Optimal redundancy against disjoint vulnerabilities in networks

Redundancy is commonly used to guarantee continued functionality in networked systems. However, often many nodes are vulnerable to the same failure or adversary. A “backup” path is not sufficient if both paths depend on nodes which share a vulnerability. For example, if two nodes of the Internet cannot be connected without using routers belonging to a given untrusted entity, then all of their communication–regardless of the specific paths utilized–will be intercepted by the controlling entity. In this and many other cases, the vulnerabilities affecting the network are disjoint: each node has exactly one vulnerability but the same vulnerability can affect many nodes. To discover optimal redundancy in this scenario, we describe each vulnerability as a color and develop a “color-avoiding percolation” which uncovers a hidden color-avoiding connectivity. We present algorithms for coloravoiding percolation of general networks and an analytic theory for random graphs with uniformly distributed colors including critical phenomena. We demonstrate our theory by uncovering the hidden color-avoiding connectivity of the Internet. We find that less well-connected countries are more likely able to communicate securely through optimally redundant paths than highly connected countries like the US. Our results reveal a new layer of hidden structure in complex systems and can enhance security and robustness through optimal redundancy in a wide range of systems including biological, economic and communications networks.