ONR MURI: NexGeNetSci From Consensus to Social Learning in Complex Networks Ali Jadbabaie Skirkanich Associate Professor of innovation Electrical & Systems Engineering and GRASP Laboratory University of Pennsylvania First Year Review, August 27, 2009 With Alireza Tahbaz-Salehi and Victor Preciado
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ONR MURI: NexGeNetSci From Consensus to Social Learning in Complex Networks Ali Jadbabaie Skirkanich Associate Professor of innovation Electrical & Systems.
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ONR MURI: NexGeNetSci
From Consensus to Social Learning in Complex Networks
Ali JadbabaieSkirkanich Associate Professor of innovation
Electrical & Systems Engineering and GRASP LaboratoryUniversity of Pennsylvania
First Year Review, August 27, 2009
With Alireza Tahbaz-Salehi and Victor Preciado
Theory DataAnalysis
Numerical Experiments
LabExperiments
FieldExercises
Real-WorldOperations
• First principles• Rigorous math• Algorithms• Proofs
• understanding network topology• simplicial homology• computing homology groups
Consensus, Flocking and Consensus, Flocking and SynchronizationSynchronization
Opinion dynamics, crowd control, synchronization and flocking
Flocking and opinion dynamics• Bounded confidence opinion model (Krause,
2000)– Nodes update their opinions
as a weighted average
of the opinion value of their friends
– Friends are those whose opinion is already close
– When will there be fragmentation and when will there be convergence of opinions?
– Dynamics changes topology
Consensus in random networks• Consider a network with n nodes and a vector of initial values, x(0)
• Consensus using a switching and directed graph Gn(t)
• In each time step, Gn(t) is a realization of a random graph where edges appear with probability, Pr(aij=1)=p, independently of each other
Random
Ensemble
)()(
)1(1
nknkk
k
IAIDW
kWk
xxConsensus dynamics
variable.random a is limx
vector,random a is where,lim
,... with ,0)(
k*
021
kx
U
WWWUUk
i
Tkk
kkkk
vv1
xxStationary behavior
Despite its easy formulation, very little is known about x* and v
Random Networks
The graphs could be correlated so long as they are stationary-ergodic.
What about the consensus value?
• Random graph sequence means that consensus value is a random variable
• Question: What is its distribution?• A relatively easy case :
– Distribution is degenerate (a Dirac) if and only if all matrices have the same left eigenvector with probability 1.
• In general:
Where is the eigenvector associated with the largest eigenvalue (Perron vector)
Can we say more?
E[WkWk] for Erdos-Renyi graphs
Define:
• For simplicity in our explanation, we illustrate the structure of E[WkWk] using the case n=4:
Random Consensus
These entries have the following expressions:
where q=1-p and H(p,n) is a special function that can be written in terms of a hypergeometric function (the detailed expression is not relevant in our exposition)
• Defining the parameter
we can finally write the left eigenvector of the expected Kronecker as:
• Furthermore, substituting the above eigenvector in our original expression for the variance (and simple algebraic simplifications) we deduce the following final expression as a function of p, n, and x(0):
where
Variance of consensus value for Erdos-Renyi graphs
• var(x*) for initial conditions uniformly distributed in [0,1], nЄ{3,6,9,12,15}, and p varying in the range (0,1]
Random Consensus (plots)
p
Var(x*)n=3 n=6 n=9 n=12 n=15
What about other random graphs?
Static Model with Prescribed Expected Degree Distribution
• Generalized static models [Chung and Lu, 2003]:– Random graph with a prescribed expected degree sequence– We can impose an expected degree wi on the i-th node
Degree distributions are useful to the extent that they tell us something about the spectral properties (at least for distributed computation/optimization)
i
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Eigenvalues of Chung-Lu Graph
• What is the eigenvalue distribution of the adjacency matrix for very large Chung-Lu random networks?
Numerical Experiment: Represent the histogram of eigenvalues for several realizations of this random graph
Limiting Spectral Density: Analytical expression only possible for very particular cases.
Contribution: Estimation of the shape of the bulk for a given expected degree sequence, (w1,…,wn).
Spectral moments of random graphs and degree distributions
• Degree distributions can reveal the moments of the spectra of graph Laplacians
• Determine synchronizability
• Speed of convergence of distributed algorithms
• Lower moments do not necessarily fix the support, but they fix the shape
• Analysis of virus spreading (depends on spectral radius of adjacency)
• Non-conservative synchronization conditions on graphs with prescribed degree distributions
• Analytic expressions for spectral moments of random geometric graphs
Consensus and Naïve Social learning
• When is consensus a good thing?• Need to make sure update converges to the
correct value
Naïve vs. Bayesian
just average
with neighbors
Fuse info with Bayes Rule
Naïve learning
Social learning
• There is a true state of the world, among countably many
• We start from a prior distribution, would like to update the distribution (or belief on the true state) with more observations
• Ideally we use Bayes rule to do the information aggregation
• Works well when there is one agent (Blackwell, Dubins’1962), become impossible when more than 2!
Locally Rational, Globally Naïve: Bayesian learning under peer pressure
Model Description
Model Description
Belief Update Rule
Why this update?
Eventually correct forecasts
Eventually-correct estimation of the output!
Why strong connectivity?
No convergence if different people interpret signals differently
N is misled by listening to the less informed agent B
Example
One can actually learn from others
Learning from others
Information in i’th signal only good for distinguishing
Convergence of beliefs and consensus on correct value!
Learning from others
Summary
Only one agent needs a positive prior on the true state!