Information, Physics, and Computation

Mézard, Marc; Montanari, Andrea

doi:10.1093/acprof:oso/9780198570837.001.0001

Cited by 1,647 publications

(2,426 citation statements)

References 252 publications

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“…Models in which the state variables are placed on the nodes of random graphs have found an enormous number of applications, including spin-glasses, satisfiability problems, error-correcting codes, and complex networks (see Refs. [25,26] and references therein), and alternative tools to study their index fluctuations would be more than welcome.…”

Section: Introductionmentioning

confidence: 99%

Index statistical properties of sparse random graphs

Metz

Stariolo

2015

Phys. Rev. E

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Using the replica method, we develop an analytical approach to compute the characteristic function for the probability P N (K,λ) that a large N × N adjacency matrix of sparse random graphs has K eigenvalues below a threshold λ. The method allows to determine, in principle, all moments of P N (K,λ), from which the typical sample-to-sample fluctuations can be fully characterized. For random graph models with localized eigenvectors, we show that the index variance scales linearly with N 1 for |λ| > 0, with a model-dependent prefactor that can be exactly calculated. Explicit results are discussed for Erdös-Rényi and regular random graphs, both exhibiting a prefactor with a nonmonotonic behavior as a function of λ. These results contrast with rotationally invariant random matrices, where the index variance scales only as ln N , with an universal prefactor that is independent of λ. Numerical diagonalization results confirm the exactness of our approach and, in addition, strongly support the Gaussian nature of the index fluctuations.

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Section: Introductionmentioning

confidence: 99%

Index statistical properties of sparse random graphs

Metz

Stariolo

2015

Phys. Rev. E

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“…Computing this distance (called the "line index of balance" in refs. 20 and 21) is a nondeterministic polynomial-time hard problem, equivalent to a series of well-known problems, such as computing the ground state of a (nonplanar) Ising spin glass (22); solving a maximumcut (MAX-CUT) problem (23,24); or finding the best solution of an overconstrained linear system over a finite field (the so-called MAX-2XORSAT problem) (25). The equivalence with energy minimization of a spin glass has, for example, been highlighted recently in ref.…”

mentioning

confidence: 99%

Computing global structural balance in large-scale signed social networks

Facchetti

Iacono

Altafini

2011

Proc. Natl. Acad. Sci. U.S.A.

363

298

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Structural balance theory affirms that signed social networks (i.e., graphs whose signed edges represent friendly/hostile interactions among individuals) tend to be organized so as to avoid conflictual situations, corresponding to cycles of negative parity. Using an algorithm for ground-state calculation in large-scale Ising spin glasses, in this paper we compute the global level of balance of very large online social networks and verify that currently available networks are indeed extremely balanced. This property is explainable in terms of the high degree of skewness of the sign distributions on the nodes of the graph. In particular, individuals linked by a large majority of negative edges create mostly "apparent disorder," rather than true "frustration."combinatorial optimization | social network theory O nline social networks are examples of large-scale communities of interacting individuals in which local ties between users (friend, fan, colleague, but also friend/foe, trust/distrust, etc.) give rise to a complex, multidimensional web of aggregated social behavior (1-4). For such complex networks, the emergence of global properties from local interactions is an intriguing subject, so far investigated mostly at structural and topological level (2,(5)(6)(7)(8). In social network theory (9-11), however, the content of the relationships is often even more important than their topology, and this calls for the development of appropriate analytical and computational tools, able to extrapolate content-related features out of the set of interactions of a social community. Obtaining efficient tools is particularly challenging when, as in social networks retrieved from online media, the size of the community is very big, of the order of 10 5 individuals or higher.A global property that has recently attracted some attention (1,(12)(13)(14) is determining the structural balance of a signed social network. Structural (or social) balance theory was first formulated by Heider (15) in order to understand the structure and origin of tensions and conflicts in a network of individuals whose mutual relationships are characterizable in terms of friendship and hostility. It was modeled in terms of signed graphs by Cartwright and Harary (16); see refs. 10 and 11 for an overview of the theory. The nodes of the graph represent users and the positive/ negative edges their friendly/hostile relationships. It has been known for some time how to interpret structural balance on such networks (16): The potential source of tensions are the cycles of the graph (i.e., the closed paths beginning and ending on the same node), notably those of negative sign (i.e., having an odd number of negative edges). It follows that the concept of balance is not related to the actual number of negative edges on the cycles but only to their parity; see Fig. 1 for an illustration on basic graphs. In particular, a signed graph is exactly balanced (i.e., tensions are completely absent) if and only if all its cycles are positive (16). As such, structural balance ...

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“…We next derive a message passing version of this iteration 3 (we refer for instance to [47,48] for background). The motivation for this modification is that message passing algorithms have appealing statistical properties.…”

Section: Message Passing Algorithmsmentioning

confidence: 99%

“…Following a general prescription from [48], we associate a factor node to each term (y a − x a , θ ) 2 /(2n) in the cost function indexed by a ∈ {1, 2, . .…”

Section: Message Passing Algorithmsmentioning

confidence: 99%

Statistical estimation: from denoising to sparse regression and hidden cliques

Tramel¹,

Kumar²,

Giurgiu³

et al. 2015

Statistical Physics, Optimization, Inference, and Message-Passing Algorithms

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Abstract. These notes review six lectures given by Prof. Andrea Montanari on the topic of statistical estimation for linear models. The first two lectures cover the principles of signal recovery from linear measurements in terms of minimax risk. Subsequent lectures demonstrate the application of these principles to several practical problems in science and engineering. Specifically, these topics include denoising of error-laden signals, recovery of compressively sensed signals, reconstruction of low-rank matrices, and also the discovery of hidden cliques within large networks.These are notes from the lecture of Andrea Montanari given at the autumn school "Statistical Physics, Optimization, Inference, and Message-Passing Algorithms", that took place in Les Houches,

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Information, Physics, and Computation

Cited by 1,647 publications

References 252 publications

Index statistical properties of sparse random graphs

Index statistical properties of sparse random graphs

Computing global structural balance in large-scale signed social networks

Statistical estimation: from denoising to sparse regression and hidden cliques

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