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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


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,..


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)


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


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)→ 13 probability of infection of nodes depends on v1 , i.e v1i

image from M. Newman, 2010


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


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


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

Model dynamics: {I + S

β−→ 2I

Iγ−→ R


SIR simulation

image from D. Easley and J. Kleinberg, 2010


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


SIR model

image from M. Newman, 2010


Stochastic modeling


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

*We will talk about social diffusion, not a physical diffusion process, whereconservation laws present


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



Time evolution of the density of stiflers (left) and spreaders (right) for RG(ER) and SF networks

from Nekovee et.al, 2007



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


Propagation tree

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

Sometimes called cascade


Online blogs

Most common cascade shapes

from Leskovec et.al, 2007


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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