## 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 xv. In the brain, this could be the potassium level; in computing applications, it could be anything the modeler chooses. Each arc has a weight we 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 dx/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: dw/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