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Micro-Simulation of Diffusion of Warnings
Cindy HuiMark Goldberg
Malik Magdon-IsmailWilliam A. Wallace
Rensselaer Polytechnic Institute
This material is based upon work partially supported by the U.S. National Science Foundation (NSF) under Grant Nos. 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 N00014-07-10150 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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Outline
• Problem
• Past Work
• Model
• Axioms
• Simulation Experiments
• Ongoing Work
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Problem
Warnings in Evacuation Situations
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Past Work
Diffusion Models
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Dynamic Social Network
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Social Network Structure
Interaction layer
Social layer
Physical layer
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Node Characteristics
Source
Individual NodesThresholds
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Characteristics
Me My Friend
My Mother
Media
Stranger
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Characteristics
Me My Friend
My Mother
Media
Stranger
t3
t4
t2t1
ts
ts
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Interactions
Me
My Friend
My Mother
Media
Stranger
{S,V}
{S,V}
{S,V}
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Node States for Evacuation
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
Believer Individual received the message and believes the value of the message
Take necessary action
Evacuated Individual has left the network No action
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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.
A B
trust{S,V}
Vkij =α( j, i)∗Vk
i , α( j, i) ∈[0,1]where α( j, i) is the function of the relationship from i to j,
Vki is the information value of source k at node i, and
Vkij is the information value received by node j
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Source Union Axiom
• The source-value pairs are updated in a receiver node when a message is received.
• The resulting source set is a union of the source sets of the incoming messages.
Let j be a receiver node and let A be the set of nodes that each sends a message to j.
Then the source set of node j is defined as:
node( j).S = node(i).Si=1
|A|
U
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Value Min-Max Axiom• When a source is found in multiple messages, the
combined information value for the source at the node is computed as follows.
S1 S2S
maxi=1
|A|
Vkij ≤Vk
j ≤min( Vkij
i=1
|A|
∑ ,1)
{S,V1} {S,V2}{S2,V2}{S1,V1}
{S,V}
node( j).InformationFusedValue=1− (1−node( j).V(i))i=1
|V|
∏
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Threshold Utility Axiom• If the node’s information fused value exceeds one of the
thresholds, the node will enter a new state.
Believer
Undecided
Uninformed
Disbelieved
1
Upper bound
Lower bound
0
Evacuated
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Experimental Network
• Erdos-Renyi Random Graph 600 nodes connected randomly with p = 0.006• Average of 3.6 neighbors for each individual node• Total of 1102 edges
• One source node connected to 60 nodes from each group (0.20 of the population receives the initial broadcast message)• Initial message sent by source has high information value of 0.95
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Experimental Population
• Population of 600 nodes consists of two equally sized groups of nodes, A and B, randomly assigned over the network
• Group A and B nodes have the same node characteristics• Thresholds
• Lower bound 0.1: low tendency to disbelieve a message• Upper bound 0.5: medium tendency to take action
• Probability of successful communication between two nodes: 75%• Social relationships, the trust values, between them are varied
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Trust Scenarios• Average trust is fixed for all scenarios 0.75• Trust differentials 0.1 and 0.3
Scenarios A A A B B A B B
1 SAME SAME SAME SAME
2 HIGH LOW LOW HIGH
3 HIGH LOW HIGH LOW
0.75LOW HIGH
0.10.3
Trust differentials
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Node: Believer State
Believer
Undecided
Uninformed
Disbelieved
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Node: Action Taken
Believer
Undecided
Uninformed
Disbelieved
5 steps later
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Proportion of Evacuated Nodes
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
Time steps
Proportion of Evacuated Nodes
Scenario 1Scenario 2 (differential 0.1)Scenario 3 (differential 0.1)Scenario 2 (differential 0.3)Scenario 3 (differential 0.3)
High trust in source 0.90High trust in same group
Equal trust
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Comparison of Scenarios
0
10
20
30
40
50
60
70
80
90
100
0 0.1 0.3
Trust Differential
Percent of Evacuated Nodes
Scenario 2
Scenario 3
High trust in source 0.90
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Proportion of Evacuated Nodes
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
Time steps
Proportion of Evacuated Nodes
Scenario 1
Scenario 2 (differential 0.1)
Scenario 3 (differential 0.1)
Scenario 2 (differential 0.3)
Scenario 3 (differential 0.3)
Moderate trust in source 0.80
High trust in same group
Equal trust
High trust in specific group
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Comparison of Scenarios
0
10
20
30
40
50
60
70
80
90
100
0 0.1 0.3
Trust Differential
Percent of Evacuated Nodes
Scenario 2
Scenario 3
Moderate trust in source 0.80
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Proportion of Evacuated Nodes
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
Time steps
Proportion of Evacuated Nodes
Scenario 1
Scenario 2 (differential 0.1)
Scenario 3 (differential 0.1)
Scenario 2 (differential 0.3)
Scenario 3 (differential 0.3)
Very high trust in source 0.99
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Comparison of Scenarios
0
10
20
30
40
50
60
70
80
90
100
0 0.1 0.3
Trust Differential
Percent of Evacuated Nodes
Scenario 2
Scenario 3
Very high trust in source 0.99
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Ongoing Work
• Explore effects of trust variants in sources
• Utilize multiple types of sources
• Vary information value of initial message
• Observe behavior in networks with different density and connectivity properties– Grid Network, Scale free Network
• Map simulation framework to actual cases
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Thank you. Questions?