  We study partition of networks into basins of attraction based on a steepest
ascent search for the node of highest degree. Each node is associated with, or
"attracted" to its neighbor of maximal degree, as long as the degree is
increasing. A node that has no neighbors of higher degree is a peak, attracting
all the nodes in its basin. Maximally random scale-free networks exhibit
different behavior based on their degree distribution exponent $\gamma$: for
small $\gamma$ (broad distribution) networks are dominated by a giant basin,
whereas for large $\gamma$ (narrow distribution) there are numerous basins,
with peaks attracting mainly their nearest neighbors. We derive expressions for
the first two moments of the number of basins. We also obtain the complete
distribution of basin sizes for a class of hierarchical deterministic
scale-free networks that resemble random nets. Finally, we generalize the
problem to regular networks and lattices where all degrees are equal, and thus
the attractiveness of a node must be determined by an assigned weight, rather
than the degree. We derive the complete distribution of basins of attraction
resulting from randomly assigned weights in one-dimensional chains.
