# Robustness of a Network Formed of Spatially Embedded Networks

###### Abstract

We present analytic and numeric results for percolation in a network formed of interdependent spatially embedded networks.
We show results for a treelike and a random regular network of networks each with unconstrained interdependent links and interdependent links restricted to a maximum length, . Analytic results are given for each network of networks with unconstrained dependency links and compared with simulations. For the case of two spatially embedded networks it was found that only for does the system undergo a first order phase transition.
We find that for treelike networks of networks significantly decreases as increases and rapidly reaches its limiting value, .
For cases where the dependencies form loops, such as in random regular networks, we show analytically and confirm through simulations, that there is a certain fraction of dependent nodes, , above which the entire network structure collapses even if a single node is removed. This decreases quickly with , the degree of the random regular network of networks.
Our results show the extreme sensitivity of coupled spatial networks and emphasize the susceptibility of these networks to sudden collapse.
The theory derived here can be used to find the robustness of any network of networks where the profile of percolation of a single network is known.

Keywords: mathematical and numerical analysis of networks, network stability under perturbation and duress, network percolation, interdependent networks

## I Introduction

As network science has expanded researchers have become aware of the fact that systems often consist of multiple interdependent networks rosato-criticalinf2008 (); parshani-prl2010 (); parshani-epl2010 (); shao-pre2011 (); leichtdsouza2009 (); cellai-pre2013 (); brummitt-pnas2012 (); xu-epl2011 (); hao2011interaction (); PhysRevE.83.056208 (); PhysRevE.84.026101 (); huang-pre2011 (); gao-prl2011 (); hu-pre2011 (); bashan-pre2011 (); parshani-pnas2011 (); bashan-jsp2011 (); PhysRevE.85.066134 (); buldyrev-nature2010 (); gao-naturephysics2012 (); vespignani-nature2010 (); dong-preprint2012 (); bashan-naturephysics2013 (); rinaldi-ieee2001 (); peerenboom-proceedings2001 (); radicchi-naturephysics2013 (); son-epl2012 (); serrano-pre2012 (); morris-prl2012 (); zhao-jstatmech2013 (); donges-epjb2011 (); gao-general-net (). Examples of such systems are power grids that depend on communication networks, individuals who participate in multiple social circles, and metabolic networks that depend on other biological functions. Previous research on networks of networks provided a mathematical framework for understanding the stability of these systems gao-prl2011 (); PhysRevE.85.066134 (); gao-naturephysics2012 (); leichtdsouza2009 (). They found that these systems undergo a first-order percolation transition rather than the second-order transition which occurs for single networks. Recent work expanded the idea of interdependent networks to a pair of spatially embedded networks wei-prl2012 (); bashan-naturephysics2013 (). This represents an important step because many interdependent systems are spatially embedded brummitt-pnas2012 (); rinaldi-ieee2001 (); agarwal-milcom2010 (); wang-proceedings2008 (); albert-pre2004 (); kinney-epjb2005 (); chassin-physa2005 (); hines-chaos2010 (); carreras-ieee2004 (); barthelemy-physicsreports2011 ().

In our model, for each pair of connected networks a fraction of nodes in network are assigned a dependent node in network . The dependencies either follow the ”no feedback condition”, where if node in network depends on node in network then depends on as well or the ”feedback condition” where such a constraint is not enforced gao-general-net (). For our simulations and theory we applied the ”no feedback condition” and note that the ”feedback condition” leads to an even more sensitive system.

When the dependencies can take on various topologies. We show some examples of possible configurations in Fig. 1.

At each step a fraction of the nodes are removed from either one or all of the networks. This leads to a dynamic cascade where each node removed causes dependent nodes to be removed in the other networks. The fraction of surviving nodes at the end of the cascade is defined as . We then find , the mutual giant connected component where all remaining nodes are in their networks’ respective giant component and where the dependencies of all nodes remaining are also still functional.

