Train track map

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 ${\mathbb{\mathbb{Z}}}^k$ or interacting particle systems when some randomness is included), as well as GDSs with infinite state space (e.g. $\mathbb{ℝ}$ as in coupled map lattices); see, for example, Wu. In the following, everything is implicitly assumed to be finite unless stated otherwise.

Formal definition

A graph dynamical system is constructed from the following components:

• A finite graph Y with vertex set v[Y] = {1,2, ... , n}. Depending on the context the graph can be directed or undirected.

• A state x_v for each vertex v of Y taken from a finite set K. The system state is the n-tuple x = (x₁, x₂, ... , x_n), and x[v] is the tuple consisting of the states associated to the vertices in the 1-neighborhood of v in Y (in some fixed order).

• A vertex function f_v for each vertex v. The vertex function maps the state of vertex v at time t to the vertex state at time t + 1 based on the states associated to the 1-neighborhood of v in Y.

• An update scheme specifying the mechanism by which the mapping of individual vertex states is carried out so as to induce a discrete dynamical system with map F: Kⁿ → Kⁿ.

The phase space associated to a dynamical system with map F: Kⁿ → Kⁿ is the finite directed graph with vertex set Kⁿ and directed edges (x, F(x)). The structure of the phase space is governed by the properties of the graph Y, the vertex functions (f_i)_i, and the update scheme. The research in this area seeks to infer phase space properties based on the structure of the system constituents. The analysis has a local-to-global character.

Generalized cellular automata (GCA)

If, for example, the update scheme consists of applying the vertex functions synchronously one obtains the class of generalized cellular automata (CA). In this case, the global map F: Kⁿ → Kⁿ is given by

$F(x{)}_v=f_v(x[v]).$

This class is referred to as generalized cellular automata since the classical or standard cellular automata are typically defined and studied over regular graphs or grids, and the vertex functions are typically assumed to be identical.

Example: Let Y be the circle graph on vertices {1,2,3,4} with edges {1,2}, {2,3}, {3,4} and {1,4}, denoted Circ₄. Let K = {0,1} be the state space for each vertex and use the function nor₃ : K³ → K defined by nor₃(x,y,z) = (1 + x)(1 + y)(1 + z) with arithmetic modulo 2 for all vertex functions. Then for example the system state (0,1,0,0) is mapped to (0, 0, 0, 1) using a synchronous update. All the transitions are shown in the phase space below.

File:Circ-4-nor.jpg

Sequential dynamical systems (SDS)

If the vertex functions are applied asynchronously in the sequence specified by a word w = (w₁, w₂, ... , w_m) or permutation $\pi$ = ( ${\pi }_1$, ${\pi }_2,\dots ,{\pi }_n$) of v[Y] one obtains the class of Sequential dynamical systems (SDS). In this case it is convenient to introduce the Y-local maps F_i constructed from the vertex functions by

$F_i(x)=(x_1,x_2,\dots ,x_{i-1},f_i(x[i]),x_{i+1},\dots ,x_n).$

The SDS map F = [F_Y , w] : Kⁿ → Kⁿ is the function composition

$[F_Y,w]=F_{w(m)}\circ F_{w(m-1)}\circ \cdots \circ F_{w(2)}\circ F_{w(1)}.$

If the update sequence is a permutation one frequently speaks of a permutation SDS to emphasize this point.

Example: Let Y be the circle graph on vertices {1,2,3,4} with edges {1,2}, {2,3}, {3,4} and {1,4}, denoted Circ₄. Let K={0,1} be the state space for each vertex and use the function nor₃ : K³ → K defined by nor₃(x, y, z) = (1 + x)(1 + y)(1 + z) with arithmetic modulo 2 for all vertex functions. Using the update sequence (1,2,3,4) then the system state (0, 1, 0, 0) is mapped to (0, 0, 1, 0). All the system state transitions for this sequential dynamical system are shown in the phase space below.

File:Circ-4-nor-1234.jpg

Stochastic graph dynamical systems

From, e.g., the point of view of applications it is interesting to consider the case where one or more of the components of a GDS contains stochastic elements. Motivating applications could include processes that are not fully understood (e.g. dynamics within a cell) and where certain aspects for all practical purposes seem to behave according to some probability distribution. There are also applications governed by deterministic principles whose description is so complex or unwieldy that it makes sense to consider probabilistic approximations.

Every element of a graph dynamical system can be made stochastic in several ways. For example, in a sequential dynamical system the update sequence can be made stochastic. At each iteration step one may choose the update sequence w at random from a given distribution of update sequences with corresponding probabilities. The matching probability space of update sequences induces a probability space of SDS maps. A natural object to study in this regard is the Markov chain on state space induced by this collection of SDS maps. This case is referred to as update sequence stochastic GDS and is motivated by, e.g., processes where "events" occur at random according to certain rates (e.g. chemical reactions), synchronization in parallel computation/discrete event simulations, and in computational paradigms described later.

This specific example with stochastic update sequence illustrates two general facts for such systems: when passing to a stochastic graph dynamical system one is generally led to (1) a study of Markov chains (with specific structure governed by the constituents of the GDS), and (2) the resulting Markov chains tend to be large having an exponential number of states. A central goal in the study of stochastic GDS is to be able to derive reduced models.

One may also consider the case where the vertex functions are stochastic, i.e., function stochastic GDS. For example, Random Boolean networks are examples of function stochastic GDS using a synchronous update scheme and where the state space is K = {0, 1}. Finite probabilistic cellular automata (PCA) is another example of function stochastic GDS. In principle the class of Interacting particle systems (IPS) covers finite and infinite PCA, but in practice the work on IPS is largely concerned with the infinite case since this allows one to introduce more interesting topologies on state space.

Applications

Graph dynamical systems constitute a natural framework for capturing distributed systems such as biological networks and epidemics over social networks, many of which are frequently referred to as complex systems.

See also

• Sequential dynamical systems
• Finite state machines
• Cellular automata
• Probabilistic Cellular Automata
• Hopfield networks
• Boolean networks
• Petri nets
• Dynamic network analysis (a social science topic)
• Chemical reaction network theory
• Kauffman networks

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

• Graph Dynamical Systems – A Mathematical Framework for Interaction-Based Systems, Their Analysis and Simulations by Henning Mortveit

Further reading

• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534