Represented as graphs, real networks are intricate combinations of order and disorder. Fixing some of the structural properties of network models to their values observed in real networks, many other properties appear as statistical consequences of these fixed observables, plus randomness in other respects. Here we employ the dk-series, a complete set of basic characteristics of the network structure, to study the statistical dependencies between different network properties. We consider six real networks—the Internet, US airport network, human protein interactions, technosocial web of trust, English word network, and an fMRI map of the human brain—and find that many important local and global structural properties of these networks are closely reproduced by dk-random graphs whose degree distributions, degree correlations and clustering are as in the corresponding real network. We discuss important conceptual, methodological, and practical implications of this evaluation of network randomness, and release software to generate dk-random graphs.

Networks representing many complex systems in nature and society share some common structural properties like heterogeneous degree distributions and strong clustering. Recent research on network geometry has shown that those real networks can be adequately modeled as random geometric graphs in hyperbolic spaces. In this paper, we present a computer program to generate such graphs. Besides real-world-like networks, the program can generate random graphs from other well-known graph ensembles, such as the soft configuration model, random geometric graphs on a circle, or Erdős-Rényi random graphs. The simulations show a good match between the expected values of different network structural properties and the corresponding empirical values measured in generated graphs, confirming the accurate behavior of the program.

In Parkinson's disease (PD), the luminance pattern electroretinogram (PERG) is reported to be abnormal, indicating dysfunction of retinal ganglion cells (RGCs). To determine the vulnerability of different subpopulations of RGCs in PD patients, the authors recorded the PERG to stimuli of chromatic (red-green [R-G] and blue-yellow [B-Y]) and achromatic (yellow-black [Y-Bk]) contrast, known to emphasize the contribution of parvocellular, koniocellular, and magnocellular RGCs, respectively. Subjects were early PD patients (n = 12; mean age, 60.1 +/- 8.3 years; range, 46 to 74 years) not undergoing treatment with levodopa and age-sex-matched controls (n = 12). Pattern electroretinograms were recorded monocularly in response to equiluminant R-G, B-Y, and Y-Bk horizontal gratings of 0.3 c/deg and 90% contrast, reversed at 1Hz, and presented at a viewing distance of 24 cm (59.2 x 59 degree field). In PD patients, the PERG amplitude was significantly reduced (by 40 to 50% on average) for both chromatic and luminance stimuli. Pattern electroretinogram latency was significantly delayed (by about 15 ms) for B-Y stimuli only. Data indicate that, in addition to achromatic PERGs, chromatic PERGs are altered in PD before levodopa therapy. Overall, chromatic PERGs to B-Y equiluminant stimuli exhibited the largest changes. Data are consistent with previous findings in PD, showing that visual evoked potentials (VEP) to B-Y chromatic stimuli are more delayed than VEPs to R-G and achromatic stimuli. The results suggest that the koniocellular subpopulation of RGCs may be particularly vulnerable in early stages of Parkinson's disease.

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