 
Summary: Subgraphs in random networks
S. Itzkovitz,1,2
R. Milo,1,2
N. Kashtan,2,3
G. Ziv,1
and U. Alon1,2
1
Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel
2
Department of Molecular Cell Biology, Weizmann Institute of Science, Rehovot 76100, Israel
3
Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel
Received 18 February 2003; published 25 August 2003
Understanding the subgraph distribution in random networks is important for modeling complex systems. In
classic Erdos networks, which exhibit a Poissonian degree distribution, the number of appearances of a
subgraph G with n nodes and g edges scales with network size as G Nn g
. However, many natural
networks have a nonPoissonian degree distribution. Here we present approximate equations for the average
number of subgraphs in an ensemble of random sparse directed networks, characterized by an arbitrary degree
sequence. We find scaling rules for the commonly occurring case of directed scalefree networks, in which the
