title: Modularity and anti-modularity in networks with arbitrary degree distribution.
abstract: BACKGROUND
  Much work in systems biology, but also in the analysis of social network and communication and transport infrastructure, involves an in-depth analysis of local and global properties of those networks, and how these properties relate to the function of the network within the integrated system. Most often, systematic controls for such networks are difficult to obtain, because the features of the network under study are thought to be germane to that function. In most such cases, a surrogate network that carries any or all of the features under consideration, while created artificially and in the absence of any selective pressure relating to the function of the network being studied, would be of considerable interest.
  
  
  RESULTS
  Here, we present an algorithmic model for growing networks with a broad range of biologically and technologically relevant degree distributions using only a small set of parameters. Specifying network connectivity via an assortativity matrix allows us to grow networks with arbitrary degree distributions and arbitrary modularity. We show that the degree distribution is controlled mainly by the ratio of node to edge addition probabilities, and the probability for node duplication. We compare topological and functional modularity measures, study their dependence on the number and strength of modules, and introduce the concept of anti-modularity: a property of networks in which nodes from one functional group preferentially do not attach to other nodes of that group. We also investigate global properties of networks as a function of the network's growth parameters, such as smallest path length, correlation coefficient, small-world-ness, and the nature of the percolation phase transition. We search the space of networks for those that are most like some well-known biological examples, and analyze the biological significance of the parameters that gave rise to them.
  
  
  CONCLUSIONS
  Growing networks with specified characters (degree distribution and modularity) provides the opportunity to create surrogates for biological and technological networks, and to test hypotheses about the processes that gave rise to them. We find that many celebrated network properties may be a consequence of the way in which these networks grew, rather than a necessary consequence of how they work or function.
authors.affiliation: Keck Graduate Institute of Applied Life Sciences, 535 Watson Drive, Claremont, CA 91711, USA. ahintze@kgi.edu
authors.forename: Arend
authors.initials: A
authors.lastname: Hintze
authors.forename: Christoph
authors.initials: C
authors.lastname: Adami
mesh_terms: D000465:Algorithms; D019295:Computational Biology; D008954:Models, Biological; D015398:Signal Transduction / Q000502:physiology; D012984:Software
references.citation: AIDS. 1989 Dec;3(12):807-17
references.pmid: 2517202
references.citation: Phys Rev Lett. 2000 Nov 20;85(21):4633-6
references.pmid: 11082614
references.citation: Proc Natl Acad Sci U S A. 2003 Feb 4;100(3):1128-33
references.pmid: 12538875
references.citation: Phys Rev E Stat Nonlin Soft Matter Phys. 2003 Feb;67(2 Pt 2):026126
references.pmid: 12636767
references.citation: Phys Rev E Stat Nonlin Soft Matter Phys. 2004 Dec;70(6 Pt 2):066111
references.pmid: 15697438
references.citation: Nat Genet. 2003 Jun;34(2):166-76
references.pmid: 12740579
references.citation: Phys Rev E Stat Nonlin Soft Matter Phys. 2004 Feb;69(2 Pt 2):026113
references.pmid: 14995526
references.citation: Phys Rev E Stat Nonlin Soft Matter Phys. 2003 Dec;68(6 Pt 2):066119
references.pmid: 14754281
references.citation: Philos Trans R Soc Lond B Biol Sci. 1986 Nov 12;314(1165):1-340
references.pmid: 22462104
references.citation: Phys Rev Lett. 2000 Jun 26;84(26 Pt 1):6114-7
references.pmid: 10991137
references.citation: J Biol. 2006;5(4):11
references.pmid: 16762047
references.citation: Phys Rev E Stat Nonlin Soft Matter Phys. 2002 Jul;66(1 Pt 1):011907
references.pmid: 12241384
references.citation: Proc Natl Acad Sci U S A. 2005 Mar 1;102(9):3192-7
references.pmid: 15728374
references.citation: Proc Natl Acad Sci U S A. 2003 Oct 14;100(21):12123-8
references.pmid: 14517352
references.citation: Nat Genet. 2004 Oct;36(10):1090-8
references.pmid: 15448693
references.citation: Proc Natl Acad Sci U S A. 2002 Apr 30;99(9):5890-5
references.pmid: 11983890
references.citation: Nature. 1998 Jun 4;393(6684):440-2
references.pmid: 9623998
references.citation: Phys Rev E Stat Nonlin Soft Matter Phys. 2001 Dec;64(6 Pt 2):066110
references.pmid: 11736239
references.citation: Proc Natl Acad Sci U S A. 2003 Oct 28;100(22):12820-4
references.pmid: 14557537
references.citation: PLoS Biol. 2005 Jul;3(7):e228
references.pmid: 16000019
references.citation: Mol Syst Biol. 2006;2:2006.0005
references.pmid: 16732191
references.citation: BMC Biol. 2004 Dec 02;2:25
references.pmid: 15574204
references.citation: Proc Natl Acad Sci U S A. 2006 Jun 6;103(23):8577-82
references.pmid: 16723398
references.citation: Phys Rev Lett. 2000 Nov 20;85(21):4629-32
references.pmid: 11082613
references.citation: Phys Rev E Stat Nonlin Soft Matter Phys. 2003 Aug;68(2 Pt 2):026121
references.pmid: 14525063
references.citation: PLoS One. 2008 Apr 30;3(4):e0002051
references.pmid: 18446219
references.citation: Dev Biol. 1983 Nov;100(1):64-119
references.pmid: 6684600
references.citation: Phys Rev Lett. 2002 Nov 11;89(20):208701
references.pmid: 12443515
references.citation: Nat Biotechnol. 2000 Nov;18(11):1121-2
references.pmid: 11062388
references.citation: Mol Biol Evol. 1998 May;15(5):583-9
references.pmid: 9580988
references.citation: Science. 1999 Oct 15;286(5439):509-12
references.pmid: 10521342
references.citation: Proc Natl Acad Sci U S A. 2006 Mar 21;103(12):4723-8
references.pmid: 16537428
references.citation: Nature. 1999 Dec 2;402(6761 Suppl):C47-52
references.pmid: 10591225
references.citation: Phys Rev Lett. 2000 Apr 3;84(14):3201-4
references.pmid: 11019047
references.citation: FEBS Lett. 2005 Mar 21;579(8):1772-8
references.pmid: 15763550
references.citation: Proc Natl Acad Sci U S A. 2000 Oct 10;97(21):11149-52
references.pmid: 11005838
references.citation: Phys Rev E Stat Nonlin Soft Matter Phys. 2003 May;67(5 Pt 2):056104
references.pmid: 12786217
references.citation: Phys Rev Lett. 2000 Dec 11;85(24):5234-7
references.pmid: 11102229
references.citation: Proc Natl Acad Sci U S A. 1998 Jul 21;95(15):8420-7
references.pmid: 9671692
references.citation: J Theor Biol. 2003 May 21;222(2):199-210
references.pmid: 12727455
references.citation: Phys Rev E Stat Nonlin Soft Matter Phys. 2001 Oct;64(4 Pt 1):041902
references.pmid: 11690047