Modeling epidemics over Networks · 2019. 11. 29. · Questions Are there network ... ib s For...

First analysis ofdynamical processesover networks

Will let us exercisesome of the ways ofthinking about theseprocesses

Modeling epidemics over Networks

1 / 72

Modeling epidemics over NetworksQuestions

Are there network properties that predict spread of infection?

Are certain network types more resilient to infection than others?

If we can intervene (vaccinate) are there nodes in the network that aremore effective to vaccinate?

2 / 72





We'll start by looking at spread over nonnetworked populations

3 / 72

Individuals in the population can bein two states

An infected individual can infect anysusceptible individual they are incontact with

If we start ( ) with somenumber of infected individuals ( ).How many infected individuals arethere at time ?

Susceptibility and infection (SI model)

t = 0

i0

t4 / 72

SI model

, average number of contacts per individual in one time step

, "rate" probability an I infects an S upon contact

I(t) = β⟨k⟩ I(t)d

dt

S(t)

N

⟨k⟩

β

5 / 72

SI model




dt

S(t)

N

⟨k⟩

β

= β⟨k⟩i(1 − i)di

dt

6 / 72

Fraction of infected individuals inpopulation

Characteristic time ( s.t. )

SI model

i(t) =i0e

β⟨k⟩t

1 − i0 + i0eβ⟨k⟩t

t

i(t) = 1/e ≈ .36

τ =1

β⟨k⟩7 / 72

SIS modelInfection ends (recovery), individual becomes susceptible again

8 / 72

SIS model

recovery rate

= β⟨k⟩i(1 − i) − μidi

dt

μ

9 / 72

SIS model

i(t) = (1 − )×μβ⟨k⟩

Ce(β⟨k⟩−μ)t

1+Ce(β⟨k⟩−μ)t

10 / 72

Endemic StatePathogen persists inpopulation aftersaturation

SIS model

R0 = > 1β⟨k⟩

μ

i(∞) = 1 − = 1 − 1/R0μ

β⟨k⟩

11 / 72

Diseasefree StatePathogen disappearsfrom population

SIS model

R0 = < 1β⟨k⟩

μ

i(∞) = 0

12 / 72

SIS modelBasic Reproductive Number

Characteristic Time

R0 =β⟨k⟩

μ

τ =1

μ(R0 − 1)

13 / 72

SIR modelIndividuals removed after infection (either death or immunity)

14 / 72

SIR model

= β⟨k⟩i(1 − r − i) − μi

= μi

= −β⟨k⟩i(1 − r − i)

didt

drdt

dsdt

15 / 72

Summary

16 / 72

Epidemic processes over networks (SI)Consider node in network:

average probability node is susceptible at time

average probability node is infected at time

i

si(t) i t

xi(t) i t

17 / 72

Epidemic processes over networks (SI)

= −siβN

∑j=1

aijxjdsi

dt

= siβN

∑j=1

aijxjdxi

dt

18 / 72


For large , and early

= −siβN

∑j=1

aijxjdsi

dt

= siβN

∑j=1

aijxjdxi

dt

N t

= βAxdx

dt

19 / 72

Any quantity over nodesin the graph can be writtenas

Eigenvalue decomposition of A

A = V TΛV

xi

x =N

∑r=1

crvr

20 / 72


Avr = λrvr

λ1 ≥ λ2 ≥ ⋯ ≥ λN

21 / 72

Eigenvalue decomposition of Let's revisit centrality

Then

A

x(t) = Atx(0) =N

∑r=1

crAtvr

x(t) =N

∑r=1

crλtrvr = λ

t1

N

∑i=1

cr( )t

vr

λr

λ1

22 / 72

Eigenvalue decomposition of As grows, first term dominates

So set centrality to be proportional to first eigenvector

A

t

x(t) = c1λt1v1

x v1

23 / 72



In which case if as desired

A

t

x(t) = c1λt1v1

x v1

x = Ax x = v11λ1

24 / 72

Back to epidemics (SI) average probability each node is infected at time

Can write as

x(t) t

= βAxdx

dt

x(t) =N

∑r=1

cr(t)vr

25 / 72

Back to epidemics (SI)

