Social Influence in Online Social Networks Toward Understanding Spatial Dependence on Epidemic Thresholds in Networks Dr. Zesheng Chen Viral marketing ( word-of-mouth ) Blog information cascading Rumor spreading Bear a resemblance to epidemic process! Indiana University Purdue University Fort Wayne 2 Epidemic Process Epidemic process is a process that information selfpropagates across networks Epidemiological Models Susceptible-infected-recovered (SIR) model for independent cascading influence spread Susceptible-infected-susceptible (SIS) model for blog information cascading Susceptible-infected-cured (SIC) model for rumor and anti-rumor propagation 3 4
Epidemic Thresholds Fundamental metric used to evaluate epidemic spread Condition on which an information will either die out or become epidemic In SIS model, Birth rate Ratio between birth rate and death rate β/ δ > epidemic threshold, become epidemic State of the Art τ= λ ( ) λ max (A) is the largest eigenvalue of the adjacency matrix A of the network Assume that the status of nodes in the network are independent of each other! β/ δ <= epidemic threshold, die out 5 6 Questions Can spatial dependence among nodes affect the epidemic threshold? If so, how significantly? How can we derive a more accurate epidemic threshold, taking into consideration a certain spatial dependence? Can the birth rate and the death rate affect the spatial dependence and thus the epidemic threshold? If so, how? 7 Outline Epidemic Thresholds in Regular Graphs Epidemic Thresholds in Arbitrary Networks 8
X i (t): status of node i at time t, ()=, = + = =) ()= + = =) + = =( =) + ( =) () 9 Outline Epidemic Thresholds in Regular Graphs Epidemic Thresholds in Arbitrary Networks Epidemic Thresholds in Regular Graphs It has been shown that the epidemic threshold proposed in previous work does not work well in regular graphs. More importantly, due to the symmetric property of regular graphs, we can derive a closed-form expression for the epidemic threshold. 2
Independent Model Assume spatial independence between nodes Markov Model Assume spatial Markov dependence Inspired by the local Markov property of Markov Random Field!, = " where k is the average nodal degree and the largest eigenvalue of adjacency matrix 3 "($%), 4 Setup Simulator is based on discrete time and random number generator Run times for each scenario Run long enough so that it reaches the steady state (e.g., 2 time steps) Assign half of nodes to be infected initially Number of infected nodes 2-2 Sample Run of Epidemic Spread in a Ring Graph δ=. and V = 5% 25% 5% 75% 95% Mean 2 4 6 8 2 Time step Number of infected nodes 2-2 5% 25% 5% 75% 95% Mean 2 4 6 8 2 Time step Number of infected nodes 2 5% 25% 5% 75% 95% Mean 2 4 6 8 2 Time step β =.3 β =.4 β =.5 5 6
Performance Evaluation Outline.6.4.8.6.4.2 τ c,mar.5.45.4.35.3.25 τ c,mar.2.2.5..5 τ c,mar Epidemic Thresholds in Regular Graphs Epidemic Thresholds in Arbitrary Networks Ring ( V = ) Lattice ( V = 25) Complete ( V = ) 7 8!, = Conjecture Epidemic Thresholds of Arbitrary Network λ ( ) λ ( )( % ) "( %).6.4.8.6.4.2 Performance Evaluation ( V = ) τ c,cor.5..5 τ c,cor.2.5..5 τ c,cor Ring (λ max (A) = 2) ER random (λ max (A) = 9.3) Power law (λ max (A) =.77) 9 2
!, = Conclusions and Future Work λ ( ) λ ( )( % ) Thanks for your attention Future work Derive the equation for epidemic thresholds in irregular graphs Apply our observations for predicting and controlling the dynamics of the epidemic spreading process 2 22 Relative Errors Relative errors.8.6.4 ε mar (Lattice) ε mar (Lattice) (Complete) ε mar (Complete).2 23 24
Correlation Coefficients at Epidemic Thresholds in Ring Graph ( V = ).7 25 Correlation coefficient.6.5.4.3.2. Empirical Markov model 26 Relative Errors Relative errors.8.6.4 ε cor (ER random) ε cor (ER random) (Power-law) ε cor (Power-law).2 27