2. 2 Introduction: phase transition phenomena Phase transition: qualitative change as a parameter crosses threshold Matter temperature magnetism demagnetism solid liquid gas Mobile agents (Vicsek et al 95; Czirok et al 99) noise level alignment nonalignment

3. 3 The model of Vicsek et al Mobile agents with constant speed in 2-D and in discrete-time Randomized initial headings

4. 4 Mobile agents with constant speed in 2-D and in discrete-time Heading update: nearest neighbor rule N i (k)  i (k) : heading of i th agent at time k N i (k): neighborhood of i th agent of given radius at time k  i (k): noise of i th agent at time k, magnitude bounded by  /2 The model of Vicsek et al

5. 5 Phase transition in Vicsek’s model Heading update: nearest neighbor rule N i (k) High noise level: nonalignment Low noise level: alignment Phase transitions are observed in simulations if noise level crosses a threshold; rigorous proof is difficult to establish Alignment in the noiseless case is proven (Jadbabaie et al 03)

6. 6 Provable phase transition with limited information Proposed simple dynamical systems models exhibiting sharp phase transitions Provided complete, rigorous analysis of phase transition behavior, with threshold found analytically Characterized the effect of information (or noise) on collective behavior noise level ≥ threshold symmetry un-consensus disagreement symmetry breaking consensus agreement noise level < threshold

7. 7 Model on fixed connected graph Update: nearest neighbor rule x i (k) time k N i (k)  : noise level Total number of agents: M Simplified noisy communication network

8. 8 Phase transition on fixed connected graph D: maximum degree in graph

9. 9 Steps of proof Define system state S(k) :=  x i (k). So For low noise level, ± M are absorbing, others are transient –Noise cannot flip the node value if the node neighborhood contains the same sign nodes; noise may flip the node value otherwise –M–M – M +2 M M -2 pr=1 0<pr<1 For high noise level, all states are transient –Noise may flip any node value with positive probability –M–M – M +2 M M -2 0<pr<1

10. 10 Model on Erdos random graph Update: nearest neighbor rule One possible realization of connections at time k Simplified noisy ad-hoc communication network Each edge forms with prob p, independent of other edges and other times  : noise level Total number of agents: M

11. 11 Phase transition on Erdos random graph Note: arbitrarily small but positive  leads to consensus, unlike the fixed connected graph case

12. 12 Steps of proof For low noise level, ± M are absorbing, others are transient –For ± M, noise cannot flip any node value –For other states, arbitrarily small noise flips any node value with pr >0, since a node connects only to another node with different sign with pr >0 –M–M – M +2 M M -2 pr=1 0<pr<1 For high noise level, all states are transient –Noise may flip any node value with pr >0 –It can be shown: E S(k) converges to zero exponentially with rate log  –M–M – M +2 M M -2 0<pr<1

13. 13 Numerical examples Fixed connected graph symmetry un-consensus disagreement symmetry breaking consensus agreement Erdos random graph Low noise level High noise level

14. 14 Conclusions and future work Discovered new phase transitions in dynamical systems on graphs Provided complete analytic study on the phase transition behavior Proposed analytic explanation to the intuition that, to reach consensus, good communication is needed