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Future Directions: Dynamic Graphs

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Control of Complex Systems

Part of the book series: Communications and Control Engineering ((CCE))

Abstract

Graph theory provides a natural framework for studying relationships between the components of a physical system. In most cases, these relationships are described by assigning directions and constant numerical values (or weights) to each edge in the graph (e.g., Harary, 1969; Ford and Fulkerson, 1962). There are, however, a number of important applications where such a model fails to capture all the relevant characteristics of the system. This potential deficiency was recognized as early as the 1940s, in the context of social interactions within groups and organizations (Radcliffe-Brown, 1940). Since that time, there have been several systematic attempts to analyze graphs whose edges are subject to change. In the late 1950s, for example, Erdös and Rènyi (1959) introduced the notion of random graphs, whose edges and vertices are assigned weights that reflect their existence probabilities. A generic problem in this context has been to determine the probability that a graph will remain connected when some of its vertices and edges are randomly eliminated.

More recently, graphs with time-varying and state-dependent edges have been considered in the context of complex systems and connective stability (Šiljak, 1978). Such graphs were used to model uncertain connections between subsystems, and to determine conditions under which the overall system remains stable when certain subsystems are temporarily disconnected. Dynamically changing graphs were also examined in (Ladde and Šiljak, 1983), where stochastic models were used to analyze multi-controller schemes for reliability enhancement (in this case, graphs were used to represent states of a Markov process).

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Authors and Affiliations

  1. Dept. Electrical Engineering, Santa Clara University, 500 EI Camino Real, Santa Clara, CA, 95053, USA

    Aleksandar I. Zečević & Dragoslav D. Šiljak

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  1. Aleksandar I. Zečević
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  2. Dragoslav D. Šiljak
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Correspondence to Aleksandar I. Zečević .

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Zečević, A.I., Šiljak, D.D. (2010). Future Directions: Dynamic Graphs. In: Control of Complex Systems. Communications and Control Engineering. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-1216-9_6

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  • DOI: https://doi.org/10.1007/978-1-4419-1216-9_6

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