The%20Influence%20Model

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The Influence Model
Presenter Michele Garetto
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References

• C. Asavathiratham, S. Roy, B. Lesieutre, G.
  Verghese. The Influence Model, IEEE Control
  Systems Magazine, Dec. 2001
• C. Asavathiratham, The influence model a
  tractable representation for the dynamics of
  networked Markov chains, Ph.D. dissertation,
  EECS Dept., MIT, Oct. 2000 http//web.media.mit.ed
  u/tanzeem/cohn/chalee_thesis.pdf
• G. Verghese, General Models of Network
  Dynamics, http//element.stanford.edu/lall/proje
  cts/architectures/kick_off/verghese.ppt

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Outline

• Motivation
• Related models
• Interactive Markov Chains
• Formulation of the Influence Model
• Application to virus modeling
• The Influence model in a nutshell
• Conclusions and discussion

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Motivation

• How can we study analitically (not just by
  simulation) the dynamic behavior of very complex
  networks, with a large number of components
  interacting together ?
• Multiple application domains

• - Communication networks
• Internet
• Energy
• Power grid
• Transportation
• Air traffic, road, rail
• Social networks
• Interactions between individuals

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Dominant Issues

• Uncertainty
• dynamic, aggregated behavior
• Events propagation
• cascading events
• transient behavior
• Resource allocation for failure mitigation
• Distributed decision and control
• Reconfiguration and recovery
• Guaranteed behavior
• Reliability
• Predictability

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Examples of particular interest

• Catastrophic outages in major infrastructures
  (such as the Internet, the power grid, the air
  traffic system ...)
• Traffic congestion (spatial and temporal
  correlations ...)
• Virus spreading (propagation speed, final size,
  effect of countermeasures, immunization ...)

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Related models of interactions on networks
Area Model Key author

Physics - Stochastic Ising Model Glauber 63
- Cellular automata Wolfram 94
- Markov chain Monte Carlo Metropolis 53
Mathematics - Infinite particle system Spitzer 70
- Voter model Holley, Liggett 75
- Contact process Harris 74
Biology - Invasion process Clifford,Sudbury 73
Sociology - Threshold model Granovetter 78
- Interactive Markov Chains Conlisk 76
Economics - Local interaction game Ellison
93
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The Interactive Markov Chain (IMC) Modeling
Framework

• Global network structure ...

but locally a Markov chain
Global Structure (the network)

• Each node is represented by a Markov chain,
  whose state transitions are influenced by the
  states of its neighbors

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Computational complexity problem

• The solution of the global Markov chain is
  feasible only for small systems (a few tens of
  nodes ?)

• It is possibile to consider very large systems

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The Influence Model (1)

• The influence model is a discrete-time Markov
  process
• Lets consider for simplicity the case of an
  ergodic system (irreducible, aperiodic)
• In a stand-alone Markov Chain the evolution of
  the status probabilities is

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The Influence Model (2)

• In the case of networked Markov Chains, the
  transitions probabilities are constrained to take
  a multilinear form

p1 (k) d1,1 p1 (k-1) P1,1 d1,2 p2 (k-1) P1,2
... d1,N pN (k-1) P1,N
( d1,1 d1,2 d1,N 1 )

• The coefficients di,j are the weights associated
  with incoming edges

• We can have self-loops

• Matrices Pi,j can be nonsquare

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The Influence Model (3)

• If we stack the status vector probabilities of
  all of the nodes into a single vector

P p1 p2 ... pN
we can write more compactly
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The Influence Model (4)

• By Perron-Frobenius theory for irreducible
  nonnegative matrices, H has a dominant real
  eigenvalue of 1 that strictly dominates all other
  eigenvalues, and the corresponding left
  eigenvector is the steady-state status
  probability vector

( M m1 mn)
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The Influence Model (5)

• The analysis of the eigenstructure of H provides
  a powerful way to capture the dynamics of the
  individual sites, but is not sufficient to
  compute the evolution of the joint probabilities
  of groups of nodes

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Example homogeneous influence model

• Consider an influence model in which each site
  can be in either of two different states, labeled
  Normal or Failed, and all of the sites have the
  same local Markov chains

• The steady-state vector of A is .833 .167

• weights di,j are identical k neighbors -gt d
  1/(k1) (including the self-loop)

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Dilemma to connect or not to connect...

