Long-Range Frustration among Unfrozen Vertices in the Ground-State Configurations of a Spin-Glass System Haijun Zhou 海军 周 Institute of Theoretical Physics, the Chinese Academy of Sciences, Beijing, China March 06, KITPC Program “Collective Dynamics in Information Systems” (March 01-April 15, 2008)

2 The vertex cover problem A simple analytical solution and stability analysis Mean-field calculation on long-range frustration within the RS cavity method Stability analysis of 1RSB cavity solution in terms of long-range frustration Conclusion

3 1. H.Z., Phys. Rev. Lett. 94: (2005) 2. H.Z., New J. Phys. 7: 123 (2005) 3. Martin Weigt and H.Z., Phys. Rev. E 74: (2006) 4. Jie Zhou, Hui Ma, H.Z., J. Stat. Mech. L06001 (2007) Collaborators: Kang LiHui MaMartin WeigtJie Zhou

4 Vertex Cover Problem

5 Minimal vertex covers A vertex cover of the global minimal size. Is a NP-hard optimization problem. Efficient algorithms for constructing near- optimal solutions for a given graph.

6 There may be many minimal vertex covers for a given graph.

7 There are three types of vertices: (1) vertices that are always covered (frozen vertices) (2) vertices that are always uncovered (frozen vertices) (3) vertices that are covered in some solutions and uncovered in the remaining solutions (unfrozen vertices)

8 Mean-field analysis of the minimal vertex cover problem on a random graph

9 Mezard, Parisi (2001) Mezard, Parisi (2003) The cavity method. Add a new vertex (0) to a system of N old vertices.

10 mean field parameters q* --- fraction of vertices that are unfrozen q^0 --- fraction of vertices that are always uncovered q^1 --- fraction of vertices that are always covered

11 = always uncovered always covered unfrozen Weigt, Hartmann, PRL (2000), PRE (2001)

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13 Mean-field theory result is lower than experimental values for c > e= Leaf-removal algorithms works in this region

14 What’s wrong with this simple mean-field theory? There exist strong correlations among vertices when c>e!

15 Long-range frustration among unfrozen vertices The spin value of an unfrozen vertex will fluctuate among different best solutions of the old graph. Spin value fluctuations of different unfrozen vertices may be correlated in the sense that certain combination of spin values may never appear in any best solutions. For example, it might be the case that the unfrozen vertices are not simultaneously covered in any single best solution.

16 Instability analysis of the RS solution in terms of long-range frustration

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18 We introduce a long-range frustration order parameter R to characterize the probability of any two unfrozen vertices being strongly correlated. R could be calculated by calculating the probability that, if one performs a perturbation to the state of an unfrozen vertices, this perturbation will propagate to influence the states of a finite fraction of other unfrozen vertices. Improving the RS solution with long-range-frustration

19 i j  Vertex i and j can be occupied simultaneously tree

20 i j  Vertex i and j can still be occupied simultaneously giant component, contain loops tree

21 i j  with probability ½, vertices i and j can be simultaneously occupied

22 calculate R f(s): the probability that a randomly chosen unfrozen vertex i, when flipped to the –1 state, will eventually fix the spin values of s unfrozen vertices, s begin finite.

23 self-consistent equations

24 Long range frustration order parameter R c ¤ = e = 2 : 7183 :::

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26 Proliferation of many macroscopic states and the Survey propagation algorithm

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28 Within a macroscopic state, a vertex is: either always covered or always uncovered or being unfrozen

29 = always uncovered always covered unfrozen

30 neighborsvertex iprobabilityVC size increase re-weighted probability all unfrozen or always covered always uncovered 0 at least one always uncovered unfrozen or always covered +1

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33 Further organization of the solution space

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35 Instability of the 1RSB solution in terms of long-range-frustration within a macrostate Some unfrozen vertices in a macrostate are supposed to be type-I initially Evolution of the fraction of type-I unfrozen vertices in a macrostate Persistence of type-I vertices  1RSB instability

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37 J. Zhou, H. Ma, H. Zhou, JSTAT, L06001 (2007)

38 Improving 1RSB solution in terms of long-range frustration? (do not go to 2RSB) This might be an interesting question.

39 Conclusion Long-range frustrations among unfrozen vertices can be used to study the possibility clustering of the zero- temperature configuration space of a spin- glass system (stability analysis) Combining LRF and the cavity method to give corrections to mean-field predictions.