leonid e. zhukov - leonid zhukov · nearest neighbours (nn), s !i 5 after ... 1v 1 leonid e. zhukov...
TRANSCRIPT
Epidemics and information spreading
Leonid E. Zhukov
School of Data Analysis and Artificial IntelligenceDepartment of Computer Science
National Research University Higher School of Economics
Social Network AnalysisMAGoLEGO course
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 1 / 40
Lecture outline
1 EpidemicsSI modelSIS modelSIR model
2 Information spread
3 Propagation trees
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 2 / 40
Epidemic dynamics models
Mathematical epidimiology
W. O. Kermack and A. G. McKendrick, 1927
Deterministic compartamental model (population classes) {S , I ,T}S(t) - succeptable, number of individuals not yet infected with thedisease at time t
I (t) - infected, number of individuals who have been infected with thedisease and are capable of spreading the disease.
R(t) - recoverd, number of individuals who have been infected andthen recovered from the disease, can’t be infected again or totransmit the infection to others.
Fully-mixing model
Closed population (no birth, death, migration)
Models: SI, SIS, SIR, SIRS,..
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 3 / 40
Network model
network of potential contacts (adjacency matrix A)
probabilistic model (state of a node):si (t) - probability that at t node i is susceptiblexi (t) - probability that at t node i is infectedri (t) - probability that at t node i is recovered
β - infection rate (probably to get infected on a contact in time δt)γ - recovery rate (probability to recover in a unit time δt)
from deterministic to probabilistic description
connected component - all nodes reachable
network is undirected (matrix A is symmetric)
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 4 / 40
Probabilistic model
Two processes:
Node infection:
Pinf = si (t)
1−∏
j∈N (i)
(1− βxj(t)δt)
≈ βsi (t)∑
j∈N (i)
xj(t)δt
Node recovery:
Prec = γxi (t)δt
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 5 / 40
SI model
SI ModelS −→ I
Probabilities that node i : si (t) - susceptible, xi (t) -infected at t
xi (t) + si (t) = 1
β - infection rate, probability to get infected in a unit time
xi (t + δt) = xi (t) + βsi∑j
Aijxjδt
infection equations
dxi (t)
dt= βsi (t)
∑j
Aijxj(t)
xi (t) + si (t) = 1
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 6 / 40
SI simulation
1 Every node at any time step is in one state {S , I}2 Initialize c nodes in state I
3 On each time step each I node has a probability β to infect itsnearest neighbors (NN), S → I
Model dynamics:
I + Sβ−→ 2I
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 7 / 40
SI model
β = 0.5
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 8 / 40
SI model
β = 0.5
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 8 / 40
SI model
β = 0.5
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 8 / 40
SI model
β = 0.5
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 8 / 40
SI model
β = 0.5
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 8 / 40
SI model
β = 0.5
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 8 / 40
SI model
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 9 / 40
SI model
1 growth rate of infections depends on λ1
2 All nodes in connected component get infected t →∞ xi (t)→ 13 probability of infection of nodes depends on v1 , i.e v1i
image from M. Newman, 2010
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 10 / 40
SIS model
SIS ModelS −→ I −→ S
Probabilites that node i : si (t) - susceptable, xi (t) -infected at t
xi (t) + si (t) = 1
β - infection rate, γ - recovery rate
infection equations:
dxi (t)
dt= βsi (t)
∑j
Aijxj(t)− γxi
xi (t) + si (t) = 1
Model dynamics: {I + S
β−→ 2I
Iγ−→ S
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 11 / 40
SIS simulation
1 Every node at any time step is in one state {S , I}2 Initialize c nodes in state I
3 Each node stays infected τγ =∫∞
0 τe−τγdτ = 1/γ time steps
4 On each time step each I node has a probability β to infect itsnearest neighbours (NN), S → I
5 After τγ time steps node recovers, I → S
image from D. Easley and J. Kleinberg, 2010
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 12 / 40
SIS model
β = 0.5, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 13 / 40
SIS model
β = 0.5, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 13 / 40
SIS model
β = 0.5, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 13 / 40
SIS model
β = 0.5, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 13 / 40
SIS model
β = 0.5, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 13 / 40
SIS model
β = 0.5, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 13 / 40
SIS model
β = 0.5, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 13 / 40
SIS model
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 14 / 40
SIS model
β = 0.2, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 15 / 40
SIS model
β = 0.2, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 15 / 40
SIS model
β = 0.2, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 15 / 40
SIS model
β = 0.2, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 15 / 40
SIS model
β = 0.2, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 15 / 40
SIS model
β = 0.2, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 15 / 40
SIS model
β = 0.2, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 15 / 40
SIS model
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 16 / 40
SIS model
Epidemic threshold R0:
if βγ < R0 - infection dies over time
if βγ > R0 - infection survives and becomes epidemic
In SIS model: R0 = 1λ1, Av1 = λ1v1
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 17 / 40
SIR model
SIR ModelS −→ I −→ R
probabilities si (t) -susceptable , xi (t) - infected, ri (t) - recovered
si (t) + xi (t) + ri (t) = 1
β - infection rate, γ - recovery rate
Infection equation:
dxidt
= βsi∑j
Aijxj − γxi
dridt
= γxi
xi (t) + si (t) + ri (t) = 1
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 18 / 40
