1. The Impact of Changes in Network Structure on Diffusion of Warnings
Cindy Hui
Malik Magdon-Ismail
William A. Wallace
Mark Goldberg
Rensselaer Polytechnic Institute

2. Diffusion on Dynamic Networks
Diffusion of warning messages through a population
Network dynamics as the result of the information flow 2

3. Diffusion Model How does information flow through the network?
How do nodes process information?
How do nodes act on the information?

4. How does information flow?
Messages are propagated when nodes interact.

5. How does information flow?
Information Loss Axiom
When a message is passed from one node to another, the information value of the message is non-increasing.
The information value of the message is a function of the social relationship between the sender and the receiver.
trust A B

6. How do nodes process information?
Source Union Axiom
The recipient node combines information from incoming messages.
Information Fusion Axiom
The combined information value is at most the sum of the individual bits of information and at least the maximum.

7. How do nodes act on the information?
Threshold Utility Axiom
If the node’s information fused value exceeds one of the thresholds, the node will enter a new state. 1
Believer
time
Action
Upper bound
Undecided
Lower bound
Disbelieved
Uninformed

8. Experiments Diffusion of evacuation warnings: Parameters:
A warning message is broadcasted to a population.
Population is a network of household nodes.
The proportion of evacuated nodes is recorded.
Parameters:
Social network structure
Seed set selection
Diffusion scenarios

9. Experimental Networks
Population Size
Density
Random
100,000
Grid
Scale-free
Blog
138,007
Barabasi-Albert (BA) model for generating random scale-free networks using preferential attachment
Degree distribution follows a power law P(k) ~ k-3
LiveJournal blog comment network
M. Goldberg, S. Kelley, M. Magdon-Ismail, K. Mertsalov, and W. A. Wallace, Communication dynamics of blog networks, in Proc. SIGKDD Workshop on Social Network Mining and Analysis, August 2008.
The edges in the networks are undirected edges where messages may flow in either direction.

10. Seed Set Selection One single information source
High information value
Broadcast message at time step 1
Initially connected to 20% of the population
Two seeding strategies
Random seed set
Highest degree set of nodes

11. Social Relation: Trust
We can use trust to differentiate the society into social groups.
We divide the population into two groups of nodes by randomly assigning each node to one of two groups, A or B.

12. Diffusion Scenario 1: No Groups
Equal trust between all nodes
Group A
Group B

13. Diffusion Scenario 2: Groups (1)
High trust between nodes in the same group
Group A
Group B
Group A
Group B
High
Low

14. Diffusion Scenario 3: Groups (2)
High trust in nodes in group A
Group A
Group B
Group A
Group B
High
Low

15. Simulation Results

16. Proportion of Evacuated Nodes in Each Trust Scenario (Infect Randomly)
Simulation Results
Proportion of Evacuated Nodes in Each Trust Scenario (Infect Randomly)
Network
No Groups
Grid
0.63
Random
0.60
Scale-free
0.56
Blog
0.58
Proportion of Evacuated Nodes in Each Trust Scenario (Infect High Degree)
Network
No Groups
Grid
0.67
Random
0.76
Scale-free
0.95
Blog
0.82

17. Proportion of Evacuated Nodes in Each Trust Scenario (Infect Randomly)
Simulation Results
Proportion of Evacuated Nodes in Each Trust Scenario (Infect Randomly)
Network
No Groups
Groups (1)
Groups (2)
0.1
0.3
Grid
0.63
0.76
0.89
0.77
Random
0.60
Scale-free
0.56
0.79
Blog
0.58
0.78
0.84
0.83
Proportion of Evacuated Nodes in Each Trust Scenario (Infect High Degree)
Network
No Groups
Groups (1)
Groups (2)
0.1
0.3
Grid
0.67
0.80
0.91
Random
0.76
0.86
0.90
Scale-free
0.95
0.98
Blog
0.82
0.87
0.88
0.89

