Weighted enumeration of spanning subgraphs in locally tree‐like graphs

Using the theory of negative association for measures and the notion of unimodularity for random weak limits of sparse graphs, we establish the validity of the cavity method for counting spanning subgraphs subject to local constraints in asymptotically tree‐like graphs. Specifically, the normalized logarithm of the associated partition function (free energy) is shown to converge along any sequence of graphs whose random weak limit is a tree, and the limit is directly expressed in terms of the unique solution to a limiting cavity equation. On a Galton–Watson tree, the latter simplifies into a recursive distributional equation which can be solved explicitly. As an illustration, we provide a new asymptotic formula for the maximum size of a b‐matching in the Erd?s–Rényi random graph with fixed average degree and diverging size, for any $b\in\mathbb{N}Using the theory of negative association for measures and the notion of unimodularity for random weak limits of sparse graphs, we establish the validity of the cavity method for counting spanning subgraphs subject to local constraints in asymptotically tree‐like graphs. Specifically, the normalized logarithm of the associated partition function (free energy) is shown to converge along any sequence of graphs whose random weak limit is a tree, and the limit is directly expressed in terms of the unique solution to a limiting cavity equation. On a Galton–Watson tree, the latter simplifies into a recursive distributional equation which can be solved explicitly. As an illustration, we provide a new asymptotic formula for the maximum size of a b‐matching in the Erd?s–Rényi random graph with fixed average degree and diverging size, for any $b\in\mathbb{N}$. To the best of our knowledge, this is the first time that correlation inequalities and unimodularity are combined together to yield a general proof of uniqueness of Gibbs measures on infinite trees. We believe that a similar argument is applicable to other Gibbs measures than those over spanning subgraphs considered here. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Intramolecular bonding in a statistical associating fluid theory of ring aggregates

The first-order thermodynamic perturbation theory of Wertheim (TPT1) is extended to treat ring aggregates, formed by inter- and intramolecular association. The expression for the residual association contribution to the Helmholtz free energy for ring aggregates, incorporating the appropriate terms in Wertheim's fundamental graph sum of the TPT1 density expansion, is derived to calculate the distribution of the molecular bonding states. This requires the introduction of two new parameters to characterise each possible ring type: the ring size τ, which is equal to one in the case of intramolecular association, and a parameter W that captures the likelihood of two ring-forming sites bonding. The resulting framework can be incorporated in equations of state that account for the residual association contribution to the free energy, such as the statistical associating fluid theory (SAFT) family, or the cubic plus association (CPA) equation of state. This extends the applicability of these equations of state to mixtures with an arbitrary number of association sites capable of hydrogen bonding to form intramolecular and intermolecular rings. The formalism is implemented within SAFT-VR Mie to calculate the fluid-phase equilibria of model chain-like molecules containing two associating sites A and B, allowing for the formation of open-chain aggregates and intramolecular bonds. The effect of adding a second component that competes for the association sites that mediate intramolecular association in the chain is also examined. Accounting for intramolecular bonding is shown to have a significant impact on the phase equilibria of such systems.     

Mining axiomatic fuzzy set association rules for classification problems

In this paper, we propose a novel method to mine association rules for classification problems namely AFSRC (AFS association rules for classification) realized in the framework of the axiomatic fuzzy set (AFS) theory. This model provides a simple and efficient rule generation mechanism. It can also retain meaningful rules for imbalanced classes by fuzzifying the concept of the class support of a rule. In addition, AFSRC can handle different data types occurring simultaneously. Furthermore, the new model can produce membership functions automatically by processing available data. An extensive suite of experiments are reported which offer a comprehensive comparison of the performance of the method with the performance of some other methods available in the literature. The experimental result shows that AFSRC outperforms most of other methods when being quantified in terms of accuracy and interpretability. AFSRC forms a classifier with high accuracy and more interpretable rule base of smaller size while retaining a sound balance between these two characteristics.     

利用有限域上非零向量的道路图结构与内积构作结合方案

Um（n,R）表示域F上n元非零向量之集．本文先确定了有限域上n≥5维非零向量的道路图结构,然后利用有限域Fq上Um（n,R）／EnR中具有相同范数的向量的道路图结构与内积构作了具有多个结合的结合方案,并计算出相应的参数．     

跨界创新联盟关联信用风险传染模型研究

跨界创新联盟是企业创新过程中资源整合与共享的一种有效形式,但在带来企业创新资源共享效益的同时,联盟中复杂的关联关系也构成了联盟关联信用风险传染的新渠道。本文结合复杂网络理论与传染病模型,根据跨界创新联盟特点,在原有资产关联研究的基础上,综合考虑行业关联、资产关联和创新合作关联三种关联关系,改进了风险传播的传染病模型,并进行了数值仿真实验。结果表明：行业关联、资产关联和创新合作关联对跨界创新联盟关联信用风险传染阈值具有综合性影响,增加行业关联和创新合作关联因素,有利于提高关联信用风险传染阈值的精度。此外,资产关联度越高则感染成员密度越大;行业关联度越低,创新合作关联度越高则感染成员密度越小。     

Association Schemes and Fusion Algebras (An Introduction)

We introduce the concept of fusion algebras at algebraic level, as a purely algebraic concept for the fusion algebras which appear in conformal field theory in mathematical physics. We first discuss the connection between fusion algebras at algebraic level and character algebras, a purely algebraic concept for Bose-Mesner algebras of association schemes. Through this correspondence, we establish the condition when the matrix S of a fusion algebra at algebraic level is unitary or symmetric. We construct integral fusion algebras at algebraic level, from association schemes, in particular from group association schemes, whose matrix S is unitary and symmetric. Finally, we consider whether the modular invariance property is satisfied or not, namely whether there exists a diagonal matrix T satisfying the condition (ST)³ = S ². We prove that this property does not hold for some integral fusion algebras at algebraic level coming from the group association scheme of certain groups of order 64, and we also prove that the (nonintegral) fusion algebra at algebraic level obtained from the Hamming association scheme H(d, q) has the modular invariance property.     

Intriguing sets in partial quadrangles

The point‐line geometry known as a partial quadrangle (introduced by Cameron in 1975) has the property that for every point/line non‐incident pair (P, ?), there is at most one line through P concurrent with ?. So in particular, the well‐studied objects known as generalized quadrangles are each partial quadrangles. An intriguing set of a generalized quadrangle is a set of points which induces an equitable partition of size two of the underlying strongly regular graph. We extend the theory of intriguing sets of generalized quadrangles by Bamberg, Law and Penttila to partial quadrangles, which gives insight into the structure of hemisystems and other intriguing sets of generalized quadrangles. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:217‐245, 2011     

A Framework for Evaluating Knowledge-Based Interestingness of Association Rules

In Knowledge Discovery in Databases (KDD)/Data Mining literature, interestingness measures are used to rank rules according to the interest a particular rule is expected to evoke. In this paper, we introduce an aspect of subjective interestingness called item-relatedness. Relatedness is a consequence of relationships that exist between items in a domain. Association rules containing unrelated or weakly related items are interesting since the co-occurrence of such items is unexpected. Item-Relatedness helps in ranking association rules on the basis of one kind of subjective unexpectedness. We identify three types of item-relatedness – captured in the structure of a fuzzy taxonomy (an extension of the classical concept hierarchy tree). An item-relatedness measure for describing relatedness between two items is developed by combining these three types. Efficacy of this measure is illustrated with the help of a sample taxonomy. We discuss three mechanisms for extending this measure from a two-item set to an association rule consisting of a set of more than two items. These mechanisms utilize the relatedness of item-pairs and other aspects of an association rule, namely its structure, distribution of items and item-pairs. We compare our approach with another method from recent literature.     

Irreducible Representations of Wreath Products of Association Schemes

The wreath product of finite association schemes is a natural generalization of the notion of the wreath product of finite permutation groups. We determine all irreducible representations (the Jacobson radical) of a wreath product of two finite association schemes over an algebraically closed field in terms of the irreducible representations (Jacobson radicals) of the two factors involved.     

Distance regularity of Kerdock codes

A code is called distance regular, if for every two codewords x, y and integers i, j the number of codewords z such that d(x, z) = i and d(y, z) = j, with d the Hamming distance, does not depend on the choice of x, y and depends only on d(x, y) and i, j. Using some properties of the discrete Fourier transform we give a new combinatorial proof of the distance regularity of an arbitrary Kerdock code. We also calculate the parameters of the distance regularity of a Kerdock code.