In order to simplify the study, we follow the method of previous studies wei-prl2012 (); bashan-naturephysics2013 () where square lattices were used as the model of choice and note that any other 2D spatially embedded network with finite connectivity links belongs to the same universality class bunde1991fractals (). In the case of spatially embedded networks the dependencies are often restricted such that dependent nodes are within some distance, , of one another wei-prl2012 (). This quantity is called the dependency length and forces two dependent nodes and , with positions and to obey and where and represent the positions of the nodes. The case is used to describe a network of networks with no restrictions on the maximum length of the dependency links. Previous work on interdependent spatially embedded networks involved only two networks and explored the interdependent fraction of the networks, , and the maximum length of the dependency links, danziger_interdependent (). For each value of there is a critical dependency length, , for which the percolation transition shifts from second order to first order. For , i.e. two fully interdependent networks it was found and for lower values of there is a higher value of danziger_interdependent (). It is of interest whether a higher number of interdependent systems could lead to a case where coupling between nearest or next nearest neighbors also leads to a first order transition.

In line with previous research on networks of networks gao-general-net (); gao-prl2011 (); PhysRevE.85.066134 (); gao-naturephysics2012 () , we show simulation results for a network of networks with treelike dependencies and for a random regular network of networks, each with no restrictions on the length of dependency links and dependency links of a maximum finite length, . We also derive a theory for the size of the giant component of an interdependent system with no restrictions on the length of dependency links as a function of the system parameters and the percolation profile of a single lattice derived numerically. In our case this profile is obtained from the percolation profile of a square lattice averaged over realizations. While we apply these equations to a lattice, we note that they can be used for any system composed of identical networks if is known for the percolation of a single network.

## Ii Interdependent Spatialy Embedded Networks with Tree-like Dependencies

### ii.1 Dynamics of cascading failures for a treelike network of networks

We begin by examining results for a treelike network of networks where the length of dependency links is unconstrained. Li et al. wei-prl2012 () derived of the cascading failure of two interdependent networks as a function of iteration count. It can be shown that if a fraction of nodes are removed from each network and is the fraction of survived nodes at the th iteration . For networks in a treelike configuration a node is in the mutual giant component if it and the nodes it depends on are all in their resepective networks’ giant components. Thus , the probability for a node to be in the giant component after fraction of nodes are removed, must be raised to the power since each node has dependencies. This gives

(1) |

Each iteration represents reducing all networks to their giant components and removing nodes which have dependencies outside the giant component. The next iteration then factors in the nodes removed due to having dependencies outside the previous giant component and again reduces each network to its giant component. The process repeats until a steady state is reached. In the limiting case of only a single network, , we get and there is no cascading effect. Further if , we obtain the known result wei-prl2012 ().

An alternate method of counting involves observing how the failures propogate across the links in the network of networks PhysRevE.85.066134 (). The initial attack on each network occurs at . A node which depends on a failed node then fails at and in general for a node that failed at , its dependent nodes fail at . Simulations of the giant component after a number of iterations and at a certain time are shown to fit well with the theoretical equations in Fig. 2.

### ii.2 Size of the giant component after the cascade

We examine what happens to when and the system reaches steady state. We define and note that represents the total fraction of nodes removed after the cascade including those removed due to interdependencies. For a given fraction of nodes, , removed from the network of networks, is the fraction that would have to be removed from a single network to obtain an equivalent giant component. Solving for we get

(2) |

In Fig. 3 we observe that the theory agrees with simulations for all values of . We also see there that close to , the percolation threshold, the system collapses through a long cascade. The number of iterations at and both increase as the number of networks increases.

To calculate we must find where the two sides of Eq. (2) are tangent at their intersection. We take the derivatives of both sides and get

(3) | ||||

(4) |

where is the derivative of .

### ii.3 Effect of , on for a starlike network of spatial networks

If we restrict the shape of the tree to be in a star formation with unrestricted dependency links (see Fig. 1) we can also give an analytic solution for any value of the coupling . In this case a fraction of the nodes are removed only from the central network. Based on Gao et al. gao-prl2011 () the equations for this system are

(8) | ||||

where the subscript refers to the central network and the subscript refers to all the other networks. Results of theory and simulations for and can be seen in Fig. 4. If Eq. (8) reduces to Eq. (2).

### ii.4 Interdependent lattices with treelike dependencies with finite

We now examine a network of networks with treelike dependencies but now with a finite maximum dependency length, i.e. . Due to spatial constrains, theory becomes a difficult task and we limit ourselves to the simulations. We remove a fraction of the nodes from each network and note that the results can be converted to a case where nodes are removed from just a single network using . Previous research on a pair of interdependent lattices found that displays rich behavior as is increased wei-prl2012 (); danziger_interdependent (). When is small, increases linearly with until it reaches a peak. This peak represents the point where the system changes from a second order transition to a first order transition. As increases past the peak decreases and approaches its limiting value at .

First we analyze the giant component as a function of for different numbers of networks in a tree. In Fig. 4(a) we observe that the system now undergoes a first order transition even for if there are a sufficient number of networks. Our simulations reveal that the results are the same (for ) regardless of whether the network of networks is in a star or a line, i.e. the results are independent of the shape of the tree. From the peak of the graph vs. in Fig. 4(b), we are able to determine , the critical dependency length where the collapse becomes first order. In the inset of Fig. 4(b) we observe that this critical dependency length decreases significantly as we increase the number of networks in the tree. When , we get which is its limiting value.

## Iii Interdependent Lattices with Random Regular Dependencies

### iii.1 Random regular network of spatial networks with random dependencies

Many real networks contain loops and thus we now derive results for a random regular network of spatial networks. Gao et al. gao-naturephysics2012 () previously showed that for such a dependency configuration the actual number of networks, is irrelevant, rather the results depend only on the number of networks each network depends on. In this case if all nodes are interdependent () the network collapses immediately due to the loops. We therefore limit our analysis to and remove a fraction of the nodes from all the networks. Gao et al. gao-naturephysics2012 () obtained

(9) |

where represents the percolation damage from all networks except from the dependency link currently being examined. The system in Eq. (9) can be solved by eliminating from the second equation and obtaining a single equation for . After substituting we obtain

### iii.2 Maximum coupling,

From the graph of in Fig. 6(a) it is clear that for each value of there is some maximum coupling for which removing even a single node will lead the entire network to collapse. We can solve for by using Eq. (10) and setting . This gives

(13) |

We can then solve for using Eq. (11). Explicitly,

### iii.3 Interdependent lattices with random regular dependencies with finite

We now analyze random regular networks where dependency links are of a finite maximum length, . We observe in Fig. 8 that the shape of the vs. curve is the same as it was for trees and the transition switches from second order to first order above which is the value of when is at a maximum. As the number of neighbors, , increases the critical dependency length , decreases significantly for any value of . Further, for high values of the system collapses even for and . Thus a system with as few as networks, fully interconnected (), is extremely sensitive if there is a high level of interdependency. In Fig. 8 we observe that decreases as increases and also as increases.

Continuing the analysis that was done in Chap. III, section B, for , we explore for finite , the values of , the maximum coupling above which the system collapses. As was the case for infinite , we find that for systems with finite drops rapidly as is increased. Further the difference for a given between with finite and with decreases as increases. For low values of introducing a finite leads to an increase in yet for there is almost no difference in for and for . Thus networks of spatial networks with many dependencies are extremely sensitive even for small and small .

## Iv Discussion

In summary, we have applied the framework for a network of networks to the case of spatially embedded networks. We provide analytic results for dependencies in a tree and a random regular configuration with no restrictions on the length of the dependency links. Further we provide simulations for these two cases and find excellent agreement with the theory. We also studied simulations for each case when dependency links are of a finite length, . For networks in a tree configuration we find that the critical dependency length for a first order transition decreases significantly as more networks are added and with enough interdependent networks the system undergoes a first order transition even for , i.e. nearest neighbor dependencies.

If the dependencies contain loops we find that there is a critical value of coupling above which the system will collapse even if a single node is removed. This decreases significantly as the number of dependencies of each network increases and for high values of this is virtually unaffected by imposing a finite length on the dependency links.

These results emphasize the vulnerability of interdependent spatially embedded networks and show that many colocalized interacting systems can collapse suddenly. Our model here can help explain sudden failures seen in many real-world systems such as powergrids.

## Acknowledgments

We acknowledge the European EPIWORK and MULTIPLEX (EU-FET project 317532) projects, the Deutsche Forschungsgemeinschaft (DFG) and the Israel Science Foundation, for providing ffinancial support.

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