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# Network Theory: Concepts and Applications

1. Alberto Garcia-Diaz1,
2. Don T. Phillips2,
3. Illya Hicks3

Published Online: 14 JAN 2011

DOI: 10.1002/9780470400531.eorms0090

## Wiley Encyclopedia of Operations Research and Management Science

#### How to Cite

Garcia-Diaz, A., Phillips, D. T. and Hicks, I. 2011. Network Theory: Concepts and Applications. Wiley Encyclopedia of Operations Research and Management Science.

#### Author Information

1. 1

The University of Tennessee, Department of Industrial and Information Engineering, Knoxville, Tennessee

2. 2

Texas A&M University, Department of Industrial and Systems Engineering, College Station, Texas

3. 3

Rice University, Department of Computational and Applied Mathematics, Houston, Texas

#### Publication History

1. Published Online: 14 JAN 2011

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### Abstract

This article summarizes fundamental concepts, definitions, notations and formulations of network-flow models, and illustrates their applicability. We focus on deterministic network-flow models of frequent use to solve significant problems in business and industry. Each topic is introduced, mathematically defined, and illustrated by means of an example. The applications presented can be classified into two broad categories. The first category includes applications where the solution is achieved by selecting a chain or cycle that optimizes a suitable objective function. Each arc is interpreted as a directed link that allows the movement of one unit of flow from the start node to the end node of the arc. The arc parameter or length is a generic cost of using the arc. Usually, these models are referred to as distance networks. In the second category, a more general interpretation of an arc is used, namely, that of a channel used for shipping multiple units of flow. The arc parameter represents the maximum amount of flow per unit time that can be shipped along the arc at any time. This parameter is called the capacity of the arc, and the network models are referred to as capacitated flow networks.

### Keywords:

• network-flow models;
• network analysis;
• maximum-flow;
• minimum cut;
• unimodularity and total unimodularity