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Spreading Process inMultilayer Networks
Luca Casini
Corso diLaurea Magistrale in Informatica
A.A. 2015/2016
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Introduction
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Introduction
Networks are used to model many real-world systems.
i.e. Transportation, Computer, (Online) social network
spreading process may involve humans, information ofvarious nature, viral agents.
Biologist first studied diffusion of pathogens and thennetwork science took over their work.
These models gained the attention in recent year for theirapplication to information over communication systemsand social networks.
Early works focused on simple network but multilayernetworks are becoming more and more important.
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Preliminaries and Definitions
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Multilayer Network
Multilayer networks are composed of monoplex networks
which are modeled as traditional graphs.We use the following notation:
V The set of nodes in a multilayer network
L The set of layers in a multilayer network
n the number of nodes
(u,lu) Node uon layerl
u
((u,lu),(v,l
v)) Edge between node uon layer l
uand node von
layer lv
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Multilayer Network
AGGIUNGERE TESTO
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Types of Multilayer Network
We can define two types of multilayer network:
Multiplex Network: All layers (almost) contain thesame nodes.
i.e. same group of people in multiple social networks
Interdependent Network:Nodes belong to just onelayer. This kind of network may be seen asinterconnected communities within a large monoplexnetwork.
i.e. power and communication infrastructure networks
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Cascade and Diffusion
We call Cascadethe trace of information diffusion starting
from a node called Seed.A cascade generates an implicit network, called DiffusionNetwork.
The multilayer network in which the cascade takes place isreferred as Underlying Network
We use the following notation:
C An Information Cascade
(u,lu,v,l
v,t)
CThe entries of the set denoted by the cascade C
D A multilayer diffusion network
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Cascade and Diffusion
cascade c1 with seed (v4,l2)
cascade c2
with seed (v4,l
1)
The diffusion networkresulting from theaggregation on c
1and
c
2
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Cascade and Diffusion
There are four possibilities of spreading:
same-node inter-layer: the cascade switches layer butnot node.i.e. User sharing content on different social networks
other-node inter-layer: the cascade goes from one node
to another on a different layer.i.e. Sharing a youtube video on Facebook
other-node intra-layer: the cascade moves inside onelayer.
i.e. simple spreading inside social network like retwitting
same-node intra-layer:Trivial, generally omitted instudies
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Variables
Since its difficult to obtain real datasets, multilayer
networks research is mostly composed of eithersimulation-based studiesor analytical studies basedon mathematical models.
Both are based on the observation of some interesting
variablesand input parameters.Many different metricscan be found in the literature, wetake a look at some of the most important.
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Input Parameters
Transmissibilityrepresents the probability of
transmitting an item from one node to another.If nodes have different types we can distinguish betweenhomogenousand heterogenoustransmissibility.
The type of underlying network(i.e. random, small
world, scale-free )
The relationship between different layers (i.e. thecorrelation between node degrees)
Variations of these two parameters can producesubstantial differences in the outcome of the spreadingprocess.
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Static Variables
Epidemic Thresholdis one of the fundamental
variable in the epidemic-like models. It indicates avalue of transmissibility above which the diffusioninvolves most of the network.
Survival Threshold indicates if a diffusion process will
survive. Absolute-Dominance Threshold indicates if adiffusion process can completely remove thecompetitor.
Infection Size(also called outbreak or cascade size) is
the number of nodes in the diffusion network. Infection Rate: the average rate of being in contact
over a link.
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Temporal Variables
The variables we introduced until now are all static.
Taking time into account we have: Epidemic Dynamics is the fraction of infected nodes
at a given time in those models that call for recovery ordeath after some time.
Cascade Velocity measure how fast a cascade reachesa certain size or some relevantnodes.
Survival Probability indicates the chances of aninfection started from a single node of still being active
at a time t.
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Target-Based Variables
Sometimes there is a subset of nodes that we consider
relevant based on some particular features.i.e. popular people on social networks
In this context we can use the measures of RecallandPrecision that can be commonly found in the field of
information retrieval. Recallis defined as the ratio of relevant nodes in the
diffusion network over the total number of nodes .
Precisionis defined as the ratio of relevant nodes in
the diffusion network over the total nodes in thediffusion network.
