1. A History Sensitive Cascade Model in Diffusion Networks
Stephen Foster1, Walt Potter1, Jiang Wu2, Bin Hu2, Yu Zhang3
March 23, 2009
ADS’09, San Diego, CA
Trinity University | Laboratory for Distributed Intelligent Agent Systems
1 Southwestern University
2 Huazhong University of Science and Technology
3 Trinity University

2. Outline Introduction History Sensitive Cascade Model (HSCM)
A Polynomial Solution of HSCM in Tree Graphs.
A Markov Solution of HSMC in General Graphs.
Experiment
Conclusion
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3. 3
Diffusion
Diffusion is a process by which information, viruses, ideas or new behavior spread over social networks.
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4. Two Basic Diffusion Models
Linear Threshold Model
A node becomes active if a predetermined fraction of the node’s neighbors are active.
Independent Cascade Model
An active node gets a one-time chance to activate each of its neighboring nodes with some probability.
Threshold=60%
Progressive
P=50%
History-Insensitive
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5. Influence Maximization Problem5
Influence Maximization Problem
Given some value k and some diffusion network with a set of nodes N, the goal is to select an initially active k-node subset from N, such that the number of nodes in N that eventually becomes active is maximized.
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6. Existing Results NP-Hard General Heuristics6
Existing Results
NP-Hard
General Heuristics
Greedy [Kempe, Kleinberg and Tardos 2003]
Submodular
A greedy strategy obtains a solution that is provably within 63% of optimal solution.
Hill Climbing [Rolfe 2004]
Simulated Annealing [Jackson Mo and Yariv 2005]
Cost-Effective Heuristic [Leskovec, Krause and Guestrin 2007]
The bound is ≥ 63%.
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7. Our Diffusion Model History Sensitive Cascade Model .99 .99 .001 .001
Alice
.99
.99
Cathy
Bobby
.001
Donald
.001
.001
Francine
Ethan
.99
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8. The HSCM Algorithm function HSCM [G=(V,E), W(ev,u)]8
function HSCM [G=(V,E), W(ev,u)]
Inputs: G = (V,E) where V is a set of vertices and E is a set of edges,
with some initially active vertices.
W(ev,u), the spreading probability, that if v is active in time
step t, then u will be active in time step t+1.
For time step = 1 to k
For each vertex v in V
If A(v) = true
For each vertex u in targets(v)
random = a random number between 0 and 1
If random < W(ev,u)
Set A(u) = true;
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9. Activation Probability Problem9
Activation Probability Problem
Given some time step k and some vertex v, what is the probability that v will be active at t=k.
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10. A Polynomial Solution in Tree Graphs
Let G=(V,E) be a graph without cycles, and let there be no two edges in E, ew,x and ey,z, such that x=z. We use the function inf(u) to denote the vertex v such that utargets(v).
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11. After 4 time steps, the activation probability for each node.
An Example 11
Initially only Node 1 is active. 1 2 3 4 5
After 4 time steps, the activation probability for each node. V
P(vt=0)
P(vt=1)
P(vt=2)
P(vt=3)
P(vt=4) 1
1.0 2
0.0
0.5
0.75
0.875
0.9375 3
0.25
0.6875 4 5
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12. Problem with Loops T=k, Pvk T=k+1, Puk+1 = Pvk × W(ev,u) < Pvk
W(eu,v)
T=k, Pvk
T=k+1, Puk+1 = Pvk × W(ev,u)
< Pvk
T=k+2, Pvk+2 = (Puk+1  Pvk) × W(eu,v) + Pvk
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13. A Markov Solution in General Graphs
Consider the graph to be a finite state system, where “state” is understood as some combination of activated vertices in V.
Define the system as a state transition matrix AN×N, where N is the size of PoweSet(V).
Update A at each time step.
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14. The principle of inclusion/exclusion
An Example
A[i,j] is defined as the probability that the network will move from state i to state j.
[] , , , ,1,2 [] 1, 3, 3,
3,1,
The principle of inclusion/exclusion
0.2×0.5=0.6
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15. Experiment Each run lasts 100 time steps
Input
Value
Definition
Nodesize
100
The total # of nodes in the network
W(ev,u)
0.6, 0.58, 0.56, 0.54, …, 0.4
The spreading probability that is Uniform Distribution
WU(0,1)
Selected-Number
1, 2, 3, …, 9
The initial selected # of nodes for targeting in cascade
Density
0.02, 0.03, 0.04, 0.06
Each run lasts 100 time steps
All results are the average of 100 runs
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16. Scale-Free Network SN,K,
P(K) decays as a power law.
P(K) ~ K-
Large networks can self-organize into a scale free state, independent of the agents.
N=100 K=6 =2.5
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17. Dynamics vs. Node-Size and Density
Selected Number=9, Nodesize=100, Threshold= 0.5, Timestep=5
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18. Cascade Time per Node Over Selected-Number and Threshold
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19. Dynamics vs. Initial Selected Number
Nodesize=100, Threshold= 0.5, Density=0.2
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20. Dynamics vs. Diffusion Threshold
Nodesize=100, Density=0.2
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21. 21
Conclusion
History Sensitive Cascade Model (HSCM) allows activated nodes to receive more than a one-time chance to activate their neighbors.
HSCM provides
a polynomial algorithm for calculating the probability of activity for any arbitrary node at any arbitrary time in tree graphs,
a Markov model for calculating the probability in general graphs.
HSCM is intractable for most general graphs.
In the future, we will study the influence maximization problems under different time constraints for HSCM.
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22. SAN ANTONIO A Case Study
With one filing for every 143 households, San Antonio, ranks 21st among the top 25 cities with the highest foreclosure rates in the U.S. city.
In the 2008 first quarter alone 3,830 foreclosures were filed with the city.
Data provided by Realtytrac on August 2008
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23. Acknowledgements NSF grants IIS 0755405 and CNS 0821585. Collaborators
Trinity: Dr. Christine Drennon in Sociology
George Mason: Dr. David Wong in Geography
Drexel: Dr. Roger McCain in Economics
Students
Lucy Elder, Stephen Foster, Jason Leezer, Patricia Perez, Will Thornton, Hudson Thrift, Aaron Welch
Trinity University | Laboratory for Distributed Intelligent Agent Systems