Targeting Influential Nodes for Recovery in Bootstrap Percolation on Hyperbolic Networks
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1 Targeting Influential Nodes for Recovery in Bootstrap Percolation on Hyperbolic Networks Christine Marshall Supervisors Colm O Riordan and James Cruickshank
2 Overview Agent based modelling of dynamic processes on complex networks Spatial effect of a network on the spread of a process Hyperbolic random geometric graphs Bootstrap percolation Introducing Bootstrap Percolation with Recovery Introducing recovery delays percolation, and the effect is more significant if we target nodes of high degree centrality over random selection.
3 Bootstrap Percolation The process where an activity spreads if the number of your active neighbours is greater than a tipping point. Can be used to model social reinforcement: Spread of opinions Voter dynamics Adoption of new trends Viral marketing
4 Simulating bootstrap percolation Bootstrap Percolation Activation Threshold Selection of active seed set Update Rule Active Inactive
5 Conceptual Framework for Bootstrap Percolation with Recovery Standard Bootstrap Inactive to Active Bootstrap Percolation with recovery Inactive to Active Active to Inactive Targeted percentage of Active nodes of highest degree centrality Percentage of randomly selected Active nodes Motivation Small scale random attack in network, which nodes can we target to obstruct the spread of activity
6 Random Geometric Graphs Distance Graphs Spread of Forest Fire Wireless ad-hoc and sensor networks
7 Different Geometric Spaces Euclidean disc Hyperbolic disc M.C. Escher Circle Limit IV 1960
8 Hyperbolic Random Geometric Graphs Hyperbolic random geometric graph, with edge density of Krioukov et al., Hyperbolic Geometry of Complex Networks, 2010
9 Application of Hyperbolic Geometric Graph Models Modelling the internet graph Snapshot of Internet connectivity K.C. Claffy
10 Research Questions In Bootstrap Percolation: As we increase the number of edges in the hyperbolic graphs, is it possible to identify a threshold between the complete spread of activity and the failure to percolate? If we modify the rules in Bootstrap Percolation and allow recovery from active to inactive state, will this impact the spread of activity? If we selectively target active nodes with high degree centrality, will this have a greater impact?
11 Experimental Set-Up Utilising the same set of hyperbolic geometric graphs for all simulations (1000 nodes) Agent based modelling of Bootstrap Percolation 20 random seeds Activation Threshold 2 10 Repeat activation mechanism at each time step until equilibrium Count number of Final Active Nodes Agent based modelling of Bootstrap Percolation with Recovery Activation followed by % recovery at each time step (10 90%) Targeted recovery based on top ranked node degree centrality Random recovery
12 Increasing Edge Density Results: Bootstrap Percolation
13 Results: Bootstrap with Recovery
14 Current Work Selectively target nodes with highly skewed graph properties for recovery In the hyperbolic graphs: centralisation measures clustering coefficients Immunisation of nodes with certain properties, to observe the effect on the spread of the activity
15 Increasing Edge Density Recap: Bootstrap Percolation
16 Number of Simulations with this outcome Bootstrap Percolation Graph 5.7_13, Bootstrap Percolation AT = AT = AT = AT = AT = 3 AT = 2 AT = 10 AT = 9 AT = 8 AT = 7 AT = 6 AT = 5 AT = 4 AT = 6 AT = 7 AT = 8 AT = 9 AT = 10 Number of Final Active at Equilibrium
17 Number of Simulations with this outcome Bootstrap percolation: Random Recovery Graph 5.7_13, 25 nodes immunised at Random AT = AT = AT = AT = AT = 3 AT = 2 AT = 10 AT = 9 AT = 8 AT = 7 AT = 6 AT = 5 AT = 4 AT = 6 AT = 7 AT = 8 AT = 9 AT = 10 Number of Final Active at Equilibrium
18 Number of Simulations with this outcome Bootstrap Percolation :Targeted recovery Graph 5.7_13, 25 nodes immunised for High Degree AT = AT = AT = AT = AT = 3 AT = 2 AT = 10 AT = 9 AT = 8 AT = 7 AT = 6 AT = 5 AT = 4 AT = 6 AT = 7 AT = 8 AT = 9 AT = 10 Number of Final Active at Equilibrium
19 Rate of Decline in Percolating Simulations
20 Future Work Repeat these experiments on more graphs at the threshold, varying target graph properties In the simulations that fail to percolate: investigate the link between the set of active seeds and the targeted nodes investigate local neighbourhood properties Repeat simulations: Euclidean random geometric graphs in the unit disc Erdős Rényi random graphs
21 Thank You
22 Top Ranked Node Degree Centrality Scores R = 5.7_13 graph properties Density Average Degree Diameter 3 WS CC Transitivity Size component 1000 Degree centralisation Betweenness centralisation Closeness centralisation Average shortest path Number of lines
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