4. P. Holme, Three faces of node importance in
network epidemiology: Exact results for small
graphs. Phys. Rev. E 96, 062305 (2017).
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.
7. 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
8. 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
9. 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
10. 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.
13. 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
18. 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).
25. 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.
26. P. Holme, L. Tupikina, Extinction in the
susceptible-infected-susceptible model: Exact
results for small graphs. arXiv:1801.????.
32. 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 . . .
33. 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 . . .
34. 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.
35. 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
36. 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