SIR simulation
1 Every node at any time step is in one state {S , I ,R}2 Initialize c nodes in state I
3 Each node stays infected τγ = 1/γ time steps
4 On each time step each I node has a prabability β to infect itsnearest neighbours (NN), S → I
5 After τγ time steps node recovers, I → R
6 Nodes R do not participate in further infection propagation
Model dynamics: {I + S
β−→ 2I
Iγ−→ R
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 19 / 40
SIR simulation
image from D. Easley and J. Kleinberg, 2010
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 20 / 40
SIR model
β = 0.5, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 21 / 40
SIR model
β = 0.5, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 21 / 40
SIR model
β = 0.5, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 21 / 40
SIR model
β = 0.5, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 21 / 40
SIR model
β = 0.5, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 21 / 40
SIR model
β = 0.5, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 21 / 40
SIR model
β = 0.5, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 21 / 40
SIR model
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 22 / 40
SIR model
β = 0.2, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 23 / 40
SIR model
β = 0.2, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 23 / 40
SIR model
β = 0.2, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 23 / 40
SIR model
β = 0.2, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 23 / 40
SIR model
β = 0.2, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 23 / 40
SIR model
β = 0.2, τ = 2
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 23 / 40
SIR model
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 24 / 40
SIR model
Epidemic threshold R0:βγ > R0 - infection survives and becomes epidemic
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 25 / 40
SIR model
Epidemic threshold R0:βγ < R0 - infection dies over time
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 26 / 40
SIR model
image from M. Newman, 2010
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 27 / 40
Stochastic modeling
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 28 / 40
5 Networks, SIR
Networks: 1) random, 2) lattice, 3) small world, 4) spatial, 5) scale-free
image from Keeling et al, 2005
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 29 / 40
5 Networks, SIR
Networks: 1) random, 2) lattice, 3) small world, 4) spatial, 5) scale-free
Keeling et al, 2005
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 30 / 40
Effective distance
J. Manitz, et.al. 2014, D. Brockman, D. Helbing, 2013
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 31 / 40
Social contagion
Social contagion phenomena refer to various processes that depend on theindividual propensity to adopt and diffuse knowledge, ideas, information.
Similar to epidemiological models:- ”susceptible” - an individual who has not learned new information- ”infected” - the spreader of the information- ”recovered” - aware of information, but no longer spreading it
Two main questions:- if the rumor reaches high number of individuals- rate of infection spread
*We will talk about social diffusion, not a physical diffusion process, whereconservation laws present
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 32 / 40
Rumor spreading model
Daley-Kendal model, 1964Maki -Thomson model, 1973
Compartmental fully mixed model:-”ignorants” - people ignorant to the rumor, I- ”spreaders” - people who heard and actively spreading rumor, S- ”stiflers” - heard, but ceased spreading it, R
Rumor spreads by pair wise contactsλ - rate of ignorant - spreader contactsα - rate of spreader-spreader and spreader-stifler contacts
I + Sλ−→ 2S
S + Rα−→ 2R
S + Sα−→ R + S
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 33 / 40
Rumor spreading model
Time evolution of the density of stiflers (left) and spreaders (right) for RG(ER) and SF networks
from Nekovee et.al, 2007
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 34 / 40
Rumor spreading model
The final rumor size as a function of spreading rate λ.Left: random graph model, RG (ER)Right: scale free network, SF
from Nekovee et.al, 2007
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 35 / 40
Propagation tree
Rumors/news/infection spreads over the edges: propagation tree (graph)
Sometimes called cascade
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 36 / 40
Online blogs
Most common cascade shapes
from Leskovec et.al, 2007
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 37 / 40
Online newsletter experiment
Online email newsletter + reward for recommending it
7,000 seed, 30,000 total recipients, around 7,000 cascades, size form2 nodes to 146 nodes, max depth 8 steps
on average k = 2.96, infection probability p = 0.00879
Suprisingly: no loops, triangles, closed paths
Iribarren et.al, 2009
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 38 / 40
Mem diffusion
Mem diffusion on Twitter
L. Weng et.al, 2012
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 39 / 40
References
Epidemic outbreaks in complex heterogeneous networks. Y. Moreno,R. Pastor-Satorras, and A. Vespignani. Eur. Phys. J. B 26, 521-529,2002.
Networks and Epidemics Models. Matt. J. Keeling and Ken.T.D.Eames, J. R. Soc. Interfac, 2, 295-307, 2005
Simulations of infections diseases on networks. G. Witten and G.Poulter. Computers in Biology and Medicine, Vol 37, No. 2, pp195-205, 2007
Dynamical processes on complex networks. A. Barrat, M. Barthelemy,A. Vespignani Eds., Cambridge University Press 2008
Dynamics of rumor spreading in complex networks. Y. Moreno, M.Nekovee, A. Pacheco, Phys. Rev. E 69, 066130, 2004
Theory of rumor spreading in complex social networks. M. Nekovee,Y. Moreno, G. Biaconi, M. Marsili, Physica A 374, pp 457-470, 2007.
Impact of Human Activity Patterns on the Dynamics of InformationDiffusion. J.L. Iribarren, E. Moro, Phys.Rev.Lett. 103, 038702, 2009.
Leonid E. Zhukov (HSE) Lecture 7 29.05.2015 40 / 40