scholarly journals A dynamical systems view of network centrality

2014 ◽  
Vol 470 (2165) ◽  
pp. 20130835 ◽  
Author(s):  
Peter Grindrod ◽  
Desmond J. Higham
Keyword(s):  
Dynamical Systems ◽  
Large Scale ◽  
Dynamic Networks ◽  
Network Evolution ◽  
Mathematical Framework ◽  
The Novel ◽  
Matrix Logarithm ◽  
The Matrix ◽  
Systems View ◽  
Logarithm Function

To gain insights about dynamic networks, the dominant paradigm is to study discrete snapshots , or timeslices , as the interactions evolve. Here, we develop and test a new mathematical framework where network evolution is handled over continuous time, giving an elegant dynamical systems representation for the important concept of node centrality. The resulting system allows us to track the relative influence of each individual. This new setting is natural in many digital applications, offering both conceptual and computational advantages. The novel differential equations approach is convenient for modelling and analysis of network evolution and gives rise to an interesting application of the matrix logarithm function. From a computational perspective, it avoids the awkward up-front compromises between accuracy, efficiency and redundancy required in the prevalent discrete-time setting. Instead, we can rely on state-of-the-art ODE software, where discretization takes place adaptively in response to the prevailing system dynamics. The new centrality system generalizes the widely used Katz measure, and allows us to identify and track, at any resolution, the most influential nodes in terms of broadcasting and receiving information through time-dependent links. In addition to the classical static network notion of attenuation across edges, the new ODE also allows for attenuation over time, as information becomes stale. This allows ‘running measures’ to be computed, so that networks can be monitored in real time over arbitrarily long intervals. With regard to computational efficiency, we explain why it is cheaper to track good receivers of information than good broadcasters. An important consequence is that the overall broadcast activity in the network can also be monitored efficiently. We use two synthetic examples to validate the relevance of the new measures. We then illustrate the ideas on a large-scale voice call network, where key features are discovered that are not evident from snapshots or aggregates.

scholarly journals On Randomized Trace Estimates for Indefinite Matrices with an Application to Determinants

2021 ◽  
Author(s):  
Alice Cortinovis ◽  
Daniel Kressner
Keyword(s):  
Large Scale ◽  
Gaussian Process Regression ◽  
Likelihood Estimation ◽  
Random Vectors ◽  
Symmetric Positive Definite ◽  
Matrix Logarithm ◽  
The Matrix ◽  
Trace Estimation ◽  
Scale Matrix ◽  
Existing Result

AbstractRandomized trace estimation is a popular and well-studied technique that approximates the trace of a large-scale matrix B by computing the average of $$x^T Bx$$ x T B x for many samples of a random vector X. Often, B is symmetric positive definite (SPD) but a number of applications give rise to indefinite B. Most notably, this is the case for log-determinant estimation, a task that features prominently in statistical learning, for instance in maximum likelihood estimation for Gaussian process regression. The analysis of randomized trace estimates, including tail bounds, has mostly focused on the SPD case. In this work, we derive new tail bounds for randomized trace estimates applied to indefinite B with Rademacher or Gaussian random vectors. These bounds significantly improve existing results for indefinite B, reducing the number of required samples by a factor n or even more, where n is the size of B. Even for an SPD matrix, our work improves an existing result by Roosta-Khorasani and Ascher (Found Comput Math, 15(5):1187–1212, 2015) for Rademacher vectors. This work also analyzes the combination of randomized trace estimates with the Lanczos method for approximating the trace of f(B). Particular attention is paid to the matrix logarithm, which is needed for log-determinant estimation. We improve and extend an existing result, to not only cover Rademacher but also Gaussian random vectors.

scholarly journals The Novel Application of Asphalt in Cross-Linked Polyethylene Composite

2021 ◽  
Author(s):  
Yi Liao ◽  
Shuangxin Lai ◽  
Shuangqiao Yang ◽  
Jinjing Liu ◽  
shibing bai
Keyword(s):  
Solid State ◽  
Large Scale ◽  
Polymer Processing ◽  
Basic Material ◽  
The Novel ◽  
Elongation At Break ◽  
Polyethylene Composite ◽  
The Matrix ◽  
The Cross ◽  
Properties Of Composites

Abstract The cross-linked polyethylene (XLPE) cables are widely used as an important basic material for power transportation. However, the cross-linked network structure caused challenge in the recycling of waste XLPE, which usually be treated by incineration and landfilling. In this research, XLPE was de-crosslinked via solid-state shear milling (S3M) and the asphalt was used as plasticizer during the thermoplastic processing. To deeply understand the influence of asphalt on the matrix, the compatibility, dispersion, and rheological properties of composites were characterized. Due to the good compatibility between de-crosslinked XLPE and asphalt, the viscosity of the composites decreased significantly. Some sea-island structure also formed in composites, which increased the toughness of the composites and the elongation at break reach as high as 322%. The use of asphalt to achieve the processing performance of de-crosslinked XLPE powder was highly effective and composites can be processed by ordinary polymer processing equipment. Furthermore, the prepared composites show potential application in the field of waterproofing which could recycle of waste XLPE cables in large-scale.

