Breakdown of the Internet under Intentional Attack

Abstract

We study the tolerance of random networks to intentional attack, whereby a fraction p of the most connected sites is removed. We focus on scale-free networks, having connectivity distribution P(k) approximately k(-alpha), and use percolation theory to study analytically and numerically the critical fraction p(c) needed for the disintegration of the network, as well as the size of the largest connected cluster. We find that even networks with alpha < or = 3, known to be resilient to random removal of sites, are sensitive to intentional attack. We also argue that, near criticality, the average distance between sites in the spanning (largest) cluster scales with its mass, M, as square root of [M], rather than as log (k)M, as expected for random networks away from criticality.

Describing and Simulating Internet Routes

“…Moreover, most of the short paths between pairs of nodes in these networks tend to pass through the highest degree nodes. Actually, almost all paths (not only short ones) tend to pass through these nodes, which make them essential for network connectivity, see for instance [26,27,28,29,30]. These observations lead us to ask how the node degree evolves along a route.…”

Networks formed from interdependent networks

“…The robustness of a network is usually either characterized by the value of the critical threshold analysed using percolation theory 52 or defined as the integrated size of the largest connected cluster during the entire attack process 53 . The percolation approach was also proved to be extremely useful in addressing other scenarios, such as efficient attacks or immunization 6,7,54,55 , and for obtaining optimal paths 56 as well as for designing robust networks 53 . Network concepts have also proven to be useful for the analysis and understanding of the spread of epidemics 57,58 , and the organizational laws of social interactions, such as friendships 59,60 or scientific collaborations 61,62 .…”

Identification of influential spreaders in complex networks

“…Indeed, people are connected according to the way they interact with one another in society and the large heterogeneity of the resulting network greatly determines the efficiency and speed of spreading. In the case of networks with a broad degree distribution (number of links per node) 6 , it is believed that the most connected people (hubs) are the key players, being responsible for the largest scale of the spreading process [6][7][8] . Furthermore, in the context of social network theory, the importance of a node for spreading is often associated with the betweenness centrality, a measure of how many shortest paths cross through this node, which is believed to determine who has more 'interpersonal influence' on others 9,10 .…”