1. Efficient Influence Maximization in Large-scale Social Networks
计算社会科学与社会计算前沿研讨会
Efficient Influence Maximization in Large-scale Social Networks
Chuan Zhou (周川)
Institute of Information Engineering, CAS
August 27, 2016

2. Efficient Influence Maximization in Large-scale Social Networks1
Introduction
Contents
Efficient Influence Maximization in Large-scale Social Networks 2
Upper
Bound
Method 4
Network Coarsening
Method 3
Subgraph Stream
Method

3. Efficient Influence Maximization in Large-scale Social Networks1
Introduction
Contents
Efficient Influence Maximization in Large-scale Social Networks 2
Upper
Bound
Method 4
Network Coarsening
Method 3
Subgraph Stream
Method

4. 1.1 Background Social networks are popularly used
Viral marketing
Information dissemination
Technology/Idea transfers
Influence propagation
Influence maximization
Community detection
Influence inference
Early warning of public opinion
Link Prediction/Friends Recommendation
Partner Recommendation/Social Cooperation/Team Formation

5. 1.1 Background Social networks are popularly used
Viral marketing
Information dissemination
Technology/Idea transfers
Influence propagation
Influence maximization
Community detection
Influence inference
Early warning of public opinion
Link Prediction/Friends Recommendation
Partner Recommendation/Social Cooperation/Team Formation

6. 1.2 Problem Formulation Challenges:
Given a directed social graph G=(V,E), a budget k, and a stochastic propagation model M, finding k nodes, such that the expected spread of the influence can be maximized [Kempe KDD’03]
Challenges:
How to measure the objective function M(S) ?
How to find the optimal solution, i.e., the subset k of the most influential nodes?

7. 1.2 Problem Formulation How to measure the influence M(S) ?.3
How to measure the influence M(S) ?
Stochastic propagation models
IC model
LT model
Other propagation models: e.g. continuous time IC or LT models
Monte-Carlo (MC) simulation
Exact calculation under IC and LT is #P-hard (Chen, KDD’ 10). .1 c .3 .1 .2 .1 a e .3 .4 f .2 .4 .1 .4 .3 h d .1 .2 .1 .2 g .4 I .4 .1
IC propagation model
#P-hard

8. 1.3 Greedy Algorithm
How to find a subset k containing the most influential nodes
Influence maximization under both IC and LT models is NP-hard . (Kemp, KDD’03)
Property 1: M(S) is monotone:
Property 2: M(S) is submodular :
The set cover problem

9. 1.3 Greedy Algorithm Advantage: Performance guarantee of 1− 1/e ≈ 63%
Disadvantage: Heavy computation cost
Inner loop： M(S) needs many Monte-Carlo simulations
Outer loop：time complexity of O(Nk), where N is network size

10. 1.4 Heuristic Algorithms Advantage: faster than the Greedy Algorithm
ShortestPath: Kimura and Saito (PKDD’06) “Tractable models for information diffusion in social networks”
DegreeDiscount: Chen et al. (KDD'09) “Efficient influence maximization in social networks”
MIA: Chen et al. (KDD'10) “Scalable influence maximization for prevalent viral marketing in large-scale social networks”
DAG: Chen et al. (ICDM’10) “Scalable influence maximization in social networks under the linear threshold model”
SIMPATH： Goyal et al. (ICDM’11)“SIMPATH: An Efficient Algorithm for Influence Maximization under the Linear Threshold Model” d e f g
Shortest Path from a to c
Node 2’s degree will shrink
Advantage: faster than the Greedy Algorithm
Disadvantage: no performance guarantee 2 5
DegreeDiscount

11. 1.5 Advanced Greedy Greedy algorithm Advanced greedy algorithms reward
CELF： Leskovec et al. (KDD'07) “Cost-effective outbreak detection in networks”
CELF++：Goyal et al. (WWW’11) “CELF++: optimizing the greedy algorithm for influence maximization in social networks”
StaticGreedy：Cheng et al. (CIKM’13) “StaticGreedy: solving the scalability-accuracy dilemma in influence maximization ”
Greedy algorithm
reward a d b b a c e c d e

12. 1.5 Advanced Greedy Greedy algorithm Advanced greedy algorithms reward
CELF： Leskovec et al. (KDD'07) “Cost-effective outbreak detection in networks”
CELF++：Goyal et al. (WWW’11) “CELF++: optimizing the greedy algorithm for influence maximization in social networks”
StaticGreedy：Cheng et al. (CIKM’13) “StaticGreedy: solving the scalability-accuracy dilemma in influence maximization ”
Greedy algorithm
reward a d b b a c e c d e