= ∑Nr=1 vr

= βA∑Nr=1 cr(t)vr = β∑

Nr=1 λrcr(t)vr

dxdt

dcrdt

26 / 72


Implying

= ∑Nr=1 vr


Nr=1 λrcr(t)vr

dxdt

dcrdt

= βλrcrdcr

dt

27 / 72


Implying

With solution

= ∑Nr=1 vr


Nr=1 λrcr(t)vr

dxdt

dcrdt

= βλrcrdcr

dt

cr(t) = cr(0)eβλrt

28 / 72

Back to epidemics (SI)As before, first term dominates so

Eigencentrality!

x(t) ∼ eβλ1tv1

29 / 72

Back to epidemics (SIR)

Similarily

= βAx − μxdx

dt

x(t) ∼ e(βλ1−μ)t

30 / 72


Similarily

Is there an epidemic? Not if

= βAx − μxdx

dt


R0 = =β

μ1λ1

31 / 72

Degree BlockApproximationAssume that allnodes of the samedegree arestatistically equivalent

Degree distributions and epidemics

32 / 72



33 / 72

Degree distributions and epidemicsSI model

Fraction of nodes of degree that are infected

With the fraction of infected neighbors for a node of degree

k

ik =Ik

Nk

= β(1 − ik)kΘkdik

dt

Θk k

34 / 72

Degree distributions and epidemicsFor early time and assuming no degree correlation

with

= βki0 et/τ SIdik

dt

⟨k⟩ − 1

⟨k⟩

τ SI =⟨k⟩

β(⟨k2⟩ − ⟨k⟩)

35 / 72

Degree distributions and epidemicsCharacteristic time

For random networks

τ SI =⟨k⟩

β(⟨k2⟩ − ⟨k⟩)

⟨k2⟩ = ⟨k⟩(⟨k⟩ + 1)

τ SI =1

β⟨k⟩

36 / 72


For power law network is finite, characteristic time is finite

τ SI =⟨k⟩

β(⟨k2⟩ − ⟨k⟩)

γ ≥ 3 ⟨k2⟩

37 / 72



For power law , does not converge as socharacteristic time goes to 0

τ SI =⟨k⟩

β(⟨k2⟩ − ⟨k⟩)

γ ≥ 3 ⟨k2⟩

γ < 3 ⟨k2⟩ N → ∞

38 / 72


39 / 72

ImmunizationSuppose a fraction of nodes is immunized (i.e. resistant)

Rate of infection in SIR model changes from

to

g

λ =β

μ

λ(1 − g)

40 / 72

ImmunizationWe can choose a fraction such that rate of infection is below epidemicthreshold

For a random network

gc

gc = 1 −μ

β

1

⟨k⟩ + 1

41 / 72

ImmunizationFor a powerlaw network

For high need to immunize almost the entire population

gc = 1 −μ

β

⟨k⟩

⟨k2⟩

⟨k2⟩

42 / 72

ImmunizationImmunization can be more effective if performed selectively.

In power law contact networks, what is the effect of immunizing highdegree nodes?

43 / 72



First, how do you find them?

Idea: choose individuals at random, ask them to nominate a neighbor incontact graph

44 / 72

What is the expected degree of thenominated neighbor?

Immunization

∝ kpk

45 / 72

SummaryEpidemic as first example of dynamical process over networkRole of eigenvalue property in understanding epidemic spreadRole of degree distribution (specifically scale) in understandingspreadRole of highdegree nodes in robustness of networks to epidemics(immunization)

46 / 72

Network RobustnessWhat if we lose nodes in the network?

47 / 72

Nodes are uniformly at randomplaced in the intersections of aregular gridNodes in adjacent intersectionsare linkedKeep track of largest component

PercolationLet's look at a simplified network growth model (related to ER)

48 / 72

As nodes are added to the graph

What is the expected size of thelargest cluster?What is the average cluster size?


