A scoring mechanism for the rank aggregation of network robustness

To date, a number of metrics have been proposed to quantify inherent robustness of network topology against failures. However, each single metric usually only offers a limited view of network vulnerability to different types of random failures and targeted attacks. When applied to certain network configurations, different metrics rank network topology robustness in different orders which is rather inconsistent, and no single metric fully characterizes network robustness against different modes of failure. To overcome such inconsistency, this work proposes a multi-metric approach as the basis of evaluating aggregate ranking of network topology robustness. This is based on simultaneous utilization of a minimal set of distinct robustness metrics that are standardized so to give way to a direct comparison of vulnerability across networks with different sizes and configurations, hence leading to an initial scoring of inherent topology robustness. Subsequently, based on the inputs of initial scoring a rank aggregation method is employed to allocate an overall ranking of robustness to each network topology. A discussion is presented in support of the presented multi-metric approach and its applications to more realistically assess and rank network topology robustness.     

Ring of simple polytopes and differential equations

Simple polytopes are a classical object of convex geometry. They play a key role in many modern fields of research, such as algebraic and symplectic geometry, toric topology, enumerative combinatorics, and mathematical physics. In this paper, the results of a new approach based on a differential ring of simple polytopes are described. This approach allows one to apply the theory of differential equations to the study of combinatorial invariants of simple polytopes.     

Stein domains in Banach algebraic geometry

In this article we give a homological characterization of the topology of Stein spaces over any valued base field. In particular, when working over the field of complex numbers, we obtain a characterization of the usual Euclidean (transcendental) topology of complex analytic spaces. For non-Archimedean base fields the topology we characterize coincides with the topology of the Berkovich analytic space associated to a non-Archimedean Stein algebra. Because the characterization we used is borrowed from a definition in derived geometry, this work should be read as a derived perspective on analytic geometry.     

On implicit systems of differential equations

This article considers implicit systems of differential equations. The implicit systems that are considered are given by polynomial relations on the coordinates of the indeterminate function and the coordinates of the time derivative of the indeterminate function. For such implicit systems of differential equations, we are concerned with computing algebraic constraints such that on the algebraic variety determined by the constraint equations the original implicit system of differential equations has an explicit representation. Our approach is algebraic. Although there have been a number of articles that approach implicit differential equations algebraically, all such approaches have relied heavily on linear algebra. The approach of this article is different, we have no linearity requirements at all, instead we rely on algebraic geometry. In particular, we use birational mappings to produce an explicit system. The methods developed in this article are easily implemented using various computer algebra systems.     

The Fractal Nature of Riem/Diff I

This paper is devoted to large scale aspects of the geometry of the space of isometry classes of Riemannian metrics, with a 2-sided curvature bound, on a fixed compact smooth manifold of dimension at least five. Using a mix of tools from logic/computer science, and differential geometry and topology, we study the diameter functional and its critical points, as well as their distribution (density) within the space and the structure of their neighborhoods.     

Global properties of integrable Hamiltonian systems

This paper deals with Lagrangian bundles which are symplectic torus bundles that occur in integrable Hamiltonian systems. We review the theory of obstructions to triviality, in particular monodromy, as well as the ensuing classification problems which involve the Chern and Lagrange class. Our approach, which uses simple ideas from differential geometry and algebraic topology, reveals the fundamental role of the integer affine structure on the base space of these bundles. We provide a geometric proof of the classification of Lagrangian bundles with fixed integer affine structure by their Lagrange class.      

Oscillator death in coupled functional differential equations near Hopf bifurcation

The stability of the equilibrium solution is analyzed for coupled systems of retarded functional differential equations near a supercritical Hopf bifurcation. Necessary and sufficient conditions are derived for asymptotic stability under general coupling conditions. It is shown that the largest eigenvalue of the graph Laplacian completely characterizes the effect of the connection topology on the stability of diffusively and symmetrically coupled identical systems. In particular, all bipartite graphs have identical stability characteristics regardless of their size. Furthermore, bipartite graphs and large complete graphs provide, respectively, lower and upper bounds for the parametric stability regions for arbitrary connection topologies. Generalizations are given for networks with asymmetric coupling. The results characterize the connection topology as a mechanism for the death of coupled oscillators near Hopf bifurcation.     

Existence and uniqueness of solutions of surface reconstruction problem

Surface reconstruction from scattered data is an important problem in such areas as reverse engineering and computer aided design.In solving partial differential equations derived from surface reconstruction problems,level-set method has been successfully used.We present in this paper a theoretical analysis on the existence and uniqueness of the solution of a partial differential equation derived from a model of surface reconstruction using the level-set approach.We give the uniqueness analysis of the cl...     

Controllability for a class of second-order evolution differential inclusions without compactness

In this paper, we discuss the existence and controllability for a class of second-order evolution differential inclusions without compactness in Banach spaces. By applying the technique of weak topology and Glicksberg–Ky Fan fixed point theorem, we prove our main results without the hypotheses of compactness on the operator generated by the linear part and any conditions on the multivalued nonlinearity expressed in terms of measures of noncompactness. Further, we extend our study to existence and controllability of second-order evolution differential inclusions with nonlocal conditions and impulses. Finally, an example is given for the illustration of the obtained theoretical results.     

Functional differential inclusions and dynamic behaviors for memristor-based BAM neural networks with time-varying delays

In this paper, we formulate and investigate a class of memristor-based BAM neural networks with time-varying delays. Under the framework of Filippov solutions, the viability and dissipativity of solutions for functional differential inclusions and memristive BAM neural networks can be guaranteed by the matrix measure approach and generalized Halanay inequalities. Then, a new method involving the application of set-valued version of Krasnoselskii’ fixed point theorem in a cone is successfully employed to derive the existence of the positive periodic solution. The dynamic analysis in this paper utilizes the theory of set-valued maps and functional differential equations with discontinuous right-hand sides of Filippov type. The obtained results extend and improve some previous works on conventional BAM neural networks. Finally, numerical examples are given to demonstrate the theoretical results via computer simulations.     

Extreme Networks

This paper is supposed to be a review on the new branch of mathematics – extreme networks theory that appeared at the crossroad of differential geometry, variational calculus and discrete mathematics. One of the starting points of the theory is the geometrical approach to the Steiner problem. The authors have selected the most impressive results of the theory demonstrating a new approach, new effects, and new constructions appearing in this context. One of the main aims of the paper is to attract the attention of the specialist from similar fields to this new and rapidly developing theory.     

Noncommutative structures

We propose a method for constructing noncommutative analogs of objects from classical calculus, differential geometry, topology, dynamical systems, etc. The standard (commutative) objects can be obtained from noncommutative ones by natural projections (a set of canonical homomorphisms). The approach is ideologically close to the noncommutative geometry of A. Connes but differs from it in technical details.     

On the Topology of Real Algebraic Plane Curves

We revisit the problem of computing the topology and geometry of a real algebraic plane curve. The topology is of prime interest but geometric information, such as the position of singular and critical points, is also relevant. A challenge is to compute efficiently this information for the given coordinate system even if the curve is not in generic position. Previous methods based on the cylindrical algebraic decomposition use sub-resultant sequences and computations with polynomials with algebraic coefficients. A novelty of our approach is to replace these tools by Gröbner basis computations and isolation with rational univariate representations. This has the advantage of avoiding computations with polynomials with algebraic coefficients, even in non-generic positions. Our algorithm isolates critical points in boxes and computes a decomposition of the plane by rectangular boxes. This decomposition also induces a new approach for computing an arrangement of polylines isotopic to the input curve. We also present an analysis of the complexity of our algorithm. An implementation of our algorithm demonstrates its efficiency, in particular on high-degree non-generic curves.     