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Models
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Models
We will now review the most important models used to
study spreading process on multilayer networksFirst we will categorize models in two groups:
Epidemic-Like
Decision-Based
The we will present some of the mathematical approachesused in the analysis of those models:
Generating Functions
Markov Chain Approximation
Mean-Field Theory
Game Theory
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Epidemic Models
Epidemic-Like Models are generally applied to either
diseases of influence diffusion.Most of the works on multilayer networks are based onthe SIR, SISor SI
1I
2Rmodels.
Those are stateful models in which a node can either be
susceptible, infected or recovered (in the SIR models).
Infectednodes diffuse the disease to their susceptibleneighbors with infection rate .
Infected nodes can recover (or return susceptible) after atime .
Transmissibility is defined as T = 1- e-
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Epidemic Models
Many variation on the SIR model can be found in the
literature. Some add new states considering the event ofbirth and death, or the effects of isolation on thespreading process.
An important one is the Independent Cascade Modelwhich is a discrete-time version of the SIR model.An infected node uat time tcan infected its neighbor v,if it succeeds vbecomes infected at time t+1
This model is often used in influence spreading studies.
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Decision-Based Models
Decision-Based Models, also called Threshold Modelsin
physics literature, are based on the idea that each agentdecides whether or not to adopt a behaviour dependingon its neighbors.
i.e. People may start smoking if their social network is
comprised of many smokers.There are two main approaches in decision-basedstudies:
Informational Effects Approach
Direct-Benefit Effects Approach
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Decision-Based Models
In the Informational Effects Approach decisions are
made based on indirect information about others choices. Linear Threshold Model: if a fraction of neighbors >
TLTM
has adopted a new behaviour then a node will takeup the same behaviour.
The Direct-Benefit Effects Approachis a game theoreticperspective of the problem where an agent takes up abehaviour if its convenient.
Ramezanian proposed a model where each node is playing
a game with its neighbors. At each round nodes updatetheir strategy (adoption on behaviour A or B) based on apayoff matrix.
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Theoretical Approaches
Widely used in the analysis of stochastic processes,
Generating Functions can uniquely determine a discretesequence of numbers, and can be useful for computing:
probability density functions moments
limit distributions solutions of linked differential-difference equations
Generating functions have also been used to studybranchingand percolationprocessesas two important
stochastic processes for modeling spread of epidemicsover networks.
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Branching Process
The branchingprocessmodel is a simple framework for
modeling epidemics on a network.While infected, an agent may spread the disease withprobability p to kotheragents (first wave). Each of thosecan then infect kother agents, spreading the disease to k2
individuals (second wave), and so on.Studying how many waves can a process survive is ofmajor interest.
When state is important (e.g. SIR model) branching
process cannot be used and bond percolation is preferred.
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Percolation
Percolation theory studies the structure of connected
clusters in random graphs.pcis the critical probability such that for p > p
cthe random
graph has a giant connected component. A percolationtransition occurs at the critical occupation probability p
c,
which is the point of appearance/disappearance of a GCC.In [102] the authors extends percolation theory tomultiplex networks by introducing the concept of weakbootstrap percolation and weak pruning percolation.
These two models are distinct and give origin to differentcritical behaviors on the emergence of critical transitions,unlike their equivalence in the case of single layer.
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Markov-Chain Approximation
The Microscopic Markov-Chain Approximation (MMA) is an
established approach to study the microscopic behavior ofepidemic dynamics.
e.g., the probability that a given node will be infected.
This approach can further be categorized as:
Discrete-time version Continuous-time version
Discrete-time version has been used to study malwarediffusion with the SIS model showing equivalence between
multilayer and single layer dynamics when the state of anode is the same in all layers.
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Mean-Field Theory
Large Markovian models may become intractable.
In Mean-field theory, a small averaged effect and anexternal field are considered instead of computing allinteractions between agents.
This allows the description of the model with a number ofnonlinear differential equations with linearly, instead ofexponentially, growing state space.
This method has been used to review and generalize
epidemic-like models.