49 / 72

PercolationThere is a critical threshold (percolation cluster )pc

50 / 72

PercolationThere is a critical threshold (percolation cluster)

Average cluster size

Order parameter ( probability node is in large cluster)

⟨s⟩ ∼ |p − pc|−γp

P∞

P∞ ∼ (p − pc)βp

51 / 72

Model failure as the reverse process(nodes are removed)

Critical Failure Threshold

52 / 72

Critical Failure ThresholdSimilar modeling strategy applied to general networks

53 / 72

Critical Failure ThresholdFirst question: is there a giant component?

MolloyReed Criterion: Yes, if

> 2⟨k2⟩

⟨k⟩

54 / 72

Critical Failure ThresholdFor ER: ⟨k2⟩ = ⟨k⟩(⟨k⟩ + 1)

55 / 72

Critical Failure ThresholdFor ER:

There is a large component if

Good, coincides with what we know

⟨k2⟩ = ⟨k⟩(⟨k⟩ + 1)

= ⟨k⟩ > 1⟨k2⟩

⟨k⟩

56 / 72

Critical Failure ThresholdSecond question: if we model failure as before, when does the giantcomponent disappear?

fc = 1 −1

− 1⟨k2⟩

⟨k⟩

57 / 72


For ER:

fc = 1 −1

− 1⟨k2⟩

⟨k⟩

fc = 1 −1

⟨k⟩

58 / 72

Critical Failure ThresholdFor power law networks

59 / 72


60 / 72

Robustness to AttacksWhat if node removal is targeted to hubs?

61 / 72


fc = 2 + kmin(fc − 1)2−γ

1−γ 2 − γ

3 − γ

3−γ

1−γ

62 / 72

Robustness to Attacks

63 / 72

Designing RobustnessHubs robustness to random attacks No hubs robustness to targetedattacks

64 / 72

Designing RobustnessMaximize

f totc = frandomc + f

targetedc

65 / 72


Mixture of nodes

fraction of nodes have degreeremaining nodes have degree


targetedc

r kmax1 − r kmin

66 / 72

Designing Robustness

67 / 72


68 / 72


69 / 72


SummaryPercolation process to understand random failure 70 / 72

SummaryPower law networks are robust to random failuresPower law networks are susceptible to targeted failuresProvable robustness to random and targeted failure using mixture ofnode degrees

71 / 72

Next timeDiffusion!!

72 / 72

Dynamical Processes over NetworksHéctor Corrada Bravo

University of Maryland, College Park, USA CMSC828O 20191002

First analysis ofdynamical processesover networks

Will let us exercisesome of the ways ofthinking about theseprocesses

Modeling epidemics over Networks

1 / 72





2 / 72





We'll start by looking at spread over nonnetworked populations

3 / 72

Individuals in the population can bein two states

An infected individual can infect anysusceptible individual they are incontact with

If we start ( ) with somenumber of infected individuals ( ).How many infected individuals arethere at time ?

Susceptibility and infection (SI model)

t = 0

i0

t4 / 72

SI model




dt

S(t)

N

⟨k⟩

β

5 / 72

SI model




dt

S(t)

N

⟨k⟩

β

= β⟨k⟩i(1 − i)di

dt

6 / 72

Fraction of infected individuals inpopulation

Characteristic time ( s.t. )

