The Combined Effect of Connectivity and Dependency Links on Percolation of Networks

Percolation theory is extensively studied in statistical physics and mathematics with applications in diverse fields. However, the research is focused on systems with only one type of links, connectivity links. We review a recently developed mathematical framework for analyzing percolation properties of realistic scenarios of networks having links of two types, connectivity and dependency links. This formalism was applied to study Erdős-Rényi (ER) networks that include also dependency links. For an ER network with average degree [`(k)]\bar{k} that is composed of dependency clusters of size s, the fraction of nodes that belong to the giant component, P <sub>∞</sub>, is given by P<sub>￥</sub>=p<sup>s-1</sup>[1-exp(-[`(k)]pP_￥) ]^sP_{\infty}=p^{s-1}[1-\exp{(-\bar{k}pP_{\infty})} ]^{s} where 1−p is the initial fraction of randomly removed nodes. Here, we apply the formalism to the study of random-regular (RR) networks and find a formula for the size of the giant component in the percolation process: P _∞=p ^s−1(1−r ^k ) ^s where r is the solution of r=p ^s (r ^k−1−1)(1−r ^k )+1, and k is the degree of the nodes. These general results coincide, for s=1, with the known equations for percolation in ER and RR networks respectively without dependency links. In contrast to s=1, where the percolation transition is second order, for s>1 it is of first order. Comparing the percolation behavior of ER and RR networks we find a remarkable difference regarding their resilience. We show, analytically and numerically, that in ER networks with low connectivity degree or large dependency clusters, removal of even a finite number (zero fraction) of the infinite network nodes will trigger a cascade of failures that fragments the whole network. Specifically, for any given s there exists a critical degree value, [`(k)]<sub>min</sub>\bar{k}_{\min}, such that an ER network with [`(k)] ￡ [`(k)]_min\bar{k}\leq \bar{k}_{\min} is unstable and collapse when removing even a single node. This result is in contrast to RR networks where such cascades and full fragmentation can be triggered only by removal of a finite fraction of nodes in the network.     

Possible origin of efficient navigation in small worlds

The small-world phenomenon is one of the most important properties found in social networks. It includes both short path lengths and efficient navigation between two individuals. It is found by Kleinberg that navigation is efficient only if the probability density distribution of an individual to have a friend at distance r scales as P(r) ～ r(-1). Although this spatial scaling is found in many empirical studies, the origin of how this scaling emerges is still missing. In this Letter, we propose the origin of this scaling law using the concept of entropy from statistical physics and show that this scaling is the result of optimization of collecting information in social networks.     

Oxygen and hydrogen profiles and electrical properties of unintentionally doped gallium nitride grown by hydride vapor phase epitaxy

Commercially available hydride vapor phase epitaxy gallium nitride (GaN) is characterized with the aim to correlate the oxygen and hydrogen secondary ion mass spectrometry profiles of a GaN wafer with the electrical properties of the sample. A GaN layer model, including doping profile and mobility, is derived, utilizing electrical (capacitance–voltage, Hall), structural (high resolution X‐ray diffraction) and optical (polarized infrared spectroscopy) methods. Oxygen and hydrogen are easily incorporated during hydride vapor phase epitaxy growth of GaN. Oxygen is an n‐type dopant in GaN, whereas hydrogen may passivate some of the donors. Electrical and optical properties correlate with a low defect concentration top GaN layer and a high defect concentration GaN interlayer.     

Evaluation of choice set generation algorithms for route choice models

This paper discusses choice set generation and route choice model estimation for large-scale urban networks. Evaluating the effectiveness of Advanced Traveler Information Systems (ATIS) requires accurate models of how drivers choose routes based on their awareness of the roadway network and their perceptions of travel time. Many of the route choice models presented in the literature pay little attention to empirical estimation and validation procedures. In this paper, a route choice data set collected in Boston is described and the ability of several different route generation algorithms to produce paths similar to those observed in the survey is analyzed. The paper also presents estimation results of some route choice models recently developed using the data set collected.     

Uniform CLT for Markov chains and its invariance principle: A martingale approach

The convergence of stochastic processes indexed by parameters which are elements of a metric space is investigated in the context of an invariance principle of the uniform central limit theorem (UCLT) for stationary Markov chains. We assume the integrability condition on metric entropy with bracketing. An eventual uniform equicontinuity result is developed which essentially gives the invariance principle of the UCLT. We translate the problem into that of a martingale difference sequence as in Gordin and Lifsic.⁽⁷⁾ Then we use the chaining argument with stratification adapted from that of Ossiander.⁽¹¹⁾ The results of this paper generalize those of Levental⁽¹⁰⁾ and Ossiander.⁽¹¹⁾     

Selective fluorination on tertiary carbon-hydrogen single bonds in the aliphatic series

Fluorine acts as an electrophile and, as such, reacts with remote and unactivated tertiary hydrogens in aliphatic alcohols and acids, resulting in the corresponding fluorine derivatives.     

The temperature dependence of the electron mobility in molecular crystals

We show that a hopping model may explain the experimental temperature dependence of the electron mobility in molecular crystals. The flat, higher temperature portion is obtained if anharmonic effects are included in a simple manner. The low temperature part is obtained if coherent effects are added to the diffusional ones.     

Sufficient conditions for continuous dependence in nonlinear hyperelasticity

Sufficient conditions are derived for the continuous dependence of C ¹-solutions of the mixed boundary value problem (with dead loading) upon the given data, within the framework of nonlinear hyperelasticity.The second author was a visitor at the Department of Mathematics, University of Strathclyde, Glasgow, Scotland, at the time that the revised version was prepared. Author's present address: Department of Mathematics, Plymouth Polytechnic, Plymouth, England.