analysis of contagions in multi-layer and multiplex ...oyagan/talks/ccs_multiplex.pdfanalysis of...

Analysis of Contagions in Multi-layerand Multiplex Networks

Osman Yagan

Department of ECE

Carnegie Mellon University

Joint work with Yong Zhuang and Alex Arenas

Supported by NSF CCF # 1422165

Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 1 / 49

Dynamical processes on complex networks

∗ Spreading of an initially localized effect throughout the whole (or, a verylarge part of the) network.

Diffusion of information, ideas, rumors, fads, etc.

Disease contagion in human and animal populations.

Cascade of failures, avalanches, sand piles.

Spread of computer viruses or worms on the Web.

Flows of data, materials, biochemicals.

Network traffic, congestion.

∗ Barrat et al. Dynamical Processes on Complex Networks, 2008


What is new?

∗ Most research focus on the limited case of a single and non-interactingnetwork.∗ Yet, many real-world systems do interact with each other.

� Social networks are coupled together:Facebook, Twitter, Google+, YouTube, etc.

Q: Dynamical processes on interacting networks?


TODAY

Information Propagation (Simple Contagion)

A music video, a news article, etc.Letting someone know of somethingReceiving one copy is enough (disease-like propagation)

• Influence Propagation (Complex contagion)

• Joining a riot, adopting a behavior, buying a product, etc.• Requires social reinforcement• Having 100 friends joining a riot will be different than having

only one who does so.


Simple Contagions/Information Propagation


Information Propagation in Multi-layer Networks

Dynamics of information spreading changed dramatically with theonline social networks; e.g., Facebook, Twitter, Instagram, etc.

A key observation: Social networks are coupled with each other andwith the physical communication network

due to people who participate in multiple networks

Q: How does the coupling affect the speed and extent of informationpropagation?


Issues with existing approaches

• Single-layer:

Individuals engage in different types of networks;e.g., Facebook, Whatsapp, Twitter, etc.

• No-clustering:

Social network has a propensity that two friends of one individualare more likely to know each other.

There is lack of study taking these two factors into consideration.

Our approach: Clustered multi-layer/multiplex networks.


An Illustration of a Multi-layer/Multiplex Network Model

Figure: An illustration of a multi-layer/multiplex network model.

Each layer is generated randomly from given degree distributions withtunable clustering coefficient.


Random Networks with Clustering (Newman and Miller)

Clustering: Informally defined as the propensity of two neighbors of a nodeto be neighbors as well.

Friend of a friend is usually a friend

Formal definitions: Strongly related with the number of triangles

Cglobal =3× (number of triangles in network)

number of connected triples, (1)

Clocal =1

n∗

∑i

Ci , (2)

where

Ci =number of triangles connected to vertex i

number of connected triples centered on vertex i, (3)


A simple example

(a) (b)

(a): global coef. = 0.2; local coef. = 0.3

(b): global coef. = 0.4; local coef. = 0.7


Clustered Random Networks

Generated randomly from given degree distributions with tunableclustering.

Configuration Model:

Stub types: only single stubs.The degree of a node: d .Degree distribution: pd

Generalized Configuration Model:

Stub types: single stubs and triangle stubs.The degree of a node: d = (ds , dt)Total degree of a node: ds + 2× dt .Degree distribution: pds ,dt


Modeling Information Propagation via SIR Epidemics

Susceptible (S): Not aware of the information

Infectious (I): Has the information and currently spreading it

Recovered (R): No longer spreading the information

Tij = 1− e−rijτi : probability that an infectious individual i transmits theinformation to a susceptible contact jrij : rate of contact over the link from i to j ; τi is recovery time for i


SIR Process → Bond Percolation in Multi-layer networks

Assume for simplicity two layers

W : Physical NetworkF : Facebook

Different transmissibility over W and F: Twij and T f

ij

Average transmissibility in W (with i.i.d. rwij and τw1 = · · · = τwn )

Tw := 〈Twij 〉 = 1−

∫ ∞0

e−rτwPw (r)dr .

Bond Percolation Model: Each edge in W (resp. F) is occupiedwith probability Tw (resp. Tf ) independently from all others.

