Efficient and exact sampling of simple graphs with given arbitrary degree sequence

Uniform sampling from graphical realizations of a given degree sequence is a fundamental component in simulation-based measurements of network observables, with applications ranging from epidemics, through social networks to Internet modeling. Existing graph sampling methods are either link-swap based (Markov-Chain Monte Carlo algorithms) or stub-matching based (the Configuration Model). Both types are ill-controlled, with typically unknown mixing times for link-swap methods and uncontrolled rejections for the Configuration Model. Here we propose an efficient, polynomial time algorithm that generates statistically independent graph samples with a given, arbitrary, degree sequence. The algorithm provides a weight associated with each sample, allowing the observable to be measured either uniformly over the graph ensemble, or, alternatively, with a desired distribution. Unlike other algorithms, this method always produces a sample, without back-tracking or rejections. Using a central limit theorem-based reasoning, we argue, that for large N, and for degree sequences admitting many realizations, the sample weights are expected to have a lognormal distribution. As examples, we apply our algorithm to generate networks with degree sequences drawn from power-law distributions and from binomial distributions.Comment: 8 pages, 3 figure

Evolving Clustered Random Networks

We propose a Markov chain simulation method to generate simple connected random graphs with a specified degree sequence and level of clustering. The networks generated by our algorithm are random in all other respects and can thus serve as generic models for studying the impacts of degree distributions and clustering on dynamical processes as well as null models for detecting other structural properties in empirical networks

Wormhole Cosmic Censorship

We analyze the properties of a Kerr-like wormhole supported by phantom matter, which is an exact solution of the Einstein-phantom field equations. It is shown that the solution has a naked ring singularity which is unreachable to null geodesics falling freely from the outside. Similarly to Roger Penrose's cosmic censorship, that states that all naked singularities in the Universe must be protected by event horizons, here we conjecture from our results that a naked singularity can also be fully protected by the intrinsic properties of a wormhole's throat

Mesoscopic organization reveals the constraints governing C. elegans nervous system

One of the biggest challenges in biology is to understand how activity at the cellular level of neurons, as a result of their mutual interactions, leads to the observed behavior of an organism responding to a variety of environmental stimuli. Investigating the intermediate or mesoscopic level of organization in the nervous system is a vital step towards understanding how the integration of micro-level dynamics results in macro-level functioning. In this paper, we have considered the somatic nervous system of the nematode Caenorhabditis elegans, for which the entire neuronal connectivity diagram is known. We focus on the organization of the system into modules, i.e., neuronal groups having relatively higher connection density compared to that of the overall network. We show that this mesoscopic feature cannot be explained exclusively in terms of considerations, such as optimizing for resource constraints (viz., total wiring cost) and communication efficiency (i.e., network path length). Comparison with other complex networks designed for efficient transport (of signals or resources) implies that neuronal networks form a distinct class. This suggests that the principal function of the network, viz., processing of sensory information resulting in appropriate motor response, may be playing a vital role in determining the connection topology. Using modular spectral analysis, we make explicit the intimate relation between function and structure in the nervous system. This is further brought out by identifying functionally critical neurons purely on the basis of patterns of intra- and inter-modular connections. Our study reveals how the design of the nervous system reflects several constraints, including its key functional role as a processor of information.Comment: Published version, Minor modifications, 16 pages, 9 figure

Holographic Anomalous Conductivities and the Chiral Magnetic Effect

We calculate anomaly induced conductivities from a holographic gauge theory model using Kubo formulas, making a clear conceptual distinction between thermodynamic state variables such as chemical potentials and external background fields. This allows us to pinpoint ambiguities in previous holographic calculations of the chiral magnetic conductivity. We also calculate the corresponding anomalous current three-point functions in special kinematic regimes. We compare the holographic results to weak coupling calculations using both dimensional regularization and cutoff regularization. In order to reproduce the weak coupling results it is necessary to allow for singular holographic gauge field configurations when a chiral chemical potential is introduced for a chiral charge defined through a gauge invariant but non-conserved chiral density. We argue that this is appropriate for actually addressing charge separation due to the chiral magnetic effect.Comment: 17 pages, 1 figure. v2: 18 pages, 1 figure, discussion clarified throughout the text, references added, version accepted for publication in JHE

Dynamics on expanding spaces: modeling the emergence of novelties

Novelties are part of our daily lives. We constantly adopt new technologies, conceive new ideas, meet new people, experiment with new situations. Occasionally, we as individuals, in a complicated cognitive and sometimes fortuitous process, come up with something that is not only new to us, but to our entire society so that what is a personal novelty can turn into an innovation at a global level. Innovations occur throughout social, biological and technological systems and, though we perceive them as a very natural ingredient of our human experience, little is known about the processes determining their emergence. Still the statistical occurrence of innovations shows striking regularities that represent a starting point to get a deeper insight in the whole phenomenology. This paper represents a small step in that direction, focusing on reviewing the scientific attempts to effectively model the emergence of the new and its regularities, with an emphasis on more recent contributions: from the plain Simon's model tracing back to the 1950s, to the newest model of Polya's urn with triggering of one novelty by another. What seems to be key in the successful modelling schemes proposed so far is the idea of looking at evolution as a path in a complex space, physical, conceptual, biological, technological, whose structure and topology get continuously reshaped and expanded by the occurrence of the new. Mathematically it is very interesting to look at the consequences of the interplay between the "actual" and the "possible" and this is the aim of this short review.Comment: 25 pages, 10 figure

