Network robustness to targeted attacks. The interplay of expansibility and degree distribution

We study the property of certain complex networks of being both sparse and highly connected, which is known as good expansion (GE). A network has GE properties if every subset S of nodes (up to 50% of the nodes) has a neighborhood that is larger than some expansion factor phi multiplied by the number of nodes in S. Using a graph spectral method we introduce here a new parameter measuring the good expansion character of a network. By means of this parameter we are able to classify 51 real-world complex networks - technological, biological, informational, biological and social - as GENs or non-GENs. Combining GE properties and node degree distribution (DD) we classify these complex networks in four different groups, which have different resilience to intentional attacks against their nodes. The simultaneous existence of GE properties and uniform degree distribution contribute significantly to the robustness in complex networks. These features appear solely in 14% of the 51 real-world networks studied here. At the other extreme we find that similar to 40% of all networks are very vulnerable to targeted attacks. They lack GE properties, display skewed DD - exponential or power-law - and their topologies are changed more dramatically by targeted attacks directed at bottlenecks than by the removal of network hubs.