Phase Transitions in Community Detection: A Solvable Toy Model

Recently, it was shown that there is a phase transition in the community detection problem. This transition was first computed using the cavity method, and has been proved rigorously in the case of $q=2$ groups. However, analytic calculations using the cavity method are challenging since they require us to understand probability distributions of messages. We study analogous transitions in so-called "zero-temperature inference" model, where this distribution is supported only on the most-likely messages. Furthermore, whenever several messages are equally likely, we break the tie by choosing among them with equal probability. While the resulting analysis does not give the correct values of the thresholds, it does reproduce some of the qualitative features of the system. It predicts a first-order detectability transition whenever $q > 2$, while the finite-temperature cavity method shows that this is the case only when $q > 4$. It also has a regime analogous to the "hard but detectable" phase, where the community structure can be partially recovered, but only when the initial messages are sufficiently accurate. Finally, we study a semisupervised setting where we are given the correct labels for a fraction $\rho$ of the nodes. For $q > 2$, we find a regime where the accuracy jumps discontinuously at a critical value of $\rho$.Comment: 6 pages, 6 figure

The physical limits of communication

It has been well-known since the pioneering work of Claude Shannon in the 1940s that a message transmitted with optimal efficiency over a channel of limited bandwidth is indistinguishable from random noise to a receiver who is unfamiliar with the language in which the message is written. In this letter we demonstrate an equivalent result about electromagnetic transmissions. We show that when electromagnetic radiation is used as the transmission medium, the most information-efficient format for a given message is indistinguishable from black-body radiation to a receiver who is unfamiliar with that format. The characteristic temperature of the radiation is set by the amount of energy used to make the transmission. If information is not encoded in the direction of the radiation, but only its timing, energy or polarization, then the most efficient format has the form of a one-dimensional black-body spectrum which is easily distinguished from the three-dimensional case.Comment: 9 pages, 1 postscript figure, typeset in LaTeX using the RevTeX macro packag

Random graph models for dynamic networks

We propose generalizations of a number of standard network models, including the classic random graph, the configuration model, and the stochastic block model, to the case of time-varying networks. We assume that the presence and absence of edges are governed by continuous-time Markov processes with rate parameters that can depend on properties of the nodes. In addition to computing equilibrium properties of these models, we demonstrate their use in data analysis and statistical inference, giving efficient algorithms for fitting them to observed network data. This allows us, for instance, to estimate the time constants of network evolution or infer community structure from temporal network data using cues embedded both in the probabilities over time that node pairs are connected by edges and in the characteristic dynamics of edge appearance and disappearance. We illustrate our methods with a selection of applications, both to computer-generated test networks and real-world examples.Comment: 15 pages, four figure

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

Characterizing Multiple Solutions to the Time - Energy Canonical Commutation Relation via Internal Symmetries

Internal symmetries can be used to classify multiple solutions to the time energy canonical commutation relation (TE-CCR). The dynamical behavior of solutions to the TE-CCR posessing particular internal symmetries involving time reversal differ significantly from solutions to the TE-CCR without those particular symmetries, implying a connection between the internal symmetries of a quantum system, its internal unitary dynamics, and the TE-CCR.Comment: Accepted for publication in Physical Review A, 10 page

Exact solutions for models of evolving networks with addition and deletion of nodes

There has been considerable recent interest in the properties of networks, such as citation networks and the worldwide web, that grow by the addition of vertices, and a number of simple solvable models of network growth have been studied. In the real world, however, many networks, including the web, not only add vertices but also lose them. Here we formulate models of the time evolution of such networks and give exact solutions for a number of cases of particular interest. For the case of net growth and so-called preferential attachment -- in which newly appearing vertices attach to previously existing ones in proportion to vertex degree -- we show that the resulting networks have power-law degree distributions, but with an exponent that diverges as the growth rate vanishes. We conjecture that the low exponent values observed in real-world networks are thus the result of vigorous growth in which the rate of addition of vertices far exceeds the rate of removal. Were growth to slow in the future, for instance in a more mature future version of the web, we would expect to see exponents increase, potentially without bound.Comment: 9 pages, 3 figure

Extreme UV QSOs

We present a sample of spectroscopically confirmed QSOs with FUV-NUV color (as measured by GALEX photometry) bluer than canonical QSO templates and than the majority of known QSOs. We analyze their FUV to NIR colors, luminosities and optical spectra. The sample includes a group of 150 objects at low redshift (z $<$ 0.5), and a group of 21 objects with redshift 1.7$<$z$<$2.6. For the low redshift objects, the "blue" FUV-NUV color may be caused by enhanced Ly$\alpha$ emission, since Ly$\alpha$ transits the GALEX FUV band from z=0.1 to z=0.47. Synthetic QSO templates constructed with Ly$\alpha$ up to 3 times stronger than in standard templates match the observed UV colors of our low redshift sample. The H$\alpha$ emission increases, and the optical spectra become bluer, with increasing absolute UV luminosity. The UV-blue QSOs at redshift about 2, where the GALEX bands sample restframe about 450-590A (FUV) and about 590-940A(NUV), are fainter than the average of UV-normal QSOs at similar redshift in NUV, while they have comparable luminosities in other bands. Therefore we speculate that their observed FUV-NUV color may be explained by a combination of steep flux rise towards short wavelengths and dust absorption below the Lyman limit, such as from small grains or crystalline carbon. The ratio of Ly$\alpha$ to CIV could be measured in 10 objects; it is higher (30% on average) than for UV-normal QSOs, and close to the value expected for shock or collisional ionization. FULL VERSION AVAILABLE FROM AUTHOR'S WEB SITE: http://dolomiti.pha.jhu.edu/papers/2009_AJ_Extreme_UV_QSOs.pdfComment: Astronomical Journal, in pres