Exact results and scaling properties of small-world networks

We study the distribution function for minimal paths in small-world networks. Using properties of this distribution function, we derive analytic results which greatly simplify the numerical calculation of the average minimal distance, $\bar{\ell}$, and its variance, $\sigma^2$. We also discuss the scaling properties of the distribution function. Finally, we study the limit of large system sizes and obtain some analytic results.Comment: RevTeX, 4 pages, 5 figures included. Minor corrections and addition

Time-reversal violating rotation of polarization plane of light in gas placed in electric field

Rotation of polarization plane of light in gas placed in electric field is considered. Different factors causing this phenomenon are investigated. Angle of polarization plane rotation for transition 6S_{1/2} - 7S_{1/2} in cesium (lambda=539 nm) is estimated. The possibility to observe this effect experimentally is discussed.Comment: 10 pages, Late

Transport Properties of Random Walks on Scale-Free/Regular-Lattice Hybrid Networks

We study numerically the mean access times for random walks on hybrid disordered structures formed by embedding scale-free networks into regular lattices, considering different transition rates for steps across lattice bonds ($F$) and across network shortcuts ($f$). For fast shortcuts ($f/F\gg 1$) and low shortcut densities, traversal time data collapse onto an universal curve, while a crossover behavior that can be related to the percolation threshold of the scale-free network component is identified at higher shortcut densities, in analogy to similar observations reported recently in Newman-Watts small-world networks. Furthermore, we observe that random walk traversal times are larger for networks with a higher degree of inhomogeneity in their shortcut distribution, and we discuss access time distributions as functions of the initial and final node degrees. These findings are relevant, in particular, when considering the optimization of existing information networks by the addition of a small number of fast shortcut connections.Comment: 8 pages, 6 figures; expanded discussions, added figures and references. To appear in J Stat Phy

Portland cement based immobilization/destruction of chemical weapon agent degradation products

The direct immobilization and destruction of two compounds relevant to chemical warfare agents, ethyl methylphosphonic acid (EMPA) and thiodiglycol (TDG), within a freshly mixed Portland cement paste was studied. Cement hydration and phase formation were analyzed to determine the upper limits on the loading of these chemicals achievable in an immobilization setting. EMPA, a degradation product of the nerve agent VX, alters the phase formation within the cements, allowing calcium aluminate dissolution while retarding hydration of calcium silicate clinker phases. This yielded ettringite, and sufficient calcium silicate hydrate for setting at 10 wt % loading, but the cohesive calcium silicate binding phase was lacking when EMPA was added at 25 wt %. The addition of TDG, a degradation product of sulfur mustard, uniformly retards the entire range of cement hydration mechanisms. Heat output was lowered and extended over a longer time frame, and less strength forming phases were produced. Up to 10% wt. TDG could be accommodated by the cement, but higher loadings caused severe disruption to the cement setting. This work demonstrates the ability of Portland cement to directly incorporate up to 10% wt. of these contaminants, and still form a stable set cement with conventional hydration phases

Range-based attack on links in scale-free networks: are long-range links responsible for the small-world phenomenon?

The small-world phenomenon in complex networks has been identified as being due to the presence of long-range links, i.e., links connecting nodes that would otherwise be separated by a long node-to-node distance. We find, surprisingly, that many scale-free networks are more sensitive to attacks on short-range than on long-range links. This result, besides its importance concerning network efficiency and/or security, has the striking implication that the small-world property of scale-free networks is mainly due to short-range links.Comment: 4 pages, 4 figures, Revtex, published versio

Relaxation Properties of Small-World Networks

Recently, Watts and Strogatz introduced the so-called small-world networks in order to describe systems which combine simultaneously properties of regular and of random lattices. In this work we study diffusion processes defined on such structures by considering explicitly the probability for a random walker to be present at the origin. The results are intermediate between the corresponding ones for fractals and for Cayley trees.Comment: 16 pages, 6 figure

Scaling Properties of Random Walks on Small-World Networks

Using both numerical simulations and scaling arguments, we study the behavior of a random walker on a one-dimensional small-world network. For the properties we study, we find that the random walk obeys a characteristic scaling form. These properties include the average number of distinct sites visited by the random walker, the mean-square displacement of the walker, and the distribution of first-return times. The scaling form has three characteristic time regimes. At short times, the walker does not see the small-world shortcuts and effectively probes an ordinary Euclidean network in $d$-dimensions. At intermediate times, the properties of the walker shows scaling behavior characteristic of an infinite small-world network. Finally, at long times, the finite size of the network becomes important, and many of the properties of the walker saturate. We propose general analytical forms for the scaling properties in all three regimes, and show that these analytical forms are consistent with our numerical simulations.Comment: 7 pages, 8 figures, two-column format. Submitted to PR

Constrained spin dynamics description of random walks on hierarchical scale-free networks

We study a random walk problem on the hierarchical network which is a scale-free network grown deterministically. The random walk problem is mapped onto a dynamical Ising spin chain system in one dimension with a nonlocal spin update rule, which allows an analytic approach. We show analytically that the characteristic relaxation time scale grows algebraically with the total number of nodes $N$ as $T \sim N^z$. From a scaling argument, we also show the power-law decay of the autocorrelation function C_{\bfsigma}(t)\sim t^{-\alpha}, which is the probability to find the Ising spins in the initial state {\bfsigma} after $t$ time steps, with the state-dependent non-universal exponent $\alpha$. It turns out that the power-law scaling behavior has its origin in an quasi-ultrametric structure of the configuration space.Comment: 9 pages, 6 figure

Synchronisation in networks of delay-coupled type-I excitable systems

We use a generic model for type-I excitability (known as the SNIPER or SNIC model) to describe the local dynamics of nodes within a network in the presence of non-zero coupling delays. Utilising the method of the Master Stability Function, we investigate the stability of the zero-lag synchronised dynamics of the network nodes and its dependence on the two coupling parameters, namely the coupling strength and delay time. Unlike in the FitzHugh-Nagumo model (a model for type-II excitability), there are parameter ranges where the stability of synchronisation depends on the coupling strength and delay time. One important implication of these results is that there exist complex networks for which the adding of inhibitory links in a small-world fashion may not only lead to a loss of stable synchronisation, but may also restabilise synchronisation or introduce multiple transitions between synchronisation and desynchronisation. To underline the scope of our results, we show using the Stuart-Landau model that such multiple transitions do not only occur in excitable systems, but also in oscillatory ones.Comment: 10 pages, 9 figure

Properties of a random attachment growing network

In this study we introduce and analyze the statistical structural properties of a model of growing networks which may be relevant to social networks. At each step a new node is added which selects 'k' possible partners from the existing network and joins them with probability delta by undirected edges. The 'activity' of the node ends here; it will get new partners only if it is selected by a newcomer. The model produces an infinite-order phase transition when a giant component appears at a specific value of delta, which depends on k. The average component size is discontinuous at the transition. In contrast, the network behaves significantly different for k=1. There is no giant component formed for any delta and thus in this sense there is no phase transition. However, the average component size diverges for delta greater or equal than one half.Comment: LaTeX, 19 pages, 6 figures. Discussion section, comments, a new figure and a new reference are added. Equations simplifie