Complex networks renormalization: flows and fixed points

Recently, it has been claimed that some complex networks are self-similar under a convenient renormalization procedure. We present a general method to study renormalization flows in graphs. We find that the behavior of some variables under renormalization, such as the maximum number of connections of a node, obeys simple scaling laws, characterized by critical exponents. This is true for any class of graphs, from random to scale-free networks, from lattices to hierarchical graphs. Therefore, renormalization flows for graphs are similar as in the renormalization of spin systems. An analysis of classic renormalization for percolation and the Ising model on the lattice confirms this analogy. Critical exponents and scaling functions can be used to classify graphs in universality classes, and to uncover similarities between graphs that are inaccessible to a standard analysis.     

n--> infinity limit of O(n) ferromagnetic models on graphs

Thirty years ago, H. E. Stanley showed that an O(n) spin model on a lattice tends to a spherical model as n-->infinity. This means that at any temperature the corresponding free energies coincide. This fundamental result is no longer valid on more general discrete structures lacking in translation invariance, i.e., on graphs. However, only the singular parts of the free energies determine the critical behavior of the two statistical models. Here we show that for ferromagnetic models such singular parts still coincide even on graphs in the thermodynamic limit. This implies that the critical exponents of O(n) models on graphs for n-->infinity tend to the spherical ones and depend only on the graph spectral dimension.     

Vertex models on Feynman diagrams

We discuss the statistical mechanics of vertex models on both generic (“thin”) and planar (“fat”) random graphs. Such models can be formulated as the N → 1 and N → ∞ limits of N × N complex matrix models, respectively. From the graph theoretic perspective one is using matrix model and field theory inspired methods to count various classes of directed graphs. For the thin random graphs we use saddle point methods to solve the models in the thermodynamic, large number of vertices limit and note that, as in the case of the eight-vertex model on the square lattice, various other models such as the Ising model appear as particular limits. The generic solution of the fat graph model is rather more elusive, but we show that for several choices of the couplings the models can be reduced to eigenvalue integrals and their critical behaviour deduced.     

Central Limit Theorem for Exponentially Quasi-local Statistics of Spin Models on Cayley Graphs

Central limit theorems for linear statistics of lattice random fields (including spin models) are usually proven under suitable mixing conditions or quasi-associativity. Many interesting examples of spin models do not satisfy mixing conditions, and on the other hand, it does not seem easy to show central limit theorem for local statistics via quasi-associativity. In this work, we prove general central limit theorems for local statistics and exponentially quasi-local statistics of spin models on discrete Cayley graphs with polynomial growth. Further, we supplement these results by proving similar central limit theorems for random fields on discrete Cayley graphs taking values in a countable space, but under the stronger assumptions of \(\alpha \)-mixing (for local statistics) and exponential \(\alpha \)-mixing (for exponentially quasi-local statistics). All our central limit theorems assume a suitable variance lower bound like many others in the literature. We illustrate our general central limit theorem with specific examples of lattice spin models and statistics arising in computational topology, statistical physics and random networks. Examples of clustering spin models include quasi-associated spin models with fast decaying covariances like the off-critical Ising model, level sets of Gaussian random fields with fast decaying covariances like the massive Gaussian free field and determinantal point processes with fast decaying kernels. Examples of local statistics include intrinsic volumes, face counts, component counts of random cubical complexes while exponentially quasi-local statistics include nearest neighbour distances in spin models and Betti numbers of sub-critical random cubical complexes.     

Structural phase transition of boron nitride compound

The full-potential band-structure scheme based on the linear combination of overlapping nonorthogonal local-orbital (FPLO) is used. The crystal potential and density are represented as a lattice sum of local overlapping nonspherical contributions. The energetic transitions of BN of zinc-blende and wurtzite structures are calculated using the band structure scheme. The energy gap at ambient pressure is found to be indirect for the two structures. The structural properties of two structures of BN are (obtained from the total energy calculations) and the total density of states are calculated. The phase transition parameter of BN is investigated. The ionicity character of BN has been calculated to test the validity of our recent models. The results are in reasonable agreement with experimental and other theoretical results.     

Directed transport of coupled systems in symmetric periodic potentials

In this paper, we discuss the damped unidirectional motions of a coupled lattice in a periodic potential. Each particle in the lattice is subject to a time-periodic ac force. Our studies reveal that a directed transport process can be observed when the ac forces acting on the coupled lattice have a phase shift (mismatch). This directed motion is a collaboration of the coupling, the substrate potential, and the periodic force, which are all symmetric. The absence of any one of these three factors will not give rise to a directed current. We discuss the complex relations between the directed current and parameters in the system. Results in this paper can be accomplished in experiments. Moreover,our results can be generalized to the studies of directed transport processes in more complicated spatially extended systems.     

