Self-similar disk packings as model spatial scale-free networks

Modified diffusive epidemic process on Apollonian networks

We present an analysis of an epidemic spreading process on an Apollonian network that can describe an epidemic spreading in a non-sedentary population. We studied the modified diffusive epidemic process using the Monte Carlo method by computational analysis. Our model may be helpful for modeling systems closer to reality consisting of two classes of individuals: susceptible (A) and infected (B). The individuals can diffuse in a network according to constant diffusion rates [Formula: see text] and [Formula: see text], for the classes A and B, respectively, and obeying three diffusive regimes, i.e., [Formula: see text], [Formula: see text], and [Formula: see text]. Into the same site i, the reaction occurs according to the dynamical rule based on Gillespie's algorithm. Finite-size scaling analysis has shown that our model exhibits continuous phase transition to an absorbing state with a set of critical exponents given by [Formula: see text], [Formula: see text], and [Formula: see text] familiar to every investigated regime. In summary, the continuous phase transition, characterized by this set of critical exponents, does not have the same exponents of the mean-field universality class in both regular lattices and complex networks.

Structure Properties of Generalized Farey graphs based on Dynamical Systems for Networks

Farey graphs are simultaneously small-world, uniquely Hamiltonian, minimally 3-colorable, maximally outerplanar and perfect. Farey graphs are therefore famous in deterministic models for complex networks. By lacking of the most important characteristics of scale-free, Farey graphs are not a good model for networks associated with some empirical complex systems. We discuss here a category of graphs which are extension of the well-known Farey graphs. These new models are named generalized Farey graphs here. We focus on the analysis of the topological characteristics of the new models and deduce the complicated and graceful analytical results from the growth mechanism used in generalized Farey graphs. The conclusions show that the new models not only possess the properties of being small-world and highly clustered, but also possess the quality of being scale-free. We also find that it is precisely because of the exponential increase of nodes' degrees in generalized Farey graphs as they grow that caused the new networks to have scale-free characteristics. In contrast, the linear incrementation of nodes' degrees in Farey graphs can only cause an exponential degree distribution.

Two-point resistances in an Apollonian network

The computation of resistance between two nodes in a resistor network is a classical problem in electric theory and graph theory. Based on the Apollonian packing, Andrade et al. introduced a deterministic growing type of networks A(k) [Phys. Rev. Lett. 94, 018702 (2005)PRLTAO0031-900710.1103/PhysRevLett.94.018702]. In this paper, first, a recursive algorithm for computing resistance between any two nodes in A(k) is given. Then as explanations, using the algorithm, explicit expressions for some resistances in A(k) are obtained.

Ferromagnetic model on the Apollonian packing

This work investigates the influence of geometrical features of the Apollonian packing (AP) on the behavior of magnetic models. The proposed model differs from previous investigations on the Apollonian network (AN), where the magnetic coupling constants depend only on the properties of the network structure defined by the packing, but not on quantitative aspects of its geometry. In opposition to the exact scale invariance observed in the AN, the circle's sizes in the AP are scaled by different factors when one goes from one generation to the next, requiring a different approach for the evaluation of the model's properties. If the nearest-neighbors coupling constants are defined by J_{i,j}∼1/(r_{i}+r_{j})^{α}, where r_{i} indicates the radius of the circle i containing the node i, the results for the correlation length ξ indicate that the model's behavior depend on α. In the thermodynamic limit, the uniform model (α=0) is characterized by ξ→∞ for all T>0. Our results indicate that, on increasing α, the system changes to an uncorrelated pattern, with finite ξ at all T>0, at a value α_{c}≃0.743. For any fixed value of α, no finite temperature singularity in the specific heat is observed, indicating that changes in the magnetic ordering occur only when α is changed. This is corroborated by the results for the magnetization and magnetic susceptibility.

Properties of kinetic transition networks for atomic clusters and glassy solids

Small-world and scale-free properties are analysed for kinetic transition networks of clusters and glassy systems.

