Criticality of forcing directions on the fragmentation and resilience of grid networks

Abstract

A general framework for probing the dynamic evolution of spatial networks comprised of nodes applying force amongst each other is presented. Aside from the already reported magnitude of forces and elongation thresholds, we show that preservation of links in a network is also crucially dependent on how nodes are connected and how edges are directed. We demonstrate that the time it takes for the networks to reach its equilibrium network structure follows a robust power law relationship consistent with Basquin's law with an exponent that can be tuned by changing only the force directions. Further, we illustrate that networks with different connection structures, node positions and edge directions have different Basquin's exponent which can be used to distinguish spatial directed networks from each other. Using an extensive waiting time simulation that spans up to over 16 orders of magnitude, we establish that the presence of memory combined with the scale-free bursty dynamics of edge breaking at the micro level leads to the evident macroscopic power law distribution of network lifetime.

Figure 1. Network architecture and index conventions.

(a) represents a lattice network with uniform force directions and (b) random force directions. In (c) and (d) nodes are arranged spatially in a grid and [(M − 1) × N] + [M × (N − 1)] edges are randomly chosen from possible connections. (c) represents an ER network where forces are directed with the highest level of uniformity and (d) randomly. Edge colors correspond to the elongation thresholds while node colors are based on the degree of incoming connections k_in. Node sizes correspond to the degree of outgoing connections k_out (largest node has higher k_out). Arrows are directed from source to target node. Figures in the last column highlight the edges of the shown selected nodes.

Figure 2. Network architecture and degree distribution of Barabasi-Albert networks with uniform and random force directions.

Almost all edges were directed outward for the uniform case while comparable number of edges were directed inward and outward for the random case. In the inset, nodes are sized according to k_out.

Figure 3. Evolution of a lattice network.

(a) For uniform force directions and F_app = 0.03, breaking starts near the top and left sides at t = 6,384.0. Failure cascades and last to be disconnected are edges near the bottom right corner where the least amount of force is felt. The network totally collapses at t = 6,602.4. No connections remain and the network elongates and moves diagonally following the net force. (b) For random force directions and F_app = 0.03, breaking starts at t = 11.0 at random positions. The network reaches its equilibrium structure at t = 238.6 with only 0.496 ± 0.007 of the edges disconnected.

Figure 4. Fraction of disconnected edges f in grid networks.

(a) Lattice networks with uniform force directions totally collapse while for those with (b) random force directions, 0.503 ± 0.006 of the edges disconnects. (c–d) Regardless of the magnitude and directions of {F_app}, an ER network experiences total system collapse. (e–f) BA networks completely fragment regardless of the magnitude and directions of {F_app}. The insets show the derivative of f or the disconnection rate of the edges. The rate of disconnection f′ at the transition point decreases with F_app in all networks. For lattice networks, randomizing force directions further decreased the rate of disconnections.

Figure 5. Evolution of an Erdos-Renyi network.

(a) For uniform force directions and F_app = 0.8, breaking starts with nodes disconnecting from the influential nodes at the bottom resulting to 77.6% of the edges to break instantly. By t = 100, 95.7% of the edges have disconnected. The remaining edges take far longer to disconnect. (b) For random force directions, breaking starts at random positions. Initial disconnections account for 79.6% of the edges. By t = 100.0, 97.0% of edges have disconnected. The remaining 3.0% breaks slower until the network completely collapses at t = 7,273.2.

Figure 6. Evolution of a Barabasi-Albert network.

(a) For uniform force directions and F_app = 0.8, 75.9% of the edges fail instantly and by t = 128.0, 96.7% of the edges has disconnected. The remaining edges have all disconnected by t = 28, 692.8. (b) For random force directions and F_app = 0.8, 75.9% of the edges fail instantly with only 1.9% of the edges remaining by t = 128.0. The network completely fragments at t = 22, 773.6.

Figure 7. The final fraction of disconnected edges in a lattice network with varying degree of force directions randomness.

The relationship is found to be linear with a slope of −0.51.

Figure 8. Lifetime of grid networks.

Dashed lines in (a) are to illustrate the Basquin's exponent of the networks. (b) shows that the macro-scale network lifetime results from microscopic scale-free waiting times between bursts. The lines shown are the range of exponents of the power-law distribution.