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Ding-Zhu Du │ University of Texas at Dallas │ Lecture 7 Rumor Blocking 0

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Least Cost Rumor Blocking in Social networks Lidan Fan, Zaixin Lu, Weili Wu, Bhavani Thuraisingham, Huan Ma, Yuanjun Bi. Published in ICDCS2013 5/11/2015

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Outline Background Motivation Problem formulation Related Works Our Contribution Two influence diffusion models Least cost rumor blocking problem Conclusions Future Works 5/11/20152

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Outline Background Motivation Problem formulation Related Works Our Contribution Two influence diffusion models Least cost rumor blocking problem Conclusions Future Works 5/11/20153

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Social networks 5/11/20154

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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 5/11/20155

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Applications Single cascade Viral marketing Recommender systems Feed ranking …… Multiple cascades Political election Multiple products promotion Rumor/misinformation controlling …… 5/11/20156

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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 structure The connections within the same community are dense and between communities are sparse. 5/11/20157

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Influence spreads fast within the same community. Influence spreads slow across different communities. 5/11/20158

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10 When misinformation or rumor spreads in social networks, what will happen?

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A misinformation said that the president of Syria is dead, and it hit the twitter greatly and was circulated fast among the population, leading to a sharp, quick increase in the price of oil. http://news.yahoo.com/blogs/technology-blog/twitter-rumor- leads-sharp-increase-price-oil-173027289.html 5/11/201511

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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. http://www.pajhwok.com/en/2012/08/20/quak e-rumour-sends-thousands-ghazni-streets 5/11/201512

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5/11/201513

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Outline Background Motivation Problem formulation Related Works Our Contribution Two influence diffusion models Least cost rumor blocking problem Conclusions Future Works 5/11/201514

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Rumors generated in a community will influence the members in the network. Find protectors to reduce the influence of rumors. 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. 5/11/201515

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Our Tasks Determine influence diffusion models. Design efficient algorithms to find protectors. Obtain real world data to evaluate our algorithms. 5/11/201516

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Outline Background Motivation Problem formulation Related Works Our Contribution Two influence diffusion models Least cost rumor blocking problem Conclusions Future Works 5/11/201517

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5/11/201518

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Outline Background Motivation Problem formulation Related Works Our Contribution Two influence diffusion models Least cost rumor blocking problem Conclusions Future Works 5/11/201519

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Deterministic One Activate Many (DOAM) Opportunistic One Activate One (OPOAO) 5/11/201520

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Common properties Two cascades: rumor and protector; Diffusion starts time: the same; Tie breaking rule: protector has priority over rumor; Status of each node: inactive, infected, protected; Monotonicity assumption: the status of infected or protected never changes. 5/11/201521

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Deterministic One Activate Many (DOAM) 5/11/201522

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Additional properties of the DOAM model When a node becomes active (infected or protected), it has a single chance to activate all of its currently inactive (not infected and not protected) neighbors. The activation attempts succeed with a probability 1. 5/11/201523

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Example 1 3 4 5 2 6 1 is a rumor, 6 is a protector. Step 1: 1--2,3; 6--2,4. 2 and 4 are protected, 3 is infected. 5/11/201524

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1 3 5 2 4 6 Step 2: 4--5. 5 is protected. Example 5/11/201525

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Opportunistic One Activate One (OPOAO) 5/11/201526

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Additional properties of the OPOAO model At each step, each active (infected or protected) node u can only choose one of its neighbors as its target, and each neighbor is chosen with a probability of 1/deg(u). Each active (infected or protected) node has unlimited chance to select the same node as its target. 5/11/201527

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Example 1 3 4 5 2 6 Step 1:1--2, 6--2. 2 is protected. 1 is a rumor, 6 is a protector. 5/11/201528

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1 3 4 5 2 6 Step 2:1--3, 6--2. 3 is infected. Example 5/11/201529

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1 3 4 5 2 6 Step 3:1--2, 3--4, 6--4. 4 is protected. Example 5/11/201530

