The Impact of Changes in Network Structure on Diffusion of Warnings

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

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
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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 nodes information fused value exceeds one
  of the thresholds, the node will enter a new
  state.

1
time
Believer
Action
Upper bound
Undecided
Lower bound
Disbelieved
0
Uninformed
8
Experiments

• Diffusion of evacuation warnings
• 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

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Experimental Networks
Network Population Size Density
Random 100,000 0.00004000
Grid 100,000 0.00003987
Scale-free 100,000 0.00003900
Blog 138,007 0.00004926

• The edges in the networks are undirected edges
  where messages may flow in either direction.

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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

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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.

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Diffusion Scenario 1 No Groups

• Equal trust between all nodes

Group A
Group B
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Diffusion Scenario 2 Groups (1)

• High trust between nodes in the same group

Group A
Group B
Group A
Group B
High Low
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Diffusion Scenario 3 Groups (2)

• High trust in nodes in group A

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

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Thank you

• Questions?

Acknowledgements This material is based upon
work partially supported by the U.S. National
Science Foundation (NSF) under Grant No.
IIS-0621303, IIS-0522672,IIS-0324947,
CNS-0323324, NSF IIS-0634875 and by the U.S.
Office of Naval Research (ONR) Contract
N00014-06-1-0466 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
N00014-07-1-0150 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.
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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 No action
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 No action
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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

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Information Fusion Axiom (a)

• 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,V11S2,V21
Node 2 S2,V22
Node 3 S1,V13 S2,V23
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Information Fusion Axiom (b)

• 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