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Game Theory
Game-theoretical approaches take into account the effect
of cooperation and competition between agents.Studies in social networks showed that communityemergence and information spreading can be explained interms of payoff maximizationand are influenced by
features of each agent: Reputation
Desire of popularity
Knowledge
Information belief
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Spreading Dynamics on
Multilayer Networks
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Interconnected Networks
Diffusion processes in interconnected networksare
affected by spectral properties of the combinatorial supra-Laplacian of underlying graph which is linked to layercoupling.
In particular changing the second eigenvalue shows two
very distinct regimes with layers either decoupled orindistinguishable.
Spreading in interconnected networks has been studied interms of:
Interaction strength between layers
Inter-layer pattern.
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Interaction Strength
second-nearest neighbors: expected number of
neighbors = k2
/k.kis the moment of the degreedistribution. In weakly coupled networks (
A
T
B) we
find a phase in which a layer may be in epidemic stateindependently of others, depending on transmissibilityand average inter-layer degree.
interconnection topology measure: quantitativemeasure of coupling given by the formula:
inter-layer link density: The ratio of existing interlayerconnection to the total possible d = m/(n
Ax n
B)
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Inter-Layer Patterns
Some studies highlighted the importance of inter-layer
links and the patterns they create.Simulation studies showed that the degree of connectionshave less impact than the density.
A new definition of Epidemic Threshold in the SIS model
was proposed, considering degree of connected nodes:
TE= 1/
1(M + N) =infection rate, M = adjacency matrix, N
inter-layer matrix
Another study observed that if correlation between intraand inter-layer degree is very strong an outbreak mayappear even below the epidemic threshold of each layer.
l
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Intra-layer structure
Epidemics dynamics depend not only on interlayer links
but also in intralayer.A study on cliques (groups of people who are close)showed how they influence epidemic threshold and
infection size and speed. They defined 3 types of link: 1
intra-clique, 2 inter-clique, and 3 online.let d
wand d
fbe the number of type 1 and 2 links per node,
there is an epidemic state when:
with E representing the moment of degree distribution
i l i il i
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inter-layer similarity
Similarity (or the lack of it) may influence the spreading
behaviour.Degree-Degree Correlation is described by factors where k is the number of intra-layer nodes in
each layer. Similarly interlayer correlation can be
measured.Average Similarity of Neighbors is defined as:
whereKA
represents number of neighbors on layer A and KC
representthe number of common neighbor.
Strong degree correlation lead to low epidemic thresholdand small infection size. Interestingly its not influenced byaverage neighbors similarity.
l i hi
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layer-switching cost
Some models must consider that diffusion on different
layer involves some kind of cost or overheadi.e. changing mean of transport or sharing content from a socialnetwork to another
A recent study considers this and observes the behaviour
of epidemic threshold in function of node degree andinfection rate.
A large difference in infection rates among layers meanhigher overhead and higher epidemic threshold.
If a layer is denser epidemic threshold lowers as spreadingbecomes easier.
Diff i V l i
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Diffusion Velocity
The presence of multiple layers impacts the speed of the
diffusion process.Intuitively multiple layers speed up the diffusion processbecause there more links to spread the information. Somestudies confirmed this showing correlation between
coupling and velocity.However some empirical studies pointed out thatinefficient topologies in monoplex networks andobstructed inter-layer links in multiplex networks lead to
decreasing speed.
P ti l O l
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Partial Overlap
In partially overlapped multilayer networksonly a
fraction of nodes is present in all layers.A study on the effect of overlapping in the SIR modeldiscovered that the epidemic threshold T
cis directly linked
to the fraction qon node present in both layers
aaaa A = branching factor
aaaaaaaa of layer A
this means that the epidemic threshold of the layer withlower diffusion capability affects the threshold of theother.
I t ti S di P
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Interacting Spreading Process
In the real world different spreading processes coexist and
interact with each other. Epidemic and games-theoreticmodels have been presented to address this.
game-theoreticstudies are mainly focused on competingrumors or companies trying to sell their product.
Epidemic models and consider two competing viruses ormemes. This two viruses can coexist or one can dominatethe other, eventually leading to extinction.
An interesting application of the model studies a virus
versus an immunization process.
All studies concluded that interaction dynamics are linkedto the underlying network topology.
