A Preconditioned Variant of the Refined Arnoldi Method for Computing PageRank Eigenvectors

The PageRank model computes the stationary distribution of a Markov random walk on the linking structure of a network, and it uses the values within to represent the importance or centrality of each node. This model is first proposed by Google for ranking web pages, then it is widely applied as a centrality measure for networks arising in various fields such as in chemistry, bioinformatics, neuroscience and social networks. For example, it can measure the node centralities of the gene-gene annotation network to evaluate the relevance of each gene with a certain disease. The networks in some fields including bioinformatics are undirected, thus the corresponding adjacency matrices are symmetry. Mathematically, the PageRank model can be stated as finding the unit positive eigenvector corresponding to the largest eigenvalue of a transition matrix built upon the linking structure. With rapid development of science and technology, the networks in real applications become larger and larger, thus the PageRank model always desires numerical algorithms with reduced algorithmic or memory complexity. In this paper, we propose a novel preconditioning approach for solving the PageRank model. This approach transforms the original PageRank eigen-problem into a new one that is more amenable to solve. We then present a preconditioned version of the refined Arnoldi method for solving this model. We demonstrate theoretically that the preconditioned Arnoldi method has higher execution efficiency and parallelism than the refined Arnoldi method. In plenty of numerical experiments, this preconditioned method exhibits noticeably faster convergence speed over its standard counterpart, especially for difficult cases with large damping factors. Besides, this superiority maintains when this technique is applied to other variants of the refined Arnoldi method. Overall, the proposed technique can give the PageRank model a faster solving process, and this will possibly improve the efficiency of researches, engineering projects and services where this model is applied.

Temporal oscillations in preference strength provide evidence for an open system model of constructed preference

The decision process is often conceptualized as a constructive process in which a decision maker accumulates information to form preferences about the choice options and ultimately make a response. Here we examine how these constructive processes unfold by tracking dynamic changes in preference strength. Across two experiments, we observed that mean preference strength systematically oscillated over time and found that eliciting a choice early in time strongly affected the pattern of preference oscillation later in time. Preferences following choices oscillated between being stronger than those without prior choice and being weaker than those without choice. To account for these phenomena, we develop an open system dynamic model which merges the dynamics of Markov random walk processes with those of quantum walk processes. This model incorporates two sources of uncertainty: epistemic uncertainty about what preference state a decision maker has at a particular point in time; and ontic uncertainty about what decision or judgment will be observed when a person has some preference state. Representing these two sources of uncertainty allows the model to account for the oscillations in preference as well as the effect of choice on preference formation.

Unsupervised Predominant Sense Detection and Its Application to Text Classification

This paper focuses on the domain-specific senses of words and proposes a method for detecting predominant sense depending on each domain. Our Domain-Specific Senses (DSS) model is an unsupervised manner and detects predominant senses in each domain. We apply a simple Markov Random Walk (MRW) model to ranking senses for each domain. It decides the importance of a sense within a graph by using the similarity of senses. The similarity of senses is obtained by using distributional representations of words from gloss texts in the thesaurus. It can capture large semantic context and thus does not require manual annotation of sense-tagged data. We used the Reuters corpus and the WordNet in the experiments. We applied the results of domain-specific senses to text classification and examined how DSS affects the overall performance of the text classification task. We compared our DSS model with one of the word sense disambiguation techniques (WSD), Context2vec, and the results demonstrate our domain-specific sense approach gains 0.053 F1 improvement on average over the WSD approach.

Distribution of the Lower Boundary Functional of the Step Process of Semi-Markov Random Walk with Delaying Barrier

In this paper, a step process of semi-Markovian random walk with delaying barrier on the zero-level is constructed and the Laplace transformation of the distribution of first crossing time of this process into the delaying barrier is obtained. Also, the expectation and standard diversion of a boundary functional of the process are given.

Ladder epochs and ladder chain of a Markov random walk with discrete driving chain

Let (Mn,Sn)n≥0 be a Markov random walk with positive recurrent driving chain (Mn)n≥0 on the countable state space 𝒮 with stationary distribution π. Suppose also that lim supn→∞Sn=∞ almost surely, so that the walk has almost-sure finite strictly ascending ladder epochs σn>. Recurrence properties of the ladder chain (Mσn>)n≥0 and a closely related excursion chain are studied. We give a necessary and sufficient condition for the recurrence of (Mσn>)n≥0 and further show that this chain is positive recurrent with stationary distribution π> and 𝔼π>σ1><∞ if and only if an associated Markov random walk (𝑀̂n,𝑆̂n)n≥0, obtained by time reversal and called the dual of (Mn,Sn)n≥0, is positive divergent, i.e. 𝑆̂n→∞ almost surely. Simple expressions for π> are also provided. Our arguments make use of coupling, Palm duality theory, and Wiener‒Hopf factorization for Markov random walks with discrete driving chain.