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

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

4. The model of Vicsek et al Mobile agents with constant speed in 2-D and in discrete-time Heading update: nearest neighbor rule qi(k): heading of ith agent at time k Ni(k):neighborhood of ith agent of given radius at time k xi(k):noise of ith agent at time k, magnitude bounded by h /2 Ni(k)

5. Ni(k) Phase transition in Vicsek’s model Heading update: nearest neighbor rule Low noise level: alignment High noise level: nonalignment • 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. 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. Model on fixed connected graph Update: nearest neighbor rule Ni(k) xi(k) • : noise level Total number of agents: M time k • Simplified noisy communication network

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

9. Steps of proof • Define system state S(k):= Sxi(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 0<pr<1 0<pr<1 0<pr<1 0<pr<1 –M –M+2 M-2 M pr=1 pr=1 0<pr<1 0<pr<1 0<pr<1 0<pr<1 • For high noise level, all states are transient • Noise may flip any node value with positive probability –M –M+2 M-2 M 0<pr<1 0<pr<1

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

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

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 0<pr<1 0<pr<1 0<pr<1 0<pr<1 –M –M+2 M-2 M pr=1 pr=1 0<pr<1 0<pr<1 0<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: ES(k) converges to zero exponentially with rate logh –M –M+2 M-2 M 0<pr<1 0<pr<1

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

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