Universität zu Köln

Lee, J. S., Hwang, S., Yeo, J., Kim, D. and Kahng, B. (2014). Ground-state energy of the q-state Potts model: The minimum modularity. Phys. Rev. E, 90 (5). COLLEGE PK: AMER PHYSICAL SOC. ISSN 1550-2376

Full text not available from this repository.

Abstract

A wide range of interacting systems can be described by complex networks. A common feature of such networks is that they consist of several communities or modules, the degree of which may quantified as the modularity. However, even a random uncorrelated network, which has no obvious modular structure, has a finite modularity due to the quenched disorder. For this reason, the modularity of a given network is meaningful only when it is compared with that of a randomized network with the same degree distribution. In this context, it is important to calculate the modularity of a random uncorrelated network with an arbitrary degree distribution. The modularity of a random network has been calculated [Reichardt and Bornholdt, Phys. Rev. E 76, 015102 (2007)]; however, this was limited to the case whereby the network was assumed to have only two communities, and it is evident that the modularity should be calculated in general with q(>= 2) communities. Here we calculate the modularity for q communities by evaluating the ground-state energy of the q-state Potts Hamiltonian, based on replica symmetric solutions assuming that the mean degree is large. We found that the modularity is proportional to <root k >/< k > regardless of q and that only the coefficient depends on q. In particular, when the degree distribution follows a power law, the modularity is proportional to < k >(-1/2). Our analytical results are confirmed by comparison with numerical simulations. Therefore, our results can be used as reference values for real-world networks.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Lee, J. S. UNSPECIFIED UNSPECIFIED UNSPECIFIED Hwang, S. UNSPECIFIED UNSPECIFIED UNSPECIFIED Yeo, J. UNSPECIFIED UNSPECIFIED UNSPECIFIED Kim, D. UNSPECIFIED UNSPECIFIED UNSPECIFIED Kahng, B. UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-423138
DOI:	10.1103/PhysRevE.90.052140
Journal or Publication Title:	Phys. Rev. E
Volume:	90
Number:	5
Date:	2014
Publisher:	AMER PHYSICAL SOC
Place of Publication:	COLLEGE PK
ISSN:	1550-2376
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language STATISTICAL-MECHANICS; COMMUNITY STRUCTURE Multiple languages Physics, Fluids & Plasmas; Physics, Mathematical Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/42313