Related papers: What is Connectivity?
In homotopy type theory we can define the join of maps as a binary operation on maps with a common co-domain. This operation is commutative, associative, and the unique map from the empty type into the common codomain is a neutral element.…
A neighborhood homotopy is an equivalence relation on spatial graphs which is generated by crossing changes on the same component and neighborhood equivalence. We give a complete classification of all 2-component spatial graphs up to…
Network or graph structures are ubiquitous in the study of complex systems. Often, we are interested in complexity trends of these system as it evolves under some dynamic. An example might be looking at the complexity of a food web as…
Networked structure emerged from a wide range of fields such as biological systems, World Wide Web and technological infrastructure. A deeply insight into the topological complexity of these networks has been gained. Some works start to pay…
A fundamental characteristic of computer networks is their topological structure. The question of the description of the structural characteristics of computer networks represents a problem that is not completely solved. Search methods for…
Multivariate graphs are prolific across many fields, including transportation and neuroscience. A key task in graph analysis is the exploration of connectivity, to, for example, analyze how signals flow through neurons, or to explore how…
We study the space of link maps, which are smooth maps from the disjoint union of manifolds P and Q to a manifold N such that the images of P and Q are disjoint. We give a range of dimensions, interpreted as the connectivity of a certain…
We define and study analogs of curve graphs for infinite type surfaces. Our definitions use the geometry of a fixed surface and vertices of our graphs are infinite multicurves which are bounded in both a geometric and a topological sense.…
Social and biological contagions are influenced by the spatial embeddedness of networks. Historically, many epidemics spread as a wave across part of the Earth's surface; however, in modern contagions long-range edges -- for example, due to…
A network embedding is a representation of a large graph in a low-dimensional space, where vertices are modeled as vectors. The objective of a good embedding is to preserve the proximity between vertices in the original graph. This way,…
The connectivity structure of graphs is typically related to the attributes of the nodes. In social networks for example, the probability of a friendship between two people depends on their attributes, such as their age, address, and…
A space Y is called an extension of a space X if Y contains X as a dense subspace. An extension Y of X is called a one-point extension if Y-X is a singleton. Compact extensions are called compactifications and connected extensions are…
A relational structure is (connected-)homogeneous if every isomorphism between finite (connected) substructures extends to an automorphism of the structure. We investigate notions which generalise (connected-)homogeneity, where…
A topology is defined on the mapping class group of a compact connected orientable surface. It is shown that a notion of "genericity" on subsets of the mapping class group arises from this definition. Many plausible results follow from this…
We introduce and use k-shell decomposition to investigate the topology of the Internet at the AS level. Our analysis separates the Internet into three sub-components: (a) a nucleus which is a small (~100 nodes) very well connected globally…
Infinite graphs are finitary in the sense that their points are connected via finite paths. So what would an infinitary generalization of finite graphs look like? Usually this question is answered with the aid of topology, e.g. in the case…
Spatial dependency and spatial embedding are basic physical properties of many phenomena modeled by networks. The most indicated computational environment to deal with spatial information is to use Georeferenced Information System (GIS) and…
Networks are a fundamental model of complex systems throughout the sciences, and network datasets are typically analyzed through lower-order connectivity patterns described at the level of individual nodes and edges. However, higher-order…
Network-based modeling of complex systems and data using the language of graphs has become an essential topic across a range of different disciplines. Arguably, this graph-based perspective derives its success from the relative simplicity…
Interactions between units in phyical, biological, technological, and social systems usually give rise to intrincate networks with non-trivial structure, which critically affects the dynamics and properties of the system. The focus of most…