A new modularity measure for Fuzzy Community detection problems based on overlap and grouping functions

One of the main challenges of fuzzy community detection problems is to be able to measure the quality of a fuzzy partition. In this paper, we present an alternative way of measuring the quality of a fuzzy community detection output based on n-dimensional grouping and overlap functions. Moreover, the proposed modularity measure generalizes the classical Girvan–Newman (GN) modularity for crisp community detection problems and also for crisp overlapping community detection problems. Therefore, it can be used to compare partitions of different nature (i.e. those composed of classical, overlapping and fuzzy communities). Particularly, as is usually done with the GN modularity, the proposed measure may be used to identify the optimal number of communities to be obtained by any network clustering algorithm in a given network. We illustrate this usage by adapting in this way a well-known algorithm for fuzzy community detection problems, extending it to also deal with overlapping community detection problems and produce a ranking of the overlapping nodes. Some computational experiments show the feasibility of the proposed approach to modularity measures through n-dimensional overlap and grouping functions.

Near-infrared and star-forming properties of local luminous infrared galaxies

We use Hubble Space Telescope (HST) NICMOS continuum and Paα observations to study the near-infrared and star formation properties of a representative sample of 30 local (d ~ 35-75 Mpc) luminous infrared galaxies (LIRGs, infrared [8-1000 μm] luminosities of log L_IR = 11-11.9 L_☉). The data provide spatial resolutions of 25-50 pc and cover the central ~3.3-7.1 kpc regions of these galaxies. About half of the LIRGs show compact (~1-2 kpc) Paα emission with a high surface brightness in the form of nuclear emission, rings, and minispirals. The rest of the sample show Paα emission along the disk and the spiral arms extending over scales of 3-7 kpc and larger. About half of the sample contains H II regions with Hα luminosities significantly higher than those observed in normal galaxies. There is a linear empirical relationship between the mid-IR 24 μm and hydrogen recombination (extinction-corrected Paα) luminosity for these LIRGs, and the H II regions in the central part of M51. This relation holds over more than four decades in luminosity, suggesting that the mid-IR emission is a good tracer of the star formation rate (SFR). Analogous to the widely used relation between the SFR and total IR luminosity of R. Kennicutt, we derive an empirical calibration of the SFR in terms of the monochromatic 24 μm luminosity that can be used for luminous, dusty galaxies.

Unpolarized transverse momentum dependent parton distribution and fragmentation functions at next-to-next-to-leading order

The transverse momentum dependent parton distribution/fragmentation functions (TMDs) are essential in the factorization of a number of processes like Drell-Yan scattering, vector boson production, semi-inclusive deep inelastic scattering, etc. We provide a comprehensive study of unpolarized TMDs at next-to-next-to-leading order, which includes an explicit calculation of these TMDs and an extraction of their matching coefficients onto their integrated analogues, for all flavor combinations. The obtained matching coefficients are important for any kind of phenomenology involving TMDs. In the present study each individual TMD is calculated without any reference to a specific process. We recover the known results for parton distribution functions and provide new results for the fragmentation functions. The results for the gluon transverse momentum dependent fragmentation functions are presented for the first time at one and two loops. We also discuss the structure of singularities of TMD operators and TMD matrix elements, crossing relations between TMD parton distribution functions and TMD fragmentation functions, and renormalization group equations. In addition, we consider the behavior of the matching coefficients at threshold and make a conjecture on their structure to all orders in perturbation theory.