13. 1.5 Advanced Greedy Greedy algorithm CELF algorithm
Advanced greedy algorithms
CELF： Leskovec et al. (KDD'07) “Cost-effective outbreak detection in networks”
CELF++：Goyal et al. (WWW’11) “CELF++: optimizing the greedy algorithm for influence maximization in social networks”
StaticGreedy：Cheng et al. (CIKM’13) “StaticGreedy: solving the scalability-accuracy dilemma in influence maximization ”
Greedy algorithm
CELF algorithm
reward
reward d a d a b b b a b a c e c e c c d d e e

14. 1.5 Advanced Greedy Greedy algorithm CELF algorithm
Advanced greedy algorithms
CELF： Leskovec et al. (KDD'07) “Cost-effective outbreak detection in networks”
CELF++：Goyal et al. (WWW’11) “CELF++: optimizing the greedy algorithm for influence maximization in social networks”
StaticGreedy：Cheng et al. (CIKM’13) “StaticGreedy: solving the scalability-accuracy dilemma in influence maximization ”
Greedy algorithm
CELF algorithm
reward
reward d a d a b b b b a a c e c e c c d d e e

15. 1.5 Advanced Greedy Greedy algorithm CELF algorithm
Advanced greedy algorithms
CELF： Leskovec et al. (KDD'07) “Cost-effective outbreak detection in networks”
CELF++：Goyal et al. (WWW’11) “CELF++: optimizing the greedy algorithm for influence maximization in social networks”
StaticGreedy：Cheng et al. (CIKM’13) “StaticGreedy: solving the scalability-accuracy dilemma in influence maximization ”
Greedy algorithm
CELF algorithm
reward
reward d a d a
Advantage: by the submodularity, CELF reduces the Monte-Carlo calls and improves the greedy algorithm by up to 700 times
Disadvantage: needs N Monte Carlo simulations to initialize the upper bound, where N is the network size. Not cater for large networks! b b b d a a b e c e c c e d c e

16. 1.6 Our Ideas
Demand for efficient and accurate solutions to influence maximization
Algorithm Level：
Upper Bound
Data Level：
Subgraph Stream
Model Level：
Network Coarsening
We explore new upper bounds to significantly reduce the number of MC simulations in GREEDY, especially at the initial step.
We break down a big networks into a series of small subgraphs and continuously estimate the influence spread on these subgraphs as data streams.
We present a new network coarsening model to learn coarsened networks, which are small and tractable for influence maximization.
Efficient Influence Maximization in Large-scale Social Networks

17. Efficient Influence Maximization in Large-scale Social Networks1
Introduction
Contents
Efficient Influence Maximization in Large-scale Social Networks 2
Upper
Bound
Method 4
Network Coarsening
Method 3
Subgraph Stream
Method

18. 2.1 Motivation
Can we initialize the upper bounds without actually computing the MC simulations ?
CELF algorithm
UBLF algorithm
UBLF algorithm
Node
upper bound MC a
2.1 1 b
1.5 c
1.1 d
1.8 e
1.2
Node
Upper bound MC a
2.3 b
1.6 c
1.2 d
1.9 e
1.3

19. 2.2 Upper Bound of M(S) Local view Global view How many heads?
Proposition 2 establishes a relationship among the activation probabilties in time t and t+1.

20. But we know its upper bound!
2.2 Upper Bound of M(S)
M(S) is bounded by a sum of series.
In what condition the series convergent? and what is the limit?
Too hard!
Its aera?
But we know its upper bound!

21. 2.2 Upper Bound of M(S)
Convergent condition：the total influence to or from any node is less than 1.
Under condition (14), we get a tractable upper bound.
+ …… =

22. 2.3 UBLF Algorithm
CELF: the first round is time-consuming, needs full MC simulations.
UBLF: the first round is analytically calculated.

23. 2.4 Example for UBLF Monte-Carlo Simulation Node 1 is selected!
(only 1 time MC simulation)

24. 2.5 Experiments Data collection Benchmark Digger Twitter Epinions
Small-world
Benchmark
CELF
Degree
DegreeDiscount
MIA
SP1M
Evaluations of upper bounds

25. 2.5 Experiments Comparison results (Numbers of MC simulations)
Observation:
The total MC calls of UBLF is significantly reduced compared to CELF.

