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Equilibrium and Dynamics on Complex Networkds
KTH, School of Computer Science and Communication (CSC), Computational Science and Technology (CST).ORCID iD: 0000-0002-1252-2899
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 [en]
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: urn:nbn:se:kth:diva-191991ISBN: 978-91-7729-058-2OAI: oai:DiVA.org:kth-191991DiVA: diva2:957675
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
List of papers
1. Cavity Method: Message Passing from a Physics perspective
Open this publication in new window or tab >>Cavity Method: Message Passing from a Physics perspective
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
cavity method, message passing, survey propagation, tree-like graphs
National Category
Physical Sciences
Research subject
Physics
Identifiers
urn:nbn:se:kth:diva-192102 (URN)
External cooperation:
Note

QC 20160905

Available from: 2016-09-05 Created: 2016-09-05 Last updated: 2016-09-05Bibliographically approved
2. On one-step replica symmetry breaking in the Edwards-Anderson spin glass model
Open this publication in new window or tab >>On one-step replica symmetry breaking in the Edwards-Anderson spin glass model
2016 (English)In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, 073305Article in journal (Refereed) Published
Abstract [en]

We consider a one-step replica symmetry breaking description of the Edwards–Anderson spin glass model in 2D. The ingredients of this description are a Kikuchi approximation to the free energy and a second-level statistical model built on the extremal points of the Kikuchi approximation, which are also fixed points of a generalized belief propagation (GBP) scheme. We show that a generalized free energy can be constructed where these extremal points are exponentially weighted by their Kikuchi free energy and a Parisi parameter y, and that the Kikuchi approximation of this generalized free energy leads to second-level, one-step replica symmetry breaking (1RSB), GBP equations. We then proceed analogously to the Bethe approximation case for tree-like graphs, where it has been shown that 1RSB belief propagation equations admit a survey propagation solution. We discuss when and how the one-step-replica symmetry breaking GBP equations that we obtain also allow a simpler class of solutions which can be interpreted as a class of generalized survey propagation equations for the single instance graph case.

Place, publisher, year, edition, pages
Institute of Physics (IOP), 2016
Keyword
Region graph approximations, generalized belief propagation, survey propagation, generalized survey propagation, low temperature
National Category
Natural Sciences Physical Sciences
Research subject
Physics
Identifiers
urn:nbn:se:kth:diva-192097 (URN)10.1088/1742-5468/2016/07/073305 (DOI)
External cooperation:
Note

QC 20160905

Available from: 2016-09-05 Created: 2016-09-05 Last updated: 2016-09-05Bibliographically approved
3. Dynamic message-passing approach for kinetic spin models with reversible dynamics
Open this publication in new window or tab >>Dynamic message-passing approach for kinetic spin models with reversible dynamics
2015 (English)In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, E-ISSN 1550-2376, Vol. 92, no 1, 010102Article in journal (Refereed) Published
Abstract [en]

A method to approximately close the dynamic cavity equations for synchronous reversible dynamics on a locally treelike topology is presented. The method builds on (a) a graph expansion to eliminate loops from the normalizations of each step in the dynamics and (b) an assumption that a set of auxilary probability distributions on histories of pairs of spins mainly have dependencies that are local in time. The closure is then effectuated by projecting these probability distributions on n-step Markov processes. The method is shown in detail on the level of ordinary Markov processes (n = 1) and outlined for higher-order approximations (n > 1). Numerical validations of the technique are provided for the reconstruction of the transient and equilibrium dynamics of the kinetic Ising model on a random graph with arbitrary connectivity symmetry.

Place, publisher, year, edition, pages
American physical society, 2015
National Category
Fusion, Plasma and Space Physics
Identifiers
urn:nbn:se:kth:diva-172162 (URN)10.1103/PhysRevE.92.010102 (DOI)000358259400001 ()
External cooperation:
Note

QC 20150818

Available from: 2015-08-18 Created: 2015-08-14 Last updated: 2016-09-05Bibliographically approved
4. A simple approach to the dynamics of Ising spin systems
Open this publication in new window or tab >>A simple approach to the dynamics of Ising spin systems
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We study a novel variational approach to solve the dynamics of Ising-like discrete spin systems. The equations are derived under mean-field approximations based on the cluster variational method. Comparison with the actual Glauber dynamics of models defined on Erdos-R\'enyi random graphs show that some of these approximations are extremely accurate, both at equilibrium and in the off-equilibrium regime, providing the same result of the Monte Carlo simulation in a much shorter time. The models studied are the ferromagnetic kinetic Ising model (both with symmetric and partially asymmetric interactions), the random field Ising model and the Viana-Bray model. Only for the latter model we find some small discrepancies in the very low temperature phase, probably due to the existence of a large number of metastable states.