Degrees and distances in random and evolving apollonian networks

In this paper we study random Apollonian networks (RANs) and evolving Apollonian networks (EANs), in d dimensions for any d≥2, i.e. dynamically evolving random d-dimensional simplices, looked at as graphs inside an initial d-dimensional simplex. We determine the limiting degree distribution in RANs and show that it follows a power-law tail with exponent τ=(2d-1)/(d-1). We further show that the degree distribution in EANs converges to the same degree distribution if the simplex-occupation parameter in the nth step of the dynamics tends to 0 but is not summable in n. This result gives a rigorous proof for the conjecture of Zhang et al. (2006) that EANs tend to exhibit similar behaviour as RANs once the occupation parameter tends to 0. We also determine the asymptotic behaviour of the shortest paths in RANs and EANs for any d≥2. For RANs we show that the shortest path between two vertices chosen u.a.r. (typical distance), the flooding time of a vertex chosen uniformly at random, and the diameter of the graph after n steps all scale as a constant multiplied by log n. We determine the constants for all three cases and prove a central limit theorem for the typical distances. We prove a similar central limit theorem for typical distances in EANs.

Prediction and Control of Slip-Free Rotation States in Sphere Assemblies

We study fixed assemblies of touching spheres that can individually rotate. From any initial state, sliding friction drives an assembly toward a slip-free rotation state. For bipartite assemblies, which have only even loops, this state has at least four degrees of freedom. For exactly four degrees of freedom, we analytically predict the final state, which we prove to be independent of the strength of sliding friction, from an arbitrary initial one. With a tabletop experiment, we show how to impose any slip-free rotation state by only controlling two spheres, regardless of the total number.

Combining solid-state NMR spectroscopy with first-principles calculations - a guide to NMR crystallography

Recent advances in the application of first-principles calculations of NMR parameters to periodic systems have resulted in widespread interest in their use to support experimental measurement. Such calculations often play an important role in the emerging field of "NMR crystallography", where NMR spectroscopy is combined with techniques such as diffraction, to aid structure determination. Here, we discuss the current state-of-the-art for combining experiment and calculation in NMR spectroscopy, considering the basic theory behind the computational approaches and their practical application. We consider the issues associated with geometry optimisation and how the effects of temperature may be included in the calculation. The automated prediction of structural candidates and the treatment of disordered and dynamic solids are discussed. Finally, we consider the areas where further development is needed in this field and its potential future impact.

Two-phase fluid flow in geometric packing

We investigate how a plug of obstacles inside a two-dimensional channel affects the drainage of high viscous fluid (oil) when the channel is invaded by a less viscous fluid (water). The plug consists of an Apollonian packing with, at most, 17 circles of different sizes, which is intended to model an inhomogeneous porous region. The work aims to quantify the amount of retained oil in the region where the flow is influenced by the packing. The investigation, carried out with the help of the computational fluid dynamics package ANSYS-FLUENT, is based on the integration of the complete set of equations of motion. The study considers the effect of both the injection speed and the number and size of obstacles, which directly affects the porosity of the system. The results indicate a complex dependence in the fraction of retained oil on the velocity and geometric parameters. The regions where the oil remains trapped is very sensitive to the number of circles and their size, which influence in different ways the porosity of the system. Nevertheless, at low values of Reynolds and capillary numbers Re<4 and n(c)≃10(-5), the overall expected result that the volume fraction of oil retained decreases with increasing porosity is recovered. A direct relationship between the injection speed and the fraction of oil is also obtained.

Generalized Open Quantum Walks on Apollonian Networks

We introduce the model of generalized open quantum walks on networks using the Transition Operation Matrices formalism. We focus our analysis on the mean first passage time and the average return time in Apollonian networks. These results differ significantly from a classical walk on these networks. We show a comparison of the classical and quantum behaviour of walks on these networks.

