## Abstract

While some network concepts (e.g., connectivity) have found important practical applications, other network concepts (e.g., network centralization) are rarely being used. Before attempting to understand why some concepts are more interesting than others, it is important to understand how network concepts relate to each other. As a step toward this goal, we explore the meaning of network concepts in approximately factorizable networks. In these types of networks, simple relationships exist between seemingly disparate network concepts. To provide a formalism for relating network concepts to each other, we define three types of network concepts: fundamental-, conformity-, and approximate conformity-based concepts. For each type, one can define intramodular and intermodular network concepts. Fundamental concepts include the standard definitions of connectivity, density, centralization, heterogeneity, clustering coefficient, and topological overlap. Approximate conformity-based analogs of fundamental network concepts have several theoretical advantages. First, they allow one to derive simple relationships between seemingly disparate network concepts. For example, we derive simple relationships between the clustering coefficient, the heterogeneity, the density, the centralization, and the topological overlap. Second, they allow one to show that fundamental network concepts can be approximated by simple functions of the connectivity in network modules. Third, they allow one to study the effects of transforming a network. We illustrate these results using protein–protein interaction networks and using block-diagonal networks. This work reviews and extends work with **Jun Dong**.

## References

- Breitkreutz BJ, Stark C, Tyers M (2003) The GRID: The general repository for interaction datasets. Genome Biol 4:R23PubMedCrossRefGoogle Scholar
- Dennis G, Sherman B, Hosack D, Yang J, Gao W, Lane H, Lempicki R (2003) DAVID: Database for annotation, visualization, and integrated discovery. Genome Biol 4(9):R60CrossRefGoogle Scholar
- Dong J, Horvath S (2007) Understanding network concepts in modules. BMC Syst Biol 1(1):24PubMedCrossRefGoogle Scholar
- Erdos P, Renyi A (1960) On the evolution of random graphs. Publ Math Inst Hung Acad Sci5: 17–60Google Scholar
- Horvath S, Dong J (2008) Geometric interpretation of gene co-expression network analysis. PLoS Comput Biol 4(8):e1000117PubMedCrossRefGoogle Scholar
- Kaufman L, Rousseeuw PJ (1990) Finding groups in data: An introduction to cluster analysis. Wiley, New YorkCrossRefGoogle Scholar
- Snijders TA (1981) The degree variance: An index of graph heterogeneity. Soc Networks 3: 163–174CrossRefGoogle Scholar