Spread influence on social networks

Detection of Opinion
Leaders and Spread of
Influence on Social
Networks
Armando Vieira

1

 Social network plays a
fundamental role as a medium
for the spread of INFLUENCE
among its members
 Opinions, ideas, information,
innovation…
 Direct Marketing takes the “word-of-
mouth” effects to significantly increase
profits (Facebook, Twitter, Youtube …)

2

Problem Setting

 Given
 a limited budget B for initial advertising (e.g. give away free
samples of product)
 estimates for influence between individuals
 Goal
 trigger a large cascade of influence (e.g. further adoptions
of a product)
 Question
 Which set of individuals should B target at?
 Application besides product marketing
 spread an innovation
 detect stories in blogs

3

What we need?

 Form models of influence in social networks.
 Obtain data about particular network (to
estimate inter-personal influence).
 Devise algorithm to maximize spread of
influence.

4

Outline

 Models of influence
 Linear Threshold
 Independent Cascade
 Influence maximization problem
 Algorithm
 Proof of performance bound
 Compute objective function
 Experiments
 Data and setting
 Results

5

Outline

 Algorithm
 Experiments
 Results

6

Models of Influence
 First mathematical models
 [Schelling '70/'78, Granovetter '78]
 Large body of subsequent work:
 [Rogers '95, Valente '95, Wasserman/Faust '94]
 Two basic classes of diffusion models: threshold
and cascade
 General operational view:
 A social network is represented as a directed graph, with
each person (customer) as a node
 Nodes start either active or inactive
 An active node may trigger activation of neighboring
nodes
 Monotonicity assumption: active nodes never deactivate

7

Outline

 Algorithm
 Experiments
 Results

8

Linear Threshold Model
 A node v has random threshold θv ~ U[0,1]
 A node v is influenced by each neighbor w according
to a weight bvw such that

∑
w neighbor of v
bv ,w ≤ 1
 A node v becomes active when at least
(weighted) θv fraction of its neighbors are active

∑
w active neighbor of v
bv ,w ≥ θ v

9

Example

Inactive Node
0.6
Active Node

0.3 0.2 0.2 Threshold

X Active neighbors
0.1
0.4 U

0.5 0.3
0.2 Stop!
0.5
w v

10

Outline

 Algorithm
 Experiments
 Results

11

Independent Cascade Model

 When node v becomes active, it has a single
chance of activating each currently inactive
neighbor w.
 The activation attempt succeeds with
probability pvw .

12

Example

0.6
Inactive Node

0.3 0.2 0.2
Active Node

Newly active
X 0.1 U
0.4 node
Successful
0.5 0.3 attempt
0.2
Unsuccessful
0.5 attempt
w
v

Stop!
13

Outline

 Algorithm
 Experiments
 Results

14

Influence Maximization
Problem
 Influence of node set S: f(S)
 expected number of active nodes at the end, if set
S is the initial active set
 Problem:
 Given a parameter k (budget), find a k-node set S
to maximize f(S)
 Constrained optimization problem with f(S) as the
objective function

15

Outline

 Algorithm
 Experiments
 Results

16

f(S): properties (to be demonstrated)

 Non-negative (obviously)
 Monotone: f (S + v) ≥ f (S )
 Submodular:

 LetN be a finite set
 A set function
f : 2 N a ℜ is submodular iff
∀S ⊂ T ⊂ N , ∀v ∈ N T ,
f ( S + v ) − f ( S ) ≥ f (T + v ) − f (T )
(diminishing returns)

17

Bad News

 For a submodular function f, if f only takes non-
negative value, and is monotone, finding a k-
element set S for which f(S) is maximized is an
NP-hard optimization problem[GFN77, NWF78].
 It is NP-hard to determine the optimum for
influence maximization for both independent
cascade model and linear threshold model.

18

Good News

 We can use Greedy Algorithm!
 Start with an empty set S
 For k iterations:
Add node v to S that maximizes f(S +v) - f(S).
 How good (bad) it is?
 Theorem: The greedy algorithm is a (1 – 1/e)
approximation.
 The resulting set S activates at least (1- 1/e) >
63% of the number of nodes that any size-k set S
could activate.
19

Outline

 Algorithm
 Experiments
 Results
20

Key 1: Prove submodularity

∀S ⊂ T ⊂ N , ∀v ∈ N T ,
f ( S + v ) − f ( S ) ≥ f (T + v ) − f (T )

21

Submodularity for Independent Cascade
0.6

 Coins for edges are
0.2 0.2
flipped during 0.3

activation attempts.
0.1
0.4

0.5 0.3

0.5

22

Submodularity for Independent Cascade
0.6

 Coins for edges are
0.2 0.2
flipped during 0.3

activation attempts.
0.1
 Can pre-flip all 0.4

coins and reveal 0.5 0.3

results immediately. 0.5
 Active nodes in the end are reachable via
green paths from initially targeted nodes.
 Study reachability in green graphs
23

Submodularity, Fixed Graph

 Fix “green graph” G.
g(S) are nodes
reachable from S in G.