Quasi-periodic events on structured earthquake models

There has been much interest in studying quasi-periodic events on earthquake models.Here we investigate quasiperiodic events in the avalanche time series on structured earthquake models by the analysis of the autocorrelation function and the fast Fourier transform.For random spatial earthquake models, quasi-periodic events are robust and we obtain a simple rule for a period that is proportional to the choice of unit time and the dissipation of the system.Moreover, computer simulations validate this rule for two-dimensional lattice models and cycle graphs, but our simulation results also show that small-world models, scale-free models, and random rule graphs do not have periodic phenomena.Although the periodicity of avalanche does not depend on the criticality of the system or the average degree of the system or the size of the system,there is evidence that it depends on the time series of the average force of the system.     

The coulomb potential in a finite slab: Orthorhombic symmetry

The vanishing exponential method is used to calculate the electrostatic potential of a rectangular crystalline slab composed of a finite, regular array of similar point charges, each charge occupying a lattice site of an orthorhombic bravais lattice. The results are used to discuss (a) the behaviour of the potential at and near the slab surface, (b) the potential of an electrically neutral crystal, (c) the relationship between the space averaged microscopic potential of (b) and the macroscopic potential and (d) the limiting case of an electrically neutral infinite slab. It is also proved for this slab system, that the limit in the vanishing exponential method does not in general commute with limit used to define an infinite crystal.     

人工智能技术的教学讨论——以粒子群算法为例

近年来人工智能技术迅速发展,各高校广泛开展了人工智能课程.但对人工智能教学平台缺乏详细的分析.为此,本文以粒子群算法为例对人工智能课程进行了阐述,并讨论了教学注意事项.研究表明,及时预习基础知识有利于学生理解人工智能模型,结合具体问题讨论人工智能算法将有利于学生掌握技术,拓展人工智能技术应用范围并引导学生对算法本身思考将有助于学生建立正确概念,建立互动式解决实验问题将有助于增加学生的学习热情.     

The theoretical analysis of the lattice hydrodynamic models for traffic flow theory

The lattice hydrodynamic model is not only a simplified version of the macroscopic hydrodynamic model, but also connected with the microscopic car following model closely. The modified Korteweg-de Vries (mKdV) equation related to the density wave in a congested traffic region has been derived near the critical point since Nagatani first proposed it. But the Korteweg-de Vries (KdV) equation near the neutral stability line has not been studied, which has been investigated in detail for the car following model. We devote ourselves to obtaining the KdV equation from the original lattice hydrodynamic models and the KdV soliton solution to describe the traffic jam. Especially, we obtain the general soliton solution of the KdV equation and the mKdV equation. We review several lattice hydrodynamic models, which were proposed recently. We compare the modified models and carry out some analysis. Numerical simulations are conducted to demonstrate the nonlinear analysis results.     

Social influencing and associated random walk models: Asymptotic consensus times on the complete graph

We investigate consensus formation and the asymptotic consensus times in stylized individual- or agent-based models, in which global agreement is achieved through pairwise negotiations with or without a bias. Considering a class of individual-based models on finite complete graphs, we introduce a coarse-graining approach (lumping microscopic variables into macrostates) to analyze the ordering dynamics in an associated random-walk framework. Within this framework, yielding a linear system, we derive general equations for the expected consensus time and the expected time spent in each macro-state. Further, we present the asymptotic solutions of the 2-word naming game and separately discuss its behavior under the influence of an external field and with the introduction of committed agents.     

Symmetric vertex models on planar random graphs

We solve a 4-(bond)-vertex model on an ensemble of 3-regular (Φ³) planar random graphs, which has the effect of coupling the vertex model to 2D quantum gravity. The method of solution, by mapping onto an Ising model in field, is inspired by the solution by Wu et.al. of the regular lattice equivalent – a symmetric 8-vertex model on the honeycomb lattice, and also applies to higher valency bond vertex models on random graphs when the vertex weights depend only on bond numbers and not cyclic ordering (the so-called symmetric vertex models).The relations between the vertex weights and Ising model parameters in the 4-vertex model on Φ³ graphs turn out to be identical to those of the honeycomb lattice model, as is the form of the equation of the Ising critical locus for the vertex weights. A symmetry of the partition function under transformations of the vertex weights, which is fundamental to the solution in both cases, can be understood in the random graph case as a change of integration variable in the matrix integral used to define the model.Finally, we note that vertex models, such as that discussed in this paper, may have a role to play in the discretisation of Lorentzian metric quantum gravity in two dimensions.     

Thermodynamic Foundations of Kinetic Theory and Lattice Boltzmann Models for Multiphase Flows

This paper demonstrates that thermodynamically consistent lattice Boltzmann models for single-component multiphase flows can be derived from a kinetic equation using both Enskog's theory for dense fluids and mean-field theory for long-range molecular interaction. The lattice Boltzmann models derived this way satisfy the correct mass, momentum, and energy conservation equations. All the thermodynamic variables in these LBM models are consistent. The strengths and weaknesses of previous lattice Boltzmann multiphase models are analyzed.     

