Random Field Ising Model on Small-World Networks

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Random Field Ising Model on Small-World Networks
Seung Woo Son, Hawoong Jeong 1 and Jae Dong Noh
2 1 Dept. Physics, Korea Advanced Institute
Science and Technology (KAIST) 2 Dept. Physics,
Chungnam National University, Daejeon, KOREA
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What is RFIM ?
Random Fields Ising Model
ex) 2D square lattice






Uniform field
Random field
cf) Diluted AntiFerromagnet in a Field
(DAFF)
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RFIM on SW networks
L number of nodesK number of out-going
linksp random rewiring probability

• Ising magnet (spin) is on each node where
  quenched random fields are applied. Spin
  interacts with the nearest-neighbor spins which
  are connected by links.

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Why should we study this problem? Just curiosity

• Critical phenomena in a stat. mech. system
  with quenched disorder.
• Applications e.g., network effect in markets

Social science
Society

• Internet telephone business
• Messenger
• IBM PC vs. Mac
• Key board (QWERTY vs. Dvorak)
• Video tape (VHS vs. Beta)
• Cyworld ?

Individuals
Selection of an item Ising spin state
Preference to a specific item random field on
each node
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Zero temperature ( T0 )

• RFIM provides a basis for understanding the
  interplay between ordering and disorder induced
  by quenched impurities.
• Many studies indicate that the ordered phase is
  dominated by a zero-temperature fixed point.
• The ground state of RFIM can be found exactly
  using optimization algorithms (Max-flow, min-cut).

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Magnetic fields distribution

• Bimodal dist.
• Hat dist.

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Finite size scaling

• Finite size scaling form
• Limiting behavior

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Results on regular networks
L ( of nodes) 100K ( of out-going edges
of each node) 5P (rewiring
probability) 0.0
Hat distribution
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Results on regular networks
Hat distribution
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Results on SW networks
L ( of nodes) 100K ( of out-going edges
of each node) 5P (rewiring
probability) 0.5
Hat distribution
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Results on SW networks
Hat distribution
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Results on SW networks
Hat distribution
Second order phase transition
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Results on SW networks
Bimodal distribution
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Results on SW networks
Bimodal field dist.
First order phase transition
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Summary

• We study the RFIM on SW networks at T0 using
  exact optimization method.
• We calculate the magnetization and obtain the
  magnetization exponent(ß) and correlation
  exponent (?) from scaling relation.
• The results shows ß/? 0.16, 1/? 0.4 under hat
  field distribution.
• From mean field theory ßMF1/2, ?MF1/2 and upper
  critical dimension of RFIM is 6. ? ? du vMF
  3 and ßMF/? 1/6 , 1/? 1/3.

R. Botet et al, Phys. Rev. Lett. 49, 478 (1982).