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12.
network fragmentation =
network dismantling =
attack vulnerability =
…
Find nodes that break a network into as
small pieces as possible.
Too slow for exact calculations on large networks.
13.
0 1 3 52
n
4
2
4
6
8
S
unconstrainedsequential
0
Networkfragmentation
WithAlexanderVertemyev,arXiv:later
22.
Ramsey numbers
The Ramsey number R(r,s) is the smallest size of a graph such that one is guaranteed to
find either a clique of r vertices or an independent set of s vertices.
R(3,3) = 6
23.
R(5,5) please. Or
we destroy Earth!
Mobilize all
computers and
mathematicians and
let’s figure it out.
24.
R(6,6) please. Or
we destroy Earth!
Leave it to the military
and hope for the best.
25.
To ask other questions
than what we usually do.
Argument #6
26.
Can we ask the same question about node
importance in epidemiology?
P. Holme, Three faces of node importance in network epidemiology:
Exact results for small graphs. Phys. Rev. E 96, 062305 (2017).
27.
Three
types
of
importance
Influence maximization:
Important = able to start large outbreak.
Vaccination:
Important = reduce outbreaks much if deleted.
Sentinel surveillance:
Important = getting infected reliably and early.
28.
Outline
1. Calculate the three node importances
exactly (as a function of infection rate (for
the standard, Markovian SIR model)).
2. Do it for every graph up to 7 nodes.
3. Find the smallest one where all three
differs, for 1,2,3 important nodes.
4. (Find structural predictors for the
important nodes.)
29.
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
34.
Predicting
important
nodes from
centralities
alone
Can we predict the importance of a node
if we just know the size of it’s graph and
its centrality values? (Not the graph
itself.)
D. Bucur, P. Holme, Beyond ranking
nodes: Predicting epidemic outbreak sizes
by network centralities, arXiv:1909.10021
Setup: Exact SIR on small (N < 11)
graphs for fixed β.
35.
Predicting
important
nodes from
centralities
alone Answer: Yeah, but it depends a bit on β.
36.
P. Holme, L. Tupikina,
Epidemic extinction in networks: Insights
from the 12,110 smallest graphs.
New J. Phys. 30, 113042 (2018).
After
(SI)R
comes
(SI)S
37.
To ask the same questions
as we usually do.
Argument #7
38.
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
Automorphically
equivalent configurations Recovery event
Infection event
SIS as a random walk in the space of configurations
Configurations
(binary coded)
41.
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)
For large β, x = uβN–1, u ≈ u0Mα.
x ≈ a(bβM)N–1, a = 126…, b = 0.0268…
42.
1. Real networks are sometimes small.
2. We use them to reason about networks.
3. Only ones that can be studied with slow algorithms.
4. To use all connected graphs as reference model.
6. To ask other questions than what we usually do.
7. To ask the same questions as we usually do.
5. Understanding small graphs is challenging.
Wrap up
43.
Thank you!
Doina Bucur
U Twente
Liubov Tupikina
CRI Paris
Alexander Veremyev
U Central Florida
Funding:
Tokyo Tech WRHI
JSPS
Sumitomo Foundation
Homepage:
petterhol.me
Collaborators: