Showing papers on "Network theory published in 1976"

Network theory of microscopic and macroscopic behavior of master equation systems

Abstract: A general microscopic and macroscopic theory is developed for systems which are governed by a (linear) master equation. The theory is based on a network representation of the master equation, and the results are obtained mostly by application of some basic theorems of mathematical graph theory. In the microscopic part of the theory, the construction of a steady state solution of the master equation in terms of graph theoretical elements is described (Kirchhoff's theorem), and it is shown that the master equation satisfies a global asymptotic Liapunov stability criterion with respect to this state. The Glansdorff-Prigogine criterion turns out to be the differential version and thus a special case of the global criterion. In the macroscopic part of the theory, a general prescription is given describing macrostates of the systems arbitrarily far from equilibrium in the language of generalized forces and fluxes of nonlinear irreversible thermodynamics. As a particular result, Onsager's reciprocity relations for the phenomenological coefficients are obtained as coinciding with the reciprocity relations of a near-to-equilibrium network.

Applications of Discrete and Continuous Network Theory to Linear Population Models

Abstract: The well-established methods of network construction and analysis are adapted to the problem of modeling single populations. A major advantage of the resulting approach is that it allows explicit incorporation of key processes in the life cycle of the organism being modeled, with feedback loops providing economy of representation where they are allowed. Thus, network structures provide heuristic vehicles by which population models can be de- veloped and modified. When a model is linear and has parameters that do not vary with time, a characteristic dynamic function can be derived by inspection from a simple transform of the network representation. The zeros of the function can be found (analytically or by commonly available numerical methods) and used directly to deduce the modeled population's dominant growth pattern and its propensity to sustain oscillations. In addition, under certain conditions (i.e., that the network model not contain both time delays and integrators), a straightforward method (partial fraction expansion) is available for deduction of the modeled population's specific responses to a variety of perturbations.