1. StaticGreedy: Solving the Scalability-Accuracy Dilemma in Influence Maximization
Suqi Cheng
Research Center of Web Data Sciences & Engineering
Institute of Computing Technology, Chinese Academy of Sciences
Authors: Suqi Cheng, Huawei Shen, Junming Huang, Guoqing Zhang, Xueqi Cheng

2. Outline Background Preliminaries Motivation StaticGreedy algorithm
Experiments

3. Information Cascade
An action or idea are adopted one by one due to social influence
cascade through social relationships
Main Applications
Word-of-Mouth marketing
Out-break detection
Popularity prediction
social network

4. Word-of-Mouth Marketing
To promote a product by seeding a few users; users adopting the product will recommend it
Advantages: efficient; cost-effective
Company
seed users
follow-up activated users
How to select the optimal seed users?
free product/
discount
influence

5. Influence Maximization for Viral Marketing
Objective function
Influence spread I(S) : expected number of activated (influenced/adpoted) nodes
Maximize I(S)
Input:
A social influence graph G=(V, E)
An information cascade model
An integer k, |S| ≤ k
Output: A seed set S

6. Information Cascade Model
Independent cascade (IC) model
each edge (u, v) has a propagation probability p(u, v)
each newly activated node u independently activates its out-neighbor v with probability p(u, v)
a discrete time model
Influence spread estimation on IC model
Monte Carlo simulation
Heuristic methods
0.2
0.1
0.1
0.3
0.1
0.5
0.2
0.5
0.1
0.4
0.3
0.4
0.4
0.2
0.1
Social influence graph
[Leskovec, 2008]

7. Difficulties in Influence Maximization
Difficulty 1: Influence maximization problem is NP-hard.[kempe, KDD’03]
Existing solutions
Heuristics
Degree
Pagerank
Betweennes
efficient
inaccurate
Greedy approximate algorithm [Kempe, KDD’03]
(1-1/e-ε)-approximation
iteratively select nodes with largest marginal influence spread
guaranteed by submodularity and montonicity properties of influence spread function
accurate
inefficient

8. Difficulties in Influence Maximization
Difficulty 2: To exactly compute influence spread is #P-hard. [Chen, KDD’10]
Monte-Carlo simulation
CELF optimization[Leskovec,KDD’07]
NewGreedy[Chen, KDD’09]
CELF++ optimization[Goyal,WWW’11]
accurate
time-consuming
Heuristic methods
DegreeDiscount[Chen, KDD’09]
CGA[Wang, KDD‘10]
PMIA[Chen,KDD’10]
IRIE[Jung, ICDM’12]
efficient
inaccurate
Existing solutions
A scalability-accuracy delimma!

9. Our works
Objective : to propose an influence maximization algorithm to solve the scalability-accuracy dilemma
Algorithm
Accuracy
Scalability
Approximate algorithms
Greedy
[Kempe, KDD’03]
gurannteed
low
CreedyCELF
[Leskovec, KDD’07]
GreedyCELF++
[Goyal, WWW’11]
NewGreedy
/MixedGreedy
[Chen, KDD’09]
StaticGreedy
[cheng, CIKM’13]
high
Heuristics
Degree
ungurannteed
PageRank
[Page, 1999]
DegreeDiscount
PMIA
[Chen, KDD’10]
IRIE
[Jung, ICDM’12]
SP1M
[Kimura, PKDD’06]
relatively low

10. Preliminaries-1 Social influence graph: G=(V, E), n=|V|, m=|E|
Influence spread: I(S)
Marginal influence spread: M(v|S)=I(S{v}) - I(S)
Properties of I(S) under independent cascade model
submodularity:
I(S{v}) - I(S)  I(T{v}) - I(S) iff vV, S  T  V
monotonicity: I(S{v})  I(S)
guarantee
Greedy approximate algorithm
iteratively select nodes with the largest marginal influence spread
provide 1-1/e-ε approximation
Influence spread estimation

11. Preliminaries-2 Monte Carlo simulation for influence spread estimation
to approximate true values of influence spread by realizations
method
An instance
Advantage
Disadvantage
simulation
modeling the information cascade process
relatively low time complexity
estimate one seed set at a time
snapshot
[Chen, KDD’09]
removing each edge (u, v) from G with probability 1-p(u, v)
can estimate any seed set simultaneously
relatively high time complexity
equivalent

