Percolation and cascade dynamics of spatial networks with partial dependency

Abstract

Recently, it has been shown that the removal of a random fraction of nodes from a system of interdependent spatial networks can lead to cascading failures which amplify the original damage and destroy the entire system, often via abrupt first-order transitions. For these distinctive phenomena to emerge, the interdependence between networks need not be total. We consider here a system of partially interdependent spatial networks (modelled as lattices) with a fraction q of the nodes interdependent and the remaining 1 − q autonomous. In our model, the dependency links between networks are of geometric length less than r. Under full dependency (q = 1), this system was shown to have a first-order percolation transition if r > r c ≈ 8. Here, we generalize this result and show that for all q > 0, there will be a first-order transition if r > r c (q). We show that r c (q) increases monotonically with decreasing q and lim q→0 + r c (q) = ∞. Additionally, we present a detailed description and explanation of the cascading failures in spatially embedded interdependent networks near the percolation threshold p c. These failures follow three mechanisms depending on the value of r. Below r c the system undergoes a continuous transition similar to standard percolation on a lattice. Above r c there are two distinct first-order transitions for finite and infinite r, respectively. The cascading failure for finite r is characterized by the emergence of a critical hole which then spreads through the system while the infinite r transition is more similar to the case of random networks. Surprisingly, we find that this spreading transition can still occur even if p < p c. We present measurements of cascade dynamics which differentiate between these phase transitions and elucidate their mechanisms. These results extend previous research on spatial networks to the more realistic case of partial dependency and shed new light on the specific dynamics of dependencydriven cascading failures.

Interdependent resistor networks with process-based dependency

“…As the system approaches criticality, the green nodes (dead ends) far outnumber the white nodes (backbone) [5,6]. than r in each direction, in line with the models used in [19][20][21], figure 1(a). This means that a node at location (i, j) in lattice A can depend on the continued functioning of a node in lattice B located at ′ ′ i j ( , )only if | − ′| ⩽ i i r and | − ′| ⩽ j j r. Of the + r (2 1) 2 nodes that fulfill this condition for any given node, one is chosen uniformly at random.…”

“…The phase transition is second-order for < r r c , and p c increases monotonically with r. For ⩾ ≈ r r 4 c , the transition is first order and p c decreases asymptotically towards its value for = ∞ r , see figures 2(a) and 3(a). In contrast, fully interdependent lattices with structural dependency undergo a first-order phase transition only when > ≈ r r 8 c [19,20]. In all cases, the p c of process-based dependency is substantially higher (more vulnerable) than structural dependency, due to the fact that the current-carrying backbone is always smaller than the GCC, see figure 3(a).…”

“…This oneto-one correspondence means that the number of functional nodes in each network is identical. We note, though, that even if only a fraction of the nodes were interdependent (as in [18,19]), the size of the backbone and giant component would be essentially the same in both systems, due to symmetry. The dependency links are assigned uniformly at random with the restriction that they connect nodes with horizontal distance not longer Rather they represent the fact that a node in one network can carry current only if the node that it depends on in the other network carries current.…”

Networks of networks – An introduction

“…When it exists, the networks preserve their functionality, and when it does not exist, the networks split into fragments so small they cannot function on their own. The percolation processes on other network of networks topologies have been presented and discussed in [7,9,12,[14][15][16][17][18][19][20].…”

Multi-Universality and Localized Attacks in Spatially Embedded Networks

“…Studies on spatially embedded interdependent networks found that in many cases they are significantly more vulnerable then non-embedded systems [27]. Further studies deal with similar models but restrict the dependency links to be of limited length [28][29][30][31].…”