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Interpretations of a program (plan) for realizing a certain complex of interrelated operations in the form of an oriented graph (cf. [[Graph, oriented|Graph, oriented]]) without circuits, reflecting the natural order of accomplishing the work in time with certain additional data of the complex of operations (cost, resources, durations, etc.). Usually network models are represented graphically on the plane, and the representation is called a network graph. A network model is fundamental to [[Network planning|network planning]], control and calendar planning. Depending on the conditions in the processing of the information, a network model can have other forms of representation: tabular, digital, etc. All forms of representation of a network model are equivalent.

At the basis of a network model is its structure, that is, the graph of the complex of operations. It is defined, as a rule, in the following way. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066390/n0663901.png" /> be a complex of operations (for example, the erection of a multi-floor building) for which the stages <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066390/n0663902.png" /> are defined: the first is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066390/n0663903.png" /> and the last is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066390/n0663904.png" />, and depending on the logic of the particular order dictated by the interrelation between the operations, each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066390/n0663905.png" /> is an intermediate stage (for example, it is impossible to finish erecting the framework of the tenth floor without completing the appropriate work on the ninth). The intermediate stages, and their number, are to some extent conditional, and are mainly defined by the corresponding stages in the realization of the complex. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066390/n0663906.png" /> be a complex of operations and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066390/n0663907.png" /> be the set of stages. If one now defines the graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066390/n0663908.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066390/n0663909.png" /> is the set of vertices, and the operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066390/n06639010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066390/n06639011.png" />, which begins at stage <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066390/n06639012.png" /> and ends at stage <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066390/n06639013.png" /> is an arc, then the resulting oriented graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066390/n06639014.png" /> without circuits is the required structure of the network model of the complex of operations in question. In the language of network models, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066390/n06639015.png" /> are called operations, and the stages <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066390/n06639016.png" /> are called events.

A network model is constructed on the basis of its structure, depending on the aims of the complex of operations. For example, if the aim is to complete the complex of operations in least time with given resources, the network model would include data about the time needed to accomplish each operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066390/n06639017.png" />; in this case it is said that the network model is constructed according to a time criterion. Network models can be constructed in terms of another criterion, or in accordance with several criteria taken simultaneously. Accordingly, a network model is called one-dimensional or multi-dimensional. One can distinguish canonical and alternative network models (see [[#References|[1]]] and [[#References|[3]]]). The first are defined by a fixed structure and the condition that any operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066390/n06639018.png" /> cannot be started until the operations that end at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066390/n06639019.png" /> are finished. The second have a variable structure and allow a certain operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066390/n06639020.png" /> to begin after the completion of a certain operation with end at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066390/n06639021.png" />. Network models can be deterministic or probabilistic, depending on the exactness of the criteria or on predictability with a certain probability.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> , ''Fundamentals in the development and application of systems of network planning and control'' , Moscow  (1974)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> , ''Encyclopaedia of cybernetics'' , '''1–2''' , Kiev  (1974)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.I. Lopatnikov,   "A short mathematical-economic dictionary" , Moscow  (1979)  (In Russian)</TD></TR></table>



====Comments====
For additional references see [[Network|Network]].