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

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

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

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

Social networks 5/11/20154

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

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

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

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

9

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

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. leads-sharp-increase-price-oil html 5/11/201511

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. e-rumour-sends-thousands-ghazni-streets 5/11/201512

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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

 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

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

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

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

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

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

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

Example 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

Step 2: is protected. Example 5/11/201525

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

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

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

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

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

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

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

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

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

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

Construct Rumor Forward Search Trees (RFST) Yellow nodes are bridge ends. 5/11/201536

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. 5/11/201537

Blue nodes are protector candidates. 5/11/ Construct Bridge End Backward Search Trees (BEBST)

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} 5/11/201539

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

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

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’.

Example of Submodular Function

Greedy Algorithm

Analysis

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’.

A General Problem

Greedy Algorithm

A General Theorem Remark:

Proof

12 3

z ek z e1 Ze2Ze2

Proof can be found in 61

Experiments  Two datasets Collaboration Network ( ): 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 Network ( ): Covers all the communications within a dataset of around half million s. Nodes: addresses Edge (i, j): Address i sends at least one to address j 5/11/201562

5/11/ DatasetsHEP-PHEnron- # of nodes # of edges Average degree # 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

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/ Experimental Results SCBGProximityMaxDegree Hep/15233/308 1% % % /36692/80 5% % % /36692/2631 1% % %

5/11/ Experiments

5/11/ 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

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/  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

Main Results 5/11/201570

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/ x.t: the influence spread of rumor x arrive a node at step t

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

Example Determine whether u is protected or infected u 5/11/ 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.

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

Greedy Algorithm G

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

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

Note that There exists i such that

Let Note that So

Note Hence,

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Experiments 5/11/201586

5/11/ 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

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

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 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 network. Our algorithm: robust and scalable 5/11/201589

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

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.

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.

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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