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, 324023 (2008) (2) Raymond & Wong, to be published (2012)

Acknowledgement: Supported by the Research Grant Council of Hong Kong (grant numbers 604008 and 605010).

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

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

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

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.

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.

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.

Simulations for 4 Colors Walksat [Bounkong et al, PRE 74, 057101 (2006)] Phase transition at C around and above 3.8 Slow convergence at C immediately below 3.8

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

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

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)

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

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

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, 324023 (2008)

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

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)

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

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

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

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

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

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).

The Physical Picture easy hard uncolorable E states E = 0 E > 0 many states connectivity Mulet et al, PRL 89, 268701 (2002)

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

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

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

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)

Simulation Fits 1RSB Better

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.

Thank you!