Efficient computation of target-oriented link criticalness centrality in uncertain graphs

We challenge the problem of efficiently identifying critical links that substantially degrade network performance if they do not function under a realistic situation where each link is probabilistically disconnected, e.g., unexpected traffic accident in a road network and unexpected server down in a communication network. To solve this problem, we utilize the bridge detection technique in graph theory and efficiently identify critical links in case the node reachability is taken as the performance measure.To be more precise, we define a set of target nodes and a new measure associated with it, Target-oriented latent link Criticalness Centrality (TCC), which is defined as the marginal loss of the expected number of nodes in the network that can reach, or equivalently can be reached from, one of the target nodes, and compute TCC for each link by use of detected bridges. We apply the proposed method to two real-world networks, one from social network and the other from spatial network, and empirically show that the proposed method has a good scalability with respect to the network size and the links our method identified possess unique properties. They are substantially more critical than those obtained by the others, and no known measures can replace the TCC measure.

Interior polynomial for signed bipartite graphs and the HOMFLY polynomial

The interior polynomial is a Tutte-type invariant of bipartite graphs, and a part of the HOMFLY polynomial of a special alternating link coincides with the interior polynomial of the Seifert graph of the link. We extend the interior polynomial to signed bipartite graphs, and we show that, in the planar case, it is equal to the maximal [Formula: see text]-degree part of the HOMFLY polynomial of a naturally associated link. Note that the latter can be any oriented link. This result fits into a program aimed at deriving the HOMFLY polynomial from Floer homology. We also establish some other, more basic properties of the signed interior polynomial. For example, the HOMFLY polynomial of the mirror image of [Formula: see text] is given by [Formula: see text]. This implies a mirroring formula for the signed interior polynomial in the planar case. We prove that the same property holds for any bipartite graph and the same graph with all signs reversed. The proof relies on Ehrhart reciprocity applied to the so-called root polytope. We also establish formulas for the signed interior polynomial inspired by the knot theoretical notions of flyping and mutation. This leads to new identities for the original unsigned interior polynomial.

An NS-3 Implementation and Experimental Performance Analysis of IEEE 802.15.6 Standard under Different Deployment Scenarios

Various simulation studies for wireless body area networks (WBANs) based on the IEEE 802.15.6 standard have recently been carried out. However, most of these studies have applied a simplified model without using any major components specific to IEEE 802.15.6, such as connection-oriented link allocations, inter-WBAN interference mitigation, or a two-hop star topology extension. Thus, such deficiencies can lead to an inaccurate performance analysis. To solve these problems, in this study, we conducted a comprehensive review of the major components of the IEEE 802.15.6 standard and herein present modeling strategies for implementing IEEE 802.15.6 MAC on an NS-3 simulator. In addition, we configured realistic network scenarios for a performance evaluation in terms of throughput, average delay, and power consumption. The simulation results prove that our simulation system provides acceptable levels of performance for various types of medical applications, and can support the latest research topics regarding the dynamic resource allocation, inter-WBAN interference mitigation, and intra-WBAN routing.

Knot invariants from Laplacian matrices

A checkerboard graph of a special diagram of an oriented link is made a directed, edge-weighted graph in a natural way so that a principal submatrix of its Laplacian matrix is a Seifert matrix of the link. Doubling and weighting the edges of the graph produces a second Laplacian matrix such that a principal submatrix is an Alexander matrix of the link. The Goeritz matrix and signature invariants are obtained in a similar way. A device introduced by Kauffman makes it possible to apply the method to general diagrams.

New presentations of a link and virtual link

An embedding presentation of a diagram is introduced, which has proved to be a unique presentation of a diagram. Let [Formula: see text] be a set of all diagrams, called also links in this paper. An algebraic system [Formula: see text] is constructed. In fact, a link in [Formula: see text] (or [Formula: see text]) is the equivalent class [Formula: see text] where [Formula: see text] is one of its embedding presentations. Based on [Formula: see text], Reduction Crossing Algorithm is proposed which is used to reduce the number of crossings in an embedding presentation by introducing a main tool called a pass replacement. For an infinite set of unknots [Formula: see text], each [Formula: see text] in [Formula: see text] can be transformed into the trivial unknot in at most [Formula: see text] by applying the algorithm where [Formula: see text] is a constant, [Formula: see text] and [Formula: see text]. As special consequences, three unknots are unknotted, which are Goeritz’s unknot, Thistlethwaite’s unknot and Haken’s unknot (image courtesy of Cameron Gordon). Moreover, an infinite family of unknots [Formula: see text] are unknotted in [Formula: see text] time. In addition, unique presentations of a virtual link, an oriented link and oriented virtual link are introduced, respectively.

The self-smoothing number of knots and links

We call smoothing a self-crossing point of an oriented link diagram self-smoothing. By self-smoothing repeatedly, we obtain an oriented link diagram without self-crossing points. In this paper, we show that every knot has an oriented diagram which becomes a two-component oriented link diagram without self-crossing points by a single self-smoothing.

Identifying the Invariants for Classical Knots and Links from the Yokonuma–Hecke Algebras

We announce the existence of a family of new 2-variable polynomial invariants for oriented classical links defined via a Markov trace on the Yokonuma–Hecke algebra of type A. Yokonuma–Hecke algebras are generalizations of Iwahori–Hecke algebras, and this family contains the HOMFLYPT polynomial, the famous 2-variable invariant for classical links arising from the Iwahori–Hecke algebra of type A. We show that these invariants are topologically equivalent to the HOMFLYPT polynomial on knots, but not on links, by providing pairs of HOMFLYPT-equivalent links that are distinguished by our invariants. In order to do this, we prove that our invariants can be defined diagrammatically via a special skein relation involving only crossings between different components. We further generalize this family of invariants to a new 3-variable skein link invariant that is stronger than the HOMFLYPT polynomial. Finally, we present a closed formula for this invariant, by W. B. R. Lickorish, that uses HOMFLYPT polynomials of sublinks and linking numbers of a given oriented link.