1. Yu Wang1, Gao Cong2, Guojie Song1, Kunqing Xie1
Community-based Greedy Algorithm for Mining Top-K Influential Nodes in Mobile Social Networks
Yu Wang1, Gao Cong2, Guojie Song1, Kunqing Xie1
1 Peking University, China
2 Nanyang Technological University, Singapore

2. Problem and Background
Problem: Given a mobile social network, we aim to mine a set of top-K influential nodes S such that R(S) is maximized using the extended Independent Cascade information diffusion model.
A mobile social network plays an essential role as the spread of information and influence in the form of "word-of-mouth“
The problem is NP-hard.
computationally expensive to run the greedy algorithm on a large network.
The previous greedy algorithms take days to finish on 723k nodes
a network with 723k nodes
in our experiments.

3. Basic Idea of the Algorithm
Dynamic
programming
Algorithm & greedy
algorithm
on selected
communities
Construct
Network from
CDR (call detailed
record)
Community
Detection: it
based on diffusion
Model on MSN

4. Step1: Extracting Mobile Social Network
Extract a Mobile Social Network from CDR data and model it as a directed weighted graph
A phone user -- a node
A directed edge u  v is established, if there exits communication from u to v
communication time -- the weight of the edge
A phone user corresponds to a node
A directed edge from node u to node v is established, if there exits communication from u to v
corresponding communication time as the weight of the edge

5. Extended Independent Cascade Model
Two states of nodes
Active & inactive
Diffusion speed λ
When an active node vi contacts an inactive node vj , the inactive node becomes active at a probability (rate) λij.

6. Extended Independent Cascade Model

7. Step2: Influential Model Based Community Detection Algorithm
Community Partition
Each node is assigned a unique community label from 1 to N
For each node compute the set of its influenced neighbors using Independent Cascade diffusion model
Iteratively propagate the labels through the network in finite iterations
for each node v ,the label of the community that the majority of its influenced neighbors belong to  the label of v
Community Combination
the difference between the node’s influence degree in its community and its influence degree in the network is smaller than a threshold.
t denotes the tth iteration
sv denotes the number of neighneighbors that are influenced by v
ui (i ∈ [1, sv]) represents an influenced neighbor of node v
ui.Ct−1 represents the community label of ui at iteration t-1
maxCMT is to compute the majority label of ui.Ct−1

8. Step3: Community-Based Greedy Algorithm
Choose communities to find the Top-1 influential node C1 C2
ΔR2=0.3
ΔR1=0.2
R[1,1]=max{R[0,1], R[3,0]+ΔR1}=0.2
s[1,1]=C1;
R[2,1]=max{R[1,1], R[3,0]+ ΔR2}=0.3
s[2,1]=C2;
R[3,1]=max{R[2,1], R[3,0]+ ΔR3}=0.3
s[3,1]=C2;
So we mine top-1 node in C2
ΔR3=0.1 C3

9. Community-Based Greedy Algorithm
Choose communities to find the Top-2 influential node C1 C2
ΔR2=0.06
ΔR1=0.2
Note ΔR2 is 0.06, but not 0.3.
R[1,2]= max{R[0,2], R[3,1]+ΔR1}=0.5
s[1,2]=C1;
R[2,2]= max{R[1,2], R[3,1]+ΔR2}=0.5
s[2,2]=C1;
R[3,2]= max{R[2,2], R[3,1]+ΔR3}=0.5
s[3,2]=C1;
We mine the second node in C1
ΔR3=0.1 C3

10. CGA:Community-Based Greedy Algorithm
Θ and Δd are two constants

11. Experiments
Data Sets
Extract a Mobile Social Network from a three-month CDR (call detailed record) data of a city from China Mobile
Node number: 723,201
Average degree: 13.4

12. Community distribution
largest community size: 95,690

13. Experiments Top-k Nodes Mining Methods Parameter study:
MixedGreedy Algorithm
NewGreedy Algorithm
DegreeDiscount
Random Method
CGA
SPCGA
Parameter study:
k, diffusion speed λ, data size

14. Results
Influence degree and time vs K

15. Results Influence degree and time vs diffusion speed λ the efficiency
of MixedGreedy drops quickly (almost exponentially) while CGA is a lot better.

16. Results Influence degree and time vs network size
the influence degree is relatively stable

17. Summary Handle large-scale networks (power-law distribution degree)
improve the efficiency of existing algorithms by an order of magnitude while the loss in approximation precision is small
Can combine with any existing algorithm to find influential nodes w.r.t. communities

18. Related work on Top-K Algorithm
Typical Greedy Algorithm( Kempel et al. KDD2003)
CELF Greedy Algorithm (Leskovec et al. KDD2007)
An improved greedy algorithm (Kimura et al. AAAI2007)
NewGreedy Algorithm, MixedGreedy, DegreeDiscount Algorithm (Chen et al. KDD2009)
MIA algorithm (Chen et al. KDD2010)
--None of them considers community property

19. Thank You !

20. Experiments
Influence degree and time with different θ

21. Influential Model Based Community Detection Algorithm
Community Combination
denotes the influence degree of node u outside the community Cm
Rm({u})denotes the influence degree of node v in its community Cm
We expect that the difference between the node’s influence degree
in its community and its influence degree in the whole network
is small. To achieve a good set of top-K influential nodes with a
good influence degree in our algorithm, we define combination entropy
to measure the connections of two communities and combine
two communities if the combination entropy between them is larger
than a threshold.
L[Cm] includes its influenced neighbors such that they will make diffusion degree of
v with regard to Cm different from diffusion degree of v with regard to the whole
network.
We set a threshold θ. If the combination entropy CoEntropy (CElm) of community
Cm to community Cl is bigger than θ, then Cm and Cl will be combined.

22. Problem Statement Influence Degree
Given a mobile social network G = (V, E, W), we aim to mine a set of top-K influential nodes S on the network
such that R(S) is maximized using the extended Independent Cascade information diffusion model.

23. Related work on Top-K Algorithm
DegreeDiscount Algorithm
NewGreedy Algorithm
CELF Greedy Algorithm
Typical Greedy Algorithm
Chen et al. KDD2009
No precision guarantee
O(KlogN+M)
Chen et al. KDD2009
(1-1/e)-approximation
O(KRM)
(1-1/e)-approximation
700 times faster
Leskovec et al. KDD2007
Kempel et al. KDD2003
(1-1/e)-approximation
O(KNRM)
--Using IC information diffusion model
--None of them consider community property

24. Outline Research Background Related Work
Preliminaries and Problem Statement
Top-K Nodes Mining Algorithm
Experiments