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Important
spreaders
in
networks
Exact
results
for
small
graphs
Network epidemiology
Step 1: Compartmental models
Network epidemiology
Step 2: Contact patterns
P. Holme, Three faces of node importance in
network epidemiology: Exact results for small
graphs. Phys. Rev. E 96, 062305 (2017).
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
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
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
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
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.
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
(37762366549514108074989296025600000000000*x**73+3314686580533426042655618661089280000000000*x**72+141438610676500742111413237916368896000000000*x**71+39
11473306168632730171826549920825344000000000*x**70+78863281455383204006473293722572552273920000000*x**69+123640329335923208553290115624080285685350400000
0*x**68+15699393806584508589027640185718259048113766400000*x**67+166047926815157089435015605011671368201261465600000*x**66+149321074471363905237290341276
3316426944533092352000*x**65+11596802000132949850753533289466811302954899065292800*x**64+78746554444636009114113624901619589833746005539712000*x**63+4721
89336478744088675459614584162673391547254471037440*x**62+2520838579589225935326332345418749193680539093862025984*x**61+1206401095019096899850350734910301
0126875638928117001472*x**60+52056956992255933979233520314531580309411479877375753088*x**59+203549678563484243489551376483389846262624546490756014656*x**
58+724337780103561588309914084265886452050608182621681489696*x**57+2354656050497226961862844168528304543472210933547719071080*x**56+701564923449842455238
4686112745652381839054053136427666012*x**55+19214703345853832788682064160120433484916446461152221422090*x**54+4850135147907566882864350378633066165848696
3404832259702487*x**53+113092671813635674828611367354176714434189107315168498776928*x**52+244103956674073644958141865118010803740486170105474665375344*x*
*51+488630192461738524748488108810018093865519958120561303765494*x**50+908600820200290624573587634392204246241182209329371400281145*x**49+157181419219371
3924618282713564682212894757971214065300446684*x**48+2533060255249516888774685502865019709652152513622605055279456*x**47+38073806110978559201473406091992
02541615833608774939794319328*x**46+5343283033579917464170385365564015741459363191088389767306398*x**45+7008169777451585300330775246188016187941023359601
556223762760*x**44+8597683922746974615895184605717665402049163153921524311480288*x**43+9873250006411367799034708683209351418045241428523260095680092*x**4
2+10619889579823429590519175999126775450296264698254861098419138*x**41+10705343531834807255308847480339463995272891980674623472362352*x**40+1011820070560
8779929491741668401657943488155430966316422413564*x**39+8970001049997570057132806423909486209951952062243977752430714*x**38+74609950378828426201573472002
24840242010916136536945407182547*x**37+5823838496430434727894405840870770563861889084747019155124352*x**36+4266664919506791073548228416349608210702346234
200930403133168*x**35+2933984529167138491682532365556698639754779991550409622936110*x**34+1893674869460884091813696066793009247613602165512467381565973*x
**33+1147053826336232131835158650815804999309595169018523338347692*x**32+651941666849429144426327239177240288949251173249669277824296*x**31+3475826020896
88677496074681434250008936206946931462543001436*x**30+173768006766411435824870505215240768500990172474739886280840*x**29+81421765454887540793714432533619
870680132300264456583686136*x**28+35737339262158151938098902744329827302836180844476824581984*x**27+14683162114404405343010776871915847702033756436713414
727552*x**26+5642689559748959313663642099659619418917821190195217427712*x**25+2026374779204279782843992193888235999388342744875618924160*x**24+6792932857
10471972392728540425835464747889906016677284352*x**23+212308973769516687666161412820684609317746546465988738048*x**22+61779935206129739114397096060550049
079944195609861388288*x**21+16711103480959516893824501251494340145597563371003242496*x**20+4194274389585662750224055987494263142843645525992972288*x**19+
974786785480329873601482567083721717912488928481935360*x**18+209288755404718390985128907159566824759100622961672192*x**17+4140027981206974066401608136057
4723512184740799807488*x**16+7522283805109866955995503020206903760121954966568960*x**15+1250984582973454622568956659303179360768874779770880*x**14+189642
358940050365501253044671260380916000122470400*x**13+26081445034209637896534715167941811495312398745600*x**12+32360118051512661806783245669231951358881431
55200*x**11+359819252450753508595254072745238613601340620800*x**10+35570023038312915426349438493625919964656435200*x**9+309579282757468843540788418050409
8172305408000*x**8+234354685053410645368523826403557664358400000*x**7+15193743364805385914791764255439847424000000*x**6+826607123241007367179342411796054
016000000*x**5+36698205721449251600489578223370240000000*x**4+1276643541446883714485459091456000000000*x**3+32631385950937894083656417280000000000*x**2+5
44847137497742374674104320000000000*x+4457869163092977175756800000000000)/
(38849174012498197907452723200000000000000*x**74+3318598191984295465296750182400000000000000*x**73+137923803520060037223899260256256000000000000*x**72+37
18836854788115338745516774129664000000000000*x**71+73187564029226708723444458146842542080000000000*x**70+112144643762880008904189426236104114176000000000