Quadratic optimal functional quantization of stochastic processes and numerical applications

In this paper, we present an overview of the recent developments of functional quantization of stochastic processes, with an emphasis on the quadratic case. Functional quantization is a way to approximate a process, viewed as a Hilbert-valued random variable, using a nearest neighbour projection on a finite codebook. A special emphasis is made on the computational aspects and the numerical applications, in particular the pricing of some path-dependent European options.Comment: 41 page

Anomalies and the chiral magnetic effect in the Sakai-Sugimoto model

In the chiral magnetic effect an imbalance in the number of left- and right-handed quarks gives rise to an electromagnetic current parallel to the magnetic field produced in noncentral heavy-ion collisions. The chiral imbalance may be induced by topologically nontrivial gluon configurations via the QCD axial anomaly, while the resulting electromagnetic current itself is a consequence of the QED anomaly. In the Sakai-Sugimoto model, which in a certain limit is dual to large-N_c QCD, we discuss the proper implementation of the QED axial anomaly, the (ambiguous) definition of chiral currents, and the calculation of the chiral magnetic effect. We show that this model correctly contains the so-called consistent anomaly, but requires the introduction of a (holographic) finite counterterm to yield the correct covariant anomaly. Introducing net chirality through an axial chemical potential, we find a nonvanishing vector current only before including this counterterm. This seems to imply the absence of the chiral magnetic effect in this model. On the other hand, for a conventional quark chemical potential and large magnetic field, which is of interest in the physics of compact stars, we obtain a nontrivial result for the axial current that is in agreement with previous calculations and known exact results for QCD.Comment: 35 pages, 4 figures, v2: added comments about frequency-dependent conductivity at the end of section 4; references added; version to appear in JHE

Correlated fragile site expression allows the identification of candidate fragile genes involved in immunity and associated with carcinogenesis

Common fragile sites (cfs) are specific regions in the human genome that are particularly prone to genomic instability under conditions of replicative stress. Several investigations support the view that common fragile sites play a role in carcinogenesis. We discuss a genome-wide approach based on graph theory and Gene Ontology vocabulary for the functional characterization of common fragile sites and for the identification of genes that contribute to tumour cell biology. CFS were assembled in a network based on a simple measure of correlation among common fragile site patterns of expression. By applying robust measurements to capture in quantitative terms the non triviality of the network, we identified several topological features clearly indicating departure from the Erdos-Renyi random graph model. The most important outcome was the presence of an unexpected large connected component far below the percolation threshold. Most of the best characterized common fragile sites belonged to this connected component. By filtering this connected component with Gene Ontology, statistically significant shared functional features were detected. Common fragile sites were found to be enriched for genes associated to the immune response and to mechanisms involved in tumour progression such as extracellular space remodeling and angiogenesis. Our results support the hypothesis that fragile sites serve a function; we propose that fragility is linked to a coordinated regulation of fragile genes expression.Comment: 18 pages, accepted for publication in BMC Bioinformatic

Influence of wiring cost on the large-scale architecture of human cortical connectivity

In the past two decades some fundamental properties of cortical connectivity have been discovered: small-world structure, pronounced hierarchical and modular organisation, and strong core and rich-club structures. A common assumption when interpreting results of this kind is that the observed structural properties are present to enable the brain's function. However, the brain is also embedded into the limited space of the skull and its wiring has associated developmental and metabolic costs. These basic physical and economic aspects place separate, often conflicting, constraints on the brain's connectivity, which must be characterized in order to understand the true relationship between brain structure and function. To address this challenge, here we ask which, and to what extent, aspects of the structural organisation of the brain are conserved if we preserve specific spatial and topological properties of the brain but otherwise randomise its connectivity. We perform a comparative analysis of a connectivity map of the cortical connectome both on high- and low-resolutions utilising three different types of surrogate networks: spatially unconstrained (‘random’), connection length preserving (‘spatial’), and connection length optimised (‘reduced’) surrogates. We find that unconstrained randomisation markedly diminishes all investigated architectural properties of cortical connectivity. By contrast, spatial and reduced surrogates largely preserve most properties and, interestingly, often more so in the reduced surrogates. Specifically, our results suggest that the cortical network is less tightly integrated than its spatial constraints would allow, but more strongly segregated than its spatial constraints would necessitate. We additionally find that hierarchical organisation and rich-club structure of the cortical connectivity are largely preserved in spatial and reduced surrogates and hence may be partially attributable to cortical wiring constraints. In contrast, the high modularity and strong s-core of the high-resolution cortical network are significantly stronger than in the surrogates, underlining their potential functional relevance in the brain