Gradient network

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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.

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 setof edges E = . Given a node i, we can define its set of neighbors in G by S<sub>i</sub><sup>(1)</sup> = .

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<sub>i</sub> associated to it.

Gradient ∇h<sub>i</sub> on a network: ∇h<sub>i</sub><math>= </math>(i, μ(i))i.e. the directed edge from i to μ(i) , where μ(i) ∈ S<sub>i</sub><sup>(1)</sup> ∪ , and h<sub>μ</sub> has the maximum value in <math></math>.

Gradient network : ∇<math>G = </math> ∇<math>G </math> <math> (V, F) </math>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...
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