1. New Results of the Color Diversity Problem Jack Raymond (英偉文) and K. Y
New Results of the Color Diversity Problem Jack Raymond (英偉文) and K. Y. Michael Wong (王國彝) Hong Kong University of Science and Technology (1) Wong & Saad, J. Phys. A 41, (2008) (2) Raymond & Wong, to be published (2012)

2. Acknowledgement: Supported by the Research Grant Council of Hong Kong (grant numbers and ).

3. Outline Color diversity problem Cavity method
Paramagnetic-glass transition
Special ratios
The many-state physical picture
Conclusions

4. A Famous Problem
How many colors are needed to color a map, so that all neighboring regions have different colors?

5. The Color Diversity Problem
Task: maximize the number of colors within one link-distance of any node
Compared with the graph coloring problem, one needs to consider next-nearest neighbor effects.
When the connectivity increases, a phase transition from incomplete to complete phase takes place in large networks.
Example: Q = 4
Fully diversified
Not diversified enough

6. Applications Distributed data storage
Files are divided into segments distributed over the network.
Nodes requesting a file collect the required segments from neighboring nodes.
If the neighborhood is not diversified enough, the left-out segments have to be retrieved from afar.

7. Phase Transitions
There are parameter regions where all constraints can be satisfied, and there are parameter regions where the violation of constraints becomes unavoidable.
When the connectivity changes, a phase transition from unsatisfied to satisfied phase takes place in large networks.

8. Linear Connectivity
To study phase transitions, we consider ensembles of graphs with a linear mixture of two consecutive values of connectivity.
For example, when C = 3.8, the fraction of nodes with connectivity 4 is 0.8, and the fraction of nodes with connectivity 3 is 0.2.

9. Simulations for 4 Colors
Walksat [Bounkong et al, PRE 74, (2006)]
Phase transition at C around and above 3.8
Slow convergence at C immediately below 3.8

10. Formulation Given a network with N nodes.
Color of node i is qi = 1, …, Q.
Minimize the energy
For example, the colors of node i and its neighbors are g, b, b, b, y

11. Antiferromagnetism
The quadratic cost function tends to equalize the colors around the neighborhood of a node.
It is equivalent to the antiferromagnetic model (apart from constant factors)
An auxiliary cost function is
However, this cost function is less robust against noise disrupting the colors of the neighbors.
Nearest neighbor
(nn)
Next nearest neighbor
(nnn)
Color count

12. The Cavity Method = k1 k2 b Fij(a,b) a i j q1 q2 Fjk1(b,q1) Fjk2(b,q2)
In sparse networks, the neighborhood of a node is locally a tree. Each generation has C  1 branches.
Consider the free energy of a tree when the colors of the vertex j and its ancestor i are b and a respectively. The free energy Fij(a,b) can be expressed as a function of the free energies of the descendents. = k1 k2 b
Fij(a,b) a i j q1 q2
Fjk1(b,q1)
Fjk2(b,q2)

13. Recursion Relation
Expressing the free energy Fij(a,b) as a function of the free energies Fjk(b,qk) of the descendents,
T   -1 = temperature
T = 0 for optimisation

14. Parameters Average energy
Edwards-Anderson order parameter: to see whether the system has frozen (like in glasses) or not
qEA = 1: completely frozen; qEA = 0: paramagnetic
Incomplete fraction

15. Paramagnetic and Potts Glass Phases
Simulation: phase transition at C around and above 3.8!!! (Bounkong et al)
glassy
spinodal point
paramagnetic
For Q = 4, phase transition at C = 3.65
C above 3.65: paramagnetic; C below 3.65: glassy phase
Glassy state ceases to exist for C above 3.65: spinodal point
Wong & Saad, J. Phys. A 41, (2008)

16. Ising Model
The case Q = 2 is equivalent to the Ising model with antiferromagnetic nn and nnn interactions.
Cost function:
 is the ratio of nn coupling to nnn coupling.
Color diversity problem:  = 1

17. Special Ratios Rewrite the cost function as Number of dimers:X 1
 = 1, ci = 3
At the special ratios Ci   = odd, the ground state is degenerate  higher entropy and lower energy
Special ratio: unlocked interaction X 1 2
 = 1, ci = 4
Zdeborova & Mezard, JStat P12004 (2008)

18. Energy at Special Ratios
 = 1: even dominated graphs
 = 2: odd dominated graphs

19. Entropy at Special Ratios
 = 2: odd dominated graphs
 = 1: even dominated graphs

20. Critical Temperature  = 1: even dominated graphs
 = 2: odd dominated graphs

21. EA Parameter Contours ( = 1)
Spin glass existence curve
 = 1
Negative entropy
Contours of EA parameters

22. EA Parameter Contours (C = 4)
Negative entropy
C = 4
Discon-tinuous transition
Con-tinuous transition
Contours of EA parameters

23. Why Negative Entropy?
We have used a recursion relation based on the physical picture when the system is dominated by a single state (replica symmetric ansatz).
The results show that the glassy state should be dominated by many states (replica symmetry-breaking).

24. The Physical Picture easy hard uncolorable E states E = 0 E > 0
many states
connectivity
Mulet et al, PRL 89, (2002)

25. The Cavity Method (Many States)
The number of states grows exponentially with N:
() is the complexity, or configurational entropy.
When new nodes are added to the network, different initial states contribute to a final state of a given free energy. Hence
() can be found by Legendre transformation of the “free energy”
Note the reweighting factor

26. The Complexity ( = 1, c = 3)  = 1, c = 3
dynamical transition: optimization may be trapped
unphysical branch
nonzero ground state energy
both quadratic and auxiliary cost functions have similar behaviors

27. The Complexity ( = 1) dynamical transition
nonzero ground state entropy
SG-PM transition
 nonzero ground state energy

28. Comparison Spin Glass Single state (RS) 2.76C3.33
Many states (1RSB)
2.69C3.45 (stable)
2.65C2.69 (metastable)
3.45C3.56 (metastable)
Many states (1RSB)

29. Simulation Fits 1RSB Better

30. Conclusions
Special ratios: When the connectivity changes, there are phase transitions from the glass phase in odd-dominated regimes to the paramagnetic phase in even-dominated regimes.
The glassy state is dominated by many states. This picture predicts a phase transition at c  2.69, agreeing with simulation results.
Expect special ratio effects also for Q colors when C + 1 = multiple of Q, but maybe only modulation of glass temperature for Q  3.

31. Thank you!