1. DAVA: Distributing Vaccines over Networks under Prior Information
Yao Zhang, B. Aditya Prakash
Department of Computer Science
Virginia Tech
SDM, Philadelphia, April 24, 2014
Zhang and Prakash, SDM 2014

2. Motivation: Epidemiology
Virus spreads over contact networks
SIR model [Anderson+ 1991]
Susceptible-Infectious-Recovered
Weights pij: propagation prob. from i to j
Recovered prob. δ for each node
(models mumps-like infections)
Zhang and Prakash, SDM2014
Zhang and Prakash, SDM 2014

3. Motivation: Social Media
Meme/Rumor spreads over friendship networks
E.g.: Twitter following network
Independent cascade model (IC) [Kempe+ KDD2003]
Each node has only one chance to infect its neighbors
Special case of SIR model
Zhang and Prakash, SDM2014

4. Immunization
Centers for Disease Control (CDC) cares about containing epidemic diseases
E.g: ~400 million dollars used for vaccines for children in 2013
Twitter tries to stop rumor spread
E.g.: rumors of victims after the Boston Marathon bombs in 2013
How to choose best nodes to vaccinate (remove)?
Zhang and Prakash, SDM2014

5. Immunization Good for baseline strategies
Pre-emptive immunization (choose nodes before the epidemic starts)
Acquaintance strategy [Cohen+ 2003]
pick a random person, immunize one of its neighbors at random
Netshield [Tong+ 2010]
Minimize the epidemic threshold
(point when the virus takes-off)
Good for baseline strategies
Zhang and Prakash, SDM2014

6. In reality ? Typically the epidemic has already started! this paper
Pre-emptive immunization (choose nodes before the epidemic starts)
Acquaintance strategy [Cohen+ 2003]
Netshield [Tong+ 2010] ?
Typically the epidemic has already started!
More realistic intervention
Which nodes to vaccinate now?
We call it Data-Aware Immunization
this paper
Zhang and Prakash, SDM2014

7. Outline Motivation Problem Definition Complexity Our Proposed Methods
Experiments
Conclusion
Zhang and Prakash, SDM2014

8. Data-Aware Vaccination Problem
Problem: Given a set of infected nodes and a contact graph,
how to distribute k vaccines (node removal)
to minimize the expected number of infected nodes
at the end of the epidemic? D D
Best solution A A E E B B
1 vaccine? F F C
Remove A, save {A, D};
Remove B, save {B};
Remove C, save {C}; C
pij =1 for all edges
Zhang and Prakash, SDM2014
Zhang and Prakash, SDM 2014

9. Outline Motivation Problem Definition Complexity Our Proposed Methods
Experiments
Conclusion
Zhang and Prakash, SDM2014

10. Complexity of DAV NP-hard Approximation algorithm?
See paper for details
NP-hard
Reduce from Maximum K-Intersection Problem (MaxKI: maximizing the intersection of k subsets)
MaxKI is NP-Complete [Vinterbo 2004]
Approximation algorithm?
Not submodular
Actually, DAV is hard to approximate within an absolute error!
Zhang and Prakash, SDM2014

11. Outline Motivation Problem Definition Complexity Our Proposed Methods
assume IC model and undirected graph
Experiments
Conclusion
Zhang and Prakash, SDM2014

12. 1: Simplify - Merging infected nodes
Idea: merge all the infected nodes into a single ‘super infected’ node I
Original Graph
Merged Graph
Super node I A A pA pA
Equivalent pX B B pB pY
Logical-OR
pB=1-(1-pX)(1-pY) pC pC C C
Zhang and Prakash, SDM2014

13. 2: DAVA-Tree Algorithm: Idea
Select nodes with the largest “benefit”
: the expected number of saved nodes after removing set S on graph G
Benefit of adding additional node j into S:
# of saved nodes
after adding j into S
Merged Infected Node
Additional number of saved nodes when adding node j into S
Benefit: 5
Benefit: 4
pij =1for all edges
Benefit: 2
Zhang and Prakash, SDM2014

14. DAVA-Tree Alg.: Optimal on Trees
For any set S:
Merged Infected Node
Fact 1: the chosen nodes in the optimal set must be neighbors of infected node I
Fact 2: the benefit of each such node is independent of the rest of the set S
Benefit: 2
Benefit: 5
pij =1for all edges
Linear Time
Benefit: 4
DAVA-tree algorithm: Select top k node from I’s neighbors with the max. benefit
Zhang and Prakash, SDM2014

15. 3: General Case – Arbitrary Graphs
Idea
We have the optimal algorithm for a tree
Extract a spanning tree, then run DAVA-tree
What kind of tree?
Minimum spanning tree
Optimal on MST by DAVA-tree
Optimal solution
Dom captures the ‘closeness’ of nodes to the infectious nodes, and importance of saving nodes.
MST
pij =1 for all edges
Zhang and Prakash, SDM2014
Zhang and Prakash, SDM 2014

16. 3: General Case – Arbitrary Graphs
Idea
We have the optimal algorithm for a tree
Build a spanning tree first
What kind of tree?
Minimum spanning tree
Software engineering
We propose to use dominator tree
u dominates v
Dom captures the ‘closeness’ of nodes to the infectious nodes, and importance of saving nodes.
every path from I to v contains u
4 dominates 8,9,10,11
pij =1 for all edges
Zhang and Prakash, SDM2014
Zhang and Prakash, SDM 2014

