Inference of hidden classes in stochastic block models is a classical problem with important applications. Most commonly used methods for this problem involve naive mean field approaches or heuristic spectral methods....
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Inference of hidden classes in stochastic block models is a classical problem with important applications. Most commonly used methods for this problem involve naive mean field approaches or heuristic spectral methods. Recently, belief propagation was proposed for this problem. In this contribution we perform a comparative study between the three methods on synthetically created networks. We show that belief propagation shows much better performance when compared to naive mean field and spectral approaches. This applies to accuracy, computational efficiency and the tendency to overfit the data.
In this paper we estimate the propagation of liquidity shocks through interbank markets when the information about the underlying credit network is incomplete. We show that techniques such as maximum entropy currently...
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In this paper we estimate the propagation of liquidity shocks through interbank markets when the information about the underlying credit network is incomplete. We show that techniques such as maximum entropy currently used to reconstruct credit networks severely underestimate the risk of contagion by assuming a trivial (fully connected) topology, a type of network structure which can be very different from the one empirically observed. We propose an efficient message-passing algorithm to explore the space of possible network structures and show that a correct estimation of the network degree of connectedness leads to more reliable estimations for systemic risk. Such an algorithm is also able to produce maximally fragile structures, providing a practical upper bound for the risk of contagion when the actual network structure is unknown. We test our algorithm on ensembles of synthetic data encoding some features of real financial networks (sparsity and heterogeneity), finding that more accurate estimations of risk can be achieved. Finally we find that this algorithm can be used to control the amount of information that regulators need to require from banks in order to sufficiently constrain the reconstruction of financial networks.
The replica extension theory has been developed for the purpose of investigating spin glass systems in the replica symmetry breaking phase. The replica extension theory is a replica theory for a specific sample and ha...
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The replica extension theory has been developed for the purpose of investigating spin glass systems in the replica symmetry breaking phase. The replica extension theory is a replica theory for a specific sample and has the scheme of the one-step replica symmetry breaking ansatz. In this paper, using the Plefka expansion, we propose the perturbative expansion of Gibbs free energies of Ising systems with the replica extension theory, and we derive the explicit form of the second- and third-order approximations. Our Plefka expansion systematically provides perturbative approximations taking the one-step replica symmetry breaking into account. In some numerical experiments, we show that our perturbative approximations are quantitatively superior to conventional perturbative approximations that do not take the one-step replica symmetry breaking into account in the spin glass phase.
In this work we introduce a novel weighted message-passing algorithm based on the cavity method for estimating volume-related properties of random polytopes, properties which are relevant in various research fields ra...
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In this work we introduce a novel weighted message-passing algorithm based on the cavity method for estimating volume-related properties of random polytopes, properties which are relevant in various research fields ranging from metabolic networks, to neural networks, to compressed sensing. We propose, as opposed to adopting the usual approach consisting in approximating the real-valued cavity marginal distributions by a few parameters, using an algorithm to faithfully represent the entire marginal distribution. We explain various alternatives for implementing the algorithm and benchmarking the theoretical findings by showing concrete applications to random polytopes. The results obtained with our approach are found to be in very good agreement with the estimates produced by the Hit-and-Run algorithm, known to produce uniform sampling.
We propose a message-passing paradigm for resource allocation problems. This serves to connect ideas from the message-passing literature, which has primarily grown out of the communications, statistical physics, and a...
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We propose a message-passing paradigm for resource allocation problems. This serves to connect ideas from the message-passing literature, which has primarily grown out of the communications, statistical physics, and artificial intelligence communities, with a problem central to operations research. This also provides a new framework for decentralized management that generalizes price-based systems by allowing incentives to vary across activities and consumption levels. We demonstrate that message-based incentives, which are characterized by a new equilibrium concept, lead to system-optimal behavior for convex resource allocation problems yet yield allocations superior to those from price-based incentives for nonconvex problems. We describe a distributed and asynchronous message-passing algorithm for computing equilibrium messages and allocations, and we demonstrate its merits in the context of a network resource allocation problem.
We consider communication over binary-input memoryless output-symmetric channels using low-density parity-check codes and message-passing decoding. The asymptotic (in the length) performance of such a combination for ...