Stable phase-locked periodic solutions in a delay differential system

For a delay differential system where the nonlinearity is motivated by applications of neural networks to spatiotemporal pattern association and can be regarded as a perturbation of a step function, we obtain the existence, stability and limiting profile of a phase-locked periodic solution using an approach very much similar to the asymptotic expansion of inner and outer layers in the analytic method of singular perturbation theory.     

Topology independent SIS process: An engineering viewpoint

A Susceptible-Infected-Susceptible (SIS) spreading process, occurring on complex networks that are characterized by a special form of contact dynamics called “acquisition exclusivity”, is studied. Assuming statistical independence of joint events, we find analytic solutions for the stationary probabilities that network nodes are infected, and more importantly, find that these solutions are independent of the network topology. We explore the possibilities to use the studied set-up as an engineering solution for controlled spreading on technological networks. In that context, an example for controlled sharing of viral countermeasures in networks is presented. Considering the high epidemic threshold that characterizes the process, “acquisition exclusivity” is suggested as a method for eradication of viral infections from networks.     

Some theoretical challenges in digital geometry: A perspective

In recent years image analysis has become a research field of exceptional significance, due to its relevance to real life problems in important societal and governmental sectors, such as medicine, defense, and security. The explicit purpose of the present Perspective is to suggest a number of strategic objectives for theoretical research, with an emphasis on the combinatorial approach in image analysis. Most of the proposed objectives relate to the need to make the theoretical foundations of combinatorial image analysis better integrated within a number of well-established subjects of theoretical computer science and discrete applied mathematics, such as the theory of algorithms and problem complexity, combinatorial optimization and polyhedral combinatorics, integer and linear programming, and computational geometry.     

二元线性样条函数插值

二元样条函数插值在计算几何与计算机辅助几何设计中有着重要的作用.本文给出了一种矩形剖分上二元线性样条函数进行Lagrange插值时插值适定结点组所满足的拓扑与几何性质,这种性质依赖于二元线性样条函数所决定的分片线性代数曲线.     

Modeling networked systems using the topologically distributed bounded rationality framework

In networked systems research, game theory is increasingly used to model a number of scenarios where distributed decision making takes place in a competitive environment. These scenarios include peer‐to‐peer network formation and routing, computer security level allocation, and TCP congestion control. It has been shown, however, that such modeling has met with limited success in capturing the real‐world behavior of computing systems. One of the main reasons for this drawback is that, whereas classical game theory assumes perfect rationality of players, real world entities in such settings have limited information, and cognitive ability which hinders their decision making. Meanwhile, new bounded rationality models have been proposed in networked game theory which take into account the topology of the network. In this article, we demonstrate that game‐theoretic modeling of computing systems would be much more accurate if a topologically distributed bounded rationality model is used. In particular, we consider (a) link formation on peer‐to‐peer overlay networks (b) assigning security levels to computers in computer networks (c) routing in peer‐to‐peer overlay networks, and show that in each of these scenarios, the accuracy of the modeling improves very significantly when topological models of bounded rationality are applied in the modeling process. Our results indicate that it is possible to use game theory to model competitive scenarios in networked systems in a way that closely reflects real world behavior, topology, and dynamics of such systems. © 2016 Wiley Periodicals, Inc. Complexity 21: 123–137, 2016     

A cross entropy approach to design of reliable networks

One of the most important parameters determining the performance of communication networks is network reliability. The network reliability strongly depends on not only topological layout of the communication networks but also reliability and availability of the communication facilities. The selection of optimal network topology is an NP-hard problem so that computation time of enumeration-based methods grows exponentially with network size. This paper presents a new solution approach based on cross-entropy method, called NCE, to design of communication networks. The design problem is to find a network topology with minimum cost such that all-terminal reliability is not less than a given level of reliability. To investigate the effectiveness of the proposed NCE, comparisons with other heuristic approaches given in the literature for the design problem are carried out in a three-stage experimental study. Computational results show that NCE is an effective heuristic approach to design of reliable networks.