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.

Outline

• Problem

• Past Work

• Model

• Axioms

• Simulation Experiments

• Ongoing Work

Problem

Warnings in Evacuation Situations

Past Work

Diffusion Models

Dynamic Social Network

Social Network Structure

Interaction layer

Social layer

Physical layer

Node Characteristics

Source

Individual NodesThresholds

Characteristics

Me My Friend

My Mother

Media

Stranger

Characteristics

Me My Friend

My Mother

Media

Stranger

t3

t4

t2t1

ts

ts

Interactions

Me

My Friend

My Mother

Media

Stranger

{S,V}

{S,V}

{S,V}

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

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

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

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|

∏

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

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

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

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

Node: Believer State

Believer

Undecided

Uninformed

Disbelieved

Node: Action Taken

Believer

Undecided

Uninformed

Disbelieved

5 steps later

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

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

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

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

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

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

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

Thank you. Questions?