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.

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. Polynomial algebra takes time
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4083656417280000000000*x**2+544847137497742374674104320000000000*x+4457869163092977175756800000000000)/
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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