I ti Diff i
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Innovation Diffusion
diffusion of innovation (new behaviours, technology,
ideas, products) received considerable interest in socialscience and economics.
The problem has been studied in using and extension of
the Watts threshold model.
A content-dependent threshold was introduced, dealingwith a specific bias each link has towards certain content.
Considering the approach of direct-benefit effects n agame-theoretic framework, a lower bound for the success
of an innovation can be found
i.e. how many people in the network adopt a specific
strategy
R C t i t
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Resource Constraints
In real life nodes of a multiplex network share limited
resources. This will impact spreading processes dynamics.i.e. people share their time between different online socialnetworks.
A variation of the SIR model called constrained SIR isintroduced. In each step, a limited number of neighborscan be infected.
The authors find that, in agreement with previous studies,in the absence of constraints, positively correlation leads
to a lower epidemic threshold than a negative correlation.However, in the presence of constraints, spreading is lessefficient in positively correlated coupling than negativelycorrelated networks.
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Applications
Applications
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Applications
Spreading processes in multilayer networks have a large
number of applications: understanding the dynamics of cascades. maximizing the influence in the context of viral
marketing .
placing sensors to detect the spreading as quickly aspossible in a network.
The application areas can be roughly categorized into twoclasses:
Forward Prediction:applications that need to steer thenetwork into a particular desired state.
Backward Prediction: applications that require to predicthow a given piece of information will diffuse in a network.
Influence Maximization
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Influence Maximization
influence maximization has the goal of spreading
information as quickly as possible.This can be achieved by choosing the most influential
nodes as a seed. these nodes are chosen according to
some measure of centrality like: pagerank, betweeness
or eigenvector centrality.On the other hand we can choose the messages that are
likely to survive longer than the others and so propagateto more nodes, obtaining the same effect.
Immunization
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Immunization
resilience to a disease can be achieved through
information dissemination. Various work investigated thisquestion using a model based on two layers:
the infection layer: where the disease spreadsthe prevention layer: where awareness spreads
Studies have observed that awareness can raise the
infection threshold and in many cases almost stop theinfection.
An important application is studying the effectiveness ofvaccination campaigns.
Delay Tolerant Networks
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Delay-Tolerant Networks
Delay-Tolerant Networks are networks which address the
problem of the lack of continuous connectivity.i.e. deep space communication, sensor networks
Routing on DTNs is more challenging than on traditionalnetworks and is an important application of forward
prediction, usually addressed using epidemic algorithmover the active connection graph.
As every sensor may have more than one communicationdevice the spreading process can be mapped with a
multilayer network where the best routing is bound bylatency and energy constraints.
Malware Propagation
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Malware Propagation
Studying malware propagation and design solution to
contain outbreak is very important and involves bothforward and backward prediction.
This problem is intrinsically multilayer; along computer wehave mobile devices that are connected through multiplewireless interfaces (3g, bluetooth, wi-fi) and the use ofapplication allows communication with device that maynot be immediate neighbors.
Each of these factor should be taken into account as aseparate layer when modelling such spreading process
Conclusions
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Conclusions
Information diffusion in multilayer networks is an active
and not yet consolidated research field, and thereforeoffers many unsolved problems to address. In some cases,phenomena that are quite well understood in monoplex
networks are comparatively not well understood in thecontext of multilayer networks; in other
cases, completely novel ideas, algorithms and analysis,specific to multilayer networks have to be developed.
Some research directions are illustrated below.
Open Problems
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Open Problems
empirical study of information diffusion:Real dataset are both
difficult to obtain and study due to their size.
metrics and measurements: New metrics specific to multilayernetwork should be researched aside from those derived from
monoplex networks.
new models: Some phenomena may require new models to be
described. An example is the Data-Mining approach for heterogeneous
networks.data visualization: Visualization is a great tool for researchers, the
muxViz project is the best contribution at the moment.
time-varying networks: Time is central to many process. Studies on
time-varying multiplex networks are yet to appear.
evolution of underlying structure: There are studies on adaptivemonoplex networks but the multilayer perspective must be deepened.
outbreak detection: Detecting as quickly as possible a spreading
processes is a field worth exploring in multilayer networks.
References
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References
[1]
[2]