0*x**69+13937137779902055729239770100875821632716800000000*x**68+144499023630993463225897485449622876114124800000000*x**67+127587886738453505303322947204
4962110670635008000000*x**66+9746370095122882492430762538399854353652121600000000*x**65+65215495588078294956304754695697556926039599349760000*x**64+38608
4747539612058845065474438915113812606522490880000*x**63+2039019743945197787806635323175732162985805873361715200*x**62+96731206795148096408228338063727564
48932826769653760000*x**61+41463182796652561547765945470954864465553792226951168000*x**60+161395707996900193288033181079336181640030323993431244800*x**59
+572985959395474247469239080818191224916889350617733529600*x**58+1862374267840213513898298449997074233654470813402085785600*x**57+55604300310863812527462
69930110412997502372446348409241600*x**56+15294884284718527165111367639524474394242463478570586521600*x**55+388609816681386665291919116321029736792678492
16163518873600*x**54+91415494941532261154338183845834115681983965207741203353600*x**53+199509970977818775632987267744641506890636069960682414182400*x**52
+404718623257332025328917625285264020089712651667338146585600*x**51+764374715419689922968327485500214314436229021737343580569600*x**50+134607566834801001
0438588949674037750529766839631942722380800*x**49+2213199045980729092667047603056735385248916780496177533440000*x**48+34015412741410347535053377730666494
07560854431902037976320000*x**47+4892103780283080871288385104328530226580371128828639113216000*x**46+6590054588677667222257130588148263220931392553656135
566592000*x**45+8321790039667741773143733704030351277421859439309217460736000*x**44+9858144123056187532156937373964423433092276532602402109568000*x**43+1
0962218518250821245562546207482370427489203414972878535065600*x**42+11448825928895319238165607419280388858411242066332743600000000*x**41+1123514915641134
6177828271421184433780217549468847670036992000*x**40+10363622140030410343302319664329679035025469377574033155174400*x**39+8988410444547910032150577099030
176615855802547117158448332800*x**38+7331310976398177080061080847301200859574935432860186655948800*x**37+562421796080909180282253163057672610489121308991
0954236620800*x**36+4058302463395757204313533365763872433751951130071815225164800*x**35+2754317595855997084920872560792429784784872162402632133836800*x**
34+1758011360746200302693219873297151407720017161990685383244800*x**33+1055073126700485634917761645630646824351646318878099638579200*x**32+59521692360865
2456635301929860947101931138207761273530828800*x**31+315529879888969797201425410449602612378143848550150479052800*x**30+157101120109380703527501506578358
977957429633622398998374400*x**29+73426181385511818150687126779106893812232809076914963968000*x**28+32193666885597296408810155132979901682101751270559157
120000*x**27+13231385437288999412775352581513090540713728729205858304000*x**26+5092979660658839210923714266447345185896748823509017728000*x**25+183413416
0579196914536664193085454779685907374914645504000*x**24+617280741609768333102889676022554200777224978653263872000*x**23+193891876858081918136769045052956
Polynomial algebra takes time
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
Small graphs
Small graphs
N no. connected
graphs
3 2
4 6
5 20
6 112
7 853
8 11,117
http://users.cecs.anu.edu.au/~bdm/data/graphs.html
Special “smallest” cases
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
[(1+√5)/2,(3+√17)/4]
[1.62..,1.78..]
β-interval
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).
Special 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
Special 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
Statistics for all graphs w N < 8
Overlap
0.1 1 10 100 0.1 1 10 100 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
Influence maximization vs. vaccination
Vaccination vs. sentinel surveillance
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.7
0.8
0.9
1
Structural explanations
0.1 1 10 100
Influence maximization Vaccination Sentinel surveillance
0.1 1 10 100 0.1 1 10 100
0.1 1 10 100 0.1 1 10 100 0.1 1 10 100
0.1 1 10 100 0.1 1 10 100 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
0.6
0.66
0.7
0.75
β β β
β β β
β β β
4.46
4.48
4.5
4.52
4.54
4.56
4.58
2.5
3
3.5
4
2.5
3
3.5
4
0.82
0.825
0.83
0.6
0.66
0.7
0.74
0.8
1.18
1.19
1.2
1.21
1.22
1
1.02
1.04
1.06
1.08
1
1.05
1.1
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
β β
(b) n = 3(a) n = 2
d d
Vaccination
Sentinel surveillance
Influence maximization
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.
P. Holme, L. Tupikina, Extinction in the
susceptible-infected-susceptible model: Exact
results for small graphs. arXiv:1801.????.
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
Automorphic configurations
Recovery event
Infection event
SIS as a random walk in the space of configurations
Configurations

(binary coded)
An example: o–o–o
Time to extinction from configuration s, xs
=
Expected time in configuration s
+
∑t
xt
× Prob(s → t)
An example: o–o–o
Yx + 1 = 0
An example: o–o–o
β
2
4
6
8
1 2 3 40
0
10
1,4
2
3 ,6
7
5 7>3,6>5>2>1,4
7>5>3,6>2>1,4
x
An example: o–o–o
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 . . .
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 . . .
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.
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
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

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