Epidemic processes on switching networks

Epidemic processes in time-
varying networks:
Commutator and concurrency
Naoki Masuda
State University of New York at Buffalo
naokimas@buffalo.edu
www.naokimasuda.net

Temporal networks
(= time-varying graphs)

From http://www.sociopatterns.org/gallery/
pupil
teacher
class
“node” =
pupil or teacher

Why does it matter?
time
A
A
B
B
C
C
D D
time
A
A
B
B
C
C
D D
“temporal network”
static (i.e. traditional)
network
✓ A → D (temporal path)
- D → A

• Key questions:
• How does time-dependence of networks change dynamical
processes on networks?
• How can we mine information from temporal network data?
Masuda & Lambiotte
(2016)
Masuda & Holme
Eds. (2017)

• Epidemic threshold βc
• Disease-free if β ≤ βc
• Outbreak/endemic if
β > βc
• βc value depends on
network structure
• On temporal networks,
the βc value depends …
Overview and aim
susceptible
(i.e. healthy)
infected
infection
β

How does temporality affect
epidemic dynamics,
particularly epidemic
thresholds?

Main results
1. When we can ignore stochastic fluctuations of the
dynamics, βc for temporal networks < βc for static
networks.(Speidel, Klemm, Eguíluz & Masuda, New J Phys, 2016).
2. Otherwise, a Markov chain approach reveals
“concurrency-induced transitions” (Onaga, Gleeson &
Masuda, Phys Rev Lett, 2017).
Temporal networks
facilitate contagion
(In the first scenario).
temporal
static
0 ˆc
⇤
c
% infected
infection rate

Two representations of
static networks
v1 2
3
4 5
v
v v v
adjacency matrix
(good for theory)
edge list (good for data handling
and numerical simulations)
(1, 2)
(1, 3)
(1, 4)
(1, 5)
(2, 4)
(4, 5)
A =
0
B
B
B
B
@
0 1 1 1 1
1 0 0 1 0
1 0 0 0 0
1 1 0 0 1
1 0 0 1 0
1
C
C
C
C
A

Two representations of
temporal networks
contact sequence sequences of networks
(good for theory)
(1, 2, t1, 1)
(2, 3, t2, 2)
(2, 4, t3, 3)
(2, 3, t4, 4)
A1, A2, . . .
time
A
A
B
B
C
C
D D
ﬁg for Masuda-Klemm-Eguiluz
(cf. Masuda & Lambiotte, A Guide to Temporal Networks, 2016)
We use this here.

Modeling temporal networks
by “switching networks”
1
3
2
4
1
3
2
time
4
1
3
2
4
1
3
2
A(0)
=
0
B
B
@
0 0 1 0
0 0 0 0
1 0 0 0
0 0 0 0
1
C
C
A A(1)
=
0
B
B
@
0 0 0 1
0 0 0 1
0 0 0 0
1 1 0 0
1
C
C
A A(2)
=
0
B
B
@
0 1 1 1
1 0 0 0
1 0 0 0
1 0 0 0
1
C
C
A A(3)
=
0
B
B
@
0 0 0 0
0 0 1 0
0 1 0 1
0 0 1 0
1
C
C
A
4
0 τ 2τ 3τ 4τ
A sequence of matrices (“snapshot” networks)

SIS model
susceptible (S)
infected (I)
infection
recovery
β
µ = 1
3β

β
fractionofinfectednodes
0
0
0.5 1 1.5 2
0.02
0.04
0.06
0.08
0.1
(a)
τ = 0
τ = 0.05
τ = 0.5
0 1 2 3 4 5
βfractionofinfectednodes
0
0.02
0.04
0.06
0.08
0.1
(b)
On an online message
temporal network
(Opsahl & Panzarasa,
Soc Netw 2009)
On a sexual contact
temporal network
(Rocha et al., PNAS 2010)
(Speidel, Klemm, Eguíluz & Masuda, New Journal of Physics, 2016)

1 2
3 4
1 2
3 4
1 2
3 4
1 2
3 4
M(1) M(2) M(3)
M * = (M(1) + M(2) + M(3)) / 3
0 τ 2τ 3τ
temporal network
static network
1
1 1 1
1
1
1
2/3
1/3
1
1/3
time
Edge (1,2) is used for
2τ “weight × time” in
total in both cases