• When a node is Failed, neighbors can help.
• When a node is Normal, neighbors can hurt

A
D
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Dilemma to connect or not to connect...

• As far as the steady-state probability of being
  Failed is concerned, it turns out that it makes
  no difference whether or not a site connects to
  the network, regardless of the network structure
  ! (the steady-state probability of being
  Failed is the same)

• What does change when sites connect together is
  the pattern of failures (joint probabilities of
  groups of nodes)

• Node failures are correlated !

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Dilemma to connect or not to connect...

• reminder the steady-state vector for each node
  is .833 .167

Failures are more likely to appear in connected
groups
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Dilemma to connect or not to connect...

• We can consider all intermediate cases with a
  tunable network matrix T specified in terms of a
  coefficient c (the self influence probability)
  T(c) c I (1-c) D

c1.0
c0.9
c0.04
Not connected - binomial distribution
Relative Frequency (obtained by simulation)

• Fully connected -
• more small failures
• more large failures

Number of Failed Nodes
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Higher order analysis (1)

• The influence matrix H provides only a partial
  and rather limited view of the behavior of the
  system. A complete description would require to
  consider the transition matrix of the giant
  global markov chain G
• but it turns out that there is an intimate
  relation between H and G

• key question what is the connection between the
  eigenvalues of H and those of G ?

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Higher order analysis (2)

• It is possible to extract intermediate
  information between that provided by H and G
  building a hierarchy of matrices H(r) , each one
  intended to study the evolution of joint
  probabilities of groups of lt r gt sites
  (at the cost of an increasing computational
  complexity)

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Higher order analysis (3)

• The eigenstructure of matrices H(r) has rich
  mathematical properties. If G has distinct
  eigenvalues, a telescoping relation exists
  among a subset of their spectrum called relevant
  eigenvalues

• Moreover, it is conjectured by the authors on
  the basis of extensive numerical experiments
  (also proved in the case of homogeneous influence
  models) that the subdominant eigenvalue (the
  eigenvalue with the second largest magnitude) of
  H equals that of G

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Application to virus modeling

• The simplest model we can build to describe the
  spreading of a virus is the following

• nodes stand for Internet hosts, and they can be
  in either of just two states infected or normal.

• weigths associated with edges stand for
  infection propagation probabilities

• we start with a network in which every node is
  normal, and we initiate a new infection turning
  the status at some node to infected.

• What happens ?

• the system evolves until it settles down to the
  configuration in which all of the nodes are
  infected or all of the nodes are normal, and
  remains that way forever

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Application to virus modeling

• an infected node turns normal again if it is
  influenced by a normal neighbor (this is not
  realistic)

• If we want a node to be influenced differently
  based on its current status, we need a
  state-dependent influence model, which is
  inherently nonlinear and does not allow a
  recursion for the state probabilities in the form

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The Influence Model in a nutshell

• The Influence Model is a stochastic model that
  provides a particular but tractable
  representation of random, dynamical interactions
  on networks
• It is based on the Interactive Markov Chain
  framework, that separates out the internal
  behavior of a node from interactions between
  nodes
• Interactions are contrained to take a
  multilinear form, that leads to a highly
  tractable model with rich mathematical structure
  

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Strenghts and weaknesses

• Strenghts

The influence model allows

• fairly general structure and high flexibility
  (freedom in choosing network matrix and local
  chains)

• scalable computation (intermediate order
  statistics)

• overall tractability and many potential areas of
  further research and applications

• Weaknesses

• The influence model is only a special case of
  Interactive Markov Chains. The imposed
  constraints lead to a highly tractable model -
  but correspondingly limit its modeling ability
• State-dependent influence models (required by
  many possible applications) are nonlinear, thus
  much more complicated to be analyzed

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The End