Abstract:
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.

Abstract:
The von Neumann entropy for an electron in periodic, disorder and quasiperiodic quantum small-world networks(QSWNs) are studied numerically. For the disorder QSWNs, the derivative of the spectrum averaged von Neumann entropy is maximal at a certain density of shortcut links p*, which can be as a signature of the localization delocalization transition of electron states. The transition point p* is agreement with that obtained by the level statistics method. For the quasiperiodic QSWNs, it is found that there are two regions of the potential parameter. The behaviors of electron states in different regions are similar to that of periodic and disorder QSWNs, respectively.

Abstract:
We introduce a model of adaptive temporal networks whose evolution is regulated by an interplay between node activity and dynamic exchange of information through links. We study the model by using a master equation approach. Starting from a homogeneous initial configuration, we show that temporal and structural heterogeneities, characteristic of real-world networks, spontaneously emerge. This theoretically tractable model thus contributes to the understanding of the dynamics of human activity and interaction networks.

Abstract:
We apply a variant of the explosive percolation procedure to large real-world networks, and show with finite-size scaling that the university class, ordinary or explosive, of the resulting percolation transition depends on the structural properties of the network as well as the number of unoccupied links considered for comparison in our procedure. We observe that in our social networks, the percolation clusters close to the critical point are related to the community structure. This relationship is further highlighted by applying the procedure to model networks with pre-defined communities.

Abstract:
A small-world topology characterizes many complex systems including the structural and functional organization of brain networks. The topology allows simultaneously for local and global efficiency in the interaction of the system constituents. However, it ignores the gradations of interactions commonly quantified by the link weight, w. Here, we identify an integrative weight organization for brain, gene, social, and language networks, in which strong links preferentially occur between nodes with overlapping neighbourhoods and the small-world properties are robust to removal of a large fraction of the weakest links. We also determine local learning rules that dynamically establish such weight organization in response to past activity and capacity demands, while preserving efficient local and global communication.

Abstract:
A dissipative sandpile model (DSM) is constructed and studied on small world networks (SWN). SWNs are generated adding extra links between two arbitrary sites of a two dimensional square lattice with different shortcut densities $\phi$. Three different regimes are identified as regular lattice (RL) for $\phi\lesssim 2^{-12}$, SWN for $2^{-12}<\phi< 0.1$ and random network (RN) for $\phi\ge 0.1$. In the RL regime, the sandpile dynamics is characterized by usual Bak, Tang, Weisenfeld (BTW) type correlated scaling whereas in the RN regime it is characterized by the mean field (MF) scaling. On SWN, both the scaling behaviors are found to coexist. Small compact avalanches below certain characteristic size $s_c$ are found to belong to the BTW universality class whereas large, sparse avalanches above $s_c$ are found to belong to the MF universality class. A scaling theory for the coexistence of two scaling forms on SWN is developed and numerically verified. Though finite size scaling (FSS) is not valid for DSM on RL as well as on SWN, it is found to be valid on RN for the same model. FSS on RN is appeared to be an outcome of super diffusive sand transport and uncorrelated toppling waves.

Abstract:
Small-world networks (SWN), obtained by randomly adding to a regular structure additional links (AL), are of current interest. In this article we explore (based on physical models) a new variant of SWN, in which the probability of realizing an AL depends on the chemical distance between the connected sites. We assume a power-law probability distribution and study random walkers on the network, focussing especially on their probability of being at the origin. We connect the results to L\'evy Flights, which follow from a mean field variant of our model.

Abstract:
We considered diffusion-driven processes on small-world networks with distance-dependent random links. The study of diffusion on such networks is motivated by transport on randomly folded polymer chains, synchronization problems in task-completion networks, and gradient driven transport on networks. Changing the parameters of the distance-dependence, we found a rich phase diagram, with different transient and recurrent phases in the context of random walks on networks. We performed the calculations in two limiting cases: in the annealed case, where the rearrangement of the random links is fast, and in the quenched case, where the link rearrangement is slow compared to the motion of the random walker or the surface. It has been well-established that in a large class of interacting systems, adding an arbitrarily small density of, possibly long-range, quenched random links to a regular lattice interaction topology, will give rise to mean-field (or annealed) like behavior. In some cases, however, mean-field scaling breaks down, such as in diffusion or in the Edwards-Wilkinson process in "low-dimensional" small-world networks. This break-down can be understood by treating the random links perturbatively, where the mean-field (or annealed) prediction appears as the lowest-order term of a naive perturbation expansion. The asymptotic analytic results are also confirmed numerically by employing exact numerical diagonalization of the network Laplacian. Further, we construct a finite-size scaling framework for the relevant observables, capturing the cross-over behaviors in finite networks. This work provides a detailed account of the self-consistent-perturbative and renormalization approaches briefly introduced in two earlier short reports.

Abstract:
Many natural and social systems develop complex networks, that are usually modelled as random graphs. The eigenvalue spectrum of these graphs provides information about their structural properties. While the semi-circle law is known to describe the spectral density of uncorrelated random graphs, much less is known about the eigenvalues of real-world graphs, describing such complex systems as the Internet, metabolic pathways, networks of power stations, scientific collaborations or movie actors, which are inherently correlated and usually very sparse. An important limitation in addressing the spectra of these systems is that the numerical determination of the spectra for systems with more than a few thousand nodes is prohibitively time and memory consuming. Making use of recent advances in algorithms for spectral characterization, here we develop new methods to determine the eigenvalues of networks comparable in size to real systems, obtaining several surprising results on the spectra of adjacency matrices corresponding to models of real-world graphs. We find that when the number of links grows as the number of nodes, the spectral density of uncorrelated random graphs does not converge to the semi-circle law. Furthermore, the spectral densities of real-world graphs have specific features depending on the details of the corresponding models. In particular, scale-free graphs develop a triangle-like spectral density with a power law tail, while small-world graphs have a complex spectral density function consisting of several sharp peaks. These and further results indicate that the spectra of correlated graphs represent a practical tool for graph classification and can provide useful insight into the relevant structural properties of real networks.

Abstract:
In this paper, the effects of various structural properties on the synchronization of coupled oscillators with local-world coupling configurations are investigated. It is found that for local world networks, the larger heterogeneity of the degree distribution, the enhanced interconnection of nodes, and the increased clustering do not improve the synchronizability of dynamical systems. On the contrary, the increase of the maximum betweenness centrality appears to be responsible for the decrease of the synchronizability.