Networks
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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