# Important spreaders in networks: exact results on small graphs

Oct. 18, 2017                      1 of 22

### Important spreaders in networks: exact results on small graphs

1. Important spreaders in networks: exact results on small graphs
2. Network epidemiology Susceptible meets Infectious Infectious With some probability or rate Susceptible or Recovered With some rate or after some time Step 1: Compartmental models
3. SIR model Was proposed by Kermack–McKendrick 1927 Is usually formulated as a differential equation system. ds dt = –βsi— di dt = βsi – νi— = νidr dt — Ω = r(∞) = 1 – exp[–R₀ Ω] where R₀ = β/ν Ω > 0 if and only if R₀ > 1 The epidemic threshold
4. time Network epidemiology Step 2: Contact patterns
5. Three types of importance Petter Holme, Three faces of node importance in network epidemiology: Exact results for small graphs, arxiv: 1708.06456. Inspiration: • F. Radicchi and C. Castellano. Fundamental difference between superblockers and superspreaders in networks. Phys. Rev. E, 95:012318 (2017). • U. Brandes and J. Hildenbrand. Smallest graphs with distinct singleton centers. Network Science, 2(3):416–418 (2014).
6. 7 susceptible infectious recovered t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 0 2 6 4 7 77 0 1 1 2 2 3 4 5 55 (a) (b) (c) (d)6 6 6 influence maximization vaccinization sentinel surveillance Three types of importance
7. Three types of importance Idea: • Search for the smallest graph with where all three notions of importance differ. • Study statistics of node importance vs centrality etc over all small graphs. To do that, I can’t use stochastic simulations.
8. susceptible infectious recovered sentinel β/(2β+1) β/(2β+1) 1/(2β+1) β/(β+1) 1/(2β+2) 1/(2β+2) β/(β+1) β/(β+1) 1/(β+1) 1/(β+1) 1/(β+1)1/(β+1) 1/(2β+2) 1/(2β+1) 1 2 3 4 5 6 7 Exact calculations probability of infection chain time of infection chain contribution to avg. time to extinction
9. 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10. Symbolic algebra Coding progress: • Started with SymPy (Python) general algebraic expressions. • Then used SymPy’s polynomial package (100 times faster). • Then FLINT (C) 10000–100000 times faster. • Then eliminating isomorphic branches of the tree (10 times faster). https://github.com/pholme/exact-importance
11. Small graphs N no. connected graphs 3 2 4 6 5 20 6 112 7 853 http://users.cecs.anu.edu.au/~bdm/data/graphs.html
12. Small graphs
13. Special “smallest” cases
14. Smallest graphs 1 6 6 6 51 12 1 4 5 6 7 3 1 2 3 4 5 6 7 0.1 1 10 0.2 0.4 0.6 0.8 1 1.2 0.1 1 10 1 2 3 4 5 0.1 1 10 β β β Influence maximization Vaccination Sentinel surveillance Ω Ω τ [(1+√5)/2,(3+√17)/4] [1.62..,1.78..] β-interval
15. Smallest graphs 2 34 14,23 12 56 3456 21 3 6 5 4 Influence maximization 3 4 5 0.1 1 10 1 1.5 2 2.5 0.1 1 10 0.1 0.2 0.3 0.4 0.5 0.6 0.1 1 10 0.0 0.7 2 6 Sentinel surveillance Vaccination β β β Ω Ω τ
16. Smallest graphs 3 7 1 6 75 1 6 751 6 1 2 3 4 5 0.1 1 10 1 2 3 4 5 6 7 0.1 1 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0.1 1 10 326 3 2 5 3 2 7 5 Sentinel surveillance VaccinationInfluence maximization Ω Ω τ 2 1 4 5 6 7 3 β β β
17. Statistics for all graphs w N < 8
18. Overlap 0.8 0.85 0.9 0.95 1 0.1 1 10 100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 1 10 100 0.4 0.5 0.6 0.7 0.8 0.1 1 10 100 Sentinel surveillance vs. influence maximization β β β (a) n = 1 (b) n = 2 (c) n = 3 J J J Influencemaximizationvs.vaccination Vaccination vs. sentinel surveillance
19. Structural explanations 3.85 3.86 3.87 3.88 3.89 3.9 3.91 3.92 0.1 1 10 100 Influence maximization Vaccination Sentinel surveillance 0.78 0.781 0.782 0.783 0.784 0.785 0.786 0.787 0.1 1 10 100 1.41 1.42 1.43 1.44 1.45 1.46 0.1 1 10 100 2 2.5 3 3.5 0.1 1 10 100 0.55 0.6 0.65 0.7 0.1 1 10 100 1 1.05 1.1 1.15 1.2 1.25 0.1 1 10 100 1.8 2 2.2 2.4 2.6 2.8 3 3.2 0.1 1 10 100 0.55 0.6 0.65 0.1 1 10 100 1 1.05 1.1 1.15 0.1 1 10 100 k k k c c c v v v (d) n = 2 (e) n = 2 (f) n = 2 (a) n = 1 (b) n = 1 (c) n = 1 (g) n = 3 (h) n = 3 (i) n = 3 β β β β β β β β β
20. Structural explanations 1.5 2 2.5 3 0.1 1 10 100 1.6 1.8 2 2.2 2.4 0.1 1 10 100 Vaccination Sentinelsurveillance β β (b) n = 3 (a) n = 2 d d
21. Summary Paper: • Found smallest connected graphs with three distinct most important nodes. • Degree is important for small β. • Vitality is important for vaccination. • With more than one active node, the separation matters for influence maximization and sentinel surveillance. Myself: • Learned efficient symbolic computation. • Graph isomorphism. • How to enumerate small graphs.
22. Thank you! Collaborators: Jari Saramäki Naoki Masuda Nelly Litvak Luis Rocha Illustrations by: Mi Jin Lee