Information propagates only through occupied edges

SIR Model is isomorphic to bond percolation processOsman Yagan (CMU) PhysPlex , Sept. 21, 2016 14 / 49

Problem Formulation and Quantities of Interest

The propagation is triggered by infecting an arbitrary node andcontinues according to SIR model.

final outbreak size: # of nodes that eventually receive the information

self-limited outbreak: outbreak sizen → 0 as n→∞

epidemic: outbreak sizen → e > 0 as n→∞

epidemic threshold : critical boundary in the space of all networkparameters that distinguishes

P[epidemic] > 0 and P[epidemic] = 0


Main Results

• Y. Zhuang and O. Yagan, “Information propagation in clusteredmultilayer networks,” IEEE Trans. Network Science and Engineering, 2016

i) the epidemic boundary;

ii) the relative final size of epidemics when P[epidemic] > 0

iii) the exact probability P[epidemic] in the super-critical regime

Main Techniques

Map SIR process to bond percolation

Explore the emergence of a giant component in the resultingmulti-layer network through a multi-type branching process

Use generating functions approach to analyze the branching process


Experiments

α : fraction of people who use Facebook

Tw = Tf

0.2 0.4 0.6 0.8 1

Epidem

icSize

0

0.2

0.4

0.6

0.8 α = 0.1− Exp

α = 0.1− Thm

α = 0.5− Exp

α = 0.5− Thm

α = 0.9− Exp

α = 0.9− Thm

(a) Poisson distribution.

Tw = Tf

0 0.2 0.4 0.6 0.8 1

Epidem

icSize

0

0.2

0.4

0.6

0.8

1

α = 0.1− Exp

α = 0.1− Thm

α = 0.5− Exp

α = 0.5− Thm

α = 0.9− Exp

α = 0.9− Thm

(b) Power-law distribution.

Perfect agreement between analysis and simulations!


Impact of clustering on information propagation

We use the following degree distributions for number of single edges andtriangles in two networks.

Network F Network W

Single Edges Distribution Poi(2λf ) 2 Poi( 4−c2 λw )

Triangle Edges Distribution Poi(λf ) Poi(c2λw

)Table: Parameters of the doubly Poisson distribution.

c ↑ ⇒ clustering ↑


Impact of clustering on epidemic size

c

0 1 2 3 4

Epidem

icSize

0

0.02

0.04

0.06

0.08

0.1α = 0.2

α = 0.3

α = 0.4

Figure: Poisson distribution

c ↑ ⇒ clustering ↑ ⇒ epidemic size ↓


Impact of clustering on epidemic boundary

Tf

0 0.1 0.2 0.3 0.4

TW

0

0.1

0.2

0.3

0.4

c = 0.01,α = 0.1

c = 2.00,α = 0.1

c = 3.99,α = 0.1

c = 0.01,α = 0.9

c = 2.00,α = 0.9

c = 3.99,α = 0.9

Figure: critical boundary of epidemics

c ↑ ⇒ clustering ↑ ⇒ Critical Threshold ↑


Lesson Learned

Clustering has an inhibitive effect on epidemics

Can be attributed to the fact that the edges used for completingwedges to triangles is redundant for information propagation


An interesting question

Figure: Which case facilitates the propagation of information??


A multi-faceted picture

α ↑ ⇒ large but loose→

{epidemic threshold ↑epidemic size ↑

More difficult to trigger an information epidemic in large online socialnetwork that is loosely connected vs. small but densely connected.

But, when information transmissibility is high, final epidemic size islarger with a large but loosely connected online social network


Complex Contagions


A dynamical process: Binary decisions with externalities

• Each individual must decide between two actions, e.g.,� To buy or not to buy a smart phone� To vote for Democrats or Republicans

Simple Threshold Model (D. Watts, PNAS, 2002)

• Nodes can be in either one of the two states: active or inactive.

• Each node is initially given a threshold τ drawn from Pth(τ).

• An inactive node with m active neighbors and k −m inactiveneighbors will turn active if m

k ≥ τ .

• Assumption: Once active, a node can not be deactivated.


An Illustration of the Simple Threshold Model

τ = 0.2

(a) mk = 1

6 < τ = 0.2: Fail to be influenced.

(b) mk = 5

6 > τ = 0.2: Influenced.


