 
Summary: Modularity from fluctuations in random graphs and complex networks
Roger Guimerà, Marta SalesPardo, and Luís A. Nunes Amaral
Department of Chemical and Biological Engineering, Northwestern University, Evanston, Illinois 60208, USA
(Received 15 March 2004; published 19 August 2004)
The mechanisms by which modularity emerges in complex networks are not well understood but recent
reports have suggested that modularity may arise from evolutionary selection. We show that finding the
modularity of a network is analogous to finding the groundstate energy of a spin system. Moreover, we
demonstrate that, due to fluctuations, stochastic network models give rise to modular networks. Specifically, we
show both numerically and analytically that random graphs and scalefree networks have modularity. We argue
that this fact must be taken into consideration to define statistically significant modularity in complex networks.
DOI: 10.1103/PhysRevE.70.025101 PACS number(s): 89.75.Hc, 02.60.Pn, 05.50. q, 89.75.Fb
Statistical, mathematical, and modelbased analysis of
complex networks have recently uncovered interesting uni
fying patterns in networks from seemingly unrelated disci
plines [15]. In spite of these advances, many properties of
complex networks remain elusive, a prominent one being
modularity [6,7]. For example, it is a matter of common
experience that social networks have communities of highly
interconnected nodes that are poorly connected to nodes in
other communities. Such modular structures have been re
