Fig 1.
Principles of network rewiring.
(A) Adaptive rewiring. The lightness of a node’s color represents the intensity of its communication with the white node. The darker the color, the more intense the communication is (B) Minimization of wiring distance. (C) Alignment to an external vector field. The red and green arrows indicate the rewiring link and the direction of the vector field respectively.
Fig 2.
Schema of a convergent-divergent unit.
In a convergent-divergent unit, a convergent hub collects inputs and passes the information to a divergent hub through a subnetwork of intermediate nodes. The nodes sending information to the convergent hub are referred as source nodes, and those receiving information from the divergent hub as target nodes. Note that typically the source and target nodes can show overlap, i.e., a node can be both a source and a target node.
Fig 3.
Schema of the adjacency matrix.
The elements of the adjacency matrix are the weights of links. Each row of the adjacency matrix contains the weights of in-links for the corresponding node, and the number of nonzero entries is its in-degree. Similarly, each column carries the weights of out-links, and the number of nonzero entries is the out-degree.
Fig 4.
Rewiring based on the functional principle develops winner-take-all configurations, based on the distance principle forms clusters, and based on the wave principle aligns the connections with the latent field.
(A) Evolution of the adjacency matrix driven by the functional principle only. (B) Evolution of the network spatial layout driven by the distance principle only. (C) Evolution of the network spatial layout driven by the wave principle only when the wave propagates laterally. In all cases, we either rewire the out-links (pin = 0 case) or the in-links (pin = 1 case). Link weights follow the normal distribution.
Fig 5.
Random rewiring enhances connectedness and increases the number of hubs when rewiring includes both advection and consensus (0<pin<1).
(A) The proportion of connected node pairs, (B) average efficiency, (C) proportion of convergent hubs, and (D) proportion of divergent hubs as a function of prandom, for different pin.
Fig 6.
prandom controls the formation, connectedness, and degree of isolation of convergent-divergent units.
(A) Proportion of steps with no convergent-divergent unit in the network, (B) number of convergent-divergent units in rewired networks, (C) proportion of source nodes, target nodes and their overlap, (D) proportion of nodes in intermediate subgraphs and (F) density of intermediate subgraphs as a function of prandom. The black horizontal line is the density of the whole digraph.
Fig 7.
Distance-based rewiring has similar effects on the connectedness and the number of hubs as random rewiring.
(A) Proportion of connected node pairs, (B) average efficiency, (C) proportion of convergent hubs, and (D) proportion of divergent hubs as a function of pdistance, for different probabilities of in-link rewiring, pin.
Fig 8.
pdistance, controls the formation, connectedness, and degree of isolation of convergent-divergent units.
(A) Proportion of steps with no convergent-divergent unit in the network, (B) number of convergent-divergent units in rewired networks, (C) proportion of source nodes, target nodes and their overlap, (D) proportion of nodes in intermediate subgraphs and (E) density of intermediate subgraphs as a function of pdistance. The black horizontal line represents the density of the whole digraph.
Fig 9.
The way the wave principle affects the formation of convergent-divergent units depends on the underlying field.
(A) Spatial layout of a network evolved with a lateral field and (E) with a radial field. Green arrows indicate the direction of the underlying field. The proportion of in-link rewiring is 0.5, and (pfunction, pdistance, pwave) is (0.4,0.3,0.3). (B-D) The proportion of connected node pairs, average efficiency, and the proportion of steps with no convergent-divergent unit in the network, as a function of the distance-based principle, pdistance, with a lateral field, and (F-H) with a radial field.
Fig 10.
Stochastic adaptive rewiring reduces the number of steps with no convergent-divergent unit is in the network.
The proportion of in-link rewiring, pin, is 0.5. The proportion of steps with no convergent-divergent unit in the network, as a function of (A) prandom, and (B) pdistance.