• For temporal networks
• For static networks
Model
˙x(t) =M(0)
x(t), 0  t  ⌧
˙x(t) =M(1)
x(t), ⌧  t  2⌧
...
˙x(t) =M(` 1)
x(t), (` 1)⌧  t  `⌧
(Masuda, Klemm & Eguíluz, Physical Review Letters, 2013)
˙x(t) = M⇤
x(t) ⌘
P` 1
`0=0 M(`0
)
`
x(t), 0  t  `⌧

“Individual-based approximation” (a.k.a.
“quenched mean-field theory”) for static
networks
Assuming all pi(t) ≈ 0
• A: adjacency matrix
• β: infection rate
• µ: recovery rate (= 1)
pi(t) = Pr(node i is infected)
˙pi(t) =
NX
j=1
Aji [1 pi(t)] pj(t) µpi(t)
⇡
NX
j=1
Ajipj(t) µpi(t)
With µ = 1, c = 1/ max(A)
˙p ⇡ ( A µI)p

In probability theory
xi(t) =
(
0 (node i is susceptible)
1 (node i is infected)
dxi =
NX
j=1
Aji(1 xi)xjd⇧
(vj ,vi)
xid⇧vi
µ
Stochastic differential equation with Poisson jumps:
Expectation:
dE[xi]
dt
=
NX
j=1
AjiE[(1 xi)xj] µE[xi]

NX
j=1
AjiE[xj] µE[xi]
Markov chain on 2N states
This slide is a mathematical side note.

dE[xi]
dt
=
NX
j=1
AjiE[(1 xi)xj] µE[xi]

NX
j=1
AjiE[xj] µE[xi]
˙p  ( A µI)p
p(t)  e( A µI)t
p(0)
= sup{✏ : pi(t)  Ce ✏t
for 9C > 0, 8i, 8p(0)}
Decay rate:
NB: γ > 0 always

p(t)  e( A µI)t
p(0)
= sup{✏ : pi(t)  Ce ✏t
for 9C > 0, 8i, 8p(0)}
Decay rate:
A quenched mean-field lower bound of the decay rate:
qMF ⌘ max( A µI)
qMF = 0 =)
✓
µ
◆
c
=
1
max(A)

Individual-based approximation
for switching networks
where
β = βc ⬌ the leading eigenvalue of T(τ,ℓ) = 1
Assume pi(t) ≈ 0
• Infection rate = β
• Recovery rate = 1
pi(t) = Pr(vertex i is infected)
or
T(⌧, `) = exp
h
( A(` 1)
I)⌧
i
· · · exp
h
( A(1)
I)⌧
i
exp
h
( A(0)
I)⌧
i
˙p(t) = ( A(`0
)
I)p(t) where `0
⌧  t < (`0
+ 1)⌧
˙pi(t) =
NX
j=1
A
(`0
)
ji pj(t) pi(t)
p(`⌧) = T(⌧, `)p(0)

Individual-based approximation
for switching networks
where
β = βc ⬌ the leading eigenvalue of T(τ,ℓ) = 1
This is to be compared with the leading eigenvalue of
where A⇤
=
1
`
` 1X
`0=0
A(`0
)
T(⌧, `) = exp
h
( A(` 1)
µI)⌧
i
· · · exp
h
( A(1)
µI)⌧
i
exp
h
( A(0)
µI)⌧
i
exp [( A⇤
µI)`⌧]

• If this holds,
• True for ℓ = 2 (Cohen et al., 1982)
• We want to know the case where φ is the largest eigenvalue.
• Counterexamples, but with negative entries (Thomson, 1965). cf.
Golden-Thompson inequality
• Supported by the analysis of two models (activity driven model,
Perra et al., Sci Rep, 2012, and its variant)
• Supported numerically
(eM(` 1)
eM(` 2)
· · · eM(0)
) (eM(` 1)
+M(` 2)
+···+M(0)
)
if all Ms have only nonnegative entries.
Conjecture:
Remarks:
ˆc  ⇤
c
time-varying static

