Important spreaders in networks:
exact results on small graphs

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

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

time
Network epidemiology
Step 2: Contact patterns

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).

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

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.

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

Polynomial algebra takes time
(37762366549514108074989296025600000000000*x**73+3314686580533426042655618661089280000000000*x**72+141438610676500742111413237916368896000000000*x**71+3
911473306168632730171826549920825344000000000*x**70+78863281455383204006473293722572552273920000000*x**69+1236403293359232085532901156240802856853504000
000*x**68+15699393806584508589027640185718259048113766400000*x**67+166047926815157089435015605011671368201261465600000*x**66+149321074471363905237290341
2763316426944533092352000*x**65+11596802000132949850753533289466811302954899065292800*x**64+78746554444636009114113624901619589833746005539712000*x**63+
472189336478744088675459614584162673391547254471037440*x**62+2520838579589225935326332345418749193680539093862025984*x**61+12064010950190968998503507349
103010126875638928117001472*x**60+52056956992255933979233520314531580309411479877375753088*x**59+2035496785634842434895513764833898462626245464907560146
56*x**58+724337780103561588309914084265886452050608182621681489696*x**57+2354656050497226961862844168528304543472210933547719071080*x**56+70156492344984
24552384686112745652381839054053136427666012*x**55+19214703345853832788682064160120433484916446461152221422090*x**54+48501351479075668828643503786330661
658486963404832259702487*x**53+113092671813635674828611367354176714434189107315168498776928*x**52+244103956674073644958141865118010803740486170105474665
375344*x**51+488630192461738524748488108810018093865519958120561303765494*x**50+908600820200290624573587634392204246241182209329371400281145*x**49+15718
14192193713924618282713564682212894757971214065300446684*x**48+2533060255249516888774685502865019709652152513622605055279456*x**47+380738061109785592014
7340609199202541615833608774939794319328*x**46+5343283033579917464170385365564015741459363191088389767306398*x**45+7008169777451585300330775246188016187
941023359601556223762760*x**44+8597683922746974615895184605717665402049163153921524311480288*x**43+98732500064113677990347086832093514180452414285232600
95680092*x**42+10619889579823429590519175999126775450296264698254861098419138*x**41+10705343531834807255308847480339463995272891980674623472362352*x**40
+10118200705608779929491741668401657943488155430966316422413564*x**39+8970001049997570057132806423909486209951952062243977752430714*x**38+74609950378828
42620157347200224840242010916136536945407182547*x**37+5823838496430434727894405840870770563861889084747019155124352*x**36+426666491950679107354822841634
9608210702346234200930403133168*x**35+2933984529167138491682532365556698639754779991550409622936110*x**34+1893674869460884091813696066793009247613602165
512467381565973*x**33+1147053826336232131835158650815804999309595169018523338347692*x**32+651941666849429144426327239177240288949251173249669277824296*x
**31+347582602089688677496074681434250008936206946931462543001436*x**30+173768006766411435824870505215240768500990172474739886280840*x**29+8142176545488
7540793714432533619870680132300264456583686136*x**28+35737339262158151938098902744329827302836180844476824581984*x**27+146831621144044053430107768719158
47702033756436713414727552*x**26+5642689559748959313663642099659619418917821190195217427712*x**25+202637477920427978284399219388823599938834274487561892
4160*x**24+679293285710471972392728540425835464747889906016677284352*x**23+212308973769516687666161412820684609317746546465988738048*x**22+6177993520612
9739114397096060550049079944195609861388288*x**21+16711103480959516893824501251494340145597563371003242496*x**20+419427438958566275022405598749426314284
3645525992972288*x**19+974786785480329873601482567083721717912488928481935360*x**18+209288755404718390985128907159566824759100622961672192*x**17+4140027
9812069740664016081360574723512184740799807488*x**16+7522283805109866955995503020206903760121954966568960*x**15+1250984582973454622568956659303179360768
874779770880*x**14+189642358940050365501253044671260380916000122470400*x**13+26081445034209637896534715167941811495312398745600*x**12+323601180515126618
