scholarly journals A Review of and Some Results for Ollivier–Ricci Network Curvature

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1416
Author(s):  
Nazanin Azarhooshang ◽  
Prithviraj Sengupta ◽  
Bhaskar DasGupta
Keyword(s):  
Ricci Curvature ◽  
Network Science ◽  
Topological Spaces ◽  
Topological Properties ◽  
Approximate Computation ◽  
Curvature Measure ◽  
Higher Dimensional ◽  
Science Domain ◽  
And Mathematics

Characterizing topological properties and anomalous behaviors of higher-dimensional topological spaces via notions of curvatures is by now quite common in mainstream physics and mathematics, and it is therefore natural to try to extend these notions from the non-network domains in a suitable way to the network science domain. In this article we discuss one such extension, namely Ollivier’s discretization of Ricci curvature. We first motivate, define and illustrate the Ollivier–Ricci Curvature. In the next section we provide some “not-previously-published” bounds on the exact and approximate computation of the curvature measure. In the penultimate section we review a method based on the linear sketching technique for efficient approximate computation of the Ollivier–Ricci network curvature. Finally in the last section we provide concluding remarks with pointers for further reading.

scholarly journals Orderability of topological spaces by continuous preferences

2001 ◽  
Vol 27 (8) ◽  
pp. 505-512 ◽  
Author(s):  
José Carlos Rodríguez Alcantud
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Topological Properties ◽  
Linear Orders ◽  
Mathematical Economics

We extend van Dalen and Wattel's (1973) characterization of orderable spaces and their subspaces by obtaining analogous results for two larger classes of topological spaces. This type of spaces are defined by considering preferences instead of linear orders in the former definitions, and possess topological properties similar to those of (totally) orderable spaces (cf. Alcantud, 1999). Our study provides particular consequences of relevance in mathematical economics; in particular, a condition equivalent to the existence of a continuous preference on a topological space is obtained.

TWO SPACES OF MINIMAL PRIMES

2012 ◽  
Vol 11 (01) ◽  
pp. 1250014 ◽  
Author(s):  
PAPIYA BHATTACHARJEE
Keyword(s):  
Compact Space ◽  
Hausdorff Space ◽  
Topological Spaces ◽  
Zariski Topology ◽  
Topological Properties ◽  
Locally Compact ◽  
Inverse Topology ◽  
Minimal Prime ◽  
Stone Spaces ◽  
Prime Elements

This paper studies algebraic frames L and the set Min (L) of minimal prime elements of L. We will endow the set Min (L) with two well-known topologies, known as the Hull-kernel (or Zariski) topology and the inverse topology, and discuss several properties of these two spaces. It will be shown that Min (L) endowed with the Hull-kernel topology is a zero-dimensional, Hausdorff space; whereas, Min (L) endowed with the inverse topology is a T1, compact space. The main goal will be to find conditions on L for the spaces Min (L) and Min (L)-1 to have various topological properties; for example, compact, locally compact, Hausdorff, zero-dimensional, and extremally disconnected. We will also discuss when the two topological spaces are Boolean and Stone spaces.

The sequential topology on is not regular

2009 ◽  
Vol 19 (5) ◽  
pp. 943-957 ◽  
Author(s):  
MATTHIAS SCHRÖDER
Keyword(s):  
Open Subset ◽  
Open Problem ◽  
Polish Space ◽  
Topological Spaces ◽  
Computable Analysis ◽  
Topological Properties ◽  
Compact Open Topology ◽  
Cartesian Closed Categories ◽  
Cartesian Closed ◽  
Continuous Functionals

The compact-open topology on the set of continuous functionals from the Baire space to the natural numbers is well known to be zero-dimensional. We prove that the closely related sequential topology on this set is not even regular. The sequential topology arises naturally as the topology carried by the exponential formed in various cartesian closed categories of topological spaces. Moreover, we give an example of an effectively open subset of that violates regularity. The topological properties of are known to be closely related to an open problem in Computable Analysis. We also show that the sequential topology on the space of continuous real-valued functions on a Polish space need not be regular.

