1. Survey on Evolving Graphs Research Speaker: Chenghui Ren Supervisors: Prof. Ben Kao, Prof. David Cheung 1

2. Motivation Evolving graphs are everywhere  Social networks  Users join social networks  Friendships are established  The Web  New Web pages are created  Hyperlinks are established 2

3. Motivation Evolving graphs are everywhere  P2P networks  New routers appear  Routing table size (vertex degree) changes  Spatio networks  Transportation cost (edge weight) changes 3

4. Research branches  Evolution of graphs  How do graphs evolve over time?  Example  The networks are becoming denser over time with the average degree increasing [J. Leskovec 2007]  Querying evolving graphs  Apply queries on evolving graphs to extract information  Example  How to update the PageRank efficiently as graphs evolve? 4

5. Roadmap  Motivation  Why we are interested in evolving graphs  Evolution of graphs  How graphs evolve over time  Macroscopic evolution  Microscopic evolution  Querying evolving graphs  How to process queries on evolving graphs  Incremental computation  Key moment detection  Find-verify-fix framework 5

6. Evolution of graphs  Macroscopic evolution of graphs  How do global properties (e.g., degree distribution, diameter) evolve?  Microscopic evolution of graphs  Example  How do a user link to other users?  Microscopic node behavior results in macroscopic behavior 6

7. Macroscopic evolution  Stable degree distributions[R. Albert 1999]  Power law distribution: P(degree = k) is proportional to 1/k^a  The major hubs are closely followed by smaller ones  The nodes tend to form communities  Examples  Social networks, including collaboration networks. An example that has been studied extensively is the collaboration of movie actors in films.  Protein-Protein interaction networks.  Sexual partners in humans, which affects the dispersal of sexually transmitted diseases.  Many kinds of computer networks, including the internet and the World Wide Web.  Semantic networks  Airline networks. 7

8. Macroscopic evolution  Densification and shrinking diameters [J. Leskovec 2007]  Densification formula  E(t) is proportional to N(t) ^ a (1 < a < 2)  Shrinking diameters 8

9. Microscopic evolution  Preferential attachment model [R. Albert 1999]  New vertices attach preferentially to sites that are already well connected  Obey the power law distribution  Global model: new vertices can connect to any vertex in the whole network 9

10. Microscopic evolution  Forest fire model [J. Leskovec 2007]  Intuition: how do authors identify references?  Find first paper and cite it  Copy a few citations from first  Continue recursively  From time to time use bibliographic tools (e.g. CiteSeer) and chase back-links 10

11. Microscopic evolution  Forest fire model [J. Leskovec 2007]  A node arrives  Randomly chooses an “ambassador”  Starts burning nodes (with probability p) and adds links to burned nodes  “Fire” spreads recursively, with exponential decay 11

12. Microscopic evolution  Forest fire model [J. Leskovec 2007]  Obey the densification, shrinking diameter and power law distribution  Local model: A newcomer will have a lot of links near the community of his/her ambassador, a few links beyond this, and significantly fewer farther away 12

13. Roadmap  Motivation  Why we are interested in evolving graphs  Evolution of graphs  How graphs evolve over time  Macroscopic evolution  Microscopic evolution  Querying evolving graphs  How to process queries on evolving graphs  Incremental computation  Key moment detection  Find-verify-fix framework 13

14. Querying evolving graphs  A number of queries in literature  PageRank queries  Diameter queries  Minimum spanning tree (MST) queries  Shortest path queries  Centrality queries  … 14

15. Querying evolving graphs Methodologies  Incremental computation  PageRank queries  Diameter queries  Key moment detection  Minimum spanning tree queries  Our work: find-verify-fix framework  Shortest path queries  Centrality queries 15

16. Incremental computation  Typically, the difference between two consecutive snapshots G1 and G2 is small  Compute the solution for G2 based on the solution for G1  The incremental algorithms are expected to be fast 16

17. PageRank queries  Rank of a web page depends on the rank of the web pages pointing to it 17

18. PageRank queries  Computing PageRank for large graphs at each time instance is expensive  Incremental algorithms are proposed [P. Desikan 2005]  Principle idea: PageRank depends only on the pages that point to it and is independent of the pages pointed by the page 18

19. PageRank queries Detect a changed portion of graph Partition the graph into scalable P and non- scalable Q such that there are no incoming links from Q to P Compute PageRank for Q Merge the rankings of the two independent partitions PageRank values of partition P are obtained by simple scaling with scaling factor n(G1)/n(G2) 19

20. Diameter queries  In a P2P network, an important and fundamental question is how many neighbors should a computer have, i.e., what size the routing table should be  Network diameter corresponds to the number of hops a query needs to travel in the worst case  If the diameter is large, the routing table size should be increased 20

21. Diameter queries  G-Scale [Y. Fujiwara 2011]  First study to address diameter detection problem that guarantees exactness and efficiency on both single big graph and evolving graphs  Weak point: It assumes that one node and its connected edges are added to a time-evolving graph at each time tick. General edge insertions and edge deletions are not considered 21

22. Key moment detection  Given an evolving graph and a query, a key moment detection algorithm tries to detect those moments at which the solution to the query changes 22

23. MST queries  MSTs can be used to solve energy-efficient problems in spatio networks  A time aggregated graph is a graph in which each edge is associated with an edge weight function  A time-sub-interval is defined as a maximal sub interval of time horizon which has a unique MST  An efficient solution to determine time-sub- intervals is available [V. Gunturi 2010] 23

24. MST queries  Methodology [V. Gunturi et al 2010]  Edge order interval: a sub interval of time horizon during which there is clear ordering of edge weight functions, i.e., none of them intersect with each other  Principle idea: An edge-order-interval has a unique MST  Inspired by Prim’s algorithm 24

25. MST queries An edge-order-interval 25

26. MST queries  V. Gunturi et al proposed methods to efficiently determine at which moments to partition the edge-order-intervals  They also provided methods to incrementally compute MST based on the MST for the preceding edge-order-interval 26

27. Our find-verify-fix framework  Given an evolving graph (G1, G2, G3, …, Gn), FVF  Find representative solutions (RS’s) for G1~Gn  Verify whether these RS’s are indeed the solution for each individual snapshot  If the verification fails, try to fix the RS’s 27

28. Our find-verify-fix framework  FVF can now handle:  Exact shortest path (SP) queries on un-weighted evolving graph  Approximate SP queries on weighted evolving graphs  Approximate centrality queries 28

29. Future work  Find more interesting queries  Incorporate the ideas of incremental algorithms and key moment detection algorithms to the FVF framework 29

30. Thanks! 30