Global cascades

• Start by activating a few nodes (give incentives, free samples)

• Global Cascades: A positive fraction of nodes (in the asymptoticlimit) eventually becomes active

∗ Condition, Probability, and Expected size of global cascades when anarbitrary (set of) node(s) is made active?


Issues with existing approaches

• Simplex: the networks with only one single link type.

Individuals engage in different types of relationships; e.g.,family, friends, office-mates, college-mates, etc.

• No-clustering:

Social network has a propensity that two friends of one individualare more likely to know each other.

There is lack of study taking these two factors into consideration.


Our approach

A content-dependent threshold model of contagion(Yagan & Gligor, Physical Review E, 2012)

Clustered (random) multiplex networks


A content-dependent threshold model for multiplexnetworks (PRE, 2012)

A multiplex network: r types of links.

Content-dependent weights c1, . . . , cr :

ci quantifying the relative bias a type-i link has

An inactive node with degree k :

mi active neighborski −mi inactive neighbors via type-i links∑r

i=1 ki = k

Perceived ratio of active neighbors :=c1m1 + c2m2 + . . .+ crmr

c1k1 + c2k2 + . . .+ crkr≥ τ


An Illustration of the Content-dependent Threshold Model

τ = 0.2.

(a) mk = 1

6 < τ = 0.2: Fail to be influenced.

(b) mk = 3×1

3×1+1×5 > 0.2: Influenced.


An Illustration of a Multiplex Network Model

Figure: An illustration of a multiplex network model.

Each link-type is generated randomly from given degree distributionswith tunable clustering coefficient.


Analysis with random clustered multiplex networks

∗ Let r = 2; i.e., assume that there are only two link types.∗ F: type-1 s single links and t triangles: {pbs,t}.∗ W: type-2 s single links and t triangles: {prs,t}.∗ H = F ∪W with colored distribution {pk}

pk = pbk1,s ,k1,t· prk2,s ,k2,t

, k = (k1,s , k1,t , k2,s , k2,t)

Q: Condition, probability, expected size of global cascades?

Y. Zhuang, A. Arenas, and O. Yagan, “Clustering determines the dynamicsof complex contagions in multiplex networks,” submitted, August 2016.


Simulation results


Agreement between the Analysis and Simulations

pbst = e−λb,1(λb,1)s

s!e−λb,2

(λb,2)t

t!, s, t = 1, 2, . . .

prst = e−λr,1(λr ,1)s

s!e−λr,2

(λr ,2)t

t!, s, t = 1, 2, . . .

(pbs,t): s single links and t triangles in type-1.

(prs,t): s single links and t triangles in type-2.

c = c1/c2 : relative importance of type-1 vs. type-2 links


Agreement between Analysis and Simulations (Cont’d)

Degree Parameter (λr,1 = λr,2 = λb,1 = λb,2)0 0.5 1 1.5 2 2.5 3

Fractional

Size

0

0.2

0.4

0.6

0.8

1S - AnlysS - Expt.Ptrigger - AnlysPtrigger - Expt.

Content Parameter (c)0 1 2 3 4 5 6

Fractional

Size

0

0.1

0.2

0.3

0.4

S - Expt.S - Anlys.Ptrigger - Anlys

Figure: Simulations for doubly Poisson degree distributions. In (a), we set thecontent parameter c = 0.25, the threshold as τ = 0.18, and α = 0.5, then varythe degree parameters. In (b), we fix τ = 0.18, λr ,1 = λr ,2 = λb,1 = λb,2 = 0.3,and α = 0.5 while varying content parameter c .

• There is a perfect agreement between our analysis and simulations.


Impact of Clustering on Influence Propagation

We use the following degree distributions for number of single edges andtriangles in two networks.

Network F Network W

Single Edges Distribution Poi(2λ) 2 Poi( 4−η2 λ)

Triangle Edges Distribution Poi(λ) Poi(η

2λ)

Table: Parameters of the doubly Poisson distribution.

η ↑ ⇒ clustering ↑


Impact of Clustering on Influence Propagation (Cont’d)

Degree Parameter (λ)0 0.5 1 1.5 2

FractionalSize

0

0.2

0.4

0.6η = 3.99η = 3η = 0.01

(a) Probability


ClusteringCoefficient

0

0.1

0.2

0.3

0.4η = 3.99η = 3η = 0.01

(b) Clustering


FractionalSize

0

0.2

0.4

0.6

0.8

1η = 3.99η = 3η = 0.01

(c) Expected Size

• η ↑ ⇒ clustering ↑.• clustering ↓ Prob. to trigger cascades (low degrees).

