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# Important spreaders in networks: exact results on small graphs

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To be able to control spreading phenomena (like the spreading of diseases and information) in networks it is important to identify influential spreaders. What "important" means depends on what is spreading and what kind of countermeasures that are available. In this work, we let the susceptible-infected-removed (SIR) model represent the spreading dynamics and contrast three different definitions of importance: Influence maximization (the expected outbreak size given a set of seed nodes), the effect of vaccination (how much deleting nodes would reduce the expected outbreak size) and sentinel surveillance (how early an outbreak could be detected with sensors at a set of nodes). We calculate the exact expressions of these quantities, as functions of the SIR parameters, for all connected graphs of three to seven nodes. We obtain the smallest graphs where the optimal node sets are not overlapping. We find that: node separation is more important than centrality for more than one active node, that vaccination and influence maximization are the most different aspects of importance, and that the three aspects are more similar when the infection rate is low. Furthermore, we discuss similar approaches to study the extinction times in the susceptible-infected- susceptible model.

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

1. 1. Important spreaders in networks: exact results on small graphs
2. 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. 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. 4. time Network epidemiology Step 2: Contact patterns
5. 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. 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. 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. 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. 9. 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10. 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. 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. 12. Small graphs
13. 13. Special “smallest” cases
14. 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. 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. 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. 17. Statistics for all graphs w N < 8
18. 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. 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. 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. 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. 22. Thank you! Collaborators: Jari Saramäki Naoki Masuda Nelly Litvak Luis Rocha Illustrations by: Mi Jin Lee