# Robustness of a Network of Networks

@article{Gao2011RobustnessOA, title={Robustness of a Network of Networks}, author={Jianxi Gao and S. Buldyrev and S. Havlin and H. Stanley}, journal={Physical review letters}, year={2011}, volume={107 19}, pages={ 195701 } }

Network research has been focused on studying the properties of a single isolated network, which rarely exists. We develop a general analytical framework for studying percolation of n interdependent networks. We illustrate our analytical solutions for three examples: (i) For any tree of n fully dependent Erdős-Rényi (ER) networks, each of average degree k, we find that the giant component is P∞ =p[1-exp(-kP∞)](n) where 1-p is the initial fraction of removed nodes. This general result coincides… Expand

#### 462 Citations

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#### References

SHOWING 1-10 OF 72 REFERENCES

Catastrophic cascade of failures in interdependent networks

- Environmental Science, Computer Science
- Nature
- 2010

This work develops a framework for understanding the robustness of interacting networks subject to cascading failures and presents exact analytical solutions for the critical fraction of nodes that, on removal, will lead to a failure cascade and to a complete fragmentation of two interdependent networks. Expand

Resilience of the internet to random breakdowns

- Physics, Medicine
- Physical review letters
- 2000

This work shows analytically and numerically that for alpha</=3 the transition never takes place, unless the network is finite, and finds that the physical structure of the Internet is impressively robust, with p(c)>0.99. Expand

Interdependent networks: reducing the coupling strength leads to a change from a first to second order percolation transition.

- Physics, Medicine
- Physical review letters
- 2010

It is shown both analytically and numerically that reducing the coupling between the networks leads to a change from a first order percolation phase transition to a second orderpercolation transition at a critical point. Expand

Critical effect of dependency groups on the function of networks

- Computer Science, Physics
- Proceedings of the National Academy of Sciences
- 2010

It is shown that a synergy exists between the failure of connectivity and dependency links that leads to an iterative process of cascading failures that has a devastating effect on the network stability. Expand

Network robustness and fragility: percolation on random graphs.

- Computer Science, Physics
- Physical review letters
- 2000

This paper studies percolation on graphs with completely general degree distribution, giving exact solutions for a variety of cases, including site percolators, bond percolations, and models in which occupation probabilities depend on vertex degree. Expand

Self-similarity of complex networks

- Medicine, Physics
- Nature
- 2005

A power-law relation is identified between the number of boxes needed to cover the network and the size of the box, defining a finite self-similar exponent to explain the scale-free nature of complex networks and suggest a common self-organization dynamics. Expand

Complex Networks: Structure, Robustness and Function

- Computer Science
- 2010

This chapter discusses random network models, which are based on the Erdos-Renyi models, and their application in the context of complex networks, where distances in scale-free networks are small and distances in complex networks are large. Expand

Statistical mechanics of complex networks

- Computer Science, Physics
- ArXiv
- 2001

A simple model based on these two principles was able to reproduce the power-law degree distribution of real networks, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network. Expand

Networks: An Introduction

- Computer Science
- 2010

This book brings together for the first time the most important breakthroughs in each of these fields and presents them in a coherent fashion, highlighting the strong interconnections between work in different areas. Expand

Structure of shells in complex networks.

- Mathematics, Medicine
- Physical review. E, Statistical, nonlinear, and soft matter physics
- 2009

A network correlation function c(rl) identical with rl/phi(rl-1) to characterize the correlations in the network is introduced, where rl is the empirical value and phi(rl)-1 is the theoretical prediction, which indicates perfect agreement between empirical results and theory. Expand