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 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.

Least cost rumor blocking in social networks

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 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.

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

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
 Opportunistic One Activate One (OPOAO)
Algorithm and experimental results under the DOAM
Algorithm and experimental results under the OPOAO
 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
2
6
1 is a rumor, 6 is a protector.
step 1: 1--2,3; 6--2,4. 2 and 4 is protected, 3 is rumored.

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

Outline
Opportunistic One Activate One (OPOAO)
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
2
6
1 is a rumor, 6 is a protector.
step 1:1--2, 6--2. 2 is protected.

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

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

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

Outline
Deterministic One Activate Many (DOAM)
 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
Deterministic One Activate Many (DOAM)
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)
7 6
5
1
3
4
2
8
9
10
11
12
14
Yellow nodes are bridge ends.

Rumor 4 Forward Search Tree
4
1 2 5
3 12
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.

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

Bridge End Backward Search Trees
7 4
5
3 4
4
11 2
9 10 3 2
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}

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.

 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
 
 A  properties: (to be proved)
 Non-negative:
 Monotone:
 Submodular:
 A 0
 A v (A)
 Let S be a finite set;
 A set function is submodular iff satisfies diminishing
  2 : S  
returns property. That is,
A B  S,vS B
 (A v)  (A)  (B  v)  (B)

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
1_x 3_x
u v
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
2_x
4_y
4_x 2_y
3_x

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.

Submodularity of function σ(A)
Fact: A non-negative linear combination of monotone and
submodular functions is still monotone and submodular.
 
A E PB A
( ( ) )


Gr Gp
prob Gr Gp are randomlygenerated PB A
 
( , ) ( )
( , )
Gr ,
Gp

Probabilities are non-negative;
|PB(A)| is submodular;
σ(A) is submodular.

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

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.

Least cost rumor blocking in social networks

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