Transformations of ray transferences of optical systems to augmented Hamiltonian matrices and the problem of the average system

2007 ◽  
Vol 66 (2) ◽  
Author(s):  
J. R. Cardoso
Keyword(s):  
Optical System ◽  
Optical Systems ◽  
Hamiltonian Matrix ◽  
First Order ◽  
Matrix Logarithm ◽  
The Matrix ◽  
Average System ◽  
Quantitative Analyses ◽  
Logarithm Function

The first-order optical nature of an optical system (including an eye) is completely characterized by a 55 ×  matrix called the ray transference.  It is known  that the image of a ray transference by the matrix logarithm function is an augmented Hamiltonian matrix.  It turns out that there are other ways of transforming transferences into augmented Hamiltonian matrices.  They include Cayley transforms and modified Cayley transforms.  This paper will describe these transforms with a view to finding the most suitable one for quantitative analyses of eyes and other systems in augmented Hamiltonian spaces.  In particular we look at the calculation of average systems.

Covalent Atomic Bridges Enable Unidirectional Enhancement of Electronic Transport in Aligned Carbon Nanotubes

2019 ◽  
Author(s):  
Mingguang Chen ◽  
Wangxiang Li ◽  
Anshuman Kumar ◽  
Guanghui Li ◽  
Mikhail Itkis ◽  
...  
Keyword(s):  
Carbon Nanotubes ◽  
Electrical Transport ◽  
Large Scale ◽  
The Novel ◽  
Electrical Measurements ◽  
Metal Atoms ◽  
Aligned Carbon Nanotubes ◽  
Transport Channels ◽  
Walled Carbon Nanotubes ◽  
Bulk Structures

<p>Interconnecting the surfaces of nanomaterials without compromising their outstanding mechanical, thermal, and electronic properties is critical in the design of advanced bulk structures that still preserve the novel properties of their nanoscale constituents. As such, bridging the p-conjugated carbon surfaces of single-walled carbon nanotubes (SWNTs) has special implications in next-generation electronics. This study presents a rational path towards improvement of the electrical transport in aligned semiconducting SWNT films by deposition of metal atoms. The formation of conducting Cr-mediated pathways between the parallel SWNTs increases the transverse (intertube) conductance, while having negligible effect on the parallel (intratube) transport. In contrast, doping with Li has a predominant effect on the intratube electrical transport of aligned SWNT films. Large-scale first-principles calculations of electrical transport on aligned SWNTs show good agreement with the experimental electrical measurements and provide insight into the changes that different metal atoms exert on the density of states near the Fermi level of the SWNTs and the formation of transport channels. </p>

Routing in Large-scale Dynamic Networks

2020 ◽  
Vol 20 (4) ◽  
pp. 1-24
Author(s):  
Weichao Gao ◽  
James Nguyen ◽  
Yalong Wu ◽  
William G. Hatcher ◽  
Wei Yu
Keyword(s):  
Large Scale ◽  
Dynamic Networks

scholarly journals Controllable and decomposable multidirectional synchronizations

2021 ◽  
Author(s):  
Gábor Bergmann
Keyword(s):  
Systems Engineering ◽  
Large Scale ◽  
Mathematical Framework ◽  
Collaborative Systems ◽  
Consistency Constraints ◽  
Restoration Mechanism ◽  
Common Goals ◽  
Access Privileges ◽  
Novel Concept ◽  
Engineering Projects

AbstractStudying large-scale collaborative systems engineering projects across teams with differing intellectual property clearances, or healthcare solutions where sensitive patient data needs to be partially shared, or similar multi-user information systems over databases, all boils down to a common mathematical framework. Updateable views (lenses) and more generally bidirectional transformations are abstractions to study the challenge of exchanging information between participants with different read access privileges. The view provided to each participant must be different due to access control or other limitations, yet also consistent in a certain sense, to enable collaboration towards common goals. A collaboration system must apply bidirectional synchronization to ensure that after a participant modifies their view, the views of other participants are updated so that they are consistent again. While bidirectional transformations (synchronizations) have been extensively studied, there are new challenges that are unique to the multidirectional case. If complex consistency constraints have to be maintained, synchronizations that work fine in isolation may not compose well. We demonstrate and characterize a failure mode of the emergent behaviour, where a consistency restoration mechanism undoes the work of other participants. On the other end of the spectrum, we study the case where synchronizations work especially well together: we characterize very well-behaved multidirectional transformations, a non-trivial generalization from the bidirectional case. For the former challenge, we introduce a novel concept of controllability, while for the latter one, we propose a novel formal notion of faithful decomposition. Additionally, the paper proposes several novel properties of multidirectional transformations.

Sign in / Sign up

Export Citation Format

Share Document