Detecting degree symmetries in networks

The surrounding of a vertex in a network can be more or less symmetric. We derive measures of a specific kind of symmetry of a vertex which we call degree symmetry -- the property that many paths going out from a vertex have overlapping degree sequences. These measures are evaluated on artificial and real networks. Specifically we consider vertices in the human metabolic network. We also measure the average degree-symmetry coefficient for different classes of real-world network. We find that most studied examples are weakly positively degree-symmetric. The exceptions are an airport network (having a negative degree-symmetry coefficient) and one-mode projections of social affiliation networks that are rather strongly degree-symmetric

Network reachability of real-world contact sequences

We use real-world contact sequences, time-ordered lists of contacts from one person to another, to study how fast information or disease can spread across network of contacts. Specifically we measure the reachability time -- the average shortest time for a series of contacts to spread information between a reachable pair of vertices (a pair where a chain of contacts exists leading from one person to the other) -- and the reachability ratio -- the fraction of reachable vertex pairs. These measures are studied using conditional uniform graph tests. We conclude, among other things, that the network reachability depends much on a core where the path lengths are short and communication frequent, that clustering of the contacts of an edge in time tend to decrease the reachability, and that the order of the contacts really do make sense for dynamical spreading processes.Comment: (v2: fig. 1 fixed

Discrete concavity and the half-plane property

Murota et al. have recently developed a theory of discrete convex analysis which concerns M-convex functions on jump systems. We introduce here a family of M-concave functions arising naturally from polynomials (over a field of generalized Puiseux series) with prescribed non-vanishing properties. This family contains several of the most studied M-concave functions in the literature. In the language of tropical geometry we study the tropicalization of the space of polynomials with the half-plane property, and show that it is strictly contained in the space of M-concave functions. We also provide a short proof of Speyer's hive theorem which he used to give a new proof of Horn's conjecture on eigenvalues of sums of Hermitian matrices.Comment: 14 pages. The proof of Theorem 4 is corrected

Neutral theory of chemical reaction networks

To what extent do the characteristic features of a chemical reaction network reflect its purpose and function? In general, one argues that correlations between specific features and specific functions are key to understanding a complex structure. However, specific features may sometimes be neutral and uncorrelated with any system-specific purpose, function or causal chain. Such neutral features are caused by chance and randomness. Here we compare two classes of chemical networks: one that has been subjected to biological evolution (the chemical reaction network of metabolism in living cells) and one that has not (the atmospheric planetary chemical reaction networks). Their degree distributions are shown to share the very same neutral system-independent features. The shape of the broad distributions is to a large extent controlled by a single parameter, the network size. From this perspective, there is little difference between atmospheric and metabolic networks; they are just different sizes of the same random assembling network. In other words, the shape of the degree distribution is a neutral characteristic feature and has no functional or evolutionary implications in itself; it is not a matter of life and death.Comment: 13 pages, 8 figure

Higher order corrections to the Newtonian potential in the Randall-Sundrum model

The general formalism for calculating the Newtonian potential in fine-tuned or critical Randall-Sundrum braneworlds is outlined. It is based on using the full tensor structure of the graviton propagator. This approach avoids the brane-bending effect arising from calculating the potential for a point source. For a single brane, this gives a clear understanding of the disputed overall factor 4/3 entering the correction. The result can be written on a compact form which is evaluated to high accuracy for both short and large distances.Comment: 12 pages, LaTeX2e with RevTeX4, 3 postscript figures; Minor corrections, references update

Majority-vote model on hyperbolic lattices

We study the critical properties of a non-equilibrium statistical model, the majority-vote model, on heptagonal and dual heptagonal lattices. Such lattices have the special feature that they only can be embedded in negatively curved surfaces. We find, by using Monte Carlo simulations and finite-size analysis, that the critical exponents $1/\nu$, $\beta/\nu$ and $\gamma/\nu$ are different from those of the majority-vote model on regular lattices with periodic boundary condition, which belongs to the same universality class as the equilibrium Ising model. The exponents are also from those of the Ising model on a hyperbolic lattice. We argue that the disagreement is caused by the effective dimensionality of the hyperbolic lattices. By comparative studies, we find that the critical exponents of the majority-vote model on hyperbolic lattices satisfy the hyperscaling relation $2\beta/\nu+\gamma/\nu=D_{\mathrm{eff}}$, where $D_{\mathrm{eff}}$ is an effective dimension of the lattice. We also investigate the effect of boundary nodes on the ordering process of the model.Comment: 8 pages, 9 figure

Relativistic Images in Randall-Sundrum II Braneworld Lensing

In this paper, we explore the properties of gravitational lensing by black holes in the Randall-Sundrum II braneworld. We use numerical techniques to calculate lensing observables using the Tidal Reissner-Nordstrom (TRN) and Garriga-Tanaka metrics to examine supermassive black holes and primordial black holes. We introduce a new way tp parameterize tidal charge in the TRN metric which results in a large increase in image magnifications for braneworld primordial black holes compared to their 4 dimensional analogues. Finally, we offer a mathematical analysis that allows us to analyze the validity of the logarithmic approximation of the bending angle for any static, spherically symmetric metric. We apply this to the TRN metric and show that it is valid for any amount of tidal charge.Comment: 13 pages, 3 figures; Accepted for Publication in Physical Review

Core-periphery organization of complex networks

Networks may, or may not, be wired to have a core that is both itself densely connected and central in terms of graph distance. In this study we propose a coefficient to measure if the network has such a clear-cut core-periphery dichotomy. We measure this coefficient for a number of real-world and model networks and find that different classes of networks have their characteristic values. For example do geographical networks have a strong core-periphery structure, while the core-periphery structure of social networks (despite their positive degree-degree correlations) is rather weak. We proceed to study radial statistics of the core, i.e. properties of the n-neighborhoods of the core vertices for increasing n. We find that almost all networks have unexpectedly many edges within n-neighborhoods at a certain distance from the core suggesting an effective radius for non-trivial network processes

Exploring the assortativity-clustering space of a network's degree sequence

Nowadays there is a multitude of measures designed to capture different aspects of network structure. To be able to say if the structure of certain network is expected or not, one needs a reference model (null model). One frequently used null model is the ensemble of graphs with the same set of degrees as the original network. In this paper we argue that this ensemble can be more than just a null model -- it also carries information about the original network and factors that affect its evolution. By mapping out this ensemble in the space of some low-level network structure -- in our case those measured by the assortativity and clustering coefficients -- one can for example study how close to the valid region of the parameter space the observed networks are. Such analysis suggests which quantities are actively optimized during the evolution of the network. We use four very different biological networks to exemplify our method. Among other things, we find that high clustering might be a force in the evolution of protein interaction networks. We also find that all four networks are conspicuously robust to both random errors and targeted attacks

On the half-plane property and the Tutte group of a matroid

A multivariate polynomial is stable if it is non-vanishing whenever all variables have positive imaginary parts. A matroid has the weak half-plane property (WHPP) if there exists a stable polynomial with support equal to the set of bases of the matroid. If the polynomial can be chosen with all of its nonzero coefficients equal to one then the matroid has the half-plane property (HPP). We describe a systematic method that allows us to reduce the WHPP to the HPP for large families of matroids. This method makes use of the Tutte group of a matroid. We prove that no projective geometry has the WHPP and that a binary matroid has the WHPP if and only if it is regular. We also prove that T_8 and R_9 fail to have the WHPP.Comment: 8 pages. To appear in J. Combin. Theory Ser.