17. Dominator Tree u is immediate dominator of v
u dominates v AND every other dominator of v dominates u
Dominator tree: add an edge between every such u and v
Optimal from DAVA-tree
Optimal solution
Linear time [Buchsbaum, Tarjan 1998]
pij =1 for
all edges
Dominator Tree
Merged Graph
Fact 1: the optimal solution should be among the children of root I in the dominator tree for any arbitrary graph
Fact 2: (for special case, k = 1, p = 1) running DAVA-tree on the dominator tree gives the optimal solution
Zhang and Prakash, SDM2014

18. Weighting the dominator tree
#P-complete
Our solution: maximum propagation path probability between nodes I and v (using Dijkstra’s algorithm) w1 p1 p3 w3 p6 w6
Dominator Tree
Merged Graph
Zhang and Prakash, SDM2014

19. DAVA algorithm Step: 1. T = Build a dominator tree
Merged Graph (pij =1 for all edges)
Step:
1. T = Build a dominator tree
2. v = Run DAVA-tree on T with
budget=1
3. Remove v from G
4. Goto Step 1 until |S|=k
Not finished
|S|=2
Iteration=1
Dominator Tree
Zhang and Prakash, SDM2014
Zhang and Prakash, SDM 2014

20. DAVA algorithm Step: 1. T = Build a dominator tree
Merged Graph
Step:
1. T = Build a dominator tree
2. v = Run DAVA-tree on T with
budget=1
3. Remove v from G
4. Goto Step 1 until |S|=k
Remove selected node
O(k(|E|+ |V|log|V|))
Too slow for large networks!
Dominator tree
Not finished
|S|=2
Iteration=2
Iteration=1
Zhang and Prakash, SDM2014
Zhang and Prakash, SDM 2014

21. DAVA-fast: a faster algorithm
Step:
1. T = Build a dominator tree
2. S = Run DAVA-tree on T
with budget=k
Merged Graph
|S|=2
In practice, the performance of DAVA-fast is very close to DAVA
Time complexity: subquadratic!
DAVA-fast: O(|V|log|V|+|E|)
Note finished
Dominator tree
Zhang and Prakash, SDM2014
Zhang and Prakash, SDM 2014

22. Extending to SIR model
See the paper
Zhang and Prakash, SDM2014

23. Outline Motivation Problem Definition Complexity Our Proposed Methods
Experiments
Conclusion
Zhang and Prakash, SDM2014

24. Experiments Virus Propagation Model
IC and SIR
Settings (See more settings in the paper)
Randomly uniformly chosen initial infected nodes
Baseline Algorithms
RANDOM: randomly uniformly chosen healthy nodes
DEGREE: choose nodes with top weighted degrees
PAGERANK: choose nodes with top pageranks
NETSHIELD
state-of-the-art pre-emptive immunization algorithm to minimize the epidemic threshold of the graph [Tong+ ICDM 2010]
Assumes no data is given before the epidemic starts
Zhang and Prakash, SDM2014

25. Experiments: datasets
Datasets are chosen from different domains
Social media (IC model)
OREGON: AS router graph
STANFORD: hyperlink network
GNUTELLA: peer-to-peer network
BRIGHTKITE: friendship network
Epidemiology (SIR model)
PORTLAND and MIAMI: large urban social-contact graph used in national smallpox modeling studies [Eubank+, 2004]
OREGON
STANFORD
GNUTELLA
BRIGHTKITE
PORTLAND
MIAMI
|V|
633
8,929
10,876
58,228
0.5 million
0.6 million
|E|
2,172
53,829
39,994
21,4078
1.6 million
2.1 million
Zhang and Prakash, SDM2014

26. Experiments: Quality GNUTELLA (IC model) PORTLAND (SIR model)
Higher is better
DAVA consistently outperforms the baseline algorithms.
Further DAVA-fast performs almost as well as DAVA.
(See more results in the paper)
Zhang and Prakash, SDM2014

27. Experiments: Scalability
did not finish within 10 hours
Running time(sec.)
Lower is better
Zhang and Prakash, SDM2014

28. Outline Motivation Problem Definition Complexity Our Proposed Methods
Experiments
Conclusion
Zhang and Prakash, SDM2014

29. Conclusion Data-Aware Vaccination problem
Given: Graph and Infected nodes
Find: ‘best’ nodes for immunization
Complexity
NP-hard
Hard to approximate within an absolute error
DAVA-tree
Optimal solution on the tree
DAVA and DAVA-fast
Merging infected nodes
Build a dominator tree, and run DAVA-tree
Running time: subquadratic
DAVA: O(k(|E|+ |V|log|V|))
DAVA-fast: O(|E|+|V|log|V|)
Graph with infected nodes
Merged graph
Dominator tree
Zhang and Prakash, SDM2014

30. Any Questions? Code at: http://people.cs.vt.edu/~yaozhang Yao Zhang
Graph with infected nodes
Code at:
Merged graph
Yao Zhang
B. Aditya Prakash
Dominator tree
Thanks for the support of NSF (Grant No. IIS ).
Zhang and Prakash, SDM2014