Sequential dynamical system. Phase space of the sequential dynamical system Sequential dynamical systems (SDSs) are a class of graph dynamical systems. They are discrete dynamical systems which generalize many aspects of for example classical cellular automata, and they provide a framework for studying asynchronous processes over graphs. The analysis of SDSs uses techniques from combinatorics, abstract algebra, graph theory, dynamical systems and probability theory. Definition[edit] An SDS is constructed from the following components: A finite graph Y with vertex set v[Y] = {1,2, ... , n}. It is convenient to introduce the Y-local maps Fi constructed from the vertex functions by The word w specifies the sequence in which the Y-local maps are composed to derive the sequential dynamical system map F: Kn → Kn as If the update sequence is a permutation one frequently speaks of a permutation SDS to emphasize this point. Example[edit] See also[edit] References[edit]
Topology. Möbius strips, which have only one surface and one edge, are a kind of object studied in topology. Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space, dimension, and transformation. Such ideas go back to Leibniz, who in the 17th century envisioned the geometria situs (Latin for "geometry of place") and analysis situs (Greek-Latin for "picking apart of place"). The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics. Topology has many subfields: See also: topology glossary for definitions of some of the terms used in topology, and topological space for a more technical treatment of the subject.
History[edit] Topology began with the investigation of certain questions in geometry. Elementary introduction[edit] Graph theory. Refer to the glossary of graph theory for basic definitions in graph theory. Definitions[edit] Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph[edit] Other senses of graph stem from different conceptions of the edge set. All of these variants and others are described more fully below. The vertices belonging to an edge are called the ends, endpoints, or end vertices of the edge. V and E are usually taken to be finite, and many of the well-known results are not true (or are rather different) for infinite graphs because many of the arguments fail in the infinite case.
(the number of vertices). For an edge {u, v}, graph theorists usually use the somewhat shorter notation uv. Applications[edit] The network graph formed by Wikipedia editors (edges) contributing to different Wikipedia language versions (nodes) during one month in summer 2013.[3] History[edit] The Königsberg Bridge problem "[...] Graph dynamical system. In mathematics, the concept of graph dynamical systems can be used to capture a wide range of processes taking place on graphs or networks. A major theme in the mathematical and computational analysis of GDSs is to relate their structural properties (e.g. the network connectivity) and the global dynamics that result. The work on GDSs considers finite graphs and finite state spaces.
As such, the research typically involves techniques from, e.g., graph theory, combinatorics, algebra, and dynamical systems rather than differential geometry. In principle, one could define and study GDSs over an infinite graph (e.g. cellular automata or Probabilistic Cellular Automata over or interacting particle systems when some randomness is included), as well as GDSs with infinite state space (e.g. as in coupled map lattices); see, for example, Wu.[1] In the following, everything is implicitly assumed to be finite unless stated otherwise.
Formal definition[edit] Generalized cellular automata (GCA)[edit] Sequential dynamical system. Gene regulatory network. Structure of a gene regulatory network Control process of a gene regulatory network In single-celled organisms, regulatory networks respond to the external environment, optimising the cell at a given time for survival in this environment. Thus a yeast cell, finding itself in a sugar solution, will turn on genes to make enzymes that process the sugar to alcohol.[1] This process, which we associate with wine-making, is how the yeast cell makes its living, gaining energy to multiply, which under normal circumstances would enhance its survival prospects. In multicellular animals the same principle has been put in the service of gene cascades that control body-shape.[2] Each time a cell divides, two cells result which, although they contain the same genome in full, can differ in which genes are turned on and making proteins.
Sometimes a 'self-sustaining feedback loop' ensures that a cell maintains its identity and passes it on. Overview[edit] Modelling[edit] Coupled ODEs[edit] nodes, and let . Boolean network. A Boolean network is a particular kind of sequential dynamical system, where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a bijection onto an integer series. Boolean networks are related to cellular automata. Usually, cellular automata are defined with an homogenous topology, i.e. a single line of nodes, a square or hexagonal grid of nodes or an even higher-dimensional structure, but each variable (node in the grid) may take on more than two values (and hence not be boolean).
A random Boolean network (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, N. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed. Classical model[edit] Attractors[edit] Topologies[edit] NK model.