⊆
Submodularity: g(T +v)
- g(T) ⊆g(S +v) - g(S)
when S T.
 g(S +v) - g(S): nodes reachable from S + v, but not from
S.
 From the picture: g(T +v) - g(T) ⊆ g(S +v) - g(S) when S
⊆ T (indeed!).

24

Submodularity of the Function

Fact: A non-negative linear
combination of submodular
functions is submodular

f ( S ) = ∑ Prob(G is green graph ) ×g G ( S )
G

 gG(S): nodes reachable from S in G.
 Each gG(S): is submodular (previous slide).
 Probabilities are non-negative.

25

Submodularity for Linear
Threshold
 Use similar “green graph” idea.
 Once a graph is fixed, “reachability” argument
is identical.
 How do we fix a green graph now?

 Assume every time nodes pick their
threshold uniformly from [0-1].
 Each node picks at most one incoming edge,
with probabilities proportional to edge
weights.
 Equivalent to linear threshold model (trickier
26

Outline

 Algorithm
 Experiments
 Results
27

Key 2: Evaluating f(S)

28

Evaluating ƒ(S)

 How to evaluate ƒ(S)?
 Still an open question of how to compute
efficiently
 But: very good estimates by simulation
 Generate green graph G’ often enough
(polynomial in n; 1/ε). Apply Greedy algorithm to
G’.
 Achieve (1± ε)-approximation to f(S).
 Generalization of Nemhauser/Wolsey proof
shows: Greedy algorithm is now a (1-1/e- ε′)-
approximation. 29

Outline

 Algorithm
 Experiments
 Results

30

Experiment Data

A collaboration graph obtained from co-
authorships in papers of the arXiv high-
energy physics theory section
 co-authorship networks arguably capture
many of the key features of social networks
more generally
 Resulting graph: 10748 nodes, 53000 distinct
edges

31

Experiment Settings
 Linear Threshold Model: multiplicity of edges as
weights
 weight(v→ω) = Cvw / dv, weight(ω→v) = Cwv / dw
 Independent Cascade Model:
 Case 1: uniform probabilities p on each edge
 Case 2: edge from v to ω has probability 1/ dω of
activating ω.
 Simulate the process 10000 times for each targeted
set, re-choosing thresholds or edge outcomes
pseudo-randomly from [0, 1] every time
 Compare with other 3 common heuristics
 (in)degree centrality, distance centrality, random nodes. 32

Outline

 Algorithm
 Experiments
 Results
33

Results: linear threshold
model

34

Independent Cascade Model –
Case 2 Reminder: linear
threshold model

35

Open Questions

 Study more general influence models. Find
trade-offs between generality and feasibility.
 Deal with negative influences.

 Model competing ideas.

 Obtain more data about how activations
occur
in real social networks.

36

Influence Maximization
When Negative Opinions
may Emerge and
Propagate
Authors: Wei Chan and lot others
Microsoft Research Technical Report 2010

38

Model of Influence

 Similar to independent cascade model.
 When node v becomes active, it has a single
chance of activating each currently inactive
neighbor w.
 The activation attempt succeeds with probability
pvw .
 If node v is negative then node w also
becomes negative.
 If node v is positive then node w becomes
positive with probability q else becomes
negative.
39

Model of Influence

 Intuition:
 Negative opinions originate from imperfect
product/service quality.
 Negative node generates the negative opinions.
 Positive and Negative opinions are asymmetric
 Negative opinions are generally much stronger.

40

Towards sub-modularity

 Generate a deterministic graph G’ from G by
flipping coin (biased according to the edge
influence probability) for each edge.
 PAP = Positive activation probability of v.
v

= q^{shortest path from S to v + 1}
 F(S, G’) = E(# positively activated nodes)

= sum(PAPv)
 F(S,G’)is monotone and submodular.
 F(S,G) = E(F(S,G’) over all possible G’)
41

Other models

 Different quality factor for every node
 Stronger negative influence probability

 Different propagation delays

None of them are sub-modular in nature!

42

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