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

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

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

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

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

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.

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

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

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

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

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.

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

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

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

Simulation Results

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

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

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

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

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

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

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.

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

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

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}

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}