相关论文: Type-0 singularities in the network flow -- Evolut…
We study the crystalline curvature flow of planar networks with a single hexagonal anisotropy. After proving the local existence of a classical solution for a rather large class of initial conditions, we classify the homothetically…
We present statistics for the structure and time-evolution of a network constructed from user activity in an Internet community. The vastness and precise time resolution of an Internet community offers unique possibilities to monitor social…
We propose an extended local-world evolving network model including a triad formation step. In the process of network evolution, random fluctuation in the number of new edges is involved. We derive analytical expressions for degree…
A mean curvature flow starting from a closed embedded hypersurface in $R^{n+1}$ must develop singularities. We show that if the flow has only generic singularities, then the space-time singular set is contained in finitely many compact…
In this study, we develop a complex network approach on a rough fracture, where evolution of elementary aperture during translational shear is characterized. In this manner, based on some hidden metric spaces (similarity measurements)…
In phylogenetics, tree-based networks are used to model and visualize the evolutionary history of species where reticulate events such as horizontal gene transfer have occurred. Formally, a tree-based network $N$ consists of a phylogenetic…
One of the main characteristics of real-world networks is their large clustering. Clustering is one aspect of a more general but much less studied structural organization of networks, i.e. edge multiplicity, defined as the number of…
As the Internet AS-level topology grows over time, some of its structural properties remain unchanged. Such time- invariant properties are generally interesting, because they tend to reflect some fundamental processes or constraints behind…
Networks are mathematical structures that are universally used to describe a large variety of complex systems such as the brain or the Internet. Characterizing the geometrical properties of these networks has become increasingly relevant…
Clustering $\unicode{x2013}$ the tendency for neighbors of nodes to be connected $\unicode{x2013}$ quantifies the coupling of a complex network to its latent metric space. In random geometric graphs, clustering undergoes a continuous phase…
Under mean curvature flow, a closed, embedded hypersurface $M(t)$ becomes singular in finite time. For certain classes of mean-convex mean curvature flows, we show the continuity of the first singular time $T$ and the limit set "$M(T)$",…
We propose a weak formulation for the binormal curvature flow of curves in $\R^3.$ This formulation is sufficiently broad to consider integral currents as initial data, and sufficiently strong for the weak-strong uniqueness property to…
In 1994 Velazquez constructed a smooth \(O(4)\times O(4)\) invariant Mean Curvature Flow that forms a type-II singularity at the origin in space-time. Stolarski very recently showed that the mean curvature on this solution is uniformly…
The co-evolution of network topology and dynamics is studied in an evolutionary Boolean network model that is a simple model of gene regulatory network. We find that a critical state emerges spontaneously resulting from interplay between…
In this paper, we prove that an ancient smooth curve shortening flow with finite-entropy embedded in $\mathbb{R}^2$ has a unique tangent flow at infinity. To this end, we show that its rescaled flows backwardly converge to a line with…
How do real graphs evolve over time? What are ``normal'' growth patterns in social, technological, and information networks? Many studies have discovered patterns in static graphs, identifying properties in a single snapshot of a large…
As individuals communicate, their exchanges form a dynamic network. We demonstrate, using time series analysis of communication in three online settings, that network structure alone can be highly revealing of the diversity and novelty of…
Transport networks are crucial to the functioning of natural and technological systems. Nature features transport networks that are adaptive over a vast range of parameters, thus providing an impressive level of robustness in supply.…
Transportation and distribution networks are a class of spatial networks that have been of interest in recent years. These networks are often characterized by the presence of complex structures such as central loops paired with peripheral…
We consider the area preserving curve shortening flow with Neumann free boundary conditions outside of a convex domain or at a straight line. We give a criterion on initial curves that guarantees the appearance of a singularity in finite…