scholarly journals SOFT α-COMPACTNESS VIA SOFT IDEALS

2016 ◽  
Vol 12 (4) ◽  
pp. 6178-6184 ◽  
Author(s):  
A A Nasef ◽  
A E Radwan ◽  
F A Ibrahem ◽  
R B Esmaeel
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Topological Properties ◽  
Soft Topological Space

In the present paper, we have continued to study the properties of soft topological spaces. We introduce new types of soft compactness based on the soft ideal Ĩ in a soft topological space (X, τ, E) namely, soft αI-compactness, soft αI-Ĩ-compactness, soft α-Ĩ-compactness, soft α-closed, soft αI-closed, soft countably α-Ĩ-compactness and soft countably αI-Ĩ-compactness. Also, several of their topological properties are investigated. The behavior of these concepts under various types of soft functions has obtained

scholarly journals On Soft Separation Axioms and Their Applications on Decision-Making Problem

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
T. M. Al-shami
Keyword(s):  
Decision Making ◽  
Topological Spaces ◽  
Topological Properties ◽  
Finite Product ◽  
Separation Axioms ◽  
Decision Making Problem ◽  
Weak Structure

In this work, we introduce new types of soft separation axioms called p t -soft α regular and p t -soft α T i -spaces i = 0,1,2,3,4 using partial belong and total nonbelong relations between ordinary points and soft α -open sets. These soft separation axioms enable us to initiate new families of soft spaces and then obtain new interesting properties. We provide several examples to elucidate the relationships between them as well as their relationships with e -soft T i , soft α T i , and t t -soft α T i -spaces. Also, we determine the conditions under which they are equivalent and link them with their counterparts on topological spaces. Furthermore, we prove that p t -soft α T i -spaces i = 0,1,2,3,4 are additive and topological properties and demonstrate that p t -soft α T i -spaces i = 0,1,2 are preserved under finite product of soft spaces. Finally, we discuss an application of optimal choices using the idea of p t -soft T i -spaces i = 0,1,2 on the content of soft weak structure. We provide an algorithm of this application with an example showing how this algorithm is carried out. In fact, this study represents the first investigation of real applications of soft separation axioms.

scholarly journals An integrated model for interdisciplinary graduate education: Computation and mathematics for biological networks

PLoS ONE ◽  
2021 ◽  
Vol 16 (9) ◽  
pp. e0257872
Author(s):  
Kelsey E. McKee ◽  
Daniel Serrano ◽  
Michelle Girvan ◽  
Gili Marbach-Ad
Keyword(s):  
Biological Networks ◽  
Network Science ◽  
Graduate Program ◽  
Interdisciplinary Training ◽  
Faculty Advisors ◽  
Social Events ◽  
Engineering Sciences ◽  
Domain 3 ◽  
And Mathematics ◽  
Program Components

The current challenges at the forefront of data-enabled science and engineering require interdisciplinary solutions. Yet most traditional doctoral programs are not structured to support successful interdisciplinary research. Here we describe the design of and students’ experiences in the COMBINE (Computation and Mathematics for Biological Networks) interdisciplinary graduate program at the University of Maryland. COMBINE focuses on the development and application of network science methods to biological systems for students from three primary domains: life sciences, computational/engineering sciences, and mathematical/physical sciences. The program integrates three established models (T-shaped, pi-shaped and shield-shaped) for interdisciplinary training. The program components largely fall into three categories: (1) core coursework that provides content expertise, communication, and technical skills, (2) discipline-bridging elective courses in the two COMBINE domains that complement the student’s home domain, (3) broadening activities such as workshops, symposiums, and formal peer-mentoring groups. Beyond these components, the program builds community through both formal and informal networking and social events. In addition to the interactions with other program participants, students engage with faculty in several ways beyond the conventional adviser framework, such as the requirement to select a second out-of-field advisor, listening to guest speakers, and networking with faculty through workshops. We collected data through post-program surveys, interviews and focus groups with students, alumni and faculty advisors. Overall, COMBINE students and alumni reported feeling that the program components supported their growth in the three program objectives of Network Science & Interdisciplinarity, Communication, and Career Preparation, but also recommended ways to improve the program. The value of the program can be seen not only through the student reports, but also through the students’ research products in network science which include multiple publications and presentations. We believe that COMBINE offers an effective model for integrated interdisciplinary training that can be readily applied in other fields.

