Constructing Networks for Cascade Resilience

Complex socio-economic networks such as transportation networks, information systems and even underground organizations are often designed for resilience to be able to function even if some of the nodes (e.g. hubs, routers, operatives etc.) are compromised by a human or natural adversary. In many cases the adversary threatens to cause a cascade where the failure of one node leads to some of the adjacent nodes being lost as well, and then many more nodes lost in a far-reaching domino e ect. Such cascades motivate the search for mechanisms and network designs that would increase network cascade resilience. This work introduces a mathematical model in which such networks are viewed as the solution to an optimization problem that trades o cascade resilience against e ciency, and describes the optimal networks under this model. The results show that of the network designs considered, a network consisting of multiple star-like cells is optimal. Also, perhaps surprisingly it was found that in many network designs and parameter values, edge density of the optimal network topology does not monotonically decrease when cascade risk increases, but may increase again when cascade risk is high. This implies that certain networks ought not to be modi ed for cascade resilience, since the cost in e ciency is too high. Understanding cascade resilience and its structural phase transitions will ultimately help identify vulnerabilities and design more durable networks in many diverse application areas.