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We consider communication over binary-input memoryless output-symmetric channels using low-density parity-check codes and message-passing decoding. The asymptotic (in the length) performance of such a combination for a fixed number of iterations is given by density evolution. Letting the number of iterations tend to infinity we get the density evolution (DE) threshold, the largest channel parameter so that the bit error probability tends to zero as a function of the iterations. In practice, we often work with short codes and perform a large number of iterations. It is, therefore, interesting to consider what happens if in the standard analysis we exchange the order in which the blocklength and the number of iterations diverge to infinity. In particular, we can ask whether both limits give the same threshold. Although empirical observations strongly suggest that the exchange of limits is valid for all channel parameters, we limit our discussion to channel parameters below the DE threshold. Specifically, we show that as long as the message reliabilities are bounded and other technical conditions are met, the bit error probability vanishes up to a nontrivial threshold regardless of how the limit is taken. This threshold is equal to the DE threshold when the minimum degree of the variable nodes is at least five and strictly less than the DE threshold for smaller degrees.
We consider a variation of the prototype combinatorial optimization problem known as graph colouring. Our optimization goal is to colour the vertices of a graph with a fixed number of colours, in a way to maximize the...
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We consider a variation of the prototype combinatorial optimization problem known as graph colouring. Our optimization goal is to colour the vertices of a graph with a fixed number of colours, in a way to maximize the number of different colours present in the set of nearest neighbours of each given vertex. This problem, which we pictorially call palette-colouring, has been recently addressed as a basic example of a problem arising in the context of distributed data storage. Even though it has not been proved to be NP-complete, random search algorithms find the problem hard to solve. Heuristics based on a naive belief propagation algorithm are observed to work quite well in certain conditions. In this paper, we build upon the mentioned result, working out the correct belief propagation algorithm, which needs to take into account the many-body nature of the constraints present in this problem. This method improves the naive belief propagation approach at the cost of increased computational effort. We also investigate the emergence of a satisfiable-to-unsatisfiable 'phase transition' as a function of the vertex mean degree, for different ensembles of sparse random graphs in the large size ('thermodynamic') limit.
message-passing algorithms based on belief propagation (BP) are implemented on a random constraint satisfaction problem (CSP) referred to as model RB, which is a prototype of hard random CSPs with growing domain size....
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message-passing algorithms based on belief propagation (BP) are implemented on a random constraint satisfaction problem (CSP) referred to as model RB, which is a prototype of hard random CSPs with growing domain size. In model RB, the number of candidate discrete values (the domain size) of each variable increases polynomially with the variable number N of the problem formula. Although the satisfiability threshold of model RB is exactly known, finding solutions for a single problem formula is quite challenging and attempts have been limited to cases of N similar to 10(2). In this paper, we propose two different kinds of message-passing algorithms guided by BP for this problem. Numerical simulations demonstrate that these algorithms allow us to find a solution for random formulas of model RB with constraint tightness slightly less than p(cr), the threshold value for the satisfiability phase transition. To evaluate the performance of these algorithms, we also provide a local search algorithm (random walk) as a comparison. Besides this, the simulated time dependence of the problem size N and the entropy of the variables for growing domain size are discussed.
We study the performance of different messagepassingalgorithms in the two-dimensional Edwards-Anderson model. We show that the standard belief propagation (BP) algorithm converges only at high temperature to a param...
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We study the performance of different messagepassingalgorithms in the two-dimensional Edwards-Anderson model. We show that the standard belief propagation (BP) algorithm converges only at high temperature to a paramagnetic solution. Then, we test a generalized belief propagation (GBP) algorithm, derived from a cluster variational method (CVM) at the plaquette level. We compare its performance with BP and with other algorithms derived under the same approximation: double loop (DL) and a two-way messagepassing algorithm (HAK). The plaquette-CVM approximation improves BP in at least three ways: the quality of the paramagnetic solution at high temperatures, a better estimate (lower) for the critical temperature, and the fact that the GBP messagepassing algorithm converges also to nonparamagnetic solutions. The lack of convergence of the standard GBP messagepassing algorithm at low temperatures seems to be related to the implementation details and not to the appearance of long range order. In fact, we prove that a gauge invariance of the constrained CVM free energy can be exploited to derive a new messagepassing algorithm which converges at even lower temperatures. In all its region of convergence this new algorithm is faster than HAK and DL by some orders of magnitude.
Most optimization problems in applied sciences realistically involve uncertainty in the parameters defining the cost function, of which only statistical information is known beforehand. Here we provide an in-depth dis...
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Most optimization problems in applied sciences realistically involve uncertainty in the parameters defining the cost function, of which only statistical information is known beforehand. Here we provide an in-depth discussion of how messagepassingalgorithms for stochastic optimization based on the cavity method of statistical physics can be constructed. We focus on two basic problems, namely the independent set problem and the matching problem, for which we display the general method and caveats for the case of the so called two-stage problem with independently distributed stochastic parameters. We compare the results with some greedy algorithms and briefly discuss the extension to more complicated stochastic multi-stage problems.
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