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Important spreaders in networks: Exact results for small graphs

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Presentation at NetSci-X Jan 7, 2018

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Important spreaders in networks: Exact results for small graphs

  1. 1. Important spreaders in networks Exact results for small graphs
  2. 2. Network epidemiology Step 1: Compartmental models
  3. 3. Network epidemiology Step 2: Contact patterns
  4. 4. P. Holme, Three faces of node importance in network epidemiology: Exact results for small graphs. Phys. Rev. E 96, 062305 (2017).
  5. 5. P. Holme. Three faces of node importance in network epidemiology: Exact results for small graphs. Phys. Rev. E 96: 062305 (2017). Inspiration •F. Radicchi & C. Castellano. Fundamental difference between superblockers and superspreaders in networks. Phys. Rev. E 95:012318 (2017). •U. Brandes & J. Hildenbrand. Smallest graphs with distinct singleton centers. Network Science 2:416–418 (2014). •H. Kim, S. H. Lee & P. Holme. Building blocks of the basin stability of power grids. Phys. Rev. E 93:062318 (2016). •Y. Bai & al. Optimizing sentinel surveillance in temporal network epidemiology. Scientific Reports 7:4804 (2017). Reference & inspiration
  6. 6. Three types of importance Influence maximization Vaccination Sentinel surveillance If removing (vaccinating) i reduces the outbreak size much, then i is important. If starting the epidemics at i tends to create large outbreaks, then i is important. If i tends to get infected early, then i is important. RATIONALES
  7. 7. Three types of importance Influence maximization Vaccination Sentinel surveillance Expected outbreak size for outbreaks starting at i. Expected outbreak size (starting anywhere) when i is removed. Expected time to extinction or reaching i. MEASURES
  8. 8. 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 6 6 6 influence maximization vaccination sentinel surveillance Three types of importance
  9. 9. 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, we can’t use stochastic simulations.
  10. 10. 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
  11. 11. 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977957429633622398998374400*x**29+73426181385511818150687126779106893812232809076914963968000*x**28+32193666885597296408810155132979901682101751270559157 120000*x**27+13231385437288999412775352581513090540713728729205858304000*x**26+5092979660658839210923714266447345185896748823509017728000*x**25+183413416 0579196914536664193085454779685907374914645504000*x**24+617280741609768333102889676022554200777224978653263872000*x**23+193891876858081918136769045052956 Polynomial algebra takes time
  12. 12. 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
  13. 13. Small graphs
  14. 14. Small graphs N no. connected graphs 3 2 4 6 5 20 6 112 7 853 8 11,117 http://users.cecs.anu.edu.au/~bdm/data/graphs.html
  15. 15. Special “smallest” cases
  16. 16. Special 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
  17. 17. Interlude n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 J. Gu, S. Lee, J. Saramäki & P. Holme. Ranking influential spreaders is an ill-defined problem. EPL 118:68002 (2017).
  18. 18. Special 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
  19. 19. Special 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
  20. 20. Statistics for all graphs w N < 8
  21. 21. Overlap 0.1 1 10 100 0.1 1 10 100 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 Influence maximization vs. vaccination Vaccination vs. sentinel surveillance 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.7 0.8 0.9 1
  22. 22. Structural explanations 0.1 1 10 100 Influence maximization Vaccination Sentinel surveillance 0.1 1 10 100 0.1 1 10 100 0.1 1 10 100 0.1 1 10 100 0.1 1 10 100 0.1 1 10 100 0.1 1 10 100 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 0.6 0.66 0.7 0.75 β β β β β β β β β 4.46 4.48 4.5 4.52 4.54 4.56 4.58 2.5 3 3.5 4 2.5 3 3.5 4 0.82 0.825 0.83 0.6 0.66 0.7 0.74 0.8 1.18 1.19 1.2 1.21 1.22 1 1.02 1.04 1.06 1.08 1 1.05 1.1
  23. 23. 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 β β (b) n = 3(a) n = 2 d d Vaccination Sentinel surveillance Influence maximization
  24. 24. 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.
  25. 25. P. Holme, L. Tupikina, Extinction in the susceptible-infected-susceptible model: Exact results for small graphs. arXiv:1801.????.
  26. 26. 1/(2β+1) 1/(β+1) 1/(β+2) 1/(β+1) 1/(β+2) 1/3 1/3 β/(β+2) β/(β+1) β/(β+1) β/(2β+1) 0 4 1 2 5 6 3 7 An example: o–o–o Absorbing state Automorphic configurations Recovery event Infection event SIS as a random walk in the space of configurations Configurations
 (binary coded)
  27. 27. An example: o–o–o Time to extinction from configuration s, xs = Expected time in configuration s + ∑t xt × Prob(s → t)
  28. 28. An example: o–o–o Yx + 1 = 0
  29. 29. An example: o–o–o
  30. 30. β 2 4 6 8 1 2 3 40 0 10 1,4 2 3 ,6 7 5 7>3,6>5>2>1,4 7>5>3,6>2>1,4 x An example: o–o–o
  31. 31. Ranking rules If all nodes are equivalent, then the extinction-time ranking is independent of β. Otherwise there are pairs of configurations that change order depending β. Except . . .
  32. 32. M=8M=12M=20 larger x for large βlarger x for small β β*=8.394…β*=3.890…β*=2.407… Ranking rules 97844723712 β28 + 2019406381056 β27 + 20485144313856 β26 + 136322491613184 β25 + 670461968908288 β24 + … 97844723712 β28 + 2019406381056 β27 + 20485144313856 β26 + 136322491613184 β25 + 670455853613056 β24 + … Exceptions . . .
  33. 33. In general N = 3 N = 4 N = 5 N = 6 N = 7N =8 0.01 0.1 1 10 100 105 202 2 3 4 5 6 7 8 9 3 4 5 6 7 8 3 4 5 6 7 8 M u N N 10–2 10–4 10–6 10–8 10–9 10–7 10–5 10–3 u0 α (a) (b) (c) Given a graph, for large β, x = uβN–1, u = u0Mα.
  34. 34. In general x ≈ a(bβM)N–1, a = 126…, b = 0.0268… Kendall’s τ Clustering coefficient –0.667 Degree assortativity 0.191 Averaged distance –0.309 S.d. of degrees –0.751
  35. 35. Thank you! My epi collaborators: Liubov Tupikina, École Polytechnique Naoki Masuda, Bristol U 白媛,吉林大学 陶丽,西南大学 Nelly Litvak, U of Twente Jari Saramäki, Aalto U Jain Gu, Sungmin Lee, Sungkyunkwan U Luis Rocha, Karolinska Institute Illustrations by: Mi Jin Lee

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