Least Cost Rumor Blocking in Social networks

1. Least Cost Rumor Blocking in Social networks Lidan Fan Computer Science Department the University of Texas at Dallas

2. Social networks

3. Social Network Social network is a social structure made up of individuals and relations between these individuals Social network provides a platform for influence diffusion

4. Applications Single cascade • Viral marketing • Recommender systems • Feed ranking • …… Multiple cascades • Political election • Multiple products promotion • Rumor/misinformation controlling • ……

5. Social network properties • Small-world effect The average distance between vertices in a network is short. • Power-law or exponential form There are many nodes with low degree and a small number with high degree. • Clustering or network transitivity Two vertices that are both neighbors of the same third vertex have a high probability of also being neighbors of one another. • Community structure The connections within the same community are dense and between communities are sparse.

6. Influence spreads fast within the same community while slow across different communities.

8. It said that the president of Syria is dead, which hit the twitter greatly and was circulated fast among the population, leading to a sharp, quick increase in the price of oil.

9. In August, 2012, thousands of people in Ghazni province left their houses in the middle of the night in panic after the rumor of earthquake.

10. Control the spread of rumors

11. Problem Setting • Rumors generated in a community will influence the members in the network. • Find protectors to reduce the influence of rumors or protect the most members in the network. • Real-world limitation: the overhead spent on protectors and protected members should be balanced. • Rumors spread very fast within their community---too much cost • Rumors spread slow across different communities---little cost • Find least number of protectors to reduce rumor influence to the members in other communities.

12. Our Tasks • Determine influence diffusion models. • Design efficient algorithms to find protectors to reduce influence from rumors. • Obtain data of particular social networks to evaluate our algorithms.

13. Outline • Model of influence diffusion • Deterministic One Activate Many (DOAM) • Opportunistic One Activate One (OPOAO) • Least cost rumor blocking problem • Algorithm and experimental results under the DOAM • Algorithm and experimental results under the OPOAO • Conclusions • Future works

14. Outline • Model of influence diffusion • Deterministic One Activate Many (DOAM) • Opportunistic OneActivateOne (OPOAO) • Least cost rumor blocking problem • Algorithm and experimental results under the DOAM • Algorithm and experimental results under the OPOAO • Conclusions • Future works

15. Our Two Influence Diffusion Models • Two cascades: rumors and protectors; • Diffusion starts time: the same; • Tie breaking rule: protectors dominate rumors; • Status of each node: inactive, rumored, protected; • Monotonicity assumption: the status of rumored or protected never change.

16. Outline • Two influence diffusion models • Deterministic One Activate Many (DOAM) • Opportunistic One Activate One (OPOAO) • Least cost rumor blocking problem • Algorithm and experimental results under the DOAM • Algorithm and experimental results under the OPOAO • Conclusions • Future works

17. Deterministic One Activate Many Model • When a node becomes active (rumored or protected) , it has a single chance to activate all of its currently inactive (not rumored and not protected) neighbors. • The activation attempts succeed with a probability 1.

18. Example 6 2 1 5 3 4 1 is a rumor, 6 is a protector. step 1: 1--2,3; 6--2,4. 2 and 4 is protected, 3 is rumored.

19. Example 6 2 1 5 3 4 step 2: 4--5. 5 is protected.

20. Outline • Two influence diffusion models • Deterministic One Activate Many (DOAM) • Opportunistic One Activate One (OPOAO) • Least cost rumor blocking problem • Algorithm and experimental results under the DOAM • Algorithm and experimental results under the OPOAO • Conclusions • Future works

21. Opportunistic One Activate One Model • At each step, each active (rumored or protected) node u can only choose one of its neighbors as its target, and each neighbor can be chosen with a probability of 1/deg(u). • Each active (rumored or protected) node has unlimited chance to select the same node as its target.

22. 6 2 1 5 3 4 1 is a rumor, 6 is a protector. step 1:1--2, 6--2. 2 is protected. Example

23. Example 6 2 1 5 3 4 step 2:1--3, 6--2. 3 is rumored.

24. Example 6 2 1 5 3 4 step 3:1--2, 3--4, 6--4. 4 is protected.

25. Example 6 2 1 5 3 4 step 4:1--3, 3--2, 6--4, 4--5. 5 is protected.

26. Outline • Two influence diffusion models • Deterministic One Activate Many (DOAM) • Opportunistic One Activate One (OPOAO) • Least cost rumor blocking problem • Algorithm and experimental results under the DOAM • Algorithm and experimental results under the OPOAO • Conclusions • Future works

27. C0 Red node is a rumor; Yellow nodes are bridges ends. C2 C1 Least Cost Rumor Blocking Problem (LCRB) Bridge ends: • form a vertex set; • belong to neigborhood communities of rumor community; • each can be reached from the rumors before others in its community.

28. Outline • Two influence diffusion models • Deterministic One Activate Many (DOAM) • Opportunistic One Activate One (OPOAO) • Least cost rumor blocking problem • Algorithm and experimental results under the DOAM • Algorithm and experimental results under the OPOAO • Conclusions • Future works

29. LCRB-D problem for the DOAM model • Given the community structure and rumors with its community, find least number of protectors to protect all of the bridge ends .