To quantify the difference
between time-varying
and static networks

Zassenhaus’ formula:
where
exp [s(M1 + M2)] = exp [sM1] exp [sM2]
Y
n 2
exp [sn
Cn(M1, M2)]
C2(M1, M2) =
1
2
[M1, M2]
C3(M1, M2) =
1
6
(2[M2, [M1, M2]] + [M1, [M1, M2]])
[M1, M2] ⌘ M1M2 M2M1
Reminder:
With τ ≪ 1 and Maclaurin expansion
time-varying
graph
static graph
T(⌧, `) = exp [( A⇤
I)`⌧]
8
<
:
I (⌧ )2
` 1X
`0=1
C2
0
@A(`0
)
,
`0
1X
`00=0
A(`00
)
1
A + O(⌧3
)
9
=
;
= I + ( A⇤
I) `⌧ +
8
<
:
( A⇤
I)
2
+
2
`2
` 1X
`0=1
`0
1X
`00=0
h
A(`0
)
, A(`00
)
i
9
=
;
(`⌧)2
2
+ O(⌧3
)
T(⌧, `) = exp
h
( A(` 1)
I)⌧
i
· · · exp
h
( A(1)
I)⌧
i
exp
h
( A(0)
I)⌧
i
A⇤
=
1
`
` 1X
`0=0
A(`0
)

C ⌘
1
(`↵⇤
max)2
` 1X
`0=1
`0
1X
`00=0
h
A(`0
)
, A(`00
)
i
2
↵⇤
max = largest eigenvalue of A⇤
where
Reminder: A⇤
=
1
`
` 1X
`0=0
A(`0
)
[A(`0
)
, A(`00
)
] =A(`0
)
A(`00
)
A(`00
)
A(`0
)

1
3
2
4
1
3
2
time
4
1
3
2
4
1
3
2
A(0)
=
0
B
B
@
0 0 1 0
0 0 0 0
1 0 0 0
0 0 0 0
1
C
C
A A(1)
=
0
B
B
@
0 0 0 1
0 0 0 1
0 0 0 0
1 1 0 0
1
C
C
A A(2)
=
0
B
B
@
0 1 1 1
1 0 0 0
1 0 0 0
1 0 0 0
1
C
C
A A(3)
=
0
B
B
@
0 0 0 0
0 0 1 0
0 1 0 1
0 0 1 0
1
C
C
A
4
0 τ 2τ 3τ 4τ
[A(0)
, A(1)
] =A(0)
A(1)
A(1)
A(0)
=
0
B
B
@
0 0 1 0
0 0 0 0
1 0 0 0
0 0 0 0
1
C
C
A
0
B
B
@
0 0 0 1
0 0 0 1
0 0 0 0
1 1 0 0
1
C
C
A
0
B
B
@
0 0 0 1
0 0 0 1
0 0 0 0
1 1 0 0
1
C
C
A
0
B
B
@
0 0 1 0
0 0 0 0
1 0 0 0
0 0 0 0
1
C
C
A
=
0
B
B
@
0 0 0 0
0 0 0 0
0 0 0 1
0 0 0 0
1
C
C
A
0
B
B
@
0 0 0 0
0 0 0 0
0 0 0 0
0 0 1 0
1
C
C
A =
0
B
B
@
0 0 0 0
0 0 0 0
0 0 0 1
0 0 1 0
1
C
C
A
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C ⌘
1
(`↵⇤
max)2
` 1X
`0=1
`0
1X
`00=0
h
A(`0
)
, A(`00
)
i
2

C ⌘
1
(`↵⇤
max)2
` 1X
`0=1
`0
1X
`00=0
h
A(`0
)
, A(`00
)
i
2
↵⇤
max = largest eigenvalue of A⇤
time-varying
static
0 ˆc
⇤
c
% infected
c ⌘
⇤
c c
⇤
c
where
Reminder:
A⇤
=
1
`
` 1X
`0=0
A(`0
)

Summary (1)
• Time-varying graphs always lessen the
epidemic threshold assuming the
conjecture.
• Individual-based approximation used
• Supported by analysis of commutators
• Quantifying the “size” of the commutator
However, the main result contradicts numerical results when
connected components of snapshot graphs are really small.

