Are the correlations assortative or disassortative (see e.g., here: http://arxiv.org/pdf/cond-mat/0205405.pdf?origin=publication_detail)? Are you concerned mainly with topology (where much of the work on network correlation measures exists)? As for comparison, there is no single measure but rather a list of features with corresponding measures that include (but are not limited to)

- local clusters (communities) and clustering coefficients (is the network governed by "small world" dynamics)?

- layers

- patterns and subpatterns of connectivity

- local "dominant" nodes or links (e.g., "hubs" or other types of local centrality)

- Computational complexity

- Structural complexity (some of the above would be subsumed under this one)

- "Information" flow (for directed complex graphs)

etc.

I know it is topic specific, but outside of books, monographs, and volumes this is the one source I could find on short notice that is freely available and addresses measures of network complexity: http://brainimaging.waisman.wisc.edu/~chung/neuro.processing/literature/rubinov.2010.NI.network.pdf

It would be helpful if you could provide a bit more information.

Thank you a lot :)

So I did a search through my files to see which papers on network complexity I had were available for free online somewhere. I then took a sample of these that addressed topics on network complexity measures, structures, dynamics, etc. Hopefully some of these are useful:

The first is a ~120 page dissertation "Dynamic-Based Structure Measures of Complex Networks" (just click "I accept" on the following page to access it: http://scholarbank.nus.edu.sg/termsofuse;jsessionid=37B8796F67CF453A96C0EEAA6D7782FA)

The rest are all papers:

Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., & Hwang, D. U. (2006). Complex networks: Structure and dynamics. Physics reports, 424(4), 175-308.

(http://www.u-cergy.fr/_attachments/complex-systems-i-article/latora_phys_rep.pdf?download=true)

Gallos, L. K., Song, C., & Makse, H. A. (2007). A review of fractality and self-similarity in complex networks. Physica A: Statistical Mechanics and its Applications, 386(2), 686-691.

(http://lisgi1.engr.ccny.cuny.edu/~lazaros/papers/30.pdf)

Song, C., Havlin, S., & Makse, H. A. (2005). Self-similarity of complex networks. Nature, 433(7024), 392-395.

http://arxiv.org/pdf/cond-mat/0503078.pdf?origin=publication_detail

Strogatz, S. H. (2001). Exploring complex networks. Nature, 410(6825), 268-276.

(http://www.nature.com/nature/journal/v410/n6825/full/410268a0.html)

(http://arxiv.org/pdf/cond-mat/0503078.pdf?origin=publication_detail)

Multi-scale Modularity and Dynamics in Complex Networks

(http://arxiv.org/pdf/1004.4268.pdf)

Dorogovtsev, S. N., Goltsev, A. V., & Mendes, J. F. (2008). Critical phenomena in complex networks. Reviews of Modern Physics, 80(4), 1275.

(http://arxiv.org/pdf/0705.0010.pdf)

Corominas-Murtra, B., Goñi, J., Solé, R. V., & Rodríguez-Caso, C. (2013). On the origins of hierarchy in complex networks. Proceedings of the National Academy of Sciences, 110(33), 13316-13321.

(http://www.pnas.org/content/110/33/13316.full)

Arenas, A., Diaz-Guilera, A., Kurths, J., Moreno, Y., & Zhou, C. (2008). Synchronization in complex networks. Physics Reports, 469(3), 93-153.

(http://arxiv.org/pdf/0805.2976.pdf)

Andrew, thank you very much :)))

You are very welcome :)

You have complex networks and would like to compare them according to various structural measures? May I suggest you try NetworKit (http://parco.iti.kit.edu/software/networkit.shtml), an open-source project I am involved in. Call the properties.overview function on the graphs, you will get a tabular view of properties like connected components, assortativity, degree distribution etc. This could help you compare their structural properties.

You may want to check this new work of ours, which seems to do exactly what you are asking:

L.K. Gallos and Nina H. Fefferman, "Revealing effective classifiers through network comparison"

http://arxiv.org/abs/1403.2668

Article Revealing effective classifiers through network comparison

Thank all of you!

Try to compare degree  distributions, weight distributions, graph properties (average degree, smallest distance, average clustering coefficient,community structure,,,), hierarchical node organization revealed in the distribution of the average clustering coefficients of the nodes with certain degree etc.