Keyword
variational approximations, spin dynamics, Glauber dynamics, cluster variation method, path probability method
National Category
Physical Sciences
Research subject
Physics
Identifiers
urn:nbn:se:kth:diva-192098 (URN)
External cooperation:
Note

QC 20160905

Available from: 2016-09-05 Created: 2016-09-05 Last updated: 2016-09-05Bibliographically approved
5. A Cavity Master Equation for the continuous time dynamics of discrete spins models
Open this publication in new window or tab >>A Cavity Master Equation for the continuous time dynamics of discrete spins models
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We present a new method to close the Master Equation representing the continuous time dynamics of Ising interacting spins. The method makes use of the the theory of Random Point Processes to derive a master equation for local conditional probabilities. We analytically test our solution studying two known cases, the dynamics of the mean field ferromagnet and the dynamics of the one dimensional Ising system. We then present numerical results comparing our predictions with Monte Carlo simulations in three different models on random graphs with finite connectivity: the Ising ferromagnet, the Random Field Ising model, and the Viana-Bray spin-glass model.

Keyword
continuous time dynamics, master equation, cavity method, cavity master equation, dynamic message passing, disordered models
National Category
Physical Sciences
Research subject
Physics
Identifiers
urn:nbn:se:kth:diva-192100 (URN)
External cooperation:
Note

QC 20160905

Available from: 2016-09-05 Created: 2016-09-05 Last updated: 2016-09-05Bibliographically approved
6. Perturbative large deviation analysis of non-equilibrium dynamics
Open this publication in new window or tab >>Perturbative large deviation analysis of non-equilibrium dynamics
2014 (English)In: Journal of the Physical Society of Japan, ISSN 0031-9015, E-ISSN 1347-4073, Vol. 83, no 8Article in journal (Refereed) Published
Abstract [en]

Macroscopic fluctuation theory has shown that a wide class of non-equilibrium stochastic dynamical systems obey a large deviation principle, but except for a few one-dimensional examples these large deviation principles are in general not known in closed form. We consider the problem of constructing successive approximations to an (unknown) large deviation functional and show that the non-equilibrium probability distribution the takes a Gibbs-Boltzmann form with a set of auxiliary (non-physical) energy functions. The expectation values of these auxiliary energy functions and their conjugate quantities satisfy a closed system of equations which can imply a considerable reduction of dimensionality of the dynamics. We show that the accuracy of the approximations can be tested self-consistently without solving the full non- equilibrium equations. We test the general procedure on the simple model problem of a relaxing 1D Ising chain. 

Place, publisher, year, edition, pages
Physical society of Japan, 2014
Keyword
Macroscopic fluctuation theory, perturbative large deviation, dynamics Ising chain.
National Category
Condensed Matter Physics
Research subject
Physics
Identifiers
urn:nbn:se:kth:diva-165789 (URN)10.7566/JPSJ.83.084001 (DOI)000339806400012 ()2-s2.0-84924898390 (ScopusID)
Projects
NETADIS
Funder
EU, FP7, Seventh Framework Programme
Note

QC 20150505

Available from: 2015-04-29 Created: 2015-04-29 Last updated: 2016-09-04
7. Causal analysis, Correlation-Response, and Dynamic cavity
Open this publication in new window or tab >>Causal analysis, Correlation-Response, and Dynamic cavity
2016 (English)In: International Meeting on High-Dimensional Data-Driven Science (HD3-2015), Institute of Physics (IOP), 2016, 012002Conference paper (Refereed)
Abstract [en]

The purpose of this note is to point out analogies between causal analysis in statistics and the correlation-response theory in statistical physics. It is further shown that for some systems the dynamic cavity offers a way to compute the stationary state of a non-equilibrium process effectively, which could then be taken an alternative starting point of causal analysis.

Place, publisher, year, edition, pages
Institute of Physics (IOP), 2016
Series
, Journal of Physics Conference Series, ISSN 1742-6588 ; 699
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-188463 (URN)10.1088/1742-6596/699/1/012002 (DOI)000376066400002 ()2-s2.0-84964835185 (ScopusID)
Conference
International Meeting on High-Dimensional Data-Driven Science (HD3), DEC 14-17, 2015, Kyoto, JAPAN
Note

QC 20160613

Available from: 2016-06-13 Created: 2016-06-10 Last updated: 2016-09-04Bibliographically approved

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