Complex networks from space-filling bearings

Two-dimensional space-filling bearings are dense packings of disks that can rotate without slip. We consider the entire first family of bearings for loops of four disks and propose a hierarchical construction of their contact network. We provide analytic expressions for the clustering coefficient and degree distribution, revealing bipartite scale-free behavior with a tunable degree exponent depending on the bearing parameters. We also analyze their average shortest path and percolation properties.

Critical behavior of the ideal-gas Bose-Einstein condensation in the Apollonian network

We show that the ideal Boson gas displays a finite-temperature Bose-Einstein condensation transition in the complex Apollonian network exhibiting scale-free, small-world, and hierarchical properties. The single-particle tight-binding Hamiltonian with properly rescaled hopping amplitudes has a fractal-like energy spectrum. The energy spectrum is analytically demonstrated to be generated by a nonlinear mapping transformation. A finite-size scaling analysis over several orders of magnitudes of network sizes is shown to provide precise estimates for the exponents characterizing the condensed fraction, correlation size, and specific heat. The critical exponents, as well as the power-law behavior of the density of states at the bottom of the band, are similar to those of the ideal Boson gas in lattices with spectral dimension d(s)=2ln(3)/ln(9/5)~/=3.74.

Locating the source of diffusion in large-scale networks

How can we localize the source of diffusion in a complex network? Because of the tremendous size of many real networks-such as the internet or the human social graph-it is usually unfeasible to observe the state of all nodes in a network. We show that it is fundamentally possible to estimate the location of the source from measurements collected by sparsely placed observers. We present a strategy that is optimal for arbitrary trees, achieving maximum probability of correct localization. We describe efficient implementations with complexity O(N(α)), where α=1 for arbitrary trees and α=3 for arbitrary graphs. In the context of several case studies, we determine how localization accuracy is affected by various system parameters, including the structure of the network, the density of observers, and the number of observed cascades.

Building complex networks with Platonic solids

We propose a unified model to build planar graphs with diverse topological characteristics which are of relevance in real applications. Here convex regular polyhedra (Platonic solids) are used as the building blocks for the construction of a variety of complex planar networks. These networks are obtained by merging polyhedra face by face on a tree-structure leading to planar graphs. We investigate two different constructions: (1) a fully deterministic construction where a self-similar fractal structure is built by using a single kind of polyhedron which is iteratively attached to every face and (2) a stochastic construction where at each step a polyhedron is attached to a randomly chosen face. These networks are scale-free, small-world, clustered, and sparse, sharing several characteristics of real-world complex networks. We derive analytical expressions for the degree distribution, the clustering coefficient, and the mean degree of nearest neighbors showing that these networks have power-law degree distributions with tunable exponents associated with the building polyhedron, and they possess a hierarchical organization that is determined by planarity.

q-state Potts model on the Apollonian network

The q-state Potts model is studied on the Apollonian network with Monte Carlo simulations and the transfer matrix method. The spontaneous magnetization, correlation length, entropy, and specific heat are analyzed as a function of temperature for different number of states, q. Different scaling functions in temperature and q are proposed. A quantitative agreement is found between results from both methods. No critical behavior is observed in the thermodynamic limit for any number of states.

Geographical networks stochastically constructed by a self-similar tiling according to population

In real communication and transportation networks, the geographical positions of nodes are very important for the efficiency and the tolerance of connectivity. Considering spatially inhomogeneous positions of nodes according to a population, we introduce a multiscale quartered (MSQ) network that is stochastically constructed by recursive subdivision of polygonal faces as a self-similar tiling. It has several advantages: the robustness of connectivity, the bounded short path lengths, and the shortest distance routing algorithm in a distributive manner. Furthermore, we show that the MSQ network is more efficient with shorter link lengths and more suitable with lower load for avoiding traffic congestion than other geographical networks which have various topologies ranging from river to scale-free networks. These results will be useful for providing an insight into the future design of ad hoc network infrastructures.

