Consensus in Random Networks SWARMS Review, February 23 2010 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)

Collective phenomena : from local decisions to global behaviors

collective phenomena in economic and social systems

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

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?

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!

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)

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.

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.

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!

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

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.

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

Rigid Body Model, and flocking in local coordinates Xw Yw Zw ri Ri

Rigid Body Model wix wiy

Deriving the inputs in body frame wix wiy

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

Collision Avoidance

Can randomness help?

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

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

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*

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).

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

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?

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.

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

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)