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- 1. Epidemic processes in time- varying networks: Commutator and concurrency Naoki Masuda State University of New York at Buffalo naokimas@buffalo.edu www.naokimasuda.net
- 2. Temporal networks (= time-varying graphs)
- 3. From http://www.sociopatterns.org/gallery/ pupil teacher class “node” = pupil or teacher
- 4. 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
- 5. • 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)
- 6. • 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
- 7. How does temporality affect epidemic dynamics, particularly epidemic thresholds?
- 8. 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
- 9. 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
- 10. 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
- 11. 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.
- 12. 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)
- 13. SIS model susceptible (S) infected (I) infection recovery β µ = 1 3β
- 14. 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)
- 15. 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
- 16. • 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 `⌧
- 17. “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
- 18. 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.
- 19. 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 This slide is a mathematical side note.
- 20. 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) This slide is a mathematical side note.
- 21. 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) (Speidel, Klemm, Eguíluz & Masuda, New Journal of Physics, 2016)
- 22. 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)`⌧]
- 23. • 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
- 24. To quantify the difference between time-varying and static networks
- 25. 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 )
- 26. 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 )
- 27. 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 <latexit sha1_base64="LvyI4kCJsKG4hRP3GiXv4qwl4o8=">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</latexit>
- 28. C ⌘ 1 (`↵⇤ max)2 ` 1X `0=1 `0 1X `00=0 h A(`0 ) , A(`00 ) i 2 (Speidel, Klemm, Eguíluz & Masuda, New Journal of Physics, 2016)
- 29. 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: (Speidel, Klemm, Eguíluz & Masuda, New Journal of Physics, 2016) A⇤ = 1 ` ` 1X `0=0 A(`0 )
- 30. 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.
- 31. 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
- 32. What’s the issue? susceptible (S) infected (I) infection recovery β µ = 1
- 33. 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
- 34. http://academiccommons.columbia.edu/catalog/ac:129217 Romantic network @ Jefferson High ✔ Polygamy vs monogamy
- 35. “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
- 36. 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?
- 37. 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)
- 38. • 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
- 39. ⇢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
- 40. 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:
- 41. 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
- 42. Summary (2) • Practical message: Temporal monogamy (low concurrency) is safer than temporal polygamy (high concurrency).
- 43. 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?