Artificial intelligence and large scale computation: A physics perspective

We study the macroscopic behavior of computation and examine both emergent collective phenomena and dynamical aspects with an emphasis on software issues, which are at the core of large scale distributed computation and artificial intelligence systems. By considering large systems, we exhibit novel phenomena which cannot be foreseen from examination of their smaller counterparts. We review both the symbolic and connectionist views of artificial intelligence, provide a number of examples which display these phenomena, and resort to statistical mechanics, dynamical systems theory and the theory of random graphs to elicit the range of possible behaviors.     

A Security-Enhanced Image Communication Scheme Using Cellular Neural Network

In the current network and big data environment, the secure transmission of digital images is facing huge challenges. The use of some methodologies in artificial intelligence to enhance its security is extremely cutting-edge and also a development trend. To this end, this paper proposes a security-enhanced image communication scheme based on cellular neural network (CNN) under cryptanalysis. First, the complex characteristics of CNN are used to create pseudorandom sequences for image encryption. Then, a plain image is sequentially confused, permuted and diffused to get the cipher image by these CNN-based sequences. Based on cryptanalysis theory, a security-enhanced algorithm structure and relevant steps are detailed. Theoretical analysis and experimental results both demonstrate its safety performance. Moreover, the structure of image cipher can effectively resist various common attacks in cryptography. Therefore, the image communication scheme based on CNN proposed in this paper is a competitive security technology method.     

Nonparametric Causal Structure Learning in High Dimensions

The PC and FCI algorithms are popular constraint-based methods for learning the structure of directed acyclic graphs (DAGs) in the absence and presence of latent and selection variables, respectively. These algorithms (and their order-independent variants, PC-stable and FCI-stable) have been shown to be consistent for learning sparse high-dimensional DAGs based on partial correlations. However, inferring conditional independences from partial correlations is valid if the data are jointly Gaussian or generated from a linear structural equation model—an assumption that may be violated in many applications. To broaden the scope of high-dimensional causal structure learning, we propose nonparametric variants of the PC-stable and FCI-stable algorithms that employ the conditional distance covariance (CdCov) to test for conditional independence relationships. As the key theoretical contribution, we prove that the high-dimensional consistency of the PC-stable and FCI-stable algorithms carry over to general distributions over DAGs when we implement CdCov-based nonparametric tests for conditional independence. Numerical studies demonstrate that our proposed algorithms perform nearly as good as the PC-stable and FCI-stable for Gaussian distributions, and offer advantages in non-Gaussian graphical models.     

On Gammelgaard’s Formula for a Star Product with Separation of Variables

We show that Gammelgaard’s formula expressing a star product with separation of variables on a pseudo-Kähler manifold in terms of directed graphs without cycles is equivalent to an inversion formula for an operator on a formal Fock space. We prove this inversion formula directly and thus offer an alternative approach to Gammelgaard’s formula which gives more insight into the question why the directed graphs in his formula have no cycles.     

Region Graph Partition Function Expansion and Approximate Free Energy Landscapes: Theory and Some Numerical Results

Graphical models for finite-dimensional spin glasses and real-world combinatorial optimization and satisfaction problems usually have an abundant number of short loops. The cluster variation method and its extension, the region graph method, are theoretical approaches for treating the complicated short-loop-induced local correlations. For graphical models represented by non-redundant or redundant region graphs, approximate free energy landscapes are constructed in this paper through the mathematical framework of region graph partition function expansion. Several free energy functionals are obtained, each of which use a set of probability distribution functions or functionals as order parameters. These probability distribution function/functionals are required to satisfy the region graph belief-propagation equation or the region graph survey-propagation equation to ensure vanishing correction contributions of region subgraphs with dangling edges. As a simple application of the general theory, we perform region graph belief-propagation simulations on the square-lattice ferromagnetic Ising model and the Edwards-Anderson model. Considerable improvements over the conventional Bethe-Peierls approximation are achieved. Collective domains of different sizes in the disordered and frustrated square lattice are identified by the message-passing procedure. Such collective domains and the frustrations among them are responsible for the low-temperature glass-like dynamical behaviors of the system.     

Space-time geometry of topological phases

The 2 + 1 dimensional lattice models of Levin and Wen (2005) [1] provide the most general known microscopic construction of topological phases of matter. Based heavily on the mathematical structure of category theory, many of the special properties of these models are not obvious. In the current paper, we present a geometrical space-time picture of the partition function of the Levin-Wen models which can be described as doubles (two copies with opposite chiralities) of underlying anyon theories. Our space-time picture describes the partition function as a knot invariant of a complicated link, where both the lattice variables of the microscopic Levin-Wen model and the terms of the Hamiltonian are represented as labeled strings of this link. This complicated link, previously studied in the mathematical literature, and known as Chain-Mail, can be related directly to known topological invariants of 3-manifolds such as the so-called Turaev-Viro invariant and the Witten-Reshitikhin-Turaev invariant. We further consider quasi-particle excitations of the Levin-Wen models and we see how they can be understood by adding additional strings to the Chain-Mail link representing quasi-particle world-lines. Our construction gives particularly important new insight into how a doubled theory arises from these microscopic models.