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1 3 4 5 2 6 Step 4:1--3, 3--2, 6--4, 4--5. 5 is protected. Example 5/11/201531

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Outline Background Motivation Problem formulation Related Works Our Contribution Two influence diffusion models Least cost rumor blocking problem Conclusions Future Works 5/11/201532

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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 own community. C0 C2 C1 Red node is a rumor; Yellow nodes are bridge ends. 5/11/201533

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LCRB-D problem for the DOAM model Given: community structure rumors rumor community Goal: Find least number of protectors to protect all of the bridge ends. 5/11/201534

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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 5/11/201535

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Construct Rumor Forward Search Trees (RFST) 67 5 1 3 4 2 8 9 10 11 12 13 14 Yellow nodes are bridge ends. 5/11/201536

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Rumor 4 Forward Search Tree 4 1 2 5 12 3 8 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. 5/11/201537

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67 5 1 3 4 2 8 9 10 11 12 13 14 Blue nodes are protector candidates. 5/11/201538 Construct Bridge End Backward Search Trees (BEBST)

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Bridge End Backward Search Trees 5 7 4 8 12 3 4 2 9 103 4 2 11 Record the protector candidate sets for each bridge end: 5: {5,7}; 8:{2,3,8,9,10,11}; 12:{2,3,12} 5/11/201539

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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}. 5/11/201540

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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 set of bridge ends. 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). 5/11/201541

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Set-Cover Given a collection C of subsets of a set E, find a minimum subcollection C’ of C such that every element of E appears in a subset in C’.

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Example of Submodular Function

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Greedy Algorithm

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Analysis

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Weighted Set Cover Given a collection C of subsets of a set E and a weight function w on C, find a minimum total- weight subcollection C’ of C such that every element of E appears in a subset in C’.

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A General Problem

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Greedy Algorithm

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A General Theorem Remark:

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Proof

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12 3

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z ek z e1 Ze2Ze2

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Proof can be found in 61

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Experiments Two datasets Collaboration Network ( http://snap.stanford.edu/data/cit-HepPh.html ): http://snap.stanford.edu/data/cit-HepPh.html Covers scientific collaborations between authors with papers submitted to High Energy Physics. Nodes: Papers Edge (i,j): Author i co-authored a paper with author j Email Network ( http://snap.stanford.edu/data/email-Enron.html ): http://snap.stanford.edu/data/email-Enron.html Covers all the email communications within a dataset of around half million emails. Nodes: Email addresses Edge (i, j): Address i sends at least one email to address j 5/11/201562

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5/11/201563 DatasetsHEP-PHEnron-Email # of nodes1523336692 # of edges58891367662 Average degree7.7310 # of selected communities 12 Description of the communities chosen Size:308 Bridge end size: 387 Size: 80 Bridge end size:135 Size: 2631 Bridge end size: 2250

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Our algorithm performs the best. The third community, which is dense and has large number of nodes, shows that our algorithm is robust and scalable. 5/11/201564 Experimental Results SCBGProximityMaxDegree Hep/15233/308 1%32.925.3140.6 5%42.174.3147.8 10%48.9133.8152.6 Email/36692/80 5%6.243.772.7 10%8.246.979.3 20%13.862.991.1 Email/36692/2631 1%20.4289.31208.8 5%50.91067.61350.2 10%68.41422.61683.8

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5/11/201565 Experiments

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5/11/201566 Experiments Our algorithm performs in all figures except Fig7(a). the network is sparse, when the number of rumors is small, it is possible that Proximity performs better than ours Proximity is better than MaxDegree in Fig7and Fig8. number of rumors is small and network is sparse MaxDegree is better than Proximity in Fig9. number of rumor is large and network is dense

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Rumor Blocking problem under the OPOAO model Given: the community structure Rumor sources R rumor community number of protectors k Goal: Find k protectors such that the expected number of bridge ends protected is maximized. Influence function σ(A) of node set A: Expected number of nodes that would be infected if A is selected as the protector seeds initially. 5/11/201567

