leonid e. zhukov - leonid zhukov · nearest neighbors (nn), s !i 5 after ... 1v 1 leonid e. zhukov...

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 26.05.2017 1 / 36

Lecture outline

1 EpidemicsSI modelSIS modelSIR model

2 Information spread

3 Propagation trees


Flu contagion

Infected - red, friends of infected - yellowN. Christakis, J. Fowler, 2010


Global contagion

Simulated epidemic:gray lines - passenger flow, red symbols epidemics location

D. Brockmann, D. Helbing, 2013


Network epidemic model

Given a network G of potential contacts

Three states model: susceptible, infected, recovered

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)

connected component - all nodes reachable

network is undirected (matrix A is symmetric)

if graph complete - fully mixing model

Based upon models from mathematical epidemiology, W.O. Kermackand McKendrick, 1927


Probabilistic model

Two processes:

Node infection:

Pinf ≈ βsi (t)∑

j∈N (i)

xj(t)δt

Node recovery:

Prec = γxi (t)δt


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


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


SI model

β = 0.5


SI model


SI model

1 growth rate of infections depends on λ1

2 All nodes in connected component get infected t →∞ xi (t)→ 1

3 probability of infection of nodes depends on v1 , i.e v1i

image from M. Newman, 2010


SIS model

SIS ModelS −→ I −→ S

Probabilities that node i : si (t) - susceptible, 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


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 neighbors (NN), S → I

5 After τγ time steps node recovers, I → S


β−→ 2I

Iγ−→ S


SIS model

β = 0.5, τ = 2


SIS model


SIS model

β = 0.2, τ = 2


SIS model


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


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


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


β−→ 2I

Iγ−→ R


SIR model

β = 0.5, τ = 2


SIR model


SIR model

β = 0.2, τ = 2


SIR model


SIR model

Epidemic threshold R0:βγ > R0 - infection survives and becomes epidemic


SIR model

Epidemic threshold R0:βγ < R0 - infection dies over time


5 Networks, SIR

Networks: 1) random, 2) lattice, 3) small world, 4) spatial, 5) scale-free

image from Keeling et al, 2005


5 Networks, SIR

Networks: 1) random, 2) lattice, 3) small world, 4) spatial, 5) scale-free

Keeling et al, 2005


Effective distance

J. Manitz, et.al. 2014, D. Brockman, D. Helbing, 2013


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


Branching process

Simple contagion model:- each infected person meets k new people- independently transmits infection with probability p

image from David Easley, Jon Kleinberg, 2010


Branching process

Galton-Watson branching random process, R0 = pk

if R0 = 1, the mean of number of infected nodes does not change

if R0 > 1, the mean grows geometrically as Rn0

if R0 < 1, the mean shrinks geometrically as Rn0

R0 = pk = 1 - point of phase transition


Propagation tree

Rumors/news/infection spreads over the edges: propagation tree (graph)

Sometimes called cascade


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


Mem diffusion

Mem diffusion on Twitter

L. Weng et.al, 2012


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.


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