 • Super-linear !

  8. Real Networks Internet AS ArXiv

  9. Densification • “Graphs over time: densification laws, shrinking diameters and possible explanations”—KDD, 05 • Densification power law • New typical property in complex networks • Like small world, scale free… • Many papers cite densification property • Some papers focus on this property • “Scale-free networks as pre-asymptotic regimes of super-linear preferential attachment”—arXiv stat-mech, CAIDA, 2008

  10. Shrinking Diameters • Diameter : max distance • Effective Diameter • 90% of total distances < Effective Diameter • Focus on effective diameter rather than diameter • Intuition and prior work say that distances between the nodes slowly grow as the network grows (like log N) • But … • Diameter Shrinks or Stabilizes over time • as the network grows the distances between nodes slowly decrease

  11. Shrinking Diameters • “Graphs over time: densification laws, shrinking diameters and possible explanations”2005 • Some researches to develop models fitting this properties • Forest Fire Model – First model in “Graphs over time …” • Positive feedback PA model – CAIDA (2007 or 2008)

  12. Microscopic Properties • How and Why • How is to explain • Unluckily Why we know less • Membership • Community growth • Giant Connector Component • Next Largest Connector Component

  13. Membership • Two Question • What properties influence a new node to join a community? • How are existing nodes influence a new node to join a community • Main Method: Decision Tree • Find some important properties: node’s neighborhood and neighborhood’s interconnections • Conclusion • More neighbors in a community more possibility to join the community (obviously) • More interconnections in neighbors more possibility to join the community(“diffusion theory” in sociology )

  14. Community Growth • It is interested to know what properties influence community’s growth • Growth rate: incremental rate of new members in a community • Method • select two kinds of communities with high growth rates and with low growth rates • find the most distinguished properties between them (also decision tree) • Conclusion • More neighbors out of communities more probability to grantee a high growth rate • closed triangles discourage communities to grow

  15. Community Growth • Closed triangles • Possible Explanation • a large density of triangles indicates a kind of “cliquishness” that makes the community less attractive to join • high triangle density is a sign of a community that stopped gaining new members at some point in the past and has subsequently been densifying, adding edges among its current set of members

  16. Giant Connected Component • Question: How does the giant connected component emerge? • Most real graphs display a gelling point, or burning off period • After gelling point, they exhibit typical behavior. This is marked by a spike in diameter. IMDB

  17. Models • Macroscopic Models • ER Random Graph Model • WS Small-World Model • Preferential Attachment Model • Community Guided Attachment Model (2005) • Forest Fire Model (2005) • Kronecker Model (2007) • Microscopic Models • Butterfly Model (2008) • Microscopic Evolution Model (2008)

  18. Community Guided Attachment Model • Idea: A model based on communities (Trees) • All members are leaf nodes • The distance h(v,w) between two nodes are the height of their nearest ancestor that is the height of the min sub-tree which contain nodes v and w • Possibility to connect depend on a function: f(h)=c^(-h)，c>=1, h is h(v,w) • Guarantee densification power law A node’s degree:

  19. Community Guided Attachment Model • So the Community Guided Attachment leads to Densification Power Law with exponent

  20. Forest Fire Model • Don’t want to have explicit communities in constructions • Densification and shrinking • Ideas: • How do we meet friends at a party?

  21. Forest Fire Model • The Forest Fire model has 2 parameters: • p: forward burning probability • r: backward burning probability • The model: • Each turn a new node v arrives • Uniformly at random chooses an “ambassador” w • Flip two geometric coins to determine the number in‐ and out‐links of w to follow (burn) • Fire spreads recursively until it dies • Node v links to all burned nodes • Easy but densification and shrinking depend on parameters

  22. Microscopic Evolution Model • Idea: concern about a node’s lifetime and its process of generating edges • “Microscopic Evolution of Social Networks”KDD08, J.Lesvokec • Three processes governing the evolution • Node arrival : node enters the network • Edge initiation :node wakes up, initiates an edge, goes to sleep • Edge destination : where to attach a new edge

  23. Microscopic Evolution Model • Apply maximum likelihood estimation to select distribution density • Node arrival stage：N(t)=exp(0.25t)。 • Edge initiation stage： “lifetime” function is p(a)=λexp(-λa) “gap” function is p(δ|d;α，β)=(1/Z)δ-αexp(-βdδ) • Edge destination stage: finish triangle edge

  24. Models

  25. Thanks!

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