Gradient network
In network science, a gradient network is a directed subnetwork of an undirected "substrate" network where each node has an associated scalar potential and one out-link that points to the node with the smallest (or largest) potential in its neighborhood, defined as the union of itself and its neighbors on the substrate network.
[1] Transport takes place on a fixed network
Given a node i, we can define its set of neighbors in G by Si(1) = {j ∈ V | (i,j)∈ E}.
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.
[3] The concept of a gradient network was first introduced by Toroczkai and Bassler (2004).
[4][5] Generally, real-world networks (such as citation graphs, the Internet, cellular metabolic networks, the worldwide airport network), which often evolve to transport entities such as information, cars, power, water, forces, and so on, are not globally designed; instead, they evolve and grow through local changes.
For example, if a router on the Internet is frequently congested and packets are lost or delayed due to that, it will be replaced by several interconnected new routers.
[2] Moreover, this flow is often generated or influenced by local gradients of a scalar.
][needs update] investigates the connection between network topology and the flow efficiency of the transportation.
[2] In a gradient network, the in-degree of a node i, ki (in) is the number of gradient edges pointing into i, and the in-degree distribution is
(independent identically distributed) the exact expression for R(l) is given by In the limit
[3] Furthermore, if the substrate network G is scale-free, like in the Barabási–Albert model, then the gradient network also follow the power-law with the same exponent as those of G.[2] The fact that the topology of the substrate network influence the level of network congestion can be illustrated by a simple example: if the network has a star-like structure, then at the central node, the flow would become congested because the central node should handle all the flow from other nodes.
However, if the network has a ring-like structure, since every node takes the same role, there is no flow congestion.
Under assumption that the flow is generated by gradients in the network, flow efficiency on networks can be characterized through the jamming factor (or congestion factor), defined as follows: where Nreceive is the number of nodes that receive gradient flow and Nsend is the number of nodes that send gradient flow.
, for an Erdős–Rényi random graph, the congestion factor becomes This result shows that random networks are maximally congested in that limit.
[7] Zonghua Liu et al. (2006) showed that congestion is more likely to occur at the nodes with high degrees in networks, and an efficient approach of selectively enhancing the message-process capability of a small fraction (e.g. 3%) of nodes is shown to perform just as well as enhancing the capability of all nodes.
[7] Ana L Pastore y Piontti et al. (2008) showed that relaxational dynamics[clarification needed] can reduce network congestion.
[8] Pan et al. (2011) studied jamming properties in a scheme where edges are given weights of a power of the scalar difference between node potentials.
[9][clarification needed] Niu and Pan (2016) showed that congestion can be reduced by introducing a correlation between the gradient field and the local network topology.
