Unified model for network dynamics exhibiting nonextensive statistics

We introduce a dynamical network model which unifies a number of network families which are individually known to exhibit $q$-exponential degree distributions. The present model dynamics incorporates static (non-growing) self-organizing networks, preferentially growing networks, and (preferentially) rewiring networks. Further, it exhibits a natural random graph limit. The proposed model generalizes network dynamics to rewiring and growth modes which depend on internal topology as well as on a metric imposed by the space they are embedded in. In all of the networks emerging from the presented model we find q-exponential degree distributions over a large parameter space. We comment on the parameter dependence of the corresponding entropic index q for the degree distributions, and on the behavior of the clustering coefficients and neighboring connectivity distributions.Comment: 11 pages 8 fig

Quantifying the connectivity of a network: The network correlation function method

Networks are useful for describing systems of interacting objects, where the nodes represent the objects and the edges represent the interactions between them. The applications include chemical and metabolic systems, food webs as well as social networks. Lately, it was found that many of these networks display some common topological features, such as high clustering, small average path length (small world networks) and a power-law degree distribution (scale free networks). The topological features of a network are commonly related to the network's functionality. However, the topology alone does not account for the nature of the interactions in the network and their strength. Here we introduce a method for evaluating the correlations between pairs of nodes in the network. These correlations depend both on the topology and on the functionality of the network. A network with high connectivity displays strong correlations between its interacting nodes and thus features small-world functionality. We quantify the correlations between all pairs of nodes in the network, and express them as matrix elements in the correlation matrix. From this information one can plot the correlation function for the network and to extract the correlation length. The connectivity of a network is then defined as the ratio between this correlation length and the average path length of the network. Using this method we distinguish between a topological small world and a functional small world, where the latter is characterized by long range correlations and high connectivity. Clearly, networks which share the same topology, may have different connectivities, based on the nature and strength of their interactions. The method is demonstrated on metabolic networks, but can be readily generalized to other types of networks.Comment: 10 figure

Inhomogeneous substructures hidden in random networks

We study the structure of the load-based spanning tree (LST) that carries the maximum weight of the Erdos-Renyi (ER) random network. The weight of an edge is given by the edge-betweenness centrality, the effective number of shortest paths through the edge. We find that the LSTs present very inhomogeneous structures in contrast to the homogeneous structures of the original networks. Moreover, it turns out that the structure of the LST changes dramatically as the edge density of an ER network increases, from scale free with a cutoff, scale free, to a starlike topology. These would not be possible if the weights are randomly distributed, which implies that topology of the shortest path is correlated in spite of the homogeneous topology of the random network.Comment: 4 pages, 4 figure

Optimization of Robustness of Complex Networks

Networks with a given degree distribution may be very resilient to one type of failure or attack but not to another. The goal of this work is to determine network design guidelines which maximize the robustness of networks to both random failure and intentional attack while keeping the cost of the network (which we take to be the average number of links per node) constant. We find optimal parameters for: (i) scale free networks having degree distributions with a single power-law regime, (ii) networks having degree distributions with two power-law regimes, and (iii) networks described by degree distributions containing two peaks. Of these various kinds of distributions we find that the optimal network design is one in which all but one of the nodes have the same degree, $k_1$ (close to the average number of links per node), and one node is of very large degree, $k_2 \sim N^{2/3}$, where $N$ is the number of nodes in the network.Comment: Accepted for publication in European Physical Journal

Giant Clusters in Random Ad Hoc Networks

The present paper introduces ad hoc communication networks as examples of large scale real networks that can be prospected by statistical means. A description of giant cluster formation based on the single parameter of node neighbor numbers is given along with the discussion of some asymptotic aspects of the giant cluster sizes.Comment: 6 pages, 5 figures; typos and correction

