The big science
of small networks
Petter Holme
7 arguments why you should start
studying small networks

Because real networks are
sometimes small.
Argument #1

Famiano,Boyd,Kajino,
Astrophys.J.57689–100(2002).

McDonald,Pizzari,Structureofsexual
networksdeterminestheoperationofsexual
selection,PNAS115,E53-E61(2018)

Pierce,Cushman&Hood(1912),Theinsect
enemiesofthecottonbollweevil.USDepartment
ofAgricultureBureauofEntomologyBulletin
100:99.

Harris&Ross(1955)

H Corley, H Chang, 1974. Finding the n most vital nodes in a
flow network. Management Science 21(3):362–364.

WWZachary,Aninformationflowmodelfor
conflictandfissioninsmallgroups,Journalof
AnthropologicalResearch33(4),452–473(1977)

Because that’s what we use to
reason about networks
Argument #2

Girvan, Newman PNAS 99, 7821–7826 (2002)

To be exact; when exact is slow.
Argument #3

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.

0 1 3 52
n
4
2
4
6
8
S
unconstrainedsequential
0
Networkfragmentation
WithAlexanderVertemyev,arXiv:later

Small
connected
graphs

N no. connected graphs
3 2
4 6
5 20
6 112
7 853
8 11,117
9 261,080
10 12,005,168
11 1,018,997,864
users.cecs.anu.edu.au/~bdm/data/graphs.html
Small
connected
graphs

1
0.1
10–3
10–4
10–5
10–2
9876543
NR=4
NR=3
NR=2
NR = 1NR = 0
N
fractionofgraphs
10–6
10–7
10
NR = 5
Network fragmentation
With Alexander Veremyev, arXiv:later

To understand the structure
imposed by simple graphs being
connected.
Argument #4

k 1 2 3 4 5 6 7 8
P(k) 0.034 0.111 0.213 0.266 0.218 0.116 0.037 0.006
degree distribution, N = 9
link density distribution, N = 8,9,10

Understanding small
networks can be hard.
Argument #5

We do these things
not because they are easy
but because they are hard
John F. Kennedy

Ishimatsuetal.,JSpacecraft&
Rockets53(2016),25–38.

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

R(5,5) please. Or
we destroy Earth!
Mobilize all
computers and
mathematicians and
let’s figure it out.

R(6,6) please. Or
we destroy Earth!
Leave it to the military
and hope for the best.

To ask other questions
than what we usually do.
Argument #6

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).

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.

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.)

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

5215240768500990172474739886280840*x^29+814217654548875
748959313663642099659619418917821190195217427712*x^25+2
1779935206129739114397096060550049079944195609861388288
09288755404718390985128907159566824759100622961672192*x
0050365501253044671260380916000122470400*x^13+260814450
9964656435200*x^9+3095792827574688435407884180504098172
70240000000*x^4+127664354144688371448545909145600000000
3+137923803520060037223899260256256000000000000*x^72+371
821632716800000000*x^68+1444990236309934632258974854496
60000*x^64+38608474753961205884506547443891511381260652
8000*x^60+161395707996900193288033181079336181640030323
6348409241600*x^56+152948842847185271651113676395244743
41506890636069960682414182400*x^52+40471862325733202532
Exact calculations

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
β-interval = [(1+√5)/2,(3+√17)/4]
[1.62..,1.78..]
Smallest graph for the case of one
important node

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
Special
graphs 2
Smallest graph for the case of one
important node

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
Special
graphs 3
The most complex behavior . . .

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.

Predicting
important
nodes from
centralities
alone Answer: Yeah, but it depends a bit on β.

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

To ask the same questions
as we usually do.
Argument #7

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)

Yx + 1 = 0
An example:
o–o–o

An example:
o–o–o
For large β, x = uβN–1

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…

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

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: