Abstract As a structural topological index, the number of subtrees has great significance for the analysis and design of hybrid locally reliable networks. In this paper, with generating function and introducing a novel two-forest dual transformation technique, we solve the subtree enumerating problems of two representatives of the self-similar networks, such as the hierarchical lattice and $(u,v)$-flower networks. Moreover, by means of the circle weight transfer technique, two linear time algorithms of computing the subtree generation functions of these two families of networks are also proposed. The subtree density of two special cases for these self-similar networks is briefly discussed as an application.

Abstract The subtree number index of a graph, defined as the number of subtrees, attracts much attention recently. Finding a proper algorithm to compute this index is an important but difficult problem for a general graph. Even for unicyclic and bicyclic graphs, it is not completely trivial, though it can be figured out by try and error. However, it is complicated for tricyclic graphs. This paper proposes path contraction carrying weights (PCCWs) algorithms to compute the subtree number index for the nontrivial case of bicyclic graphs and all 15 cases of tricyclic graphs, based on three techniques: PCCWs, generating function and structural decomposition. Our approach provides a foundation and useful methods to compute subtree number index for graphs with more complicated cycle structures and can be applied to investigate the novel structural property of some important nanomaterials such as the pentagonal carbon nanocone.

Detecting self-organized coalitions from functional networks is one of the most important ways to uncover functional mechanisms in the brain. Determining these raises well-known technical challenges in terms of scale imbalance, outliers and hard-examples. In this article, we propose a novel self-adaptive skeleton approach to detect coalitions through an approximation method based on probabilistic mixture models. The nodes in the networks are characterized in terms of robust k -order complete subgraphs ( k -clique ) as essential substructures. The k -clique enumeration algorithm quickly enumerates all k -cliques in a parallel manner for a given network. Then, the cliques, from max -clique down to min -clique, of each order k , are hierarchically embedded into a probabilistic mixture model. They are self-adapted to the corresponding structure density of coalitions in the brain functional networks through different order k . All the cliques are merged and evolved into robust skeletons to sustain each unbalanced coalition by eliminating outliers and separating overlaps. We call this the k -CLIque Merging Evolution (CLIME) algorithm. The experimental results illustrate that the proposed approaches are robust to density variation and coalition mixture and can enable the effective detection of coalitions from real brain functional networks. There exist potential cognitive functional relations between the regions of interest in the coalitions revealed by our methods, which suggests the approach can be usefully applied in neuroscientific studies.

Mining the relationship structures among the investors plays a vital role in promoting economic development as well as preventing financial risks, especially in the context of big data. This article proposes fast networking approaches from investment big data to explore three underlying structures, namely, investment pedigrees, investment groups, and structural holes. Inspired by disjoint sets and path compression, we first present a pedigree classification algorithm to identify investment pedigrees. Second, through introducing a pruning strategy and a data structure termed as “2-tuple list,” we develop a novel linear-time structure mining algorithm in network (SMAN) for investigating investment groups and structural holes from the investment pedigree. Finally, we show that our SMAN has higher clustering accuracy and efficiency than other existing algorithms on a variety of real-world tasks in terms of normalized mutual information (NMI) values. Our method is particularly well suited for mining the underlying structures from investment big data.

Sequence transfer learning is of interest in both academia and industry with the emergence of numerous new text domains from Twitter and other social media tools. In this paper, we put forward the data-sensitive granularity for transfer learning, and then, a novel substructural regularization transfer learning model (STLM) is proposed to preserve target domain features at substructural granularity in the light of the condition of labeled data set size. Our model is underpinned by hidden Markov model and regularization theory, where the substructural representation can be integrated as a penalty after measuring the dissimilarity of substructures between target domain and STLM with relative entropy. STLM can achieve the competing goals of preserving the target domain substructure and utilizing the observations from both the target and source domains simultaneously. The estimation of STLM is very efficient since an analytical solution can be derived as a necessary and sufficient condition. The relative usability of substructures to act as regularization parameters and the time complexity of STLM are also analyzed and discussed. Comprehensive experiments of part-of-speech tagging with both Brown and Twitter corpora fully justify that our model can make improvements on all the combinations of source and target domains.

As one of the counting-based topological indices, the number of subtrees and its variations has received much attention in recent years. In this paper, using generating functions, we investigate and derive formulas for this index of hexagonal and phenylene chains. We also present graph-theoretical algorithms for enumerating subtrees of these two chains. Extremal values and graphs with respect to the subtree number among all hexagonal and phenylene chains with n hexagons are also determined. As an application, we briefly examine the subtree densities of these two chains.