Important spreaders in networks: exact results on small graphs
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The document discusses network epidemiology, focusing on the SIR model and the importance of node roles in epidemic spreading, emphasizing superblockers and superspreaders. It also highlights the need for exact calculations on small graphs to understand infection dynamics and node importance versus centrality. Additionally, advancements in symbolic algebra coding for improved computational efficiency in analyzing small graphs are outlined.
![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](https://image.slidesharecdn.com/myworkonsiretc-171018000948/85/Important-spreaders-in-networks-exact-results-on-small-graphs-3-320.jpg)
![Smallest graphs 1
6 6
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51
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1
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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](https://image.slidesharecdn.com/myworkonsiretc-171018000948/85/Important-spreaders-in-networks-exact-results-on-small-graphs-14-320.jpg)
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
- 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