SI model

i(t) =i0e

β⟨k⟩t

1 − i0 + i0eβ⟨k⟩t

t

i(t) = 1/e ≈ .36

τ =1

β⟨k⟩7 / 72

SIS modelInfection ends (recovery), individual becomes susceptible again

8 / 72

SIS model

recovery rate

= β⟨k⟩i(1 − i) − μidi

dt

μ

9 / 72

SIS model

i(t) = (1 − )×μβ⟨k⟩

Ce(β⟨k⟩−μ)t

1+Ce(β⟨k⟩−μ)t

10 / 72

Endemic StatePathogen persists inpopulation aftersaturation

SIS model

R0 = > 1β⟨k⟩

μ

i(∞) = 1 − = 1 − 1/R0μ

β⟨k⟩

11 / 72

Diseasefree StatePathogen disappearsfrom population

SIS model

R0 = < 1β⟨k⟩

μ

i(∞) = 0

12 / 72

SIS modelBasic Reproductive Number

Characteristic Time

R0 =β⟨k⟩

μ

τ =1

μ(R0 − 1)

13 / 72

SIR modelIndividuals removed after infection (either death or immunity)

14 / 72

SIR model

= β⟨k⟩i(1 − r − i) − μi

= μi

= −β⟨k⟩i(1 − r − i)

didt

drdt

dsdt

15 / 72

Summary

16 / 72

Epidemic processes over networks (SI)Consider node in network:

average probability node is susceptible at time

average probability node is infected at time

i

si(t) i t

xi(t) i t

17 / 72


= −siβ

N

∑j=1

aijxjdsi

dt

= siβ

N

∑j=1

aijxjdxi

dt

18 / 72


For large , and early

= −siβ

N

∑j=1

aijxjdsi

dt

= siβ

N

∑j=1

aijxjdxi

dt

N t

= βAxdx

dt

19 / 72

Any quantity over nodesin the graph can be writtenas


A = VT

ΛV

xi

x =

N

∑r=1

crvr

20 / 72


Avr = λrvr

λ1 ≥ λ2 ≥ ⋯ ≥ λN

21 / 72

Eigenvalue decomposition of Let's revisit centrality

Then

A

x(t) = Atx(0) =N

∑r=1

crAtvr

x(t) =N

∑r=1

crλtrvr = λ

t

1

N

∑i=1

cr( )t

vr

λr

λ1

22 / 72



A

t

x(t) = c1λt

1v1

x v1

23 / 72



In which case if as desired

A

t

x(t) = c1λt

1v1

x v1

x = Ax x = v11λ1

24 / 72

Back to epidemics (SI) average probability each node is infected at time

Can write as

x(t) t

= βAxdx

dt

x(t) =N

∑r=1

cr(t)vr

25 / 72


= ∑N

r=1 vr

= βA∑N

r=1 cr(t)vr = β∑N

r=1 λrcr(t)vr

dxdt

dcrdt

26 / 72


Implying

= ∑N

r=1 vr

= βA∑N


r=1 λrcr(t)vr

dxdt

dcrdt

= βλrcrdcr

dt

27 / 72


Implying

With solution

= ∑N

r=1 vr

= βA∑N


r=1 λrcr(t)vr

dxdt

dcrdt

= βλrcrdcr

dt

cr(t) = cr(0)eβλrt

28 / 72

Back to epidemics (SI)As before, first term dominates so

Eigencentrality!

x(t) ∼ eβλ1tv1

29 / 72


Similarily

= βAx − μxdx

dt


30 / 72


Similarily

Is there an epidemic? Not if

= βAx − μxdx

dt


R0 = =β

μ1λ1

31 / 72



32 / 72



33 / 72

Degree distributions and epidemicsSI model

Fraction of nodes of degree that are infected

With the fraction of infected neighbors for a node of degree

k

ik =Ik

Nk

= β(1 − ik)kΘkdik

dt

Θk k

34 / 72

Degree distributions and epidemicsFor early time and assuming no degree correlation

with

= βki0 et/τ SIdik

dt

⟨k⟩ − 1

⟨k⟩

τ SI =⟨k⟩

β(⟨k2⟩ − ⟨k⟩)

35 / 72


For random networks

τ SI =⟨k⟩

β(⟨k2⟩ − ⟨k⟩)

⟨k2⟩ = ⟨k⟩(⟨k⟩ + 1)

τ SI =1

β⟨k⟩

36 / 72



τ SI =⟨k⟩

β(⟨k2⟩ − ⟨k⟩)