0678324566923195135888143155200*x**11+359819252450753508595254072745238613601340620800*x**10+35570023038312915426349438493625919964656435200*x**9+309579
2827574688435407884180504098172305408000*x**8+234354685053410645368523826403557664358400000*x**7+15193743364805385914791764255439847424000000*x**6+82660
7123241007367179342411796054016000000*x**5+36698205721449251600489578223370240000000*x**4+1276643541446883714485459091456000000000*x**3+3263138595093789
4083656417280000000000*x**2+544847137497742374674104320000000000*x+4457869163092977175756800000000000)/
(38849174012498197907452723200000000000000*x**74+3318598191984295465296750182400000000000000*x**73+137923803520060037223899260256256000000000000*x**72+3
718836854788115338745516774129664000000000000*x**71+73187564029226708723444458146842542080000000000*x**70+1121446437628800089041894262361041141760000000
000*x**69+13937137779902055729239770100875821632716800000000*x**68+144499023630993463225897485449622876114124800000000*x**67+127587886738453505303322947
2044962110670635008000000*x**66+9746370095122882492430762538399854353652121600000000*x**65+65215495588078294956304754695697556926039599349760000*x**64+3
86084747539612058845065474438915113812606522490880000*x**63+2039019743945197787806635323175732162985805873361715200*x**62+967312067951480964082283380637
2756448932826769653760000*x**61+41463182796652561547765945470954864465553792226951168000*x**60+161395707996900193288033181079336181640030323993431244800
*x**59+572985959395474247469239080818191224916889350617733529600*x**58+1862374267840213513898298449997074233654470813402085785600*x**57+5560430031086381
252746269930110412997502372446348409241600*x**56+15294884284718527165111367639524474394242463478570586521600*x**55+3886098166813866652919191163210297367
9267849216163518873600*x**54+91415494941532261154338183845834115681983965207741203353600*x**53+199509970977818775632987267744641506890636069960682414182
400*x**52+404718623257332025328917625285264020089712651667338146585600*x**51+764374715419689922968327485500214314436229021737343580569600*x**50+13460756
68348010010438588949674037750529766839631942722380800*x**49+2213199045980729092667047603056735385248916780496177533440000*x**48+340154127414103475350533
7773066649407560854431902037976320000*x**47+4892103780283080871288385104328530226580371128828639113216000*x**46+6590054588677667222257130588148263220931
392553656135566592000*x**45+8321790039667741773143733704030351277421859439309217460736000*x**44+98581441230561875321569373739644234330922765326024021095
68000*x**43+10962218518250821245562546207482370427489203414972878535065600*x**42+11448825928895319238165607419280388858411242066332743600000000*x**41+11
235149156411346177828271421184433780217549468847670036992000*x**40+10363622140030410343302319664329679035025469377574033155174400*x**39+8988410444547910
032150577099030176615855802547117158448332800*x**38+7331310976398177080061080847301200859574935432860186655948800*x**37+56242179608090918028225316305767
26104891213089910954236620800*x**36+4058302463395757204313533365763872433751951130071815225164800*x**35+275431759585599708492087256079242978478487216240
2632133836800*x**34+1758011360746200302693219873297151407720017161990685383244800*x**33+1055073126700485634917761645630646824351646318878099638579200*x*
*32+595216923608652456635301929860947101931138207761273530828800*x**31+315529879888969797201425410449602612378143848550150479052800*x**30+15710112010938
0703527501506578358977957429633622398998374400*x**29+73426181385511818150687126779106893812232809076914963968000*x**28+321936668855972964088101551329799
01682101751270559157120000*x**27+13231385437288999412775352581513090540713728729205858304000*x**26+50929796606588392109237142664473451858967488235090177

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
N
no. connected
graphs
3 2
4 6
5 20
6 112
7 853
http://users.cecs.anu.edu.au/~bdm/data/graphs.html

Small graphs

Special “smallest” cases

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

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

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

Statistics for all graphs w N < 8

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

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

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

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.

Thank you!
Collaborators:
Jari Saramäki
Naoki Masuda
Nelly Litvak
Luis Rocha
Illustrations by:
Mi Jin Lee