Graph dynamical system
Encyclopedia
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 over
or interacting particle systems), as well as GDSs with infinite state space (e.g. 
as in coupled map lattices); see, e.g., Wu. In the following everything is implicitly assumed to be finite unless stated otherwise.
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 (fi)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.
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.
= ( 
, 
) of v[Y] one obtains the class of Sequential dynamical system
s (SDS). In this case it is convenient to introduce the Y-local maps Fi constructed from the vertex functions by
The SDS map F = [FY , w] : Kn → Kn is the function composition
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.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
, combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
, algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
, and dynamical systems rather than differential geometry. In principle, one could define and study GDSs over an infinite graph (e.g. cellular automata over
or interacting particle systems), as well as GDSs with infinite state space (e.g. as in coupled map lattices); see, e.g., 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 xv for each vertex v of Y taken from a finite set K. The system state is the n-tuple x = (x1, x2, ... , xn), 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 fv 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 (fi)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 byThis class is referred to as generalized cellular automata since the classical or standard cellular automata
Cellular automaton
A cellular automaton is a discrete model studied in computability theory, mathematics, physics, complexity science, theoretical biology and microstructure modeling. It consists of a regular grid of cells, each in one of a finite number of states, such as "On" and "Off"...
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.
Sequential dynamical systems (SDS)
If the vertex functions are applied asynchronously in the sequence specified by a word w = (w1, w2, ... , wm) or permutation = ( , ) of v[Y] one obtains the class of Sequential dynamical systemSequential dynamical system
Sequential dynamical systems 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...
s (SDS). In this case it is convenient to introduce the Y-local maps Fi constructed from the vertex functions by
The SDS map F = [FY , w] : Kn → Kn is the function composition
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.
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 systemSequential dynamical systemSequential dynamical systems 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...
s - Finite state machineFinite state machineA finite-state machine or finite-state automaton , or simply a state machine, is a mathematical model used to design computer programs and digital logic circuits. It is conceived as an abstract machine that can be in one of a finite number of states...
s - Cellular automataCellular automatonA cellular automaton is a discrete model studied in computability theory, mathematics, physics, complexity science, theoretical biology and microstructure modeling. It consists of a regular grid of cells, each in one of a finite number of states, such as "On" and "Off"...
- Hopfield netHopfield netA Hopfield network is a form of recurrent artificial neural network invented by John Hopfield. Hopfield nets serve as content-addressable memory systems with binary threshold units. They are guaranteed to converge to a local minimum, but convergence to one of the stored patterns is not guaranteed...
works - Boolean networkBoolean networkA Boolean network consists of a set of Boolean variables whose state is determined by other variables in the network. They are a particular case of discrete dynamical networks, where time and states are discrete, i.e. they have a bijection onto an integer series...
s - Petri netPetri netA Petri net is one of several mathematical modeling languages for the description of distributed systems. A Petri net is a directed bipartite graph, in which the nodes represent transitions and places...
s - Chemical reaction networks
- Kauffman networks