12. R: number of Monte Carlo simulations for estimation
Motivation
In existing greedy algorithms
a risk of unguaranteed submodularity and monotonicity of influence spread function
caused by using different results of Monte Carlo simulation across different influence spread estimation
a very large value of R is required, e.g. R=20000
R: number of Monte Carlo simulations for estimation
iteration 1
iteration 2
Submodularity is breaked!
influence graph
snapshot1
snapshot 2

13. StaticGreedy algorithm
Core idea: to always use the same snapshots for influence spread estimation
influence spread function is submodular and monotone
a small value of R is required, e.g. R=100
Part1: Generate R static snapshots
Part 2: Greedy selection

14. Performance analysis: Convergence rate
provide (1-1/e-ε)-approximation with a small value of R
seed set size = 50
dR,k
log R
NetHEPT: a benchmark network
uniform independent cascade (UIC) model: p(u, v) = p = 0.01
weighted independent cascade (WIC) model: p(u, v) = 1/(# of in-neighbors of v)

15. Performance analysis: Scalability
Minimal R required
Running time
≈102 times
≈103 times
log Rmin
log running time (sec)
seed set size
seed set size
R is significantly reduced
Running time is significantly reduced

16. Performance analysis: Complexity
n: number of nodes in social influence graph
m: number of edges in social influence graph
m’: expected number of edges in a snapshot

17. Speed up StaticGreedy A dynamic update strategy
calculates the marginal gain in an efficient incremental manner
at each step t, for each snapshot: M(v)  M(v) - |R(v)R(vt*)|, R(v)  R(v) - R(v)R(vt*)
trades space for time
R(v): reachable nodes from v in the snapshot
initial v1 v1 v2
M(v1)=4
M(v2)=3
M(v3)=2
M(v4)=1
M(v5)=1
M(v6)=1
M(v7)=2
M(v8)=1 v3 v4 v5 v6 v7 v8
snapshot

18. Speed up StaticGreedy X X X X A dynamic update strategy
calculates the marginal gain in an efficient incremental manner
at each step t, for each snapshot: M(v)  M(v) - |R(v)R(vt*)|, R(v)  R(v) - R(v)R(vt*)
trades space for time
R(v): reachable nodes from v in the snapshot
after select v* = v1 X -4 v1 v1 v2 -1
M(v1)=4
M(v2)=3
M(v3)=2
M(v4)=1
M(v5)=1
M(v6)=1
M(v7)=2
M(v8)=1
M(v1)=0
M(v2)=2
M(v3)=0
M(v4)=0
M(v5)=1
M(v6)=0
M(v7)=2
M(v8)=1 X X -2 v3 -1 v4 v5
directly
update X -1 v6 v7 v8
snapshot

19. Experiments: setup Algorithms: Tested datasets
Our algorithms: StaticGreedyCELF, StaticGreedyDU
Baselines: CELFGreedy, SP1M, PMIA, Degree, DegreeDiscount
Tested datasets
Independent cascade models
uniform independent cascade(UIC) model: p(u, v) = p = 0.01
weighted independent cascade(WIC) model: p(u, v) = 1/(# of in-neighbors of v)
Metrics: Influence spread, running time

20. Experiments: influence spread
StaticGreedy achieves better accuracy than other heuristics
NetPHY
UIC model
WIC model
DBLP
UIC model
WIC model

21. Experiments: running time
StaticGreedy runs >103 times faster than CELFGreedy
StaticGreedy has comparable scalability to state-of-the-art heuristics
StaticGreedyDU always runs faster than StaticGreedyCELF
log running time (sec)
UIC model WIC model

22. conclusion
Essential reason of the inefficiency of existing greedy algorithms
a risk of unguaranteed submodularity and monotonicity
caused by different Monte Carlo simulations across different estimations
a very large value of R is required  guaranteed accuracy + inefficiency
StaticGreedy algorithm
guaranteed submodularity and monotonicity
using the same Monte Carlo simulations across different estimations
a small value of R is required  guaranteed accuracy + high scalability
runs >103 times quicker than conventional greedy algorithms
A dynamic update strategy to speed up StaticGreedy
about 10 times faster

23. Thank you!
Q & A