# Cluster Structure of Optimal Solutions in Bipartitioning of Small Worlds

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Model and Simulated Annealing

## 3. Results

#### 3.1. d = 1

#### 3.2. d = 2

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Hartmann, A.K.; Weigt, M. Phase Transitions in Combinatorial Optimization Problems: Basics, Algorithms and Statistical Mechanics; John Wiley & Sons: Chichester, UK, 2006. [Google Scholar]
- Krzakala, F.; Ricci-Tersenghi, F.; Zdeborova, L.; Zecchina, R.; Tramel, E.W.; Cugli, L.F. (Eds.) Statistical Physics, Optimization, Inference, and Message-Passing Algorithms; Number 2013 in Lecture Notes of the Les Houches; Oxford University Press: Oxford, UK, 2016. [Google Scholar]
- Karypis, G.; Aggarwal, R.; Kumar, V.; Shekhar, S. Multilevel hypergraph partitioning: Applications in VLSI domain. IEEE Trans. Very Large Scale Integr. VLSI Syst.
**1999**, 7, 69–79. [Google Scholar] [CrossRef] - Pothen, A. Graph partitioning algorithms with applications to scientific computing. In Parallel Numerical Algorithms; Springer: Berlin/Heidelberger, Germany, 1997; pp. 323–368. [Google Scholar]
- Kolmogorov, V.; Zabin, R. What energy functions can be minimized via graph cuts? IEEE Trans. Pattern Anal. Mach. Intell.
**2004**, 26, 147–159. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Banavar, J.R.; Sherrington, D.; Sourlas, N. Graph bipartitioning and statistical mechanics. J. Phys. A
**1987**, 20, L1. [Google Scholar] [CrossRef] - Schreiber, G.R.; Martin, O.C. Cut size statistics of graph bisection heuristics. SIAM J. Optim.
**1999**, 10, 231–251. [Google Scholar] [CrossRef] [Green Version] - Boettcher, S.; Percus, A.G. Extremal optimization for graph partitioning. Phys. Rev. E
**2001**, 64, 026114. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Fu, Y.; Anderson, P.W. Application of statistical mechanics to NP-complete problems in combinatorial optimisation. J. Phys. A
**1986**, 19, 1605. [Google Scholar] [CrossRef] - Liao, W. Graph bipartitioning problem. Phys. Rev. Lett.
**1987**, 59, 1625. [Google Scholar] [CrossRef] [PubMed] - Mézard, M.; Parisi, G. Mean-field theory of randomly frustrated systems with finite connectivity. EPL
**1987**, 3, 1067. [Google Scholar] [CrossRef] - Percus, A.G.; Istrate, G.; Gonçalves, B.; Sumi, R.Z.; Boettcher, S. The peculiar phase structure of random graph bisection. J. Math. Phys.
**2008**, 49, 125219. [Google Scholar] [CrossRef] [Green Version] - Šulc, P.; Zdeborová, L. Belief propagation for graph partitioning. J. Phys. A
**2010**, 43, 285003. [Google Scholar] [CrossRef] - Lipowski, A.; Ferreira, A.L.; Lipowska, D.; Barroso, M.A. Bipartitioning of directed and mixed random graphs. J. Stat. Mech. Theory Exp.
**2019**, 2019, 083301. [Google Scholar] [CrossRef] [Green Version] - Watts, D.J.; Strogatz, S.H. Collective dynamics of ‘small world’ networks. Nature
**1998**, 393, 440–442. [Google Scholar] [CrossRef] [PubMed] - Newman, M.E.; Barabási, A.L.E.; Watts, D.J. The Structure and Dynamics of Networks; Princeton University Press: Princeton, NJ, USA, 2006. [Google Scholar]
- Lopes, J.V.; Pogorelov, Y.G.; Lopes dos Santos, J.M.B.; Toral, R. Exact solution of Ising model on a small world network. Phys. Rev. E
**2004**, 70, 026112. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Capraro, V.; Perc, M.; Vilone, D. Lying on networks: The role of structure and topology in promoting honesty. Phys. Rev. E
**2020**, 101, 032305. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Liu, M.; Li, D.; Qin, P.; Liu, C.; Wang, H.; Wang, F. Epidemics in interconnected small world networks. PLoS ONE
**2015**, 10, e0120701. [Google Scholar] [CrossRef] [PubMed] - Kirkpatrick, S.; Gelatt, C.D.; Vecchi, M.P. Optimization by simulated annealing. Science
**1983**, 220, 671–680. [Google Scholar] [CrossRef] [PubMed] - Marinari, E.; Parisi, G.; Ricci-Tersenghi, F.; Ruiz-Lorenzo, J.J.; Zuliani, F. Replica symmetry breaking in short-range spin glasses: Theoretical foundations and numerical evidences. J. Stat. Phys.
**2000**, 98, 973–1074. [Google Scholar] [CrossRef] [Green Version] - Katzgraber, H.G.; Palassini, M.; Young, A. Monte Carlo simulations of spin glasses at low temperatures. Phys. Rev. B
**2001**, 63, 184422. [Google Scholar] [CrossRef] [Green Version] - Krzakała, F.; Montanari, A.; Ricci-Tersenghi, F.; Semerjian, G.; Zdeborová, L. Gibbs states and the set of solutions of random constraint satisfaction problems. Proc. Natl. Acad. Sci. USA
**2007**, 104, 10318–10323. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Monasson, R. Optimization problems and replica symmetry breaking in finite connectivity spin glasses. J. Phys. A
**1998**, 31, 513. [Google Scholar] [CrossRef] - Stauffer, D.; Aharony, A. Introduction to Percolation Theory; CRC Press: New York, NY, USA, 2018. [Google Scholar]
- Huang, K. Statistical Mechanics; John Wiley & Sons: Chichester, UK, 1987. [Google Scholar]

