  Attacks on networks is a very important issue in developing strategies of
eradicating spreads of malicious phenomena in networks, such as epidemics and
fake information, which are generally named in the research networks
immunization. The traditional approach of evaluating the effectiveness of
attacks on networks, is focused on measuring some macro parameters related to
an entire attack, such as the critical probability of a percolation occurrence
in the network $p_c$ and the relative size of the largest component in the
network - the giant component, but not considering the attack on a micro
perspective, which is the analysis of the node removals, during an attack,
themselves, their characteristics and results. In this paper we present and
apply the last method of focusing on the micro scale of an attack. Based on the
theory of percolation in networks, we analyze the phenomenon of an avalanche
results due to a single node removal from a network, that is a state in which a
removal of one node belongs to the network's giant component causes other nodes
to be also disconnected from it, which is a much greater contribution to the
fragmentation (immunization) of the network than only the single original node
removal itself. Specifically, we focus ourselves on the parameter of the size
of an avalanche, which is the number of nodes that are disconnected from the
giant component due to a single node removal. Relating to a random attack on a
network of the type of Cayley tree, we derive analytically the distribution of
the sizes of avalanches occur during the entire attack on it, until the network
is dismantled (immunized) and the attack is terminated.