26. 2.5 Experiments Comparison results (Influence spread) Observations:
The spreads of UBLF and CELF are completely identical, which explains again that UBLF and CELF share the same logic in selecting nodes.

27. 2.5 Experiments Comparison results (Time cost) Observation:
UBLF is 2-10 times faster than CELF.

28. Efficient Influence Maximization in Large-scale Social Networks1
Introduction
Contents
Efficient Influence Maximization in Large-scale Social Networks 2
Upper
Bound
Method 4
Network Coarsening
Method 3
Subgraph Stream
Method

29. 3.1 Motivations Can we deal with the whole network like data streams?
Can we restore results of original network from results on the subgraph streams?

30. 3.2 Expression of Influence Spread
How to obtain strongly connected
components of the original networks
from those of subgraphs?

31. 3.3 Joins of SCCs
(A)
(A)
(B)
(B)
(C)
(C)
(D)
(D)

32. 3.4 Influence Spread Estimation
More Combinations ! ?
Reasonable?

33. 3.4 Influence Spread Estimation
Only keep those random
variables with orthogonal paths
How many?

34. 3.4 Influence Spread Estimation

35. 3.5 Subgraph Incremental Algorithm

36. 3.6 Experiments Data collection Benchmark Statistics of data
ego-Facebook
ca-HepPh
ca-CondMat
-Enron
Benchmark
CELF
Staticgreedy
DegreeDiscount
PageRank
PMIA

37. 3.6 Experiments Comparison results (Influence spread) Observations:
The spreads of Ours, StaticGreedy and CELF are nearly the same, which explains that our method to estimate the influence spread is accurate.

38. 3.6 Experiments Comparison results (Time cost) Observation:
Ours is more than 100 times faster than CELF and is nearly 5-10 times faster than the StaticGreedy.

39. Efficient Influence Maximization in Large-scale Social Networks1
Introduction
Contents
Efficient Influence Maximization in Large-scale Social Networks 4
Network Coarsening
Method 2
Upper
Bound
Method 3
Subgraph Stream
Method

40. 4.1 Motivations
Can we coarsen the networks in order to obtain a smaller equivalent representation?
Can we solve the influence maximization problem using the smaller representation?

41. 4.1 Motivations
Can we coarsen the networks in order to obtain a smaller equivalent representation?
Can we solve the influence maximization problem using the smaller representation? 9 1 2 4 3 5 8 6 7 13 12 10 11
0.221
0.128
0.097
0.184
0.204
0.216
0.265
0.305
0.051
0.141
0.241
0.028
0.045
2,4,10,9,12
1,3
5,8 2’ 5’ 1’
(1) coarsen
(2) solve
(3) project

42. 4.2 Problem Statement
Given a network G=(V,E) and a network coarsening rate 0< α < 1, the proposed Network Coarsening aims to infer a smaller coarsened network Gcoarsen =(V’,E’) by merging the nodes that tightly connect each other, such that
and to find a smaller equivalent network to describe the original large yet sparse network.

43. 4.2 Problem Statement
Original large-scale network
Static network structures
Coarsening small network
Dynamic network information spreading data
Ideas: if two nodes tightly connect with each other and co-occur frequently in the same information cascade, they are likely to be combined together
We called it “semi-data-driven coarsening method”

44. 4.3 Challenges
C1: Network structure data and network information spreading data are heterogeneous data that need to be modeled jointly. How to maintain the diffusive characteristics after nodes merging?
C2: Network Coarsening leads to a complicated optimization problem which requires efficient algorithm.

45. 4.4 Proposed Method

46. 4.4 Proposed Method: Heterogeneous data
1st step: Model dynamic information spreading cascades

47. 4.4 Proposed Method: Heterogeneous data
2nd step: Model static network structures (graph regularization)
3rd step: Formulate node label distribution learning function
It simultaneously minimize the graph regularization on the static network structures and maximize the network coarsening likelihood on information spreading data.

48. 4.4 Proposed Method: Efficient algorithm
Here we adapt the Accelerated Proximal Gradient (APG) [Beck and Teboulle, 2009] to learn the optimal node label distribution Y.
Based on optimal Y, we merge nodes with same label.
Assign new weights.

