Least cost rumor blocking in social networks

Least Cost Rumor Blocking in

Social networks

Lidan Fan

Computer Science Department

the University of Texas at Dallas

Social networks

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

Applications

Single cascade

Viral marketing

Recommender systems

Feed ranking

……

Multiple cascades

Political election

Multiple products promotion

Rumor/misinformation controlling

……

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.

Influence spreads fast within the same

community while slow across different

communities.

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.

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.

Problem SettingRumors 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.

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.

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

Outline

Model of influence diffusion






Conclusions

Future works

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.

Outline

Two influence diffusion models






Conclusions

Future works

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.

Example

1

3

4

5

26

1 is a rumor, 6 is a protector.

step 1: 1--2,3; 6--2,4. 2 and 4 is protected, 3 is rumored.

1

3

5

2

4

6

step 2: 4--5. 5 is protected.

Example

Outline







Conclusions

Future works

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.

Example

1

3

4

5

26

step 1:1--2, 6--2. 2 is protected.

1 is a rumor, 6 is a protector.

1

3

4

5

26

step 2:1--3, 6--2. 3 is rumored.

Example

1

3

4

5

26

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

Example

1

3

4

5

26

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

Example

Outline







Conclusions

Future works

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.

Outline







Conclusions

Future works

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 .

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

Construct Rumor Forward Search Trees (RFST)

67

5

1

34

2

8

9

10

11

12

14

Yellow nodes are bridge ends.

Rumor 4 Forward Search Tree

4

1 2 5

123

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.

67

5

1

34

2

8

9

10

11

12

14

Blue nodes are

protector candidates.

Bridge End Backward Search Trees

5

7 4

8 12

3 4

29 10 3

4

211

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

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}.

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).

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.

Hep:

community size 308,

bridge end size 387.

Email:

• community size 80,


• community size 2631,


Our algorithm performs the best,

especially in the third community.

Outline







Conclusions

Future works

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

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,

A

)(AvA

2: S

0A

BSvSBA \,

)()()()( BvBAvA

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.

Proof of Submodularity

Timestamp assignment of rumor diffusion

x y

u v

w z

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_x1_x

2_x

4_y4_x 2_y

3_x

Proof of SubmodularityProve 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.

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.

)(),(

))((

,

),(

APBgeneratedrandomlyareGpGrprob

APBEA

GpGr

GpGr

Experiments

Outline







Conclusions

Future works

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 email 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 email network

Outline







Conclusions

Future works

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.

Documents

Least cost rumor blocking in social networks