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269
Community detection in graphs
, 2009
"... The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of th ..."
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Cited by 801 (1 self)
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The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Such
Broken replica symmetry bounds in the mean field spin glass model
 Comm. Math Phys
, 2003
"... By using a simple interpolation argument, in previous work we have proven the existence of the thermodynamic limit, for mean field disordered models, including the SherringtonKirkpatrick model, and the Derrida pspin model. Here we extend this argument in order to compare the limiting free energy w ..."
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Cited by 146 (15 self)
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By using a simple interpolation argument, in previous work we have proven the existence of the thermodynamic limit, for mean field disordered models, including the SherringtonKirkpatrick model, and the Derrida pspin model. Here we extend this argument in order to compare the limiting free energy with the expression given by the Parisi Ansatz, and including full spontaneous replica symmetry breaking. Our main result is that the quenched average of the free energy is bounded from below by the value given in the Parisi Ansatz, uniformly in the size of the system. Moreover, the difference between the two expressions is given in the form of a sum rule, extending our previous work on the comparison between the true free energy and its replica symmetric SherringtonKirkpatrick approximation. We give also a variational bound for the infinite volume limit of the ground state energy per site.
Landscapes and Their Correlation Functions
, 1996
"... Fitness landscapes are an important concept in molecular evolution. Many important examples of landscapes in physics and combinatorial optimation, which are widely used as model landscapes in simulations of molecular evolution and adaptation, are "elementary", i.e., they are (up to an addi ..."
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Cited by 105 (16 self)
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Fitness landscapes are an important concept in molecular evolution. Many important examples of landscapes in physics and combinatorial optimation, which are widely used as model landscapes in simulations of molecular evolution and adaptation, are "elementary", i.e., they are (up to an additive constant) eigenfuctions of a graph Laplacian. It is shown that elementary landscapes are characterized by their correlation functions. The correlation functions are in turn uniquely determined by the geometry of the underlying configuration space and the nearest neighbor correlation of the elementary landscape. Two types of correlation functions are investigated here: the correlation of a time series sampled along a random walk on the landscape and the correlation function with respect to a partition of the set of all vertex pairs.
Discrete scale invariance and complex dimensions
 Physics Reports
, 1998
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General properties of overlap probability distributions in disordered spin systems
 Toward Parisi ultrametricity, J. Phys. A: Math. Gen
, 1998
"... For a very general class of probability distributions in disordered Ising spin systems, in the thermodynamical limit, we prove the following property for overlaps among real replicas. Consider the overlaps among s replicas. Add one replica s + 1. Then, the overlap qa,s+1 between one of the first s r ..."
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Cited by 64 (1 self)
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For a very general class of probability distributions in disordered Ising spin systems, in the thermodynamical limit, we prove the following property for overlaps among real replicas. Consider the overlaps among s replicas. Add one replica s + 1. Then, the overlap qa,s+1 between one of the first s replicas, let us say a, and the added s + 1 is either independent of the former ones, or it is identical to one of the overlaps qab, with b running among the first s replicas, excluding a. Each of these cases has equal probability 1/s. 1
Threshold values of random kSAT from the cavity method
, 2005
"... Using the cavity equations of Mézard, Parisi, and Zecchina [Science 297 (2002), 812; Mézard and Zecchina, Phys Rev E 66 (2002), 056126] we derive the various threshold values for the number of clauses per variable of the random Ksatisfiability problem, generalizing the previous results to K ≥ 4. We ..."
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Cited by 46 (4 self)
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Using the cavity equations of Mézard, Parisi, and Zecchina [Science 297 (2002), 812; Mézard and Zecchina, Phys Rev E 66 (2002), 056126] we derive the various threshold values for the number of clauses per variable of the random Ksatisfiability problem, generalizing the previous results to K ≥ 4. We also give an analytic solution of the equations, and some closed expressions for these thresholds, in an expansion around large K. The stability of the solution is also computed. For any K, the satisfiability threshold is found to be in the stable region of the solution, which adds further credit to the conjecture that this computation gives the exact satisfiability threshold.
The high temperature region of the VianaBray diluted spin glass model
, 2008
"... In this paper, we study the high temperature or low connectivity phase of the VianaBray model. This is a diluted version of the well known SherringtonKirkpatrick mean field spin glass. In the whole replica symmetric region, we obtain a complete control of the system, proving annealing for the infi ..."
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Cited by 42 (2 self)
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In this paper, we study the high temperature or low connectivity phase of the VianaBray model. This is a diluted version of the well known SherringtonKirkpatrick mean field spin glass. In the whole replica symmetric region, we obtain a complete control of the system, proving annealing for the infinite volume free energy, and a central limit theorem for the suitably rescaled fluctuations of the multioverlaps. Moreover, we show that free energy fluctuations, on the scale 1/N, converge in the infinite volume limit to a nonGaussian random variable, whose variance diverges at the boundary of the replicasymmetric region. The connection with the fully connected SherringtonKirkpatrick model is discussed.
Survey propagation: an algorithm for satisfiability
, 2002
"... ABSTRACT: We study the satisfiability of randomly generated formulas formed by M clauses of exactly K literals over N Boolean variables. For a given value of N the problem is known to be most difficult when α = M/N is close to the experimental threshold αc separating the region where almost all form ..."
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Cited by 41 (1 self)
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ABSTRACT: We study the satisfiability of randomly generated formulas formed by M clauses of exactly K literals over N Boolean variables. For a given value of N the problem is known to be most difficult when α = M/N is close to the experimental threshold αc separating the region where almost all formulas are SAT from the region where all formulas are UNSAT. Recent results from a statistical physics analysis suggest that the difficulty is related to the existence of a clustering phenomenon of the solutions when α is close to (but smaller than) αc. We introduce a new type of message passing algorithm which allows to find efficiently a satisfying assignment of the variables in this difficult region. This algorithm is iterative and composed of two main parts. The first is a messagepassing procedure which generalizes the usual methods like SumProduct or Belief Propagation: It passes messages that may be thought of as surveys over clusters of the ordinary messages. The second part uses the detailed probabilistic information obtained from the surveys in order to fix variables and simplify the problem. Eventually, the simplified problem that remains is solved by a conventional
Computational transition at the uniqueness threshold
 In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
, 2010
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Graphical Models Concepts in Compressed Sensing
"... This paper surveys recent work in applying ideas from graphical models and message passing algorithms to solve large scale regularized regression problems. In particular, the focus is on compressed sensing reconstruction via ℓ1 penalized leastsquares (known as LASSO or BPDN). We discuss how to deri ..."
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Cited by 38 (2 self)
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This paper surveys recent work in applying ideas from graphical models and message passing algorithms to solve large scale regularized regression problems. In particular, the focus is on compressed sensing reconstruction via ℓ1 penalized leastsquares (known as LASSO or BPDN). We discuss how to derive fast approximate message passing algorithms to solve this problem. Surprisingly, the analysis of such algorithms allows to prove exact highdimensional limit results for the LASSO risk. This paper will appear as a chapter in a book on ‘Compressed Sensing ’ edited by Yonina Eldar and Gitta Kutynok. 1