49. 4.5 Algorithm Flowchart
Step 1: Coarsen the network to get a quick picture
Step 2: Solve the influence maximization problem in the coarsened network. (such as Greedy algorithm) 12
0.221 9 9 13 13
0.128 13 11
1,3 10 11
0.097
(1) coarsen 1’
0.204
0.045
0.051
0.184 1 2
0.305 7 2’ 7
0.216 4
0.265
0.241 3
2,4,10,9,12
0.141
0.097 6 5 6 5’
0.216 8
0.028
5,8
(3) project
(2) solve

50. 4.5 Algorithm Flowchart
Step 1: Coarsen the network to get a quick picture
Step 2: Solve the influence maximization problem in the coarsened network. (such as Greedy algorithm) 12
0.221 9 9 13 13
0.128 13 11
1,3 10 11
0.097
(1) coarsen 1’
0.204
0.045
0.051
0.184 1 2
0.305 7 2’ 7
0.216 4
0.265
0.241 3
2,4,10,9,12
0.141
0.097 6 5 6 5’
0.216 8
0.028
5,8
(3) project
(2) solve

51. 4.5 Algorithm Flowchart
Step 1: Coarsen the network to get a quick picture
Step 2: Solve the influence maximization problem in the coarsened network. (such as Greedy algorithm) 12
0.221 9 9 13 13
0.128 13 11
1,3 10 11
0.097
(1) coarsen 1’
0.204
0.045
0.051
0.184 1 2
0.305 7 2’ 7
0.216 4
0.265
0.241 3
2,4,10,9,12
0.141
0.097 6 5 6 5’
0.216 8
0.028
5,8
(3) project
(2) solve

52. 4.5 Algorithm Flowchart
Step 3: Project the solutions back to the original network.
(i) Randomly select
(ii) Select node as
(iii) Select node as , where 12
0.221 9 13
0.128 13 11
1,3 10 11
0.097
(1) coarsen 1’
0.204
0.045
0.051
0.184 1 2
0.305 7 2’ 7
0.216 4
0.265
0.241 3
2,4,10,9,12
0.141
0.097 6 5 6 5’
0.216 8
0.028
5,8
(3) project
(2) solve

53. 4.6 Experiments: Influence estimation
Metrics:
(1) the average influence spread error rate
(2) the running time

54. 4.6 Experiments: Influence estimation
Observations:
The average influence spread error rate of ours is much smaller than the baselines
Our method uses more time than its peers
To sum up, our method outperforms baseline methods in terms of accuracy without significantly raising time cost

55. 4.6 Experiments: Influence maximization
Baseline methods:
CFSInflu method proposed in the paper
Random-based
CoarseNet-based [Purohit et al., 2014]
Data-driven-based, InfluLearner-based [Du et al., 2014]
PMIA [Chen et al., 2010]
InfluLearner-based and PMIA methods are carried out directly on the original network; Random-based, CoarseNet-based and Data-driven-based methods work on the coarsened network.
How do we perform influence maximization analysis?
What can we obtain compared with the methods conducted on the original network?

56. 4.6 Experiments: Influence maximization
Observations:
Influence spread derived from the ours approximates the result on the original network. Meanwhile, ours outperforms all the baseline methods that conducted on the coarsened networks.
Our method runs orders of magnitude faster than the InfluLearner-based or PMIA methods while maintaining the influence spread results.

57. Conclusions Background Problem Formulation Greedy Algorithm
Heuristic Algorithms: MIA,
SimplePath,
PageRank, et al.
Advanced Greedy Algorithms:
CELF, CELF++,
et al.
Upper
Bound
Subgraph Stream
Our Solutions
Network
Coarsening

58. References
(TKDE-15) Chuan Zhou, Peng Zhang, Wenyu Zang, and Li Guo. On the Upper Bounds of Spread for Greedy Algorithms in Social Network Influence Maximization. IEEE Transactions on Knowledge and Data Engineering, Vol. 27, No.10, 2015, (CCF-A)
(IJCAI-15) Wei-Xue Lu, Peng Zhang, Chuan Zhou, Chunyi Li, and Li Gao. Big Network Influence Maximization: An Incremental Algorithm for Streaming Subgraph Influence Spread Estimation. In Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence, (CCF-A)
(IJCAI-16) Li Gao, Jia Wu, Hong Yang, Zhi Qiao, Chuan Zhou*, and Yue Hu. Semi-Data-Driven Network Coarsening. In Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence, (CCF-A)

59. Thank You! 周川
Tel:
地址：北京市海淀区闵庄路甲89号