18. Proportion of Evacuated Nodes in Each Trust Scenario (Infect Randomly)
Simulation Results
Proportion of Evacuated Nodes in Each Trust Scenario (Infect Randomly)
Network
No Groups
Groups (1)
Groups (2)
0.1
0.3
Grid
0.63
0.76
0.89
0.77
Random
0.60
Scale-free
0.56
0.79
Blog
0.58
0.78
0.84
0.83
Proportion of Evacuated Nodes in Each Trust Scenario (Infect High Degree)
Network
No Groups
Groups (1)
Groups (2)
0.1
0.3
Grid
0.67
0.80
0.91
Random
0.76
0.86
0.90
Scale-free
0.95
0.98
Blog
0.82
0.87
0.88
0.89

19. Proportion of Evacuated Nodes in Each Trust Scenario (Infect Randomly)
Simulation Results
Proportion of Evacuated Nodes in Each Trust Scenario (Infect Randomly)
Network
No Groups
Groups (1)
Groups (2)
No Groups (diff)
0.1
0.3
Grid
0.63
0.76
0.89
0.77
0.54
0.82
Random
0.60
0.52
0.85
Scale-free
0.56
0.79
0.49
Blog
0.58
0.78
0.84
0.83
0.51
0.81
Proportion of Evacuated Nodes in Each Trust Scenario (Infect High Degree)
Network
No Groups
Groups (1)
Groups (2)
No Groups (diff)
0.1
0.3
Grid
0.67
0.80
0.91
0.58
0.84
Random
0.76
0.86
0.90
Scale-free
0.95
0.98
0.85
0.92
Blog
0.82
0.87
0.88
0.89
0.74

20. Proportion of Evacuated Nodes in Each Trust Scenario (Infect Randomly)
Simulation Results
Proportion of Evacuated Nodes in Each Trust Scenario (Infect Randomly)
Network
No Groups
Groups (1)
Groups (2)
No Groups (diff)
0.1
0.3
Grid
0.63
0.76
0.89
0.77
0.54
0.82
Random
0.60
0.52
0.85
Scale-free
0.56
0.79
0.49
Blog
0.58
0.78
0.84
0.83
0.51
0.81
Proportion of Evacuated Nodes in Each Trust Scenario (Infect High Degree)
Network
No Groups
Groups (1)
Groups (2)
No Groups (diff)
0.1
0.3
Grid
0.67
0.80
0.91
0.58
0.84
Random
0.76
0.86
0.90
Scale-free
0.95
0.98
0.85
0.92
Blog
0.82
0.87
0.88
0.89
0.74

21. Conclusion Presented a model for information propagation
Nodes process and act on the information
Group structure by assigning trust between nodes
Social groups are important for diffusion
Diffusion was more efficient when based on social group than in an unstructured way
Increasing trust differentials led to larger proportions of evacuated nodes
Trust differential alone does not accomplish the same as organized trust differentials (social groups)
Diffusion process and effectiveness depends on
Network structure
Seeding mechanism

22. Thank you
Questions?
Acknowledgements: This material is based upon work partially supported by the U.S. National Science Foundation (NSF) under Grant No. IIS , IIS ,IIS , CNS , NSF IIS and by the U.S. Office of Naval Research (ONR) Contract N and by the U.S. Department of Homeland Security (DHS) through the Center for Dynamic Data Analysis for Homeland Security administered through ONR grant number N to Rutgers University.The content of this paper does not necessarily reflect the position or policy of the U.S. Government, no official endorsement should be inferred or implied.

24. Node States State Description Behavior Uninformed
Individual has not received the message
No action
Disbelieved
Individual received the message, but does not understand or has not personalized the message
Undecided
Individual received the message and is uncertain of what to do
Query neighbors in network
Believer
Individual received the message and believes the value of the message
Propagate the message
Evacuated
Individual has left the network

25. Node Parameters Node thresholds: Lower bound 0.1, Upper bound 0.3
Once a node enters believer state, they will evacuate from the network after 5 time steps
Nodes have high trust in the source (0.90)
Probability of successful communication on a link (0.75)
Information fusion
Source appears in multiple messages, take the maximum
Information fused value at the node, take the sum

26. Information Fusion Axiom (a)
When a source S is found in multiple messages with information values V1,V2,…, the information value from source S is fused into a single value V*, where
Node 1 {S1,V11;S2,V21}
Node 2 {S2,V22}
Node 3 {S1,V13; S2,V23}

27. Information Fusion Axiom (b)
Suppose that the sources (S1, S2,…) have information values (V1, V2,…).
The fused information value at the node is
Node 3 {S1,V13; S2,V23}