Determination of multifractal dimensions of complex networks by means of the sandbox algorithm

Complex networks have attracted much attention in diverse areas of science and technology. Multifractal analysis (MFA) is a useful way to systematically describe the spatial heterogeneity of both theoretical and experimental fractal patterns. In this paper, we employ the sandbox (SB) algorithm proposed by T\'{e}l et al. (Physica A, 159 (1989) 155-166), for MFA of complex networks. First we compare the SB algorithm with two existing algorithms of MFA for complex networks: the compact-box-burning (CBB) algorithm proposed by Furuya and Yakubo (Phys. Rev. E, 84 (2011) 036118), and the improved box-counting (BC) algorithm proposed by Li et al. (J. Stat. Mech.: Theor. Exp., 2014 (2014) P02020) by calculating the mass exponents tau(q) of some deterministic model networks. We make a detailed comparison between the numerical and theoretical results of these model networks. The comparison results show that the SB algorithm is the most effective and feasible algorithm to calculate the mass exponents tau(q) and to explore the multifractal behavior of complex networks. Then we apply the SB algorithm to study the multifractal property of some classic model networks, such as scale-free networks, small-world networks, and random networks. Our results show that multifractality exists in scale-free networks, that of small-world networks is not obvious, and it almost does not exist in random networks.Comment: 17 pages, 2 table, 10 figure

Multifractal analysis of weighted networks by a modified sandbox algorithm

Complex networks have attracted growing attention in many fields. As a generalization of fractal analysis, multifractal analysis (MFA) is a useful way to systematically describe the spatial heterogeneity of both theoretical and experimental fractal patterns. Some algorithms for MFA of unweighted complex networks have been proposed in the past a few years, including the sandbox (SB) algorithm recently employed by our group. In this paper, a modified SB algorithm (we call it SBw algorithm) is proposed for MFA of weighted networks.First, we use the SBw algorithm to study the multifractal property of two families of weighted fractal networks (WFNs): "Sierpinski" WFNs and "Cantor dust" WFNs. We also discuss how the fractal dimension and generalized fractal dimensions change with the edge-weights of the WFN. From the comparison between the theoretical and numerical fractal dimensions of these networks, we can find that the proposed SBw algorithm is efficient and feasible for MFA of weighted networks. Then, we apply the SBw algorithm to study multifractal properties of some real weighted networks ---collaboration networks. It is found that the multifractality exists in these weighted networks, and is affected by their edge-weights.Comment: 15 pages, 6 figures. Accepted for publication by Scientific Report

ProtoEM: A Prototype-Enhanced Matching Framework for Event Relation Extraction

Event Relation Extraction (ERE) aims to extract multiple kinds of relations among events in texts. However, existing methods singly categorize event relations as different classes, which are inadequately capturing the intrinsic semantics of these relations. To comprehensively understand their intrinsic semantics, in this paper, we obtain prototype representations for each type of event relation and propose a Prototype-Enhanced Matching (ProtoEM) framework for the joint extraction of multiple kinds of event relations. Specifically, ProtoEM extracts event relations in a two-step manner, i.e., prototype representing and prototype matching. In the first step, to capture the connotations of different event relations, ProtoEM utilizes examples to represent the prototypes corresponding to these relations. Subsequently, to capture the interdependence among event relations, it constructs a dependency graph for the prototypes corresponding to these relations and utilized a Graph Neural Network (GNN)-based module for modeling. In the second step, it obtains the representations of new event pairs and calculates their similarity with those prototypes obtained in the first step to evaluate which types of event relations they belong to. Experimental results on the MAVEN-ERE dataset demonstrate that the proposed ProtoEM framework can effectively represent the prototypes of event relations and further obtain a significant improvement over baseline models.Comment: Work in progres

A well-balanced lattice Boltzmann model for binary fluids based on the incompressible phase-field theory

Spurious velocities arising from the imperfect offset of the undesired term at the discrete level are frequently observed in numerical simulations of equilibrium multiphase flow systems using the lattice Boltzmann equation (LBE) method. To capture the physical equilibrium state of two-phase fluid systems and eliminate spurious velocities, a well-balanced LBE model based on the incompressible phase-field theory is developed. In this model, the equilibrium distribution function for the Cahn-Hilliard (CH) equation is designed by treating the convection term as a source to avoid the introduction of undesired terms, enabling achievement of possible discrete force balance. Furthermore, this approach allows for the attainment of a divergence-free velocity field, effectively mitigating the impact of artificial compression effects and enhancing numerical stability. Numerical tests, including a flat interface problem, a stationary droplet, and the coalescence of two droplets, demonstrate the well-balanced properties and improvements in the stability of the present model

(1-Oxo-2,6,7-trioxa-1-phosphabicyclo­[2.2.2]octan-4-yl)methyl 4-methyl­benzene­sulfonate

In the title compound, C12H15O7PS, the P atom has a distorted tetra­hedral environment. The P—O—C—C torsion angles deviate significantly from zero [average = 12.0 (3)°], indicating that the bicyclic OP(OCH2)3C cage is strained. In the crystal, weak C—H⋯O inter­actions consolidate the packing

A computationally-efficient sandbox algorithm for multifractal analysis of large-scale complex networks with tens of millions of nodes

Multifractal analysis (MFA) is a useful tool to systematically describe the spatial heterogeneity of both theoretical and experimental fractal patterns. One of the widely used methods for fractal analysis is box-covering. It is known to be NP-hard. More severely, in comparison with fractal analysis algorithms, MFA algorithms have much higher computational complexity. Among various MFA algorithms for complex networks, the sandbox MFA algorithm behaves with the best computational efficiency. However, the existing sandbox algorithm is still computationally expensive. It becomes challenging to implement the MFA for large-scale networks with tens of millions of nodes. It is also not clear whether or not MFA results can be improved by a largely increased size of a theoretical network. To tackle these challenges, a computationally-efficient sandbox algorithm (CESA) is presented in this paper for MFA of large-scale networks. Our CESA employs the breadth-first search (BFS) technique to directly search the neighbor nodes of each layer of center nodes, and then to retrieve the required information. Our CESA's input is a sparse data structure derived from the compressed sparse row (CSR) format designed for compressed storage of the adjacency matrix of large-scale network. A theoretical analysis reveals that the CESA reduces the time complexity of the existing sandbox algorithm from cubic to quadratic, and also improves the space complexity from quadratic to linear. MFA experiments are performed for typical complex networks to verify our CESA. Finally, our CESA is applied to a few typical real-world networks of large scale.Comment: 19 pages, 9 figure

An In-Context Schema Understanding Method for Knowledge Base Question Answering

The Knowledge Base Question Answering (KBQA) task aims to answer natural language questions based on a given knowledge base. As a kind of common method for this task, semantic parsing-based ones first convert natural language questions to logical forms (e.g., SPARQL queries) and then execute them on knowledge bases to get answers. Recently, Large Language Models (LLMs) have shown strong abilities in language understanding and may be adopted as semantic parsers in such kinds of methods. However, in doing so, a great challenge for LLMs is to understand the schema of knowledge bases. Therefore, in this paper, we propose an In-Context Schema Understanding (ICSU) method for facilitating LLMs to be used as a semantic parser in KBQA. Specifically, ICSU adopts the In-context Learning mechanism to instruct LLMs to generate SPARQL queries with examples. In order to retrieve appropriate examples from annotated question-query pairs, which contain comprehensive schema information related to questions, ICSU explores four different retrieval strategies. Experimental results on the largest KBQA benchmark, KQA Pro, show that ICSU with all these strategies outperforms that with a random retrieval strategy significantly (from 12\% to 78.76\% in accuracy)