Mean Hitting Time for Random Walks on a Class of Sparse Networks

For random walks on a complex network, the configuration of a network that provides optimal or suboptimal navigation efficiency is meaningful research. It has been proven that a complete graph has the exact minimal mean hitting time, which grows linearly with the network order. In this paper, we present a class of sparse networks G(t) in view of a graphic operation, which have a similar dynamic process with the complete graph; however, their topological properties are different. We capture that G(t) has a remarkable scale-free nature that exists in most real networks and give the recursive relations of several related matrices for the studied network. According to the connections between random walks and electrical networks, three types of graph invariants are calculated, including regular Kirchhoff index, M-Kirchhoff index and A-Kirchhoff index. We derive the closed-form solutions for the mean hitting time of G(t), and our results show that the dominant scaling of which exhibits the same behavior as that of a complete graph. The result could be considered when designing networks with high navigation efficiency.

Thumbs up and down: The cultural technique of thumb-typing

This paper explores thumb-typing as a cultural technique stemming from the mutual development of typing interfaces and practices. Focusing on the work of the typing fingers, it examines how the assignment of thumbs to be the primary writing digits is an innovation that correlates—and in some respects causes—textual and social changes that are central to digital culture. It argues that thumb-typing embodies recursive relations between behavioral patterns, technological infrastructure, and textual creation. The analysis shows how the invention of the typewriter keyboard introduced the fingers to typing, and how developments of digital media refined the finger-work in interacting with the device, resulting in thumb-typing. The new functionality of the thumb as an executing rather than supporting finger, promotes a novel equivalency and interchangeability in finger employment to typing. This, I propose, problematizes traditional concepts of textuality, its performance, and authorship.

On Certain Properties and Applications of the Perturbed Meixner–Pollaczek Weight

This paper deals with monic orthogonal polynomials orthogonal with a perturbation of classical Meixner–Pollaczek measure. These polynomials, called Perturbed Meixner–Pollaczek polynomials, are described by their weight function emanating from an exponential deformation of the classical Meixner–Pollaczek measure. In this contribution, we investigate certain properties such as moments of finite order, some new recursive relations, concise formulations, differential-recurrence relations, integral representation and some properties of the zeros (quasi-orthogonality, monotonicity and convexity of the extreme zeros) of the corresponding perturbed polynomials. Some auxiliary results for Meixner–Pollaczek polynomials are revisited. Some applications such as Fisher’s information, Toda-type relations associated with these polynomials, Gauss–Meixner–Pollaczek quadrature as well as their role in quantum oscillators are also reproduced.

An Incremental and Backward-Conflict Guided Method for Unfolding Petri Nets

The unfolding technique of Petri net can characterize the real concurrency and alleviate the state space explosion problem. Thus, it is greatly suitable to analyze/check some potential errors in concurrent systems. During the unfolding process of a Petri net, the calculations of configurations, cuts, and cut-off events are the key factors for the unfolding efficiency. However, most of the unfolding methods do not specify a highly efficient calculations on them. In this paper, we reveal some recursive relations and structural properties of these factors. Subsequently, we propose an improved method for computing configurations and cuts. Meanwhile, backward conflicts are used to guide the calculations of cut-off events. Moreover, a case study and a series of experiments are done to illustrate the effectiveness and application scenarios of our methods.

A user’s guide to basic knot and link theory

We define simple invariants of knots or links (linking number, Arf-Casson invariants and Alexander-Conway polynomials) motivated by interesting results whose statements are accessible to a non-specialist or a student. The simplest invariants naturally appear in an attempt to unknot a knot or unlink a link. Then we present certain ‘skein’ recursive relations for the simplest invariants, which allow us to introduce stronger invariants. We state the Vassiliev–Kontsevich theorem in a way convenient for calculating the invariants themselves, not only the dimension of the space of the invariants. No prerequisites are required; we give rigorous definitions of the main notions in a way not obstructing intuitive understanding.

Real-World Networks Are Not Always Fast Mixing

The mixing time of random walks on a graph has found broad applications across both theoretical and practical aspects of computer science, with the application effects depending on the behavior of mixing time. It is extensively believed that real-world networks, especially social networks, are fast mixing with their mixing time at most $O(\log N)$ where $N$ is the number of vertices. However, the behavior of mixing time in the real-life networks has not been examined carefully, and exactly analytical research for mixing time in models mimicking real networks is still lacking. In this paper, we first experimentally evaluate the mixing time of various real-world networks with scale-free small-world properties and show that their mixing time is much higher than anticipated. To better understand the behavior of the mixing time for real-world networks, we then analytically study the mixing time of the Apollonian network, which is simultaneously scale-free and small-world. To this end, we derive the recursive relations for all eigenvalues, especially the second largest eigenvalue modulus of the transition matrix, based on which we deduce a lower bound for the mixing time of the Apollonian network, which approximately scales sublinearly with $N$. Our results indicate that real-world networks are not always fast mixing, which has potential implications in the design of algorithms related to mixing time.

Certain Laguerre-based Generalized Apostol Type Polynomials

A variety of polynomials, their extensions and variants have been extensively investigated, due mainly to their potential applications in diverse research areas. In this paper, we aim to introduce Laguerre-based generalized Apostol type polynomials and investigate some properties and identities involving them. Among them, some differential-recursive relations for the Hermite-Laguerre polynomials, which are expressed in terms of generalized Apostol type numbers and the Laguerre-based generalized Apostol type polynomials, an implicit summation formula and addition-symmetry identities for the Laguerre-based generalized Apostol type polynomials are presented. The results presented here, being very general, are pointed out to reduce to yield some known or new formulas and identities for relatively simple polynomials and numbers.

The Reliability of a System Involving Change Points

The reliability of a system having some change points is presented. The technique of calculation is based on a previously developed TFCF procedure for evaluating the reliability for i.i.d. component. It involves the use of some auxiliary functions to set up a set of recursive relations. The resultant equations are solved numerically. An extension to the more general TSCSTFCF procedure and its application to start-up demonstration tests is given. Also, in case of testing, the possibility of carrying out simultaneous tests on a set of units is considered.