Gathering Spheres of Influence As US influence spread our appetite for more respect grew.
Maximizing the Spread of Influence through a Social Network
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Maximizing the Spread of Influence through a Social Network
Authors: David Kempe, Jon Kleinberg, Éva TardosKDD 2003
Source:http://www.cs.cmu.edu/~xiaonanz/Maximizing-the-Spread-of-Influence.ppt
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Social Network and Spread of Influence 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 (Gmail, Tupperware popularization, Microsoft Origami …)
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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
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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.
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Outline Models of influence
Linear Threshold Independent Cascade
Influence maximization problem Algorithm Proof of performance bound Compute objective function
Experiments Data and setting Results
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Outline Models of influence
Linear Threshold Independent Cascade
Influence maximization problem Algorithm Proof of performance bound Compute objective function
Experiments Data and setting Results
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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
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Outline Models of influence
Linear Threshold Independent Cascade
Influence maximization problem Algorithm Proof of performance bound Compute objective function
Experiments Data and setting Results
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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
A node v becomes active when at least (weighted) θv fraction of its neighbors are active
, active neighbor of
v w vw v
b
, neighbor of
1v ww v
b
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Example
Inactive Node
Active Node
Threshold
Active neighbors
vw 0.5
0.30.20.5
0.10.4
0.3 0.2
0.6
0.2
Stop!
U
X
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Outline Models of influence
Linear Threshold Independent Cascade
Influence maximization problem Algorithm Proof of performance bound Compute objective function
Experiments Data and setting Results
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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 .
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Example
vw 0.5
0.3 0.20.5
0.10.4
0.3 0.2
0.6
0.2Inactive Node
Active Node
Newly active node
Successful attemptUnsuccessfulattempt
Stop!
UX
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Outline Models of influence
Linear Threshold Independent Cascade
Influence maximization problem Algorithm Proof of performance bound Compute objective function
Experiments Data and setting Results
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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
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Outline Models of influence
Linear Threshold Independent Cascade
Influence maximization problem Algorithm Proof of performance bound Compute objective function
Experiments Data and setting Results
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f(S): properties (to be demonstrated)
Non-negative (obviously) Monotone: Submodular:
Let N be a finite set A set function is submodular iff
(diminishing returns)
: 2Nf , \ ,
( ) ( ) ( ) ( )S T N v N T
f S v f S f T v f T
( ) ( )f S v f S
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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.
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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.
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Outline Models of influence
Linear Threshold Independent Cascade
Influence maximization problem Algorithm Proof of performance bound Compute objective function
Experiments Data and setting Results
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Key 1: Prove submodularity
, \ ,( ) ( ) ( ) ( )S T N v N T
f S v f S f T v f T
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Submodularity for Independent Cascade Coins for edges are
flipped during activation attempts.
0.5
0.30.5
0.10.4
0.3 0.2
0.6
0.2
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Submodularity for Independent Cascade Coins for edges are
flipped during activation attempts.
Can pre-flip all coins and reveal results immediately. 0.5
0.30.5
0.10.4
0.3 0.2
0.6
0.2
Active nodes in the end are reachable via green paths from initially targeted nodes.
Study reachability in green graphs
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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.
V
S
T
g(S)
g(T)
g(v)
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!).
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Submodularity of the Function
gG(S): nodes reachable from S in G. Each gG(S): is submodular (previous slide). Probabilities are non-negative.
Fact: A non-negative linear combination of submodular functions is submodular
( ) Prob( ) ( )GG
f S G is green graph g S
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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 proof).
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Outline Models of influence
Linear Threshold Independent Cascade
Influence maximization problem Algorithm Proof of performance bound Compute objective function
Experiments Data and setting Results
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Key 2: Evaluating f(S)
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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.
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Outline Models of influence
Linear Threshold Independent Cascade
Influence maximization problem Algorithm Proof of performance bound Compute objective function
Experiments Data and setting Results
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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
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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.
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Outline Models of influence
Linear Threshold Independent Cascade
Influence maximization problem Algorithm Proof of performance bound Compute objective function
Experiments Data and setting Results
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Results: linear threshold model
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Independent Cascade Model – Case 2 Reminder: linear
threshold model
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More in the Paper A broader framework that simultaneously
generalizes the two models Non-progressive process: active nodes CAN
deactivate. More realistic marketing:
different marketing actions increase likelihoodof initial activation, for several nodes at once.
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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.
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Thanks!
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Influence Maximization When Negative Opinions may Emerge and Propagate
Authors: Wei Chan and lot othersMicrosoft Research Technical Report 2010
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Outline Model of influence f(S) Sub-modular Other models
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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. q is the product quality factor.
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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.
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Towards sub-modularity Generate a deterministic graph G’ from G by
flipping coin (biased according to the edge influence probability) for each edge.
PAPv = Positive activation probability of 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’)
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Other models Different quality factor for every node Stronger negative influence probability Different propagation delays
None of them are sub-modular in nature!
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