![㠢㠸㠧㠳ã å ã 㠼㠿 2007ç · (9 ÷]9]W]d] ][]/] ] ]O]2\ / ÷ ñ ® ³ .]']d] ]+]0 ÿ º 3 ò 1\» 60$57 9 + Title: 㠢㠸㠧㠳ã å ã 㠼㠿_2007ç](https://static.fdocuments.in/doc/165x107/60325704c2c96c22f606dc61/-2007-9-9wd-o2-.jpg)
Languages
Pages
Legal
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
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?
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
, 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
= 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
Epidemic processes over networks (SI)
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
Eigenvalue decomposition of A
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
Eigenvalue decomposition of As grows, first term dominates
So set centrality to be proportional to first eigenvector
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
Back to epidemics (SI)
Implying
= ∑Nr=1 vr
= βA∑Nr=1 cr(t)vr = β∑
Nr=1 λrcr(t)vr
dxdt
dcrdt
= βλrcrdcr
dt
27 / 72
Back to epidemics (SI)
Implying
With solution
= ∑Nr=1 vr
= βA∑Nr=1 cr(t)vr = β∑
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
Back to epidemics (SIR)
Similarily
Is there an epidemic? Not if
= βAx − μxdx
dt
x(t) ∼ e(βλ1−μ)t
R0 = =β
μ1λ1
31 / 72
Degree BlockApproximationAssume that allnodes of the samedegree arestatistically equivalent
Degree distributions and epidemics
32 / 72
Degree BlockApproximationAssume that allnodes of the samedegree arestatistically equivalent
Degree distributions and epidemics
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
Degree distributions and epidemicsCharacteristic time
For power law network is finite, characteristic time is finite
τ SI =⟨k⟩
β(⟨k2⟩ − ⟨k⟩)
γ ≥ 3 ⟨k2⟩
37 / 72
Degree distributions and epidemicsCharacteristic time
For power law network is finite, characteristic time is finite
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
Degree distributions and epidemics
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
ImmunizationImmunization can be more effective if performed selectively.
In power law contact networks, what is the effect of immunizing highdegree nodes?
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?
PercolationLet's look at a simplified network growth model (related to ER)
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
Critical Failure ThresholdSecond question: if we model failure as before, when does the giantcomponent disappear?
For ER:
fc = 1 −1
− 1⟨k2⟩
⟨k⟩
fc = 1 −1
⟨k⟩
58 / 72
Critical Failure ThresholdFor power law networks
59 / 72
Critical Failure ThresholdFor power law networks
60 / 72
Robustness to AttacksWhat if node removal is targeted to hubs?
61 / 72
Robustness to AttacksWhat if node removal is targeted to hubs?
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
Designing RobustnessMaximize
Mixture of nodes
fraction of nodes have degreeremaining nodes have degree
f totc = frandomc + f
targetedc
r kmax1 − r kmin
66 / 72
Designing Robustness
67 / 72
Designing Robustness
68 / 72
Designing Robustness
69 / 72
Designing Robustness
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
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
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?
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
, 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
= 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)
= −siβ
N
∑j=1
aijxjdsi
dt
= siβ
N
∑j=1
aijxjdxi
dt
18 / 72
Epidemic processes over networks (SI)
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
Eigenvalue decomposition of A
A = VT
ΛV
xi
x =
N
∑r=1
crvr
20 / 72
Eigenvalue decomposition of A
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
Eigenvalue decomposition of As grows, first term dominates
So set centrality to be proportional to first eigenvector
A
t
x(t) = c1λt
1v1
x v1
23 / 72
Eigenvalue decomposition of As grows, first term dominates
So set centrality to be proportional to first eigenvector
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
Back to epidemics (SI)
= ∑N
r=1 vr
= βA∑N
r=1 cr(t)vr = β∑N
r=1 λrcr(t)vr
dxdt
dcrdt
26 / 72
Back to epidemics (SI)
Implying
= ∑N
r=1 vr
= βA∑N
r=1 cr(t)vr = β∑N
r=1 λrcr(t)vr
dxdt
dcrdt
= βλrcrdcr
dt
27 / 72
Back to epidemics (SI)
Implying
With solution
= ∑N
r=1 vr
= βA∑N
r=1 cr(t)vr = β∑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
Back to epidemics (SIR)
Similarily
= βAx − μxdx
dt
x(t) ∼ e(βλ1−μ)t
30 / 72
Back to epidemics (SIR)
Similarily
Is there an epidemic? Not if
= βAx − μxdx
dt
x(t) ∼ e(βλ1−μ)t
R0 = =β
μ1λ1
31 / 72
Degree BlockApproximationAssume that allnodes of the samedegree arestatistically equivalent
Degree distributions and epidemics
32 / 72
Degree BlockApproximationAssume that allnodes of the samedegree arestatistically equivalent
Degree distributions and epidemics
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
Degree distributions and epidemicsCharacteristic time
For power law network is finite, characteristic time is finite
τ SI =⟨k⟩
β(⟨k2⟩ − ⟨k⟩)
γ ≥ 3 ⟨k2⟩
37 / 72
Degree distributions and epidemicsCharacteristic time
For power law network is finite, characteristic time is finite
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
Degree distributions and epidemics
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
ImmunizationImmunization can be more effective if performed selectively.
In power law contact networks, what is the effect of immunizing highdegree nodes?
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?
PercolationLet's look at a simplified network growth model (related to ER)
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
Critical Failure ThresholdSecond question: if we model failure as before, when does the giantcomponent disappear?
For ER:
fc = 1 −1
− 1⟨k2⟩
⟨k⟩
fc = 1 −1
⟨k⟩
58 / 72
Critical Failure ThresholdFor power law networks
59 / 72
Critical Failure ThresholdFor power law networks
60 / 72
Robustness to AttacksWhat if node removal is targeted to hubs?
61 / 72
Robustness to AttacksWhat if node removal is targeted to hubs?
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
Designing RobustnessMaximize
Mixture of nodes
fraction of nodes have degreeremaining nodes have degree
f totc = frandomc + f
targetedc
r kmax1 − r kmin
66 / 72
Designing Robustness
67 / 72
Designing Robustness
68 / 72
Designing Robustness
69 / 72
Designing Robustness
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
Top Related