Estimate for the fractal dimension of the Apollonian gasket in d dimensions

We adapt a recent theory for the random close packing of polydisperse spheres in three dimensions [R. S. Farr and R. D. Groot, J. Chem. Phys. 131, 244104 (2009)] in order to predict the Hausdorff dimension dA of the Apollonian gasket in dimensions 2 and above. Our approximate results agree with published values in two and three dimensions to within 0.05% and 0.6%, respectively, and we provide predictions for dimensions 4-8.

Bose-Einstein condensation in the Apollonian complex network

We demonstrate that a topology-induced Bose-Einstein condensation (BEC) takes place in a complex network. As a model topology, we consider the deterministic Apollonian network which exhibits scale-free, small-world, and hierarchical properties. Within a tight-binding approach for noninteracting bosons, we report that the BEC transition temperature and the gap between the ground and first excited states follow the same finite-size scaling law. An anomalous density dependence of the transition temperature is reported and linked to the structure of gaps and degeneracies of the energy spectrum. The specific heat is shown to be discontinuous at the transition, with low-temperature modulations as a consequence of the fragmented density of states.

Infinitely robust order and local order-parameter tulips in Apollonian networks with quenched disorder

For a variety of quenched random spin systems on an Apollonian network, including ferromagnetic and antiferromagnetic bond percolation and the Ising spin glass, we find the persistence of ordered phases up to infinite temperature over the entire range of disorder. We develop a renormalization-group technique that yields highly detailed information, including the exact distributions of local magnetizations and local spin-glass order parameters, which turn out to exhibit, as function of temperature, complex and distinctive tulip patterns.

Standard random walks and trapping on the Koch network with scale-free behavior and small-world effect

A vast variety of real-life networks display the ubiquitous presence of scale-free phenomenon and small-world effect, both of which play a significant role in the dynamical processes running on networks. Although various dynamical processes have been investigated in scale-free small-world networks, analytical research about random walks on such networks is much less. In this paper, we will study analytically the scaling of the mean first-passage time (MFPT) for random walks on scale-free small-world networks. To this end, we first map the classical Koch fractal to a network, called Koch network. According to this proposed mapping, we present an iterative algorithm for generating the Koch network; based on which we derive closed-form expressions for the relevant topological features, such as degree distribution, clustering coefficient, average path length, and degree correlations. The obtained solutions show that the Koch network exhibits scale-free behavior and small-world effect. Then, we investigate the standard random walks and trapping issue on the Koch network. Through the recurrence relations derived from the structure of the Koch network, we obtain the exact scaling for the MFPT. We show that in the infinite network order limit, the MFPT grows linearly with the number of all nodes in the network. The obtained analytical results are corroborated by direct extensive numerical calculations. In addition, we also determine the scaling efficiency exponents characterizing random walks on the Koch network.

Uncovering the properties of energy-weighted conformation space networks with a hydrophobic-hydrophilic model

The conformation spaces generated by short hydrophobic-hydrophilic (HP) lattice chains are mapped to conformation space networks (CSNs). The vertices (nodes) of the network are the conformations and the links are the transitions between them. It has been found that these networks have "small-world" properties without considering the interaction energy of the monomers in the chain, i. e. the hydrophobic or hydrophilic amino acids inside the chain. When the weight based on the interaction energy of the monomers in the chain is added to the CSNs, it is found that the weighted networks show the "scale-free" characteristic. In addition, it reveals that there is a connection between the scale-free property of the weighted CSN and the folding dynamics of the chain by investigating the relationship between the scale-free structure of the weighted CSN and the noted parameter Z score. Moreover, the modular (community) structure of weighted CSNs is also studied. These results are helpful to understand the topological properties of the CSN and the underlying free-energy landscapes.