Random graph model with power-law distributed triangle subgraphs

Clustering is well-known to play a prominent role in the description and understanding of complex networks, and a large spectrum of tools and ideas have been introduced to this end. In particular, it has been recognized that the abundance of small subgraphs is important. Here, we study the arrangement of triangles in a model for scale-free random graphs and determine the asymptotic behavior of the clustering coefficient, the average number of triangles, as well as the number of triangles attached to the vertex of maximum degree. We prove that triangles are power-law distributed among vertices and characterized by both vertex and edge coagulation when the degree exponent satisfies $2<\beta<2.5$; furthermore, a finite density of triangles appears as $\beta=2+1/3$.Comment: 4 pages, 2 figure; v2: major conceptual change

Robustness of interdependent networks under targeted attack

When an initial failure of nodes occurs in interdependent networks, a cascade of failure between the networks occurs. Earlier studies focused on random initial failures. Here we study the robustness of interdependent networks under targeted attack on high or low degree nodes. We introduce a general technique and show that the {\it targeted-attack} problem in interdependent networks can be mapped to the {\it random-attack} problem in a transformed pair of interdependent networks. We find that when the highly connected nodes are protected and have lower probability to fail, in contrast to single scale free (SF) networks where the percolation threshold $p_c=0$, coupled SF networks are significantly more vulnerable with $p_c$ significantly larger than zero. The result implies that interdependent networks are difficult to defend by strategies such as protecting the high degree nodes that have been found useful to significantly improve robustness of single networks.Comment: 11 pages, 2 figure

Microscopic analysis of multipole susceptibility of actinide dioxides: A scenario of multipole ordering in AmO2_2

By evaluating multipole susceptibility of a seven-orbital impurity Anderson model with the use of a numerical renormalization group method, we discuss possible multipole states of actinide dioxides at low temperatures. In particular, here we point out a possible scenario for multipole ordering in americium dioxide. For Am$^{4+}$ ion with five $5f$ electrons, it is considered that the ground state is $\Gamma_7^{-}$ doublet and the first excited state is $\Gamma_8^{-}$ quartet, but we remark that the $f^5$ ground state is easily converted due to the competition between spin-orbit coupling and Coulomb interactions. Then, we find that the $\Gamma_8^-$ quartet can be the ground state of AmO$_2$ even for the same crystalline electric field potential. In the case of $\Gamma_8^-$ quartet ground state, the numerical results suggest that high-order multipoles such as quadrupole and octupole can be relevant to AmO$_2$.Comment: 8 pages, 4 figures. To appear in Phys. Rev.

Analytical Solution of a Stochastic Content Based Network Model

We define and completely solve a content-based directed network whose nodes consist of random words and an adjacency rule involving perfect or approximate matches, for an alphabet with an arbitrary number of letters. The analytic expression for the out-degree distribution shows a crossover from a leading power law behavior to a log-periodic regime bounded by a different power law decay. The leading exponents in the two regions have a weak dependence on the mean word length, and an even weaker dependence on the alphabet size. The in-degree distribution, on the other hand, is much narrower and does not show scaling behavior. The results might be of interest for understanding the emergence of genomic interaction networks, which rely, to a large extent, on mechanisms based on sequence matching, and exhibit similar global features to those found here.Comment: 13 pages, 5 figures. Rewrote conclusions regarding the relevance to gene regulation networks, fixed minor errors and replaced fig. 4. Main body of paper (model and calculations) remains unchanged. Submitted for publicatio

Coherence in scale-free networks of chaotic maps

We study fully synchronized states in scale-free networks of chaotic logistic maps as a function of both dynamical and topological parameters. Three different network topologies are considered: (i) random scale-free topology, (ii) deterministic pseudo-fractal scale-free network, and (iii) Apollonian network. For the random scale-free topology we find a coupling strength threshold beyond which full synchronization is attained. This threshold scales as $k^{-\mu}$, where $k$ is the outgoing connectivity and $\mu$ depends on the local nonlinearity. For deterministic scale-free networks coherence is observed only when the coupling strength is proportional to the neighbor connectivity. We show that the transition to coherence is of first-order and study the role of the most connected nodes in the collective dynamics of oscillators in scale-free networks.Comment: 9 pages, 8 figure