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Property of Submodularity 5/11/201569 Submodularity : PB(A): the set of nodes that can be protected by set A. PB(A+v)-PB(A): can be protected by A+v can not be protected by A A PB(v) B PB(A) PB(B) v

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Main Results 5/11/201570

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Proof of Submodularity Timestamp assignment of rumor diffusion x y u v wz x y vu w z x.1 x.2 x.4 x.3 y.1 y.2 y.3 y.4 y.2 y.4 y.3 x.2 x.3 x.4 y.4 x.4 x.3 y.1 y.3 x.3 x.1 x.2 y.4 x.4 y.2 x.3 5/11/2015 71 x.t: the influence spread of rumor x arrive a node at step t

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Proof of Submodularity Prove the submodularity of cardinality function |PB(A)| The nodes in PB(A) satisfies: infected if the set of protectors is empty not infected if the set of protectors is A Create rumor(protector) random diffusion graph-Gr(Gp). Among the incoming edges of bridge end u in Gr and Gp: find the oldest timestamp in Gr and Gp respectively compare them if the oldest one in Gp is older than the one in Gr then u can be protected otherwise then u will be infected 5/11/201572

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Example Determine whether u is protected or infected u 5/11/2015 73 p w r r.1 r.2 r.3 p.1 p.2 p.3 Graph G r: rumor p: protector u w r r.1 r.3 u p w p.1 p.3 Random protector diffusion graph Gp Random rumor diffusion graph Gr Since p.1 is older than r.3, then u is protected.

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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. 5/11/201574

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A general result on greedy algorithm With non-integer potential function Consider a monotone increasing, submodular function Consider the following problem: whereis a nonnegative cost function

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Greedy Algorithm G

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Theorem Suppose in Greedy Algorithm G, selected x always satisfies Then its p.r. where

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Proof. Letbe obtained by Greedy Algorithm G. Denote Let be an optimal solution. Denote

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Note that There exists i such that

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Let Note that So

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Note Hence,

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Experiments 5/11/201586

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5/11/201587 Experiments In Fig4, Fig5 and Fig6, our algorithm performs the best except in several early hops. number of rumors is small Proximity is better than MaxDegree in Fig4, Fig5 and Fig6. stochastic selection mechanism The difference between Proximity and MaxDegree in Fig4 is larger than that in Fig5 and Fig6. network in Fig4 is sparse

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Outline Background Motivation Problem formulation Related Works Our Contribution Two influence diffusion models Least cost rumor blocking problem Conclusions Future Works 5/11/201588

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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 two models LCRB-D problem under the DOAM—protect all the bridge ends Algorithm: Set Cover Based Greedy (SCBG) Data: collaboration network and email network Our algorithm: robust and scalable. LCRB-P problem under the OPOAO—protect α fraction of the bridge ends Influence function σ(A): submodularity Method: timestamp assignment Algorithm: Greedy Data: collaboration network and email network. Our algorithm: robust and scalable 5/11/201589

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Outline Background Motivation Problem formulation Related Works Our Contribution Two influence diffusion models Least cost rumor blocking problem Conclusions Future Works 5/11/201590

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2015-5-1191 Future Works Establish continuous time influence propagation model In real world, under most situations, influence diffuses in continuous time. Measure the diffusion time based on factors such as individual attributes, information properties, strength of relations, etc.

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2015-5-1192 Future Works Study rumor blocking and influence diffusion under dynamic social structures Under most cases, the relations between individuals change along with time, that is, social structures change along with time, what results can we get for rumor blocking and influence diffusion in dynamic situation.

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2015-5-11 93 Future Works Detect rumor sources Previous works in controlling rumor diffusion assume that rumor sources are known. However, in reality, it is hard to know the accurate rumor sources. Estimate rumor sources accurately using existing information.

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