Scale-free (SF) networks exhibiting a power-law degree distribution can be grouped into the assortative, dissortative, and neutral networks according to the behavior of the degree-degree correlation coefficient. Here we investigate the betweenness centrality (BC) correlation for each type of SF networks. While the BC-BC correlation coefficients behave similarly to the degree-degree correlation coefficients for the dissortative and neutral networks, the BC correlation is nontrivial for the assortative ones found mainly in social networks. The mean BC of neighbors of a vertex with BC $g_i$ is almost independent of $g_i,$ implying that each person is surrounded by almost the same influential environments of people no matter how influential the person may be.

Fractal scaling—a power-law behavior of the number of boxes needed to tile a given network with respect to the lateral size of the box—is studied. We introduce a box-covering algorithm that is a modified version of the original algorithm introduced by Song et al. [Nature (London) 433, 392 (2005)]; this algorithm enables easy implementation. Fractal networks are viewed as comprising a skeleton and shortcuts. The skeleton, embedded underneath the original network, is a special type of spanning tree based on the edge betweenness centrality; it provides a scaffold for the fractality of the network. When the skeleton is regarded as a branching tree, it exhibits a plateau in the mean branching number as a function of the distance from a root. For nonfractal networks, on the other hand, the mean branching number decays to zero without forming a plateau. Based on these observations, we construct a fractal network model by combining a random branching tree and local shortcuts. The scaffold branching tree can be either critical or supercritical, depending on the small worldness of a given network. For the network constructed from the critical (supercritical) branching tree, the average number of vertices within a given box grows with the lateral size of the box according to a power-law (an exponential) form in the cluster-growing method. The critical and supercritical skeletons are observed in protein interaction networks and the World Wide Web, respectively. The distribution of box masses, i.e., the number of vertices within each box, follows a power law $P_m\left(M\right)\sim M^{-\eta }$. The exponent $\eta$ depends on the box lateral size ${\ell }_B$. For small values of ${\ell }_B$, $\eta$ is equal to the degree exponent $\gamma$ of a given scale-free network, whereas $\eta$ approaches the exponent $\tau =\gamma ∕(\gamma -1)$ as ${\ell }_B$ increases, which is the exponent of the cluster-size distribution of the random branching tree. Finally, we study the perimeter $H_{\alpha }$ of a given box $\alpha$, i.e., the number of edges connected to different boxes from a given box $\alpha$ as a function of the box mass $M_{B,\alpha }$. It is obtained that the average perimeter over the boxes with box mass $M_B$ is likely to scale as $⟨H\left(M_B\right)⟩\sim M_B$, irrespective of the box size ${\ell }_B$.

We study the synchronization transition (ST) of a modified Kuramoto model on two different types of modular complex networks. It is found that the ST depends on the type of intermodular connections. For the network with decentralized (centralized) intermodular connections, the ST occurs at finite coupling constant (behaves abnormally). Such distinct features are found in the yeast protein interaction network and the Internet, respectively. Moreover, by applying the finite-size scaling analysis to an artificial network with decentralized intermodular connections, we obtain the exponent associated with the order parameter of the ST to be $\beta \approx 1$ different from ${\beta }_{\mathrm{MF}}\approx 1∕2$ obtained from the scale-free network with the same degree distribution but the absence of modular structure, corresponding to the mean field value.

We have studied the effects of an external electric field on photoluminescence spectra and carrier lifetimes in strained ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}/\mathrm{G}\mathrm{a}\mathrm{N}$ quantum wells embedded in $p-i-n$ light-emitting diode (LED) structures. Two sample structures with $x=0.15\mathrm{}$ for blue LED’s and 0.2 for green LED’s have been investigated, with increasing reverse bias up to $-30\hspace{0.5em}\mathrm{V}.$ From spectrum-resolved photoluminescence, we observed region of blueshift and redshift in photoluminescence peak energies. From the energy shift, the strength of piezoelectric field was estimated to be $2.1\pm 0.2\hspace{0.5em}\mathrm{M}\mathrm{V}/\mathrm{c}\mathrm{m}.$ Within our bias range, we observed three orders and one order of magnitude changes in carrier lifetime, for blue and green LED’s, respectively. These time-domain results are explained by escape tunneling and thermionic emission, together with carrier recombination which depends on the electron-hole wave function overlap change.

Continuous redshift of an emission from 3.23 to 2.85 eV with decreasing excitation density is observed in the photoluminescence spectra of highly doped GaN:Mg. It has been explained by the formation of minibands originating from the overlap of deep levels and shallow acceptor levels. For nominally undoped samples the intensity ratio of a donor-bound exciton line and a donor–acceptor pair line changes dramatically with excitation density due to the limited number of acceptors.