Related papers: Scale-Invariant Random Spatial Networks
A random network is grown by introducing at unit rate randomly selected nodes on the Euclidean space. A node is randomly connected to its $i$-th predecessor of degree $k_i$ with a directed link of length $\ell$ using a probability…
We introduce a broad class of analytically solvable processes on networks. In the special case, they reduce to random walk and consensus process - two most basic processes on networks. Our class differs from previous models of interactions…
We study the minimal spanning arborescence which is the directed analogue of the minimal spanning tree, with a particular focus on its infinite volume limit and its geometric properties. We prove that in a certain large class of transient…
Scale-free percolation is a stochastic model for complex networks. In this spatial random graph model, vertices $x,y\in\mathbb{Z}^d$ are linked by an edge with probability depending on i.i.d.\ vertex weights and the Euclidean distance…
Interconnected road lanes are a central concept for navigating urban roads. Currently, most autonomous vehicles rely on preconstructed lane maps as designing an algorithmic model is difficult. However, the generation and maintenance of such…
In real-world applications, observations are often constrained to a small fraction of a system. Such spatial subsampling can be caused by the inaccessibility or the sheer size of the system, and cannot be overcome by longer sampling.…
Networks of disparate phenomena-- be it the global ecology, human social institutions, within the human brain, or in micro-scale protein interactions-- exhibit broadly consistent architectural features. To explain this, we propose a new…
We study spatial networks constructed by randomly placing nodes on a manifold and joining two nodes with an edge whenever their distance is less than a certain cutoff. We derive the general expression for the connectivity distribution of…
Let $G$ be a directed graph on finitely many vertices and edges, and assign a positive weight to each edge on $G$. Fix vertices $u$ and $v$ and consider the set of paths that start at $u$ and end at $v$, self-intersecting in any number of…
Treating neural network inputs and outputs as random variables, we characterize the structure of neural networks that can be used to model data that are invariant or equivariant under the action of a compact group. Much recent research has…
We consider two different variational models of transport networks, the so-called branched transport problem and the urban planning problem. Based on a novel relation to Mumford-Shah image inpainting and techniques developed in that field,…
Scale-free networks contain many small cliques and cycles. We model such networks as inhomogeneous random graphs with regularly varying infinite-variance weights. For these models, the number of cliques and cycles have exact integral…
In the study of dynamical processes on networks, there has been intense focus on network structure -- i.e., the arrangement of edges and their associated weights -- but the effects of the temporal patterns of edges remains poorly…
Dating back to two famous experiments by the social-psychologist, Stanley Milgram, in the 1960s, the small-world phenomenon is the idea that all people are connected through a short chain of acquaintances that can be used to route messages.…
We analyze the statistics of the shortest and fastest paths on the road network between randomly sampled end points. To a good approximation, these optimal paths are found to be directed in that their lengths (at large scales) are linearly…
Small world models are networks consisting of many local links and fewer long range `shortcuts'. In this paper, we consider some particular instances, and rigorously investigate the distribution of their inter--point network distances. Our…
This paper studies real-world road networks from an algorithmic perspective, focusing on empirical studies that yield useful properties of road networks that can be exploited in the design of fast algorithms that deal with geographic data.…
The body plan of the fruit fly is determined by the expression of just a handful of genes. We show that the spatial patterns of expression for several of these genes scale precisely with the size of the embryo. Concretely, discrete…
Many real-world scale-free networks, such as neural networks and online communication networks, consist of a fixed number of nodes but exhibit dynamic edge fluctuations. However, traditional models frequently overlook scenarios where the…
Networks with underlying metric spaces attract increasing research attention in network science, statistical physics, applied mathematics, computer science, sociology, and other fields. This attention is further amplified by the current…