  We study the statistical behavior under random sequential
renormalization(RSR) of several network models including Erd"os R'enyi (ER)
graphs, scale-free networks and an annealed model (AM) related to ER graphs. In
RSR the network is locally coarse grained by choosing at each renormalization
step a node at random and joining it to all its neighbors. Compared to previous
(quasi-)parallel renormalization methods [C.Song et.al], RSR allows a more
fine-grained analysis of the renormalization group (RG) flow, and unravels new
features, that were not discussed in the previous analyses. In particular we
find that all networks exhibit a second order transition in their RG flow. This
phase transition is associated with the emergence of a giant hub and can be
viewed as a new variant of percolation, called agglomerative percolation. We
claim that this transition exists also in previous graph renormalization
schemes and explains some of the scaling laws seen there. For critical trees it
happens as N/N0 -> 0 in the limit of large systems (where N0 is the initial
size of the graph and N its size at a given RSR step). In contrast, it happens
at finite N/N0 in sparse ER graphs and in the annealed model, while it happens
for N/N0 -> 1 on scale-free networks. Critical exponents seem to depend on the
type of the graph but not on the average degree and obey usual scaling
relations for percolation phenomena. For the annealed model they agree with the
exponents obtained from a mean-field theory. At late times, the networks
exhibit a star-like structure in agreement with the results of Radicchi et. al.
While degree distributions are of main interest when regarding the scheme as
network renormalization, mass distributions (which are more relevant when
considering 'supernodes' as clusters) are much easier to study using the fast
Newman-Ziff algorithm for percolation, allowing us to obtain very high
statistics.
