Continuous graph

This article is about sets of vertices and edges (graphs) defined on a continuous space. For graphs of continuous functions, see Continuous function.

For connected graphs, see Connectivity (graph theory).

A continuous graph is a graph whose set of vertices is a continuous space X. Edges are then defined by a function from the cartesian product X² to the set {0, 1}. This could represent 1 for an edge between two vertices, and 0 for no edge, or it could represent a complete graph with a 2-color edge coloring. In this context, the set {0,1} is often denoted by 2, so we have f(X²)→2. For multi-colorings of edges we would have f(X²)→n.^[1][2][3] Continuous graphs have applications to peer-to-peer systems.^[4]

A graph limit or graphon is the limit of a sequence of graphs. Such a limit is a symmetric measurable function in two variables^[5], that can often be written f(S²)→[0,1] which is the same as a complete continuous graph where the edges have values in the interval [0,1].

For any sets X and Y, the two-variable function f(X²)→Y is a complete graph with edges labelled with elements of Y. For multi-variable functions we have f(Xⁿ)→Y for the complete hypergraph with edges labelled with elements of Y.^[6]

Given a discrete-time dynamical system, the trajectories, or orbits (state space) of all the points form a (possibly disconnected) directed graph which is a continuous graph if the system is defined on a continuous space. The trajectories of a continuous-time dynamical system would form a collection of curved paths (phase space) rather than a collection of piece-wise linear paths and so is not a graph in the traditional sense.

See also

• Tree (set theory)
• Petri Net#Discrete, continuous, and hybrid Petri nets

References

1. ^ CONTINUOUS GRAPHS AND C*-ALGEBRAS, VALENTIN DEACONU, Operator theoretical methods: 17th International Conference on Operator Theory, Timișoara (Romania), July 23-26, 1998, ISBN 978-973-99097-2-3.
2. ^ CONTINUOUS RAMSEY THEORY ON POLISH SPACES AND COVERING THE PLANE BY FUNCTIONS, STEFAN GESCHKE, MARTIN GOLDSTERN, AND MENACHEM KOJMAN, Journal of Mathematical Logic, 2004.
3. ^ INFINITE RAMSEY THEORY, STEFAN GESCHKE.
4. ^ Novel architectures for P2P applications: the continuous-discrete approach, M Naor, U Wieder - ACM Transactions on Algorithms (TALG), 2007.
5. ^ Graph Limits and Parameter Testing, Christian Borgs, Jennifer Chayes, László Lovász, Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, 2006
6. ^ Testability and repair of hereditary hypergraph properties, T Austin, T Tao - Random Structures and Algorithms, 2010

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