• clustering ↑ Prob. to trigger a cascades (high degrees).

• clustering ↓ Expected cascade size (low degrees).

• clustering ↑ Expected cascade size (high degrees).


Simplex Networks vs. Multiplex Networks

We focus on the case of

Non-clustered networks

Standard threshold model, i.e., c = 1


Degree Distribution

The degree distributions of red and blue edges.

pbk = e−λbλkbk!, k = 0, . . . , (4)

prk = αe−λrλkrk!

+ (1− α)1[k = 0], k = 0, . . . .

where

α: the relative size of nodes that have red and blue edges to that onlyhave blue edges.

λr and λb: the degree parameter of red and blue edges respectively.


Simplex Projected Theory vs. Multiplex Theory

multiplex theory:

Constraint: red stubs are connected to red ones. blue stubs areconnected to blue ones.

projected theory:

No Constraint: ignores the color of the edges and matches allstubs randomly with each other.

assortativity: the Pearson correlation coefficient of degree between pairsof linked nodes


Simplex Projected Theory vs. Multiplex Theory

multiplex theory:

Constraint: red stubs are connected to red ones. blue stubs areconnected to blue ones.high assortativity when α is small.

projected theory:

No Constraint: ignores the color of the edges and matches allstubs randomly with each other.the assortativity is negligible.

assortativity: the Pearson correlation coefficient of degree between pairsof linked nodes


Multiplex Networks with Limited Assortativity

We enforce

λr = λb. (5)

projected theory:

the assortativity is negligible.

multiplex theory:

the assortativity is negligible when α is large


Multiplex Networks with Limited Assortativity (Cont’d)

Degree Parameter (λr = λb)0 2 4 6 8

Fractional

Size

0

0.2

0.4

0.6

0.8

1

projected theory: α = 0.99multiplex theory: α = 0.99simulations: α = 0.99

Degree Parameter (λr = λb)0 2 4 6 8

Fractional

Size

0

0.2

0.4

0.6

0.8

1


projected theory:

α = 0.99: negligible assortativityα = 0.1: negligible assortativity

multiplex theory:

α = 0.99: negligible assortativityα = 0.1: assortativity ≈ 0.21

Slight difference when α = 0.1

Probable suspect: assortativity .


Multiplex Networks with Assortativity

To better observe the impact of assortativity, we set

αλr = λb. (6)

If α = 0.1, then λr is 10 times as large as λb.

projected theory:

the assortativity is still negligible.

multiplex theory:

significant assortativity.


Multiplex Networks with Assortativity (Cont’d)

Degree Parameter (αλr = λb)0 2 4 6 8

Fractional

Size

0

0.2

0.4

0.6

0.8

1



Fractional

Size

0

0.2

0.4

0.6

0.8

1


projected theory:

α = 0.99: negligible assortativityα = 0.1: negligible assortativity

multiplex theory:

α = 0.99: negligible assortativityα = 0.1: assortativity ∈ [0.19, 0.79].


Two vs. Four Phase Transitions


Fractional

Size

0

0.2

0.4

0.6

0.8

1

multiplex theory: α = 0.1multiplex theory: α = 0.166multiplex theory: α = 0.99

Figure: The demonstration of multiplex transition phases.

α = 0.1: Cascade only in nodes with red edge.α = 0.166: Cascade in nodes regardless of colors.α = 0.99: Cascade only in nodes with blue edge.


Conclusions

We analyze the diffusion of influence in clustered multiplex networks.

We solve analytically for the probability of global cascades andexpected cascade size.

We demonstrate how clustering affects the probability of triggering aglobal cascade and the expected cascade size.

We also make a comparison between influence propagation in simplexand multiplex networks.


THANKS

See www.ece.cmu.edu/~oyagan for references


www.ece.cmu.edu/~oyagan

analysis of contagions in multi-layer and multiplex ...oyagan/talks/ccs_multiplex.pdfanalysis of...

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