## Networks

Site under construction ...### Definition of Networks

From an economic point of view, it is to be stated that all communicating activities are scattered though space and time. Networks are considered to be a complex instrument which enables the interaction of these separated items.

In the theory of graphs, a graph consists of a set of junctions points
called **nodes**, with certain pairs of the nodes being joined by
**arcs**, **links**, or **edges**. A **network** is considered
to be a graph with a flow of some type on its branches. Cf.
Hillier, Liebermann (1995).

### Examples of Networks

road networks, railroad networks, shipping networks, airline networks

water networks, power supply systems

telecommunications networks, data networks

### Neural Networks

Neural networks determine an important class of "dynamic" networks.
A neural network consists of nodes that correspond to neurons and arcs that
correspond to synaptic connections in the biological metaphor. Each node
has a neural state x_{v}. In the brain, this could be the potassium
level; in computing applications, it could be anything the modeler
chooses. Each arc has a weight w_{e} that affects the state of its
neighboring nodes when firing. If the weight is positive, it is said to be
excitatory; if it is negative, it is inhibitory. The neural states **x**
change by some differential (or difference) equation depending on
prevailing arc weights **w**, say d**x**/dt = F(**x**, **w**, t).
Typically (but not necessarily), - F is the gradient of an energy function
(in keeping with the biological metaphor) so that **x**(t) follows a
path of steepest descent towards a minimum energy state. A learning
mechanism L could consist of equations to change the weights:
d**w**/dt = L(**x**, **w**, t).
Various learning mechanisms are represented this way, including a form of
supervised learning that uses a training set to provide feedback on
errors. Other elements can be learned besides the arc weights, including
the topology of the network.

Literature: Greenberg, H. J., Mathematical Programming Glossary. online