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

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Social networks

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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

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

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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 structureCommunity structure The connections within the same community are dense and between communities are sparse.

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

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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.

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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.

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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.

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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.

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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

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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

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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.

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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

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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.

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

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step 2: is protected. Example

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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

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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.

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Example step 1:1--2, is protected. 1 is a rumor, 6 is a protector.

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step 2:1--3, is rumored. Example

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

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

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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

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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. C0 C2 C1 Red node is a rumor; Yellow nodes are bridges ends.

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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

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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.

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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

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Construct Rumor Forward Search Trees (RFST) Yellow nodes are bridge ends.

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Rumor 4 Forward Search Tree 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.

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Blue nodes are protector candidates.

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

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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}.

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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).

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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. Network: covers all the communications within a dataset of around half million s. Nodes of the network are addresses and if an address i sends at least one to address j, a directed edge from i to j is added in the graph, 10.0 average degree.

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

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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

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

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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,

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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.

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Proof of Submodularity Timestamp assignment of rumor diffusion x y u v wz x y vu w z 1_x 2_x 4_x 3_x 1_y 2_y 3_y 4_y 2_y 4_y 3_y 2_x 3_x 4_x 4_y 4_x 3_x 1_y 3_y 3_x 1_x 2_x 4_y 4_x 2_y 3_x

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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.

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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.

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Experiments

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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

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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—protect all the bridge ends Design set cover based greedy algorithm (SCBG) Run experiments over collaboration network and network LCRB-P problem under the OPOAO—protect α 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 network

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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

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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.

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