the results for continuum phase-field models have discrete analogues for graphs, just replacing calculus with calculus on graphs. Fracture in solids is often...

Phase-field models on graphs are a discrete analogue to phase-field models, defined on a graph. They are used in image analysis (for feature identification)...

variational models on graphs which can be interpreted as discrete versions of partial differential equations or continuum variational models. Such equations...

the mathematical field of graph theory, the Erdős–Rényi model refers to one of two closely related models for generating random graphs or the evolution...

models. Michael Aizenman and coauthors used it to study the phase boundaries in 1D Ising and Potts models. Tutte polynomial Ising model Random graph Swendsen–Wang...

Rényi, A. (1959). "On random graphs I.". Publ. Math. (6): 290–297. Erdős, P. & Rényi, A. (1960). "The evolution of random graphs". Publ. Math. Inst....

computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context...

In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability...

different structural phases.The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition. Though...

"Expander graphs are globally synchronising". arXiv:2210.12788 [math.CO]. Sloman, Leila (24 July 2023). "New Proof Shows That 'Expander' Graphs Synchronize"...

Alec Wolman, also in 2003. Skip graphs have the full functionality of a balanced tree in a distributed system. Skip graphs are mostly used in searching peer-to-peer...

lattice field theory are areas of study. Lattice models are also used to simulate the structure and dynamics of polymers. A number of lattice models can be...

simplest models that shows a phase transition. Lars Onsager famously calculated exactly the free energy of an infinite-sized square-lattice Ising model at zero...

than logical as for simple graphs, values, explaining the difference in the results - for simple graphs, the symmetrized graph still needs to be simple...

others. These two types of mixtures result in very different graphs. Another type of binary phase diagram is a boiling-point diagram for a mixture of two components...

{N}{2}}=N(N-1)/2} ; for directed graphs (with no self-connected nodes), E max = N ( N − 1 ) {\displaystyle E_{\max }=N(N-1)} ; for directed graphs with self-connections...

The stochastic block model is a generative model for random graphs. This model tends to produce graphs containing communities, subsets of nodes characterized...

which the graphs are arranged in concentric circles around some starting node and to three-dimensional layered drawings of graphs. In layered graph drawings...

{\displaystyle x_{e}=0} . A graph with one or more such open edges is referred to as an open graph. Quantum graphs are metric graphs equipped with a differential...

speaking only the same on the level of infinitesimal symmetries) because the diagonal phase factor also acts on other fields – quarks in particular....

undirected random graphs are given in the table below: Erdős–Rényi model – Two closely related models for generating random graphs Fractals – Infinitely...

integral models Some notable hysteretic models are listed below, along with their associated fields. Bean's critical state model (magnetism) Bouc–Wen model (structural...

(ER) model – Two closely related models for generating random graphs Local World Evolving Network Models Percolation theory – Mathematical theory on behavior...

In physics, chemistry, and other related fields like biology, a phase transition (or phase change) is the physical process of transition between one state...

Ecosystem Graphs: The Social Footprint of Foundation Models, arXiv:2303.15772 Ha, David; Schmidhuber, Jürgen (3 December 2018). "Recurrent world models facilitate...

network is a graph (network) with non-trivial topological features—features that do not occur in simple networks such as lattices or random graphs but often...

also called field-oriented control (FOC), is a variable-frequency drive (VFD) control method in which the stator currents of a three-phase AC motor are...

these applications, the models often relied on force fields or potentials derived empirically or with simulations. These models thus remained confined...

succeeded by VulkanSceneGraph project, a Vulkan-based library. OpenSceneGraph is in maintenance phase since 2019. The OpenSceneGraph project was initiated...

In classical and quantum mechanics, geometric phase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic...