Fractal and multifractal properties of a family of fractal networks

In this work, we study the fractal and multifractal properties of a family of fractal networks introduced by Gallos {\it et al.} ({\it Proc. Natl. Acad. Sci. U.S.A.}, 2007, {\bf 104}: 7746). In this fractal network model, there is a parameter $e$ which is between $0$ and $1$, and allows for tuning the level of fractality in the network. Here we examine the multifractal behavior of these networks, dependence relationship of fractal dimension and the multifractal parameters on the parameter $e$. First, we find that the empirical fractal dimensions of these networks obtained by our program coincide with the theoretical formula given by Song {\it et al.} ( {\it Nat. Phys}, 2006, {\bf 2}: 275). Then from the shape of the $\tau(q)$ and $D(q)$ curves, we find the existence of multifractality in these networks. Last, we find that there exists a linear relationship between the average information dimension $$ and the parameter $e$.Comment: 12 pages, 7 figures, accepted by J. Stat. Mec

Multifractal analysis of weighted networks by a modified sandbox algorithm

Complex networks have attracted growing attention in many fields. As a generalization of fractal analysis, multifractal analysis (MFA) is a useful way to systematically describe the spatial heterogeneity of both theoretical and experimental fractal patterns. Some algorithms for MFA of unweighted complex networks have been proposed in the past a few years, including the sandbox (SB) algorithm recently employed by our group. In this paper, a modified SB algorithm (we call it SBw algorithm) is proposed for MFA of weighted networks.First, we use the SBw algorithm to study the multifractal property of two families of weighted fractal networks (WFNs): "Sierpinski" WFNs and "Cantor dust" WFNs. We also discuss how the fractal dimension and generalized fractal dimensions change with the edge-weights of the WFN. From the comparison between the theoretical and numerical fractal dimensions of these networks, we can find that the proposed SBw algorithm is efficient and feasible for MFA of weighted networks. Then, we apply the SBw algorithm to study multifractal properties of some real weighted networks ---collaboration networks. It is found that the multifractality exists in these weighted networks, and is affected by their edge-weights.Comment: 15 pages, 6 figures. Accepted for publication by Scientific Report

Dicyanido[tris­(2-pyridyl­meth­yl)amine]­cobalt(III) hexa­fluorido­phosphate

In the title complex, [Co(CN)2(C18H18N4)]PF6, the CoIII atom together with one of the pyridyl rings and two cyanide anions are located on a mirror plane, while the P atom is located on an inversion centre. The CoIII atom exhibits an octa­hedral geometry, coordinated by four N atoms from the tris­(2-pyridyl­meth­yl)amine ligand with an average Co—N distance of 1.953 (2) Å, and two cyanide C atoms with an average Co—C distance of 1.895 (2) Å. The crystal packing is stabilized by inter­molecular C—H⋯N and C—H⋯F inter­actions

Probing WL′WHW^\prime_L WH and WR′WHW^\prime_R W H Interaction at LHC

Many new physics models predict the existence of TeV-scale charged gauge boson $W^\prime$ together with Higgs boson(s). We study the $W^\prime WH$ interaction and explore the angular distribution of charged lepton to distinguish $W_R^\prime WH$ from $W_L^\prime WH$ in $pp\to HW\to b \bar b l \nu$ process at the LHC. It is found that a new type forward-backward asymmetry($A_{FB}$) relating to the angle between the direction of the charged lepton in $W$ rest frame and that of the reconstructed $W^\prime$ in laboratory frame is useful to investigate the properties of $W^\prime W H$ interaction. We analyze the Standard Model backgrounds and develop a set of cuts to highlight the signal and suppress the backgrounds at LHC. We find that $A_{FB}$ can reach 0.03(-0.07) for $W_R^\prime$($W_L^\prime$) production at $\sqrt{S}=14$ TeV