Related papers: Mean-field theory of vector spin models on network…
Mean-field analysis is an important tool for understanding dynamics on complex networks. However, surprisingly little attention has been paid to the question of whether mean-field predictions are accurate, and this is particularly true for…
A general expansion scheme based on the concept of linked cluster expansion from the theory of classical spin systems is constructed for models of interacting electrons. It is shown that with a suitable variational formulation of mean-field…
We discuss the mean-field theory of spin-glass models with frustrated long-range random spin exchange. We analyze the reasons for breakdown of the simple mean-field theory of Sherrington and Kirkpatrick. We relate the replica-symmetry…
Dynamical mean-field theory is a powerful physics tool used to analyze the typical behavior of neural networks, where neurons can be recurrently connected, or multiple layers of neurons can be stacked. However, it is not easy for beginners…
Recent advances in experimental techniques enable the simultaneous recording of activity from thousands of neurons in the brain, presenting both an opportunity and a challenge: to build meaningful, scalable models of large neural…
Models of coupled oscillator networks play an important role in describing collective synchronization dynamics in biological and technological systems. The Kuramoto model describes oscillator's phase evolution and explains the transition…
A recent dynamic mean-field theory for sequence processing in fully connected neural networks of Hopfield-type (During, Coolen and Sherrington, 1998) is extended and analized here for a symmetrically diluted network with finite connectivity…
The incoherent state of the Kuramoto model of coupled oscillators exhibits marginal modes in mean field theory. We demonstrate that corrections due to finite size effects render these modes stable in the subcritical case, i.e. when the…
The study of the normalized sum of random variables and its asymptotic behaviour has been and continues to be a central chapter in probability and statistical mechanics. When those variables are independent the central limit theorem ensures…
In this work we study the dynamics of systems composed of numerous interacting elements interconnected through a random weighted directed graph, such as models of random neural networks. We develop an original theoretical approach based on…
Mean field theory is a device to analyze the collective behavior of a dynamical system comprising many interacting particles. The theory allows to reduce the behavior of the system to the properties of a handful of parameters. In neural…
Machine learning algorithms relying on deep neural networks recently allowed a great leap forward in artificial intelligence. Despite the popularity of their applications, the efficiency of these algorithms remains largely unexplained from…
We develop a unified theory that encompasses the macroscopic dynamics of recurrent interactions of binary units within arbitrary network architectures. Using the martingale theory, our mathematical analysis provides a complete description…
The Kuramoto model (KM) of coupled phase oscillators on graphs provides the most influential framework for studying collective dynamics and synchronization. It exhibits a rich repertoire of dynamical regimes. Since the work of Strogatz and…
Employing the Kuramoto model as an illustrative example, we show how the use of the mean field approximation can be applied to large networks of phase oscillators with assortativity. We then use the ansatz of Ott and Antonsen [Chaos 19,…
Many complex dynamical systems in the real world, including ecological, climate, financial, and power-grid systems, often show critical transitions, or tipping points, in which the system's dynamics suddenly transit into a qualitatively…
We present the results of numerical studies of superconductivity and antiferromagnetism in a strongly correlated electron system. To do this we construct a Hubbard model on a lattice of self-consistently embedded multi-site clusters by…
The concept of replica symmetry breaking found in the solution of the mean-field Sherrington-Kirkpatrick spin-glass model has been applied to a variety of problems in science ranging from biological to computational and even financial…
Originally arising in the context of interacting particle systems in statistical physics, dynamical systems and differential equations on networks/graphs have permeated into a broad number of mathematical areas as well as into many…
Clustering is the propensity of nodes that share a common neighbour to be connected. It is ubiquitous in many networks but poses many modelling challenges. Clustering typically manifests itself by a higher than expected frequency of…