Structural Inference of Hierarchies in Networks

One property of networks that has received comparatively little attention is hierarchy, i.e., the property of having vertices that cluster together in groups, which then join to form groups of groups, and so forth, up through all levels of organization in the network. Here, we give a precise definition of hierarchical structure, give a generic model for generating arbitrary hierarchical structure in a random graph, and describe a statistically principled way to learn the set of hierarchical features that most plausibly explain a particular real-world network. By applying this approach to two example networks, we demonstrate its advantages for the interpretation of network data, the annotation of graphs with edge, vertex and community properties, and the generation of generic null models for further hypothesis testing.Comment: 8 pages, 8 figure

Hierarchical structure and the prediction of missing links in networks

Networks have in recent years emerged as an invaluable tool for describing and quantifying complex systems in many branches of science(1-3). Recent studies suggest that networks often exhibit hierarchical organization, in which vertices divide into groups that further subdivide into groups of groups, and so forth over multiple scales. In many cases the groups are found to correspond to known functional units, such as ecological niches in food webs, modules in biochemical networks ( protein interaction networks, metabolic networks or genetic regulatory networks) or communities in social networks(4-7). Here we present a general technique for inferring hierarchical structure from network data and show that the existence of hierarchy can simultaneously explain and quantitatively reproduce many commonly observed topological properties of networks, such as right- skewed degree distributions, high clustering coefficients and short path lengths. We further show that knowledge of hierarchical structure can be used to predict missing connections in partly known networks with high accuracy, and for more general network structures than competing techniques(8). Taken together, our results suggest that hierarchy is a central organizing principle of complex networks, capable of offering insight into many network phenomena.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/62623/1/nature06830.pd

Hawkes process as a model of social interactions: a view on video dynamics

We study by computer simulation the "Hawkes process" that was proposed in a recent paper by Crane and Sornette (Proc. Nat. Acad. Sci. USA 105, 15649 (2008)) as a plausible model for the dynamics of YouTube video viewing numbers. We test the claims made there that robust identification is possible for classes of dynamic response following activity bursts. Our simulated timeseries for the Hawkes process indeed fall into the different categories predicted by Crane and Sornette. However the Hawkes process gives a much narrower spread of decay exponents than the YouTube data, suggesting limits to the universality of the Hawkes-based analysis.Comment: Added errors to parameter estimates and further description. IOP style, 13 pages, 5 figure

Power-law distributions in empirical data

Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution -- the part of the distribution representing large but rare events -- and by the difficulty of identifying the range over which power-law behavior holds. Commonly used methods for analyzing power-law data, such as least-squares fitting, can produce substantially inaccurate estimates of parameters for power-law distributions, and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all. Here we present a principled statistical framework for discerning and quantifying power-law behavior in empirical data. Our approach combines maximum-likelihood fitting methods with goodness-of-fit tests based on the Kolmogorov-Smirnov statistic and likelihood ratios. We evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches. We also apply the proposed methods to twenty-four real-world data sets from a range of different disciplines, each of which has been conjectured to follow a power-law distribution. In some cases we find these conjectures to be consistent with the data while in others the power law is ruled out.Comment: 43 pages, 11 figures, 7 tables, 4 appendices; code available at http://www.santafe.edu/~aaronc/powerlaws

Ortho-semantic learning of novel words: An event-related potential study of grade 3 children

Introduction: As children become independent readers, they regularly encounter new words whose meanings they must infer from context, and whose spellings must be learned for future recognition. The self-teaching hypothesis proposes orthographic learning skills are critical in the transition to fluent reading, while the lexical quality hypothesis further emphasizes the importance of semantics. Event-related potential (ERP) studies of reading development have focused on effects related to the N170 component—print tuning (letters vs. symbols) and lexical tuning (real words vs. consonant strings)—as well as the N400 reflecting semantic processing, but have not investigated the relationship of these components to word learning during independent reading. Methods: In this study, children in grade 3 independently read short stories that introduced novel words, then completed a lexical decision task from which ERPs were derived. Results: Like real words, newly-learned novel words evoked a lexical tuning effect, indicating rapid establishment of orthographic representations. Both real and novel words elicited significantly smaller N400s than pseudowords, suggesting that semantic representations of the novel words were established. Further, N170 print tuning predicted accuracy on identifying the spellings of the novel words, while the N400 effect for novel words was associated with reading comprehension. Discussion: Exposure to novel words during self-directed reading rapidly establishes neural markers of orthographic and semantic processing. Furthermore, the ability to rapidly filter letter strings from symbols is predictive of orthographic learning, while rapid establishment of semantic representations of novel words is associated with stronger reading comprehension

