how community structure aects epidemics - clara stegehuis

How community structure affects epidemicsClara Stegehuis

joint work with Remco van der Hofstad and Johan van Leeuwaarden

Eindhoven University of Technology

How does community structure affect epidemics?

Hierarchical configuration model

Epidemics

Configuration model [Bollobas 1980]

Community structure is lost!

Original network Configuration model


Instead of pairing vertices, pair communities with half-edges

Start Possible result

Easy to analyze

Use that on community level still a configuration model to obtain:

I clustering

I component sizes

I degree correlations

.........

...

· · · · · · · · ·

......

HCM*

Randomize also edges within communities

Start Possible result

Different models

CM

Fixed degrees

HCM*

Fixed sets ofcommunities

HCM

Fixed communitystructure

Increasing randomness


Epidemics

Bond percolation

Infect neighbor with probability p

S : fraction of vertices in largest component

Bond percolation

Infect neighbor with probability pS : fraction of vertices in largest component

Bond percolation results

Email network

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

edge occupation probability p

Siz

eof

larg

estc

ompo

nent

dataHCMHCM*CM

Internet router network

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

edge occupation probability pS

ize

ofla

rges

tcom

pone

nt

dataHCMHCM*CM

SIR epidemic

Email network

0 5 10 15 20 25 300.00

0.05

0.10

0.15

time

infe

cted

node

s

dataHCMHCM*CM

Internet router network

0 5 10 15 20 25 300.00

0.10

0.20

0.30

timein

fect

edno

des

dataHCMHCM*CM

Conclusion

I Communities are important for how epidemics spread acrossnetworks

I Exact community shapes are less important

I Communities may inhibit or enforce epidemics

Open problem

Overlapping communities

how community structure aects epidemics - clara stegehuis

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