What’s the issue?
susceptible (S)
infected (I)
infection
recovery
β
µ = 1

What’s the issue?
• Individual-based approximation is valid only for large m (=
dhub), where stochastic effects are negligible.
• We found the opposite results in our numerical
simulations for small m (not published).
• Small m is relevant to applications (sexually transmitted
infections, conversations in small groups or dyads).
• Let’s look at small m with the activity-driven model.
• e.g., m=1 for monogamous sexual relationships
m=3

http://academiccommons.columbia.edu/catalog/ac:129217
Romantic network @ Jefferson High
✔ Polygamy vs monogamy

“Concurrency”
• Polygamy vs (a sequence of) monogamy?
• Concept coming from mathematical epidemiology and HIV/AIDs
studies in mid 1990s
• Measured in field
• There are mathematical models, but its implications remain unclear.
time

concurrent (polygamous), m = 4
non-concurrent (serial monogamous), m = 1
static
“activity-driven model” (Perra et al., Scientific Reports, 2012)
m = concurrency
Which case enhances infection more?

• ρ1: prob that a hub with activity potential a is infected
after applying the star graph for time τ
• c1: prob that the hub is infected at time t+τ when the
hub is the only infected node at time t (complicated
expressions but doable)
• c2: prob that the hub is infected at time t+τ when only
a single leaf is infected at time t.
• Assumptions
• segregated stars in each time window
• Near the epidemic threshold

⇢2(a, a0
, t + ⌧) = c3⇢(a, t) + c4⇢(a0
, t) + c5(m 1)h⇢(t)i
⇢(a, t + ⌧) =a⇢1(a, t + ⌧) +
Z
da0
F(a0
)ma0
⇢2(a, a0
, t + ⌧)
+(1 a mhai)e ⌧
⇢(a, t)
⇥(z, t + ⌧) =c0
1⇥(1)
(z, t) + c0
2⇥(1, t)g(1)
(z) + c0
3⇥(z, t)
+
h
c0
4⇥(1)
(1, t) + c0
5⇥(1, t)
i
g(z)
where c0
1 ⌘ c1 e ⌧
, c0
2 ⌘ mc2, . . .
g(z) ⌘
Z
daF(a)za
, ⇥(z, t) ⌘
Z
daF(a)⇢(a, t)za
A leaf is infected
An arbitrary vertex is infected
⇢1(a, t + ⌧) = c1⇢(a, t) + c2mh⇢(t)i
A hub is infected
0 τ τ2 τ3 t

Maclaurin series: ⇢(a, t) =
1X
n=1
wn(t)an 1
w(t + ⌧) = T (⌧)w(t)
T =
0
B
B
B
B
B
B
B
@
c0
3 + haic0
4 + c0
5 ha2
ic0
4 + haic0
5 ha3
ic0
4 + ha2
ic0
5 ha4
ic0
4 + ha3
ic0
5 ha5
ic0
4 + ha4
ic0
5 · · ·
c0
1 + c0
2 haic0
2 + c0
3 ha2
ic0
2 ha3
ic0
2 ha4
ic0
2 · · ·
0 c0
1 c0
3 0 0 · · ·
0 0 c0
1 c0
3 0 · · ·
0 0 0 c0
1 c0
3 · · ·
...
...
...
...
...
...
1
C
C
C
C
C
C
C
A
→ An eigenvalue problem → Epidemic threshold
• Carefully model cases of different concurrency
• Formulate a Markov chain
• Generating functions
• Maclaurin series
• Matrix algebra
Linear map:

Phase diagrams
✔ Also for scale-free networks
In the “regular graph” case,
mc =
3
1 4a
⌧⇤ = ln
1 (1 + m)a
1 (1 + m)2a
(Onaga, Gleeson & Masuda, Physical Review Letters, 2017)
(concu
rrency)
time-varying
graph
static
graph
m = 4, polygamy
m = 1, serial monogamy
5
100
βc
10
50
τ
1
5
mc
10
0 0.2 0.80.60.4 1
m die out
enhanced
suppressed
0 τ τ2 τ3 t

Summary (2)
• Practical message: Temporal monogamy
(low concurrency) is safer than temporal
polygamy (high concurrency).

Conclusions
• Individual-based approximation and stochastic dynamics
• At high concurrency, temporal networks boost infection.
• At low concurrency, temporal networks suppress infection.
• A “commutator norm” tells how much the epidemic threshold
is moved by the temporality of the network. Analysis of
stochastic dynamics
• Further questions:
• Mobility
• Applications? Interventions? Design of “smart” interaction
orders?
• More on coevolutionary networks
• Similar theory for SIR (cf. Rocha & Masuda, Sci Rep 2016)
and other processes?

Epidemic processes on switching networks

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