An Inquiry into the Practice of Proving in Low-Dimensional Topology

The aim of this article is to investigate specific aspects connected with visualization in the practice of a mathematical subfield: low-dimensional topology. Through a case study, it will be established that visualization can play an epistemic role. The background assumption is that the consideration of the actual practice of mathematics is relevant to address epistemological issues. It will be shown that in low-dimensional topology, justifications can be based on sequences of pictures. Three theses will be defended. First, the representations used in the practice are an integral part of the mathematical reasoning. As a matter of fact, they convey in a material form the relevant transitions and thus allow experts to draw inferential connections. Second, in low-dimensional topology experts exploit a particular type of manipulative imagination which is connected to intuition of two- and three-dimensional space and motor agency. This imagination allows recognizing the transformations which connect different pictures in an argument. Third, the epistemic—and inferential—actions performed are permissible only within a specific practice: this form of reasoning is subject-matter dependent. Local criteria of validity are established to assure the soundness of representationally heterogeneous arguments in low-dimensional topology

SPECTRUM-BASED AND COLLABORATIVE NETWORK TOPOLOGY ANALYSIS AND VISUALIZATION

Networks are of significant importance in many application domains, such as World Wide Web and social networks, which often embed rich topological information. Since network topology captures the organization of network nodes and links, studying net- work topology is very important to network analysis. In this dissertation, we study networks by analyzing their topology structure to explore community structure, the relationship among network members and links as well as their importance to the belonged communities. We provide new network visualization methods by studying network topology through two aspects: spectrum-based and collaborative visualiza- tion techniques. For the spectrum-based network visualization, we use eigenvalues and eigenvectors to express network topological features instead of using network datasets directly. We provide a visual analytics approach to analyze unsigned networks based on re- cent achievements on spectrum-based analysis techniques which utilize the features of node distribution and coordinates in the high dimensional spectral space. To assist the interactive exploration of network topologies, we have designed network visual- ization and interactive analysis methods allowing users to explore the global topology structure. Further, to address the question of real-life applications involving of both positive and negative relationships, we present a spectral analysis framework to study both signed and unsigned networks. Our framework concentrates on two problems of net- work analysis - what are the important spectral patterns and how to use them to study signed networks. Based on the framework, we present visual analysis methods, which guide the selection of k-dimensional spectral space and interactive exploration of network topology. With the increasing complexity and volume of dynamic networks, it is important to adopt strategies of joint decision-making through developing collaborative visualiza- tion approaches. Thus, we design and develop a collaborative detection mechanism with matrix visualization for complex intrusion detection applications. We establish a set of collaboration guidelines for team coordination with distributed visualization tools. We apply them to generate a prototype system with interactions that facilitates collaborative visual analysis. In order to evaluate the collaborative detection mechanism, a formal user study is presented. The user study monitored participants to collaborate under co-located and distributed collaboration environments to tackle the problems of intrusion detection. We have observed participants’ behaviors and collected their performances from the aspects of coordination and communication. Based on the results, we conclude several coordination strategies and summarize the values of communication for collaborative visualization. Our visualization methods have been demonstrated to be efficient topology explo- ration with both synthetic and real-life datasets in spectrum-based and collaborative exploration. We believe that our methods can provide useful information for future design and development of network topology visualization system

A combined measure for quantifying and qualifying the topology preservation of growing self-organizing maps

The Self-OrganizingMap (SOM) is a neural network model that performs an ordered projection of a high dimensional input space in a low-dimensional topological structure. The process in which such mapping is formed is defined by the SOM algorithm, which is a competitive, unsupervised and nonparametric method, since it does not make any assumption about the input data distribution. The feature maps provided by this algorithm have been successfully applied for vector quantization, clustering and high dimensional data visualization processes. However, the initialization of the network topology and the selection of the SOM training parameters are two difficult tasks caused by the unknown distribution of the input signals. A misconfiguration of these parameters can generate a feature map of low-quality, so it is necessary to have some measure of the degree of adaptation of the SOM network to the input data model. The topologypreservation is the most common concept used to implement this measure. Several qualitative and quantitative methods have been proposed for measuring the degree of SOM topologypreservation, particularly using Kohonen's model. In this work, two methods for measuring the topologypreservation of the Growing Cell Structures (GCSs) model are proposed: the topographic function and the topology preserving ma