**Figure 1.**(

**a**) When the number of additional links (dashed lines) is small, the cost of the optimal configuration composed of two clusters of length $N/2$ is $B=2$. Note the periodic boundary conditions. (

**b**) For a larger number of additional links, an optimal configuration composed of smaller clusters has the cost $B=4$.

**Figure 2.**The average partition cost B determined using simulated annealing as a function of the number of additional links M. Straight lines correspond to the two cluster estimation, where half of the additional links contribute to the partition cost.

**Figure 3.**The average cluster size S as a function of $M/N$. A power-law diverging fit to our data (line) for $N=1000$ and $M/N>0.1$ suggests that the possible divergence of S takes place at $M/N=0$. Hence, for any $M/N>0$, the optimal solution consists of finite size clusters. The inset shows our data on the log-log scale, and the dotted straight line has a slope −0.67. For small $M/N$, we observe a deviation from the power-law behavior, and we attribute it to finite-size effects.

**Figure 4.**The probability distribution $P\left(q\right)$ of the overlap q. The calculations were made for the $d=1$ small world with $N=100,\phantom{\rule{4pt}{0ex}}M=50$ (

**upper panel**), $N=100,M=100$ (

**middle**), and the random graph with $N=100$, $z=8$ (

**bottom**). A small but nonzero value of $P\left(q\right)$ at $q=0$ might indicate that in all cases, the replica symmetry is broken. In the case of random graphs, there are some independent arguments and calculations that support such a claim [12,13].

**Figure 5.**Exemplary optimal configurations for a two-dimensional 30 × 30 graph. Upon increasing the number of additional links, around $M=1000$, the stripe-like solutions turn into disordered clusters.

**Figure 7.**The probability distributions $P\left(q\right)$ and ${P}_{res}\left(q\right)$. Calculations are made for the $d=2$ small world with $L=10$ and $M=50$.

**Figure 8.**Th probability distributions $P\left(q\right)$ and ${P}_{res}\left(q\right)$. Calculations are made for the $d=2$ small world with $L=10$ and $M=500$.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lipowski, A.; Ferreira, A.L.; Lipowska, D.
Cluster Structure of Optimal Solutions in Bipartitioning of Small Worlds. *Entropy* **2020**, *22*, 1319.
https://doi.org/10.3390/e22111319

**AMA Style**

Lipowski A, Ferreira AL, Lipowska D.
Cluster Structure of Optimal Solutions in Bipartitioning of Small Worlds. *Entropy*. 2020; 22(11):1319.
https://doi.org/10.3390/e22111319

**Chicago/Turabian Style**

Lipowski, Adam, António L. Ferreira, and Dorota Lipowska.
2020. "Cluster Structure of Optimal Solutions in Bipartitioning of Small Worlds" *Entropy* 22, no. 11: 1319.
https://doi.org/10.3390/e22111319