1. Consensus in Random Networks
SWARMS Review, February
Consensus in Random Networks
Ali Jadbabaie
Department of Electrical and Systems Engineering
and GRASP Laboratory
With Alireza Tahbaz Salehi (MIT), Victor Preciado (Penn)
Nader Motee (Caltech), Nima Moshtagh (SSCI), Antonis Papachristodoulou (Oxford)

2. Collective phenomena : from local decisions to global behaviors

3. collective phenomena in economic and social systems

4. Pretty simulations, few proofs
Emergence of Consensus,
synchronization, and flocking
Pretty simulations, few proofs

5. Example: Flocking and Motion Coordination
How can a group of moving agents collectively decide on direction, based on nearest neighbor interaction?
neighbors of
agent i r
agent i
How does global behavior (herding) emerge from local interactions?

6. Distributed consensus algorithm for
kinematic agents
= heading
= speed
MAIN QUESTION : Under what conditions do all headings converge to the same value and agents reach a consensus on where to go?
For angles in (-/2,/2), the nonlinear problem becomes
linear with a coordinate change!

7. Conditions for reaching consensus
Theorem (Jadbabaie et al. 2003): If there is a sequence of bounded, non-overlapping time intervals Tk, such that over any interval of length Tk, the network of agents is “jointly connected ”, then all agents will reach consensus on their velocity vectors.
This happens to be both necessary and sufficient for
exponential coordination, boundedness of intervals not required for asymptotic coordination. (see Moreau ’04)

8. Vicsek’s Model with Periodic Boundary Condition
Autonomous agents with constant speed and adjustable headings
Local interaction rule:
A consensus algorithm if the graphs are jointly connected over time
Periodic boundary conditions (Motion over a flattened unit torus) to avoid boundary effects.
Location on a torus can be seen as the fractional part of the location of an agent moving in an infinite plane.
Reasonable assumption for statistical physics simulations, for large populations
Vicsek’s simulations: velocity alignment without any assumption on connectivity.

9. Kronecker’s Theorem, Weyl’s Theorem and Number Theory
Kronecker’s Theorem: The numbers are dense in the unit interval if is an irrational number.
Weyl’s Theorem: If the sequence grows fast enough, then for almost every the numbers are equidistributed in the unit interval
Equidistributedness implies denseness and more!
Weyl’s theorem can be generalized to higher dimensions.

10. Velocity alignment in Vicsek’s simulations: Proof sketch
If the graphs stay jointly connected infinitely-often then
earlier results imply asymptotic alignment.
Suppose the graphs do not stay jointly connected infinitely often, then:
At least two jointly connected components after a certain point in time.
such that there is no path between and over time.
The headings in each jointly connected component reach consensus.
Difference between components of and in infinite plane.
For almost all , the numbers
are dense in unit square. Therefore, for almost all initial conditions ,
get arbitrarily close, and graphs become connected, contradiction!

11. Synchronization Fireflies Flashing Jadbabaie et al 2004
Continuous version of Vicsek’s model
Jadbabaie et al 2004

12. 2. Heterogeneously Delayed Case 2. Heterogeneously Delayed Case
Consensus, agreement and synchronization with nonlinear dynamics, network change, time delay
Theorem (Papachristodoulou & J2010): If switching with dwell time,
and interaction graph contains a spanning tree over time then
consensus set is asymptotically attracting attracting
2. Heterogeneously Delayed Case
2. Heterogeneously Delayed Case
Thm: If fij are locally passive and graph contains a spanning tree, then consensus set is as. attracting.
Thm: If fij are locally passive and graph contains a spanning tree, then consensus set is as. attracting.

13. Endogenous change of network: Simplest example not solved!
Bounded confidence opinion model (Krause, 2000, Hendrix et al. 2008)
Nodes update their opinions as a weighted average of the opinion value of their friends
Friends are those whose opinion is already close (e.g. within 1 unit)
When will there be fragmentation and when will there be convergence of opinions?
Node dynamics changes topology
Special case: Gossiping: each node only talks to one neighbor at a time
Simulations informative but not enough
Close to 2

14. Rigid Body Model, and flocking in local coordinatesXw Yw Zw ri Ri

15. Rigid Body Model
wix
wiy

16. Deriving the inputs in body frame
wix
wiy

17. Nonholonomic Constraint
Moshtagh et al. 2009 Yi Xi vi vj wi

18. Collision Avoidance

19. Can randomness help?

20. Consensus in Random Networks
The graphs could even be correlated so long as they are stationary-ergodic.
Furthermore, reaching consensus is a trivial event
Also Hatano & Mesbahi 2006; Wu 2006; Picci & Taylor 2007;
Fagnani & Zampieri 2008; Porfiri & Stilwell 2007

21. What about consensus value?
Almost Surely
Can we say more?

22. Switching Erdos-Renyi graphs
Consider a network with n nodes and a vector of initial values, x(0)
Repeated local averaging using a switching and directed graph
In each time step, is a realization of a random graph where edges appear with probability, Pr(aij=1)=p, independently of each other
Random
Ensemble
Consensus dynamics
Stationary behavior
We can find a close form expression for the mean & variance of x*

23. Mean and variance in E-R graphs
Remember, for any IID graph sequence
Expected weight matrix is symmetric! Therefore mean is just average of initial condiitons!
Computing the variance of x* is more complicated
Involves the Perron vector of the matrix E[WkWk] .
( it is not Kronecker product of two eigenvectors!)
Can derive a closed form expression of the left eigenvector of E[WkWk] for any network size n, link probability p, and initial condition x(0).

24. What does E[WkWk] look like?
E[WkWk] is Not E[Wk ]  E[Wk ] , but it almost is!
Only n of the n2 entries are different!
q=1-p and H(p,n) can be written
in terms of a hypergeometric function

25. A surprising result Theorem:
No Kronecker products or hyper-geometric functions
As network size grows, variance of consensus value goes to zero!
What about other random graph models?

26. Random Consensus on incomplete graphs
Consider a network Gc with n nodes and a set of possible directed communication links Ec
We perform consensus in a randomly switching directed graph Gn(t), where in each time step, Gn(t) is a realization of a random graph where a comunication link (i,j)ЄEc appear with probability, Pr(aij=1)=pj
We have the following result for the mean of x*
Lemma: The dominant left eigenvector of EWk is given by
where s is the stationary distribution of the random walk on the directed graph Gc , S=diag(M(di,pi)) with
and r is a normalizing scalar.

27. Randomized Directed Consensus
Hence, the expected consensus value is equal to
Note that the above expectation is, in general, different to the initial average of the initial conditions x(0)
For a given Gc and probability profile {pi}, one can put weights the initial conditions to obtain an asymptotic consensus unbiased with respect to the initial average. The weighted initial conditions are
which gives the following expectation for the asymptotic consensus

28. Other related work
Closed form solutions for moments of adjacency/Laplacian matrices of random geometric graphs (all moments in 1d, first 3 moments in 2d)
(Preciado & J., CDC 09)
Given the moments, we can estimated the shape of eigenvalue distribution, and estimate the spread
Can predict synchronizability, speed of convergence once we know the spread of eigenvalues (Preciado & J., CDC 09)