COP 5725 Advanced Database System Presentation

1. COP 5725 Advanced Database System Presentation Sheng Tan

2. PageRank on an Evolving Graph BahmanBahmani, Ravi Kumar, Mohammad Mahdian and Eli Upfal. In KDD, pages 24-32, 2012.

3. Algorithmic paradigms Classical ▪ Stationary data set ▪ Unrestricted Access to data Contemporary ▪ Online algorithm ▪ Streaming algorithms ▪ Sub-linear time algorithms ▪ Algorithmic game theory

4. Motivating examples Web pages ▪ Millions of hyperlinks modified each day Social networks ▪ Millions of social links modified each day Public opinion

7. Evolving data The data changes over time ▪ Slow changes ▪ Changes can occur anywhere Keep up with changes and adjusting solution based on new observations ▪ Not feasible to track all the changes ▪ We care about the quality of the solution

8. Algorithms for evolving data A general model for algorithm design on evolving data ▪ Data slowly changes over time ▪ Goal: compute a function of the data at each point of time ▪ Algorithm can probe specific parts of the data to know about some of the changes First introduced by A. Anagnostopoulos et al, ICALP, 2009 applied on sorting

9. Formal description At time t, true data = Xt Goal is to compute yt=f(Xt) Input changes slowly(stochastically) such that d(Xt+1,Xt) is small Algorithm can make limited probes into Xt each t to compute a solution y’t Goal is to have y’t ≈ yt

10. Evolving graph model A sequence of directed graphs over time ▪ Gt = (V,Et) = graph at time t ▪ Nodes do not change(for simplicity) Assume |Et+1 - Et| is small ▪ Choose t fine enough ▪ No change model assumed At time t, algorithm can probe a node u V to get N(u),i.e., all edges in Et of the form(u,v)

11. Example: Evolving graph G1 G’1

12. Example(contd) G2 G’2

13. Example(contd) G3 G’3

14. Example(contd) G4 G’4

15. PageRank: Recap Stationary distribution of the following random walk on a directed graph G= (V,E) If u is the current node ▪ 1 – ε, pick one of the neighbors v of u uniformly at random and move to v ▪ ε, move to a uniformly chosen node from v V π= PageRank of G, π(u) = PageRank of u V Based on assumption authoritative pages usually contain links to good pages

16. PageRank on evolving graph Probing depends on function and metric ▪ Function = PageRank ▪ Error metric = L ∞ A high PageRank node has more effect on the PageRank of other nodes ▪ πt(v) is large and v adds or deletes an edge at time t => |πt+1(u) - πt(u)| is large Reminder: we do not have access to πt

17. Proportional probing Embodies the previous intuition Pretend π’tlooks like πt At time t ▪ Pick a node v to probe with probability proportional to π’t-1(v), obtain N(v) ▪ Update G’t using G’t-1 and N(v) ▪ Output π’t, the PageRank vector G’t

18. Priority probing A deterministic variant of proportional probing Initialize: priorityu=0.0, u V At time t ▪ v= argmaxupriorityu ▪ Probe v and obtain N(v) ▪ Update G’t using G’t-1 and N(v) ▪ Output π’t, the PageRank vector G’t ▪ priorityv=0.0 ▪ priorityu= priorityu+π’t(u), u≠ v

19. Baseline probing methods Random probing ▪ Probe a node v chosen uniformly at random at each time step Round-robin probing ▪ Cycle through all nodes and probe each node in a round-robin manner We can vary the ratio of changes rate and probing rate

20. Experiments

21. Conclusion Evolving graphs is an interesting and useful model for dynamic and massive data By applying priority probing technique we can compute approximate result for PageRank Experiments confirm priority probing achieve better result than random and round-robin probing

22. Future Direction Can we improve accuracy of priority probing by using existing PageRank information Can we apply evolving graph model to other algorithm such as clustering

23. Reference BahmanBahmani, Ravi Kumar, Mohammad Mahdianand Eli Upfal. PageRank on an Evolving Graph. In KDD, pages 24-32, 2012.

24. Thank you ! Questions/Comments