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A gradient network is a directed subnetwork of an undirected "substrate" network in which each node has an associated scalar potential and one out-link that point to the node with the smallest (or largest) potential in its neighborhood, defined as the reunion of itself and its nearest neighbors on the substrate networks.〔Bogdan Danila, Congestion-gradient driven transport on complex networks, PHYSICAL REVIEW E 74, 046114(2006)〕 Let us consider that transport takes place on a fixed network ''G'' = ''G''(''V'',''E'') called the substrate graph. It has N nodes, V = and the set of edges ''E'' = . Given a node ''i'', we can define its set of neighbors in G by Si(1) = . Let us also consider a scalar field, ''h'' = defined on the set of nodes V, so that every node i has a scalar value ''h''''i'' associated to it. Gradient ∇''h''''i'' on a network: ∇h''i(i, μ(i))'' i.e. the directed edge from ''i'' to ''μ(i)'', where ''μ''(''i'') ∈ Si(1) ∪ , and hμ has the maximum value in . ''Gradient network'' : ''∇ ∇ '' where ''F'' is the set of gradient edges on ''G''. In general, the scalar field depends on time, due to the flow, external sources and sinks on the network. Therefore, the gradient network ∇ will be dynamic. == Motivation == Real-world networks evolve to fulfill a main function, which is often to transport entities such as information, cars, power, water, etc. All these large-scale networks mentioned above are non-globally designed. They evolve and grow through local changes, through a natural selection-like dynamics. For example, if a router on the Internet is frequently congested and packets are lost or delayed due to that, it will get replaced by several interconnected new routers. Recent research investigate the connection between network topology and the flow efficiency of the transportation.〔http://cnls.lanl.gov/External/people/highlights/Toroczkai_net.pdf〕 The flow is often generated or influenced by local gradients of a scalar, for example: electric current driven by a gradient of electric potential; in the information networks, properties of nodes will generate a bias in the way of information is transmitted from a node to its neighbors. This idea motivated the approach through gradient networks which studies flow efficiency on the network when the flow is driven by gradients of a scalar field distributed on the network〔Z. Toroczkai, B. Kozma, K.E. Bassler, N.W. Hengartner and G. Korniss. Gradient Networks, cond-mat/0408262.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Gradient network」の詳細全文を読む スポンサード リンク
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