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Important spreaders in networks:
exact results on small graphs
Network epidemiology
Susceptible
meets
Infectious
Infectious
With some probability or rate
Susceptible or
Recovered
With s...
SIR model
Was proposed by Kermack–McKendrick 1927
Is usually formulated as a differential equation system.
ds
dt
= –βsi—
d...
time
Network epidemiology
Step 2: Contact patterns
Three types of importance
Petter Holme, Three faces of node importance in network
epidemiology: Exact results for small gr...
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
...
Three types of importance
Idea:
• Search for the smallest graph with where all three
notions of importance differ.
• Study...
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/(...
Polynomial algebra takes time
(37762366549514108074989296025600000000000*x**73+3314686580533426042655618661089280000000000...
Symbolic algebra
Coding progress:
• Started with SymPy (Python) general algebraic
expressions.
• Then used SymPy’s polynom...
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
Small graphs
Special “smallest” cases
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
I...
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....
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 ...
Statistics for all graphs w N < 8
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...
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 su...
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...
Summary
Paper:
• Found smallest connected graphs with three distinct
most important nodes.
• Degree is important for small...
Thank you!
Collaborators:
Jari Saramäki
Naoki Masuda
Nelly Litvak
Luis Rocha
Illustrations by:
Mi Jin Lee
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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. Polynomial algebra takes time (37762366549514108074989296025600000000000*x**73+3314686580533426042655618661089280000000000*x**72+141438610676500742111413237916368896000000000*x**71+3 911473306168632730171826549920825344000000000*x**70+78863281455383204006473293722572552273920000000*x**69+1236403293359232085532901156240802856853504000 000*x**68+15699393806584508589027640185718259048113766400000*x**67+166047926815157089435015605011671368201261465600000*x**66+149321074471363905237290341 2763316426944533092352000*x**65+11596802000132949850753533289466811302954899065292800*x**64+78746554444636009114113624901619589833746005539712000*x**63+ 472189336478744088675459614584162673391547254471037440*x**62+2520838579589225935326332345418749193680539093862025984*x**61+12064010950190968998503507349 103010126875638928117001472*x**60+52056956992255933979233520314531580309411479877375753088*x**59+2035496785634842434895513764833898462626245464907560146 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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

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