γ ≥ 3 ⟨k2⟩

37 / 72



For power law , does not converge as socharacteristic time goes to 0

τ SI =⟨k⟩

β(⟨k2⟩ − ⟨k⟩)

γ ≥ 3 ⟨k2⟩

γ < 3 ⟨k2⟩ N → ∞

38 / 72


39 / 72

ImmunizationSuppose a fraction of nodes is immunized (i.e. resistant)

Rate of infection in SIR model changes from

to

g

λ =β

μ

λ(1 − g)

40 / 72

ImmunizationWe can choose a fraction such that rate of infection is below epidemicthreshold

For a random network

gc

gc = 1 −μ

β

1

⟨k⟩ + 1

41 / 72

ImmunizationFor a powerlaw network

For high need to immunize almost the entire population

gc = 1 −μ

β

⟨k⟩

⟨k2⟩

⟨k2⟩

42 / 72



43 / 72



First, how do you find them?

Idea: choose individuals at random, ask them to nominate a neighbor incontact graph

44 / 72

What is the expected degree of thenominated neighbor?

Immunization

∝ kpk

45 / 72

SummaryEpidemic as first example of dynamical process over networkRole of eigenvalue property in understanding epidemic spreadRole of degree distribution (specifically scale) in understandingspreadRole of highdegree nodes in robustness of networks to epidemics(immunization)

46 / 72

Network RobustnessWhat if we lose nodes in the network?

47 / 72

Nodes are uniformly at randomplaced in the intersections of aregular gridNodes in adjacent intersectionsare linkedKeep track of largest component


48 / 72

As nodes are added to the graph

What is the expected size of thelargest cluster?What is the average cluster size?


49 / 72

PercolationThere is a critical threshold (percolation cluster )pc

50 / 72

PercolationThere is a critical threshold (percolation cluster)

Average cluster size

Order parameter ( probability node is in large cluster)

⟨s⟩ ∼ |p − pc|−γp

P∞

P∞ ∼ (p − pc)βp

51 / 72

Model failure as the reverse process(nodes are removed)

Critical Failure Threshold

52 / 72

Critical Failure ThresholdSimilar modeling strategy applied to general networks

53 / 72

Critical Failure ThresholdFirst question: is there a giant component?

MolloyReed Criterion: Yes, if

> 2⟨k2⟩

⟨k⟩

54 / 72

Critical Failure ThresholdFor ER: ⟨k2⟩ = ⟨k⟩(⟨k⟩ + 1)

55 / 72

Critical Failure ThresholdFor ER:

There is a large component if

Good, coincides with what we know

⟨k2⟩ = ⟨k⟩(⟨k⟩ + 1)

= ⟨k⟩ > 1⟨k2⟩

⟨k⟩

56 / 72


fc = 1 −1

− 1⟨k2⟩

⟨k⟩

57 / 72


For ER:

fc = 1 −1

− 1⟨k2⟩

⟨k⟩

fc = 1 −1

⟨k⟩

58 / 72


59 / 72


60 / 72


61 / 72


fc = 2 + kmin(fc − 1)2−γ

1−γ 2 − γ

3 − γ

3−γ

1−γ

62 / 72

Robustness to Attacks

63 / 72

Designing RobustnessHubs robustness to random attacks No hubs robustness to targetedattacks

64 / 72



targetedc

65 / 72


Mixture of nodes

fraction of nodes have degreeremaining nodes have degree


targetedc

r kmax1 − r kmin

66 / 72


67 / 72


68 / 72


69 / 72


SummaryPercolation process to understand random failure 70 / 72

SummaryPower law networks are robust to random failuresPower law networks are susceptible to targeted failuresProvable robustness to random and targeted failure using mixture ofnode degrees

71 / 72

Next timeDiffusion!!

72 / 72

Modeling epidemics over Networks · 2019. 11. 29. · Questions Are there network ... ib s For...

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