Introduction: phase transition phenomena
Phase transition: qualitative change as a parameter crosses threshold
• Matter
temperature
solid
temperature
temperature
liquid
gas
magnetism
demagnetism
• Mobile agents (Vicsek et al 95; Czirok et al 99)
noise level
alignment
nonalignment
2
The model of Vicsek et al
Mobile agents with constant speed in 2-D and in discrete-time
Randomized initial headings
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The model of Vicsek et al
Mobile agents with constant speed in 2-D and in discrete-time
Heading update: nearest neighbor rule
i (k  1) 
1
| N i (k ) |

 j (k )   i (k )
j N i ( k )
i(k): heading of i th agent at time k
Ni(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 h /2
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Phase transition in Vicsek’s model
Heading update: nearest neighbor rule
i (k  1) 
1
| N i (k ) |

 j (k )   i (k )
j N i ( k )
Ni(k)
High noise level:
nonalignment
k 
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)
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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
symmetry
un-consensus
disagreement
noise level ≥ threshold
noise level < threshold
symmetry breaking
consensus
agreement
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Model on fixed connected graph
Update: nearest neighbor rule
Ni(k)
xi(k)
 1
xi (k  1)  sgn 
 | Ni |

j N i

x j (k )   i (k ) 

xi (k )    1,1
 i (k )  h / 2,h / 2
h: noise level
Total number of agents: M
time k
• Simplified noisy communication network
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Phase transition on fixed connected graph
h  (1  2 / D, 1]
k 
k 0
h 1
D: maximum degree in graph
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Steps of proof
• Define system state S(k):= S xi(k). So S (k )   M ,M  2,, M  2, M 
• 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
–M
pr=1
0<pr<1
0<pr<1
–M+2
0<pr<1
0<pr<1
M-2
0<pr<1
0<pr<1
0<pr<1
M
pr=1
• For high noise level, all states are transient
– Noise may flip any node value with positive probability
–M
0<pr<1
–M+2
M-2
M
0<pr<1
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Model on Erdos random graph
Each edge forms with prob p, independent of other edges and other times
Update: nearest neighbor rule
 1
xi (k  1)  sgn 
 | N i (k ) |


x
(
k
)


(
k
)

j
i

j N i ( k )

xi (k )    1,1
 i (k )  h / 2,h / 2
h: noise level
Total number of agents: M
One possible realization of connections at time k
• Simplified noisy ad-hoc communication network
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Phase transition on Erdos random graph
h  (0, 1]
k 
k 0
h 1
Note: arbitrarily small but positive h leads
to consensus, unlike the fixed
connected graph case
k 
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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
–M
pr=1
0<pr<1
0<pr<1
–M+2
0<pr<1
0<pr<1
M-2
0<pr<1
0<pr<1
0<pr<1
M
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
0<pr<1
–M+2
M-2
M
0<pr<1
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Numerical examples
Low noise level
High noise level
Fixed
connected
graph
Erdos
random
graph
symmetry breaking
consensus
agreement
symmetry
un-consensus
disagreement
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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
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