2

Introduction: phase transition phenomena

Phase transition: qualitative change as a parameter crosses threshold

• Mattertemperature temperature temperature

magnetism demagnetismsolid liquid gas

• Mobile agents (Vicsek et al 95; Czirok et al 99)

noise levelalignment nonalignment

3

The model of Vicsek et al

Mobile agents with constant speed in 2-D and in discrete-time

Randomized initial headings

4

Mobile agents with constant speed in 2-D and in discrete-time

Heading update: nearest neighbor rule

)()(|)(|

1)1(

)(

kkkN

k ikNj

ji

i

i

Ni(k)

i(k): heading of i th agent at time k

Ni(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

Phase transition in Vicsek’s modelHeading update: nearest neighbor rule

)()(|)(|

1)1(

)(

kkkN

k ikNj

ji

i

i

Ni(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)

k

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 ≥ thresholdsymmetry

un-consensusdisagreement

symmetry breakingconsensusagreement

noise level < threshold

7

Model on fixed connected graph

Update: nearest neighbor rule

)()(||

1sgn)1( kkx

Nkx i

Njj

ii

i

xi(k)

time k

Ni(k)

2/,2/)(

1,1)(

k

kx

i

i

: noise levelTotal number of agents: M

• Simplified noisy communication network

8

Phase transition on fixed connected graph

]1,/21( D

1

k

k

0k

D: maximum degree in graph

9

Steps of proof• Define system state S(k):= xi(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

MMMMkS ,2,,2,)(

–M –M+2 MM-2

pr=1 pr=1

0<pr<1 0<pr<1 0<pr<1 0<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 MM-2

0<pr<1 0<pr<1

10

Model on Erdos random graph

Update: nearest neighbor rule

)()(|)(|

1sgn)1(

)(

kkxkN

kx ikNj

ji

i

i

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

2/,2/)(

1,1)(

k

kx

i

i

: noise levelTotal number of agents: M

11

Phase transition on Erdos random graph

]1,0(

1

k

k

0k

Note: arbitrarily small but positive 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

–M –M+2 MM-2

pr=1 pr=1

0<pr<1 0<pr<1 0<pr<1 0<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 log

–M –M+2 MM-2

0<pr<1 0<pr<1

13

Numerical examples

Fixed connectedgraph

symmetry un-consensusdisagreement

symmetry breakingconsensusagreement

Erdos randomgraph

Low noise level High noise level

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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