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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
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 2 / 49
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?
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 3 / 49
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.
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 4 / 49
Simple Contagions/Information Propagation
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 5 / 49
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?
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 6 / 49
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 7 / 49
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.
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 8 / 49
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.
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 9 / 49
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)
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 10 / 49
A simple example
(a) (b)
(a): global coef. = 0.2; local coef. = 0.3
(b): global coef. = 0.4; local coef. = 0.7
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 11 / 49
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
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 12 / 49
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
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 13 / 49
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
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 15 / 49
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
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 16 / 49
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!
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 17 / 49
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 ↑
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 18 / 49
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 ↓
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 19 / 49
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 ↑
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 20 / 49
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
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 21 / 49
An interesting question
Figure: Which case facilitates the propagation of information??
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 22 / 49
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
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 23 / 49
Complex Contagions
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 24 / 49
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.
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 25 / 49
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.
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 26 / 49
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?
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 27 / 49
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.
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 28 / 49
Our approach
A content-dependent threshold model of contagion(Yagan & Gligor, Physical Review E, 2012)
Clustered (random) multiplex networks
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 29 / 49
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
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 30 / 49
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.
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 31 / 49
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.
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 32 / 49
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.
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 33 / 49
Simulation results
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 34 / 49
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
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 35 / 49
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.
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 36 / 49
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 ↑
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 37 / 49
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
Degree Parameter (λ)0 0.5 1 1.5 2
ClusteringCoefficient
0
0.1
0.2
0.3
0.4η = 3.99η = 3η = 0.01
(b) Clustering
Degree Parameter (λ)0 0.5 1 1.5 2
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).
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 38 / 49
Simplex Networks vs. Multiplex Networks
We focus on the case of
Non-clustered networks
Standard threshold model, i.e., c = 1
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 39 / 49
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.
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 40 / 49
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
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 41 / 49
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
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 42 / 49
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
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 43 / 49
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 = 0.1multiplex theory: α = 0.1simulations: α = 0.1
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 .
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 44 / 49
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.
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 45 / 49
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
projected theory: α = 0.99multiplex theory: α = 0.99simulations: α = 0.99
Degree Parameter (αλr = λb)0 2 4 6 8
Fractional
Size
0
0.2
0.4
0.6
0.8
1
projected theory: α = 0.1multiplex theory: α = 0.1simulations: α = 0.1
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].
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 46 / 49
Two vs. Four Phase Transitions
Degree Parameter (αλr = λb)0 2 4 6 8
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.
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 47 / 49
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.
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 48 / 49
THANKS
See www.ece.cmu.edu/~oyagan for references
Osman Yagan (CMU) PhysPlex , Sept. 21, 2016 49 / 49
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