30. Set Cover Based Greedy (SCBG) Algorithm Main idea • Convert to set cover problem using Breadth First Search (BFS) method. • Three stages: • construct Rumor Forward Search Trees(RFST)--bridge ends • construct Bridge End Backward Search Trees(BEBST)--protector candidates • construct vertex sets used in set cover problem

31. 7 6 9 5 8 3 10 4 11 2 1 Yellow nodes are bridge ends. 12 14 13 Construct Rumor Forward Search Trees(RFST)

32. 4 The minimal hops: 1 hop between 4 and 5; 2 hops between 4 and 12; 3 hops between 4 and 8. 5,8,12 are the bridge ends. 1 2 5 3 12 8 Rumor 4 Forward Search Tree

33. 7 6 9 5 8 3 10 4 11 2 1 Blue nodes are protector candidates. 12 14 13

34. Bridge End Backward Search Trees 4 4 3 7 4 11 2 9 2 10 3 5 8 12 • Record the protector candidate sets for each bridge end: 5: {5,7}; 8:{2,3,8,9,10,11}; 12:{2,3,12}

35. Construct vertex sets in set cover problem • Find the bridge ends that each candidate can protect: 2:{8,12}; 3:{8,12} ; 5:{5}; 7:{5}; 8:{8}; 9:{8}; 10:{8};11{8}; 12{12} Apply the Greedy algorithm • choose 2 or 3 , bridge ends 8 and 12 are protected; • choose 5 or 7, bridge end 5 is protected; • the output is {2,5} or {2,7} or {3,5} or {3,7}.

36. Theoretical Results • There is a polynomial time O(ln n)−approximation algorithm for the LCRB-D problem, where n is the number of vertices in the bridge end set. • If the LCRB-D problem has an approximation algorithm with ratio k(n) if and only if the set cover problem has an approximation algorithm with ratio k(n).

37. Experiments • Two Social networks • Collaboration Network: is from the e-print arXiv and covers scientific collaborations between authors with papers submitted to High Energy • Physics. If an author i co-authored a paper with author j, then the graph • contains an undirected directed edge between i to j,7.73 average degree. • Email Network: covers all the email communications within a dataset of around half million emails. Nodes of the network are email addresses and • if an address i sends at least one email to address j, a directed edge from i • to j is added in the graph, 10.0 average degree.

38. Hep: • community size 308, • bridge end size 387. • Email: • community size 80, • bridge end size 135. • community size 2631, • bridge end size 2250. Our algorithm performs the best, especially in the third community.

39. Outline • Two influence diffusion models • Deterministic One Activate Many (DOAM) • Opportunistic One Activate One (OPOAO) • Least cost rumor blocking problem • Algorithm and experimental results under the DOAM • Algorithm and experimental results under the OPOAO • Conclusions • Future works

40. LCRB-P problem for OPOAO model • Given the community structure and rumors with its community, find least number of protectors to protectα fraction of the bridge ends, where 0 <α <=1. • Influence function σ(A) of node set A: • expectednumber of nodes that would be rumored if set A is not selected as the protector seed initially

41. Results • properties: (to be proved) • Non-negative: • Monotone: • Submodular: • Let S be a finite set; • A set function is submodular iff satisfies diminishing returns property. That is,

42. The Greedy Algorithm • Start with an empty set A; • While the number of protected bridge ends has not reach α fraction of the number of all the bridge ends: • Add node v to S such that σ(A+v)-σ(A) is maximized.

43. x y x y 1_y 2_y 3_y 4_y 1_x 2_x 4_x 3_x 1_x 3_x 1_y 3_y u v 3_y u v 2_x 3_x 4_x 4_y 2_y 4_y 2_x 4_y 4_x 4_x 2_y 3_x 3_x w z z w Proof of Submodularity • Timestamp assignment of rumor diffusion

44. Proof of Submodularity • Prove the submodularity of cardinality function |PB(A)| • PB(A): the protector blocking set on bridge ends, in which individuals will be rumored if the protector seed set is empty but is not rumored if the protector seed set is A. • Rumor/protector random diffusion graphs-Gr/Gp. • Find the oldest (smallest) timestamp among the incoming edges of each bridge end u in Gr and Gp, and compare them, if the oldest one in Gp is older than the one in Gr, then u can be protected, otherwise, it will be rumored.

45. Submodularity of function σ(A) • Fact: A non-negative linear combination of monotone and submodular functions is still monotone and submodular. • Probabilities are non-negative; • |PB(A)| is submodular; • σ(A) is submodular.

46. Experiments

47. Outline • Two influence diffusion models • Deterministic One Activate Many (DOAM) • Opportunistic One Activate One (OPOAO) • Least cost rumor blocking problem • Algorithm and experimental results under the DOAM • Algorithm and experimental results under the OPOAO • Conclusions • Future works

48. Conclusions • Introduce two influence diffusion models • Deterministic One Activate Many --DOAM • Opportunistic One Activate One--OPOAO • The least cost rumor blocking (LCRB) problem in those two models • LCRB-D problem under the DOAM—protectallthe bridge ends • Design set cover based greedy algorithm (SCBG) • Run experiments over collaboration network and email network • LCRB-P problem under the OPOAO—protectα fraction of the bridge ends • Prove the submodularity of influence function σ(A); using timestamp assaignment strategy • Design greedy algorithm • Run experiments over collaboration network and email network

49. Outline • Two influence diffusion models • Deterministic One Activate Many (DOAM) • Opportunistic One Activate One (OPOAO) • Least cost rumor blocking problem • Algorithm and experimental results under the DOAM • Algorithm and experimental results under the OPOAO • Conclusions • Future works

50. Future Works • The greedy algorithm in the OPOAO model is time consuming, explore efficient algorithms for the LCRB-P problem. • Time is an important factor in rumor diffusion, consider the rumor blocking problem with time constraint. • It is hard to locate rumor sources, find algorithms to estimate rumor sources to control rumor diffusion efficiently.