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Cavity Method: Message Passing from a Physics perspective
KTH, School of Computer Science and Communication (CSC), Computational Science and Technology (CST).ORCID iD: 0000-0002-1252-2899
2016 (English)In: Statistical Physics, Optimization, Inference, and Message-Passing Algorithms: Lecture Notes of the Les Houches School of Physics - Special Issue October 2013 / [ed] F. Krzakala et al., Oxford University Press, 2016Chapter in book (Other academic)
Abstract [en]

In this three-sections lecture cavity method is introduced as heuristic framework from a Physics perspective to solve probabilistic graphical models and it is presented both at the replica symmetric (RS) and 1-step replica symmetry breaking (1RSB) level. This technique has been applied with success on a wide range of models and problems such as spin glasses, random constrain satisfaction problems (rCSP), error correcting codes etc. Firstly, the RS cavity solution for Sherrington-Kirkpatrick model---a fully connected spin glass model---is derived and its equivalence to the RS solution obtained using replicas is discussed. Then, the general cavity method for diluted graphs is illustrated both at RS and 1RSB level. The latter was a significant breakthrough in the last decade and has direct applications to rCSP. Finally, as example of an actual problem, K-SAT is investigated using belief and survey propagation.

Place, publisher, year, edition, pages
Oxford University Press, 2016.
Keyword [en]
cavity method, message passing, survey propagation, tree-like graphs
National Category
Physical Sciences
Research subject
Physics
Identifiers
URN: urn:nbn:se:kth:diva-192102OAI: oai:DiVA.org:kth-192102DiVA: diva2:958026
Note

QC 20160905

Available from: 2016-09-05 Created: 2016-09-05 Last updated: 2016-09-05Bibliographically approved
In thesis
1. Equilibrium and Dynamics on Complex Networkds
Open this publication in new window or tab >>Equilibrium and Dynamics on Complex Networkds
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Complex networks are an important class of models used to describe the behaviour of a very broad category of systems which appear in different fields of science ranging from physics, biology and statistics to computer science and other disciplines. This set of models includes spin systems on a graph, neural networks, decision networks, spreading disease, financial trade, social networks and all systems which can be represented as interacting agents on some sort of graph architecture.

In this thesis, by using the theoretical framework of statistical mechanics, the equilibrium and the dynamical behaviour of such systems is studied.

For the equilibrium case, after presenting the region graph free energy approximation, the Survey Propagation method, previously used to investi- gate the low temperature phase of complex systems on tree-like topologies, is extended to the case of loopy graph architectures.

For time-dependent behaviour, both discrete-time and continuous-time dynamics are considered. It is shown how to extend the cavity method ap- proach from a tool used to study equilibrium properties of complex systems to the discrete-time dynamical scenario. A closure scheme of the dynamic message-passing equation based on a Markovian approximations is presented. This allows to estimate non-equilibrium marginals of spin models on a graph with reversible dynamics. As an alternative to this approach, an extension of region graph variational free energy approximations to the non-equilibrium case is also presented. Non-equilibrium functionals that, when minimized with constraints, lead to approximate equations for out-of-equilibrium marginals of general spin models are introduced and discussed.

For the continuous-time dynamics a novel approach that extends the cav- ity method also to this case is discussed. The main result of this part is a Cavity Master Equation which, together with an approximate version of the Master Equation, constitutes a closure scheme to estimate non-equilibrium marginals of continuous-time spin models. The investigation of dynamics of spin systems is concluded by applying a quasi-equilibrium approach to a sim- ple case. A way to test self-consistently the assumptions of the method as well as its limits is discussed.

In the final part of the thesis, analogies and differences between the graph- ical model approaches discussed in the manuscript and causal analysis in statistics are presented.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2016. 189 p.
Series
TRITA-CSC-A, ISSN 1653-5723 ; 2016:17
Keyword
Statistical mechanics, complex networks, spin systems, non equilibrium dynamics, generalized belief propagation, message passing, cavity method, variational approaches
National Category
Other Physics Topics Physical Sciences
Research subject
Physics
Identifiers
urn:nbn:se:kth:diva-191991 (URN)978-91-7729-058-2 (ISBN)
Public defence
2016-09-09, Kollegiesalen, Brinellvägen 8, plan 4, Stockholm, 10:00 (English)
Opponent
Supervisors
Note

QC 20160904

Available from: 2016-09-04 Created: 2016-09-03 Last updated: 2016-09-05Bibliographically approved

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