Exploring Watts’ Cascade Boundary

! z . In this paper we study cascades on finite networks in this upper boundary region via simulations. We distinguish two kinds of cascades: total network cascades (or TNCs) that essentially consume the entire finite network, and cascades corresponding to those predicted by percolation theory that consume only the initially struck vulnerable clusters and possibly a few others. We show that the experimental upper boundary found by Watts corresponds to TNCs. TNCs can start when, for example, as few as 2 vulnerable nodes in a network of 10000 nodes are flipped initially by a single seed and no cluster of vulnerable nodes larger than 21 exists. The mechanism by which these cascades start and grow near the upper boundary is not described by percolation theory because the nodes involved form a densely connected subnetwork that is not tree-like. Instead a different mechanism is involved in which a particular motif comprising patterns of linkages between vulnerable and stable nodes must be present in sufficient quantity to allow the cascade to hop from the initially struck clusters to others. While the clusterhopping mechanism is necessary to start the cascade, the later emergence of a TNC requires the presence of relatively large clusters of vulnerable nodes. The shrinkage of the size of these largest clusters relative to network size is a major factor in the disappearance of TNCs as finite networks with the same