I-Rough Topological Spaces

2016 ◽  
Vol 3 (1) ◽  
pp. 98-113 ◽  
Author(s):  
Boby P. Mathew ◽  
Sunil Jacob John
Keyword(s):  
Topological Spaces ◽  
Comparison Study ◽  
Topological Properties ◽  
Point Set

This paper extends the concept of topology on a point set to a topology of an arbitrary rough universe. The concept of I- rough topology on an arbitrary rough universe is introduced and some topological properties of the resultant I-rough topological spaces are studied. Also a comparison study is carried out between topological spaces and I-rough topological spaces.

Topological Properties of a Class of Higher-dimensional Self-affine Tiles

2019 ◽  
Vol 62 (4) ◽  
pp. 727-740
Author(s):  
Guotai Deng ◽  
Chuntai Liu ◽  
Sze-Man Ngai
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Topological Properties ◽  
Higher Dimensional ◽  
Necessary And Sufficient

AbstractWe construct a family of self-affine tiles in $\mathbb{R}^{d}$ ($d\geqslant 2$) with noncollinear digit sets, which naturally generalizes a class studied originally by Q.-R. Deng and K.-S. Lau in $\mathbb{R}^{2}$, and its extension to $\mathbb{R}^{3}$ by the authors. We obtain necessary and sufficient conditions for the tiles to be connected and for their interiors to be contractible.

scholarly journals On topological properties of dominating David derived networks

2016 ◽  
Vol 94 (2) ◽  
pp. 137-148 ◽  
Author(s):  
Muhammad Imran ◽  
Abdul Qudair Baig ◽  
Haidar Ali
Keyword(s):  
Interconnection Networks ◽  
Network Science ◽  
Chemical Properties ◽  
Chemical Compounds ◽  
Topological Indices ◽  
Molecular Topology ◽  
Topological Properties ◽  
Graph Invariant ◽  
Physico Chemical ◽  
Atom Bond

Topological indices are numerical parameters of a graph that characterize its molecular topology and are usually graph invariant. In a QSAR/QSPR study, the physico-chemical properties and topological indices such as the Randić, atom–bond connectivity (ABC), and geometric–arithmetic (GA) indices are used to predict the bioactivity of chemical compounds. Graph theory has found a considerable use in this important area of research. All of the studied interconnection networks in this paper are constructed by the Star of David network. In this paper, we study the general Randić, first Zagreb, ABC, GA, ABC4 and GA5, indices for the first, second, and third types of dominating David derived networks and give closed formulas of these indices for these networks. These results are useful in network science to understand the underlying topologies of these networks.

scholarly journals A simple differential geometry for complex networks

2020 ◽  
pp. 1-28
Author(s):  
Emil Saucan ◽  
Areejit Samal ◽  
Jürgen Jost
Keyword(s):  
Differential Geometry ◽  
Complex Networks ◽  
Ricci Curvature ◽  
Real World ◽  
Large Scale ◽  
Metric Spaces ◽  
Geodesic Curvature ◽  
Higher Dimensional ◽  
Large Scale Networks ◽  
Bonnet Theorem

Abstract We introduce new definitions of sectional, Ricci, and scalar curvatures for networks and their higher dimensional counterparts, derived from two classical notions of curvature for curves in general metric spaces, namely, the Menger curvature and the Haantjes curvature. These curvatures are applicable to unweighted or weighted and undirected or directed networks and are more intuitive and easier to compute than other network curvatures. In particular, the proposed curvatures based on the interpretation of Haantjes definition as geodesic curvature allow us to give a network analogue of the classical local Gauss–Bonnet theorem. Furthermore, we propose even simpler and more intuitive proxies for the Haantjes curvature that allow for even faster and easier computations in large-scale networks. In addition, we also investigate the embedding properties of the proposed Ricci curvatures. Lastly, we also investigate the behavior, both on model and real-world networks, of the curvatures introduced herein with more established notions of Ricci curvature and other widely used network measures.

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