Dependence of Galaxy Quenching on Halo Mass and Distance from its Centre

We study the dependence of star-formation quenching on galaxy mass and environment, in the SDSS (z~0.1) and the AEGIS (z~1). It is crucial that we define quenching by low star-formation rate rather than by red colour, given that one third of the red galaxies are star forming. We address stellar mass M*, halo mass Mh, density over the nearest N neighbours deltaN, and distance to the halo centre D. The fraction of quenched galaxies appears more strongly correlated with Mh at fixed M* than with M* at fixed Mh, while for satellites quenching also depends on D. We present the M*-Mh relation for centrals at z~1. At z~1, the dependence of quenching on M* at fixed Mh is somewhat more pronounced than at z~0, but the quenched fraction is low (10%) and the haloes are less massive. For satellites, M*-dependent quenching is noticeable at high D, suggesting a quenching dependence on sub-halo mass for recently captured satellites. At small D, where satellites likely fell in more than a few Gyr ago, quenching strongly depends on Mh, and not on M*. The Mh-dependence of quenching is consistent with theoretical wisdom where virial shock heating in massive haloes shuts down accretion and triggers ram-pressure stripping, causing quenching. The interpretation of deltaN is complicated by the fact that it depends on the number of observed group members compared to N, motivating the use of D as a better measure of local environment.Comment: 23 pages, 13 figures, accepted by MNRA

On the Evolution of the Velocity-Mass-Size Relations of Disk-Dominated Galaxies over the Past 10 Billion Years

We study the evolution of the scaling relations between maximum circular velocity, stellar mass and optical half-light radius of star-forming disk-dominated galaxies in the context of LCDM-based galaxy formation models. Using data from the literature combined with new data from the DEEP2 and AEGIS surveys we show that there is a consistent observational and theoretical picture for the evolution of these scaling relations from z\sim 2 to z=0. The evolution of the observed stellar scaling relations is weaker than that of the virial scaling relations of dark matter haloes, which can be reproduced, both qualitatively and quantitatively, with a simple, cosmologically-motivated model for disk evolution inside growing NFW dark matter haloes. In this model optical half-light radii are smaller, both at fixed stellar mass and maximum circular velocity, at higher redshifts. This model also predicts that the scaling relations between baryonic quantities evolve even more weakly than the corresponding stellar relations. We emphasize, though, that this weak evolution does not imply that individual galaxies evolve weakly. On the contrary, individual galaxies grow strongly in mass, size and velocity, but in such a way that they move largely along the scaling relations. Finally, recent observations have claimed surprisingly large sizes for a number of star-forming disk galaxies at z \sim 2, which has caused some authors to suggest that high redshift disk galaxies have abnormally high spin parameters. However, we argue that the disk scale lengths in question have been systematically overestimated by a factor \sim 2, and that there is an offset of a factor \sim 1.4 between H\alpha sizes and optical sizes. Taking these effects into account, there is no indication that star forming galaxies at high redshifts (z\sim 2) have abnormally high spin parameters.Comment: 19 pages, 10 figures, accepted to MNRAS, minor changes to previous versio

The DEEP3 Galaxy Redshift Survey: The Impact of Environment on the Size Evolution of Massive Early-type Galaxies at Intermediate Redshift

Using data drawn from the DEEP2 and DEEP3 Galaxy Redshift Surveys, we investigate the relationship between the environment and the structure of galaxies residing on the red sequence at intermediate redshift. Within the massive (10 < log(M*/Msun) < 11) early-type population at 0.4 < z <1.2, we find a significant correlation between local galaxy overdensity (or environment) and galaxy size, such that early-type systems in higher-density regions tend to have larger effective radii (by ~0.5 kpc or 25% larger) than their counterparts of equal stellar mass and Sersic index in lower-density environments. This observed size-density relation is consistent with a model of galaxy formation in which the evolution of early-type systems at z < 2 is accelerated in high-density environments such as groups and clusters and in which dry, minor mergers (versus mechanisms such as quasar feedback) play a central role in the structural evolution of the massive, early-type galaxy population.Comment: 11 pages, 5 figures, 2 tables; resubmitted to MNRAS after addressing referee's comments (originally submitted to journal on August 16, 2011