Ising model on the Apollonian network with node-dependent interactions

This work considers an Ising model on the Apollonian network, where the exchange constant J(i,j) approximately 1/(k(i)k(j))(mu) between two neighboring spins (i,j) is a function of the degree k of both spins. Using the exact geometrical construction rule for the network, the thermodynamical and magnetic properties are evaluated by iterating a system of discrete maps that allows for very precise results in the thermodynamic limit. The results can be compared to the predictions of a general framework for spin models on scale-free networks, where the node distribution P(k) approximately k(-gamma) , with node-dependent interacting constants. We observe that, by increasing mu , the critical behavior of the model changes from a phase transition at T=infinity for a uniform system (mu=0) to a T=0 phase transition when mu=1 : in the thermodynamic limit, the system shows no true critical behavior at a finite temperature for the whole mu > or = 0 interval. The magnetization and magnetic susceptibility are found to present noncritical scaling properties.

Free-electron gas in the Apollonian network: multifractal energy spectrum and its thermodynamic fingerprints

We study the free-electron gas in an Apollonian network within the tight-binding framework. The scale-free and small-world character of the underlying lattice is known to result in a quite structured energy spectrum with deltalike singularities, gaps, and minibands. After an exact numerical diagonalization of the corresponding adjacency matrix of the network with a finite number of generations, we employ a scaling analysis of the moments of the density of states to characterize its multifractality and report the associated singularity spectrum. The fractal nature of the energy spectrum is also shown to be reflected in the thermodynamic behavior by logarithmic modulations on the temperature dependence of the specific heat. The absence of chiral symmetry of the Apollonian network leads to distinct thermodynamic behaviors due to electrons and holes thermal excitations.

Finite-size effects for percolation on Apollonian networks

We study the percolation problem on the Apollonian network model. The Apollonian networks display many interesting properties commonly observed in real network systems, such as small-world behavior, scale-free distribution, and a hierarchical structure. By taking advantage of the deterministic hierarchical construction of these networks, we use the real-space renormalization-group technique to write exact iterative equations that relate percolation network properties at different scales. More precisely, our results indicate that the percolation probability and average mass of the percolating cluster approach the thermodynamic limit logarithmically. We suggest that such ultraslow convergence might be a property of hierarchical networks. Since real complex systems are certainly finite and very commonly hierarchical, we believe that taking into account finite-size effects in real-network systems is of fundamental importance.

Coherent transport on Apollonian networks and continuous-time quantum walks

We study the coherent exciton transport on Apollonian networks generated by simple iterative rules. The coherent exciton dynamics is modeled by continuous-time quantum walks and we calculate the transition probabilities between two nodes of the networks. We find that the transport depends on the initial nodes of the excitation. For networks up to the second generation the coherent transport shows perfect recurrences when the initial excitation starts at the central node. For networks of higher generation, the transport only shows partial revivals. Moreover, we find that the excitation is most likely to be found at the initial nodes while the coherent transport to other nodes has a very low probability. In the long time limit, the transition probabilities show characteristic patterns with identical values of limiting probabilities. Finally, the dynamics of quantum transport are compared with the classical transport modeled by continuous-time random walks.

Analytical solution of average path length for Apollonian networks

With the help of recursion relations derived from the self-similar structure, we obtain the solution of average path length, d[over ]_(t) , for Apollonian networks. In contrast to the well-known numerical result d[over ]_{t} proportional, variant(ln N_(t));(3/4) [J. S. Andrade, Jr., Phys. Rev. Lett. 94, 018702 (2005)], our rigorous solution shows that the average path length grows logarithmically as d[over ]_(t) proportional, variantln N_(t) in the infinite limit of network size N_(t) . The extensive numerical calculations completely agree with our closed-form solution.

Network of inherent structures in spin glasses: scaling and scale-free distributions

The local minima (inherent structures) of a system and their associated transition links give rise to a network. Here we consider the topological and distance properties of such a network in the context of spin glasses. We use steepest descent dynamics, determining for each disorder sample the transition links appearing within a given barrier height. We find that differences between linked inherent structures are typically associated with local clusters of spins; we interpret this within a framework based on droplets in which the characteristic "length scale" grows with the barrier height. We also consider the network connectivity and the degrees of its nodes. Interestingly, for spin glasses based on random graphs, the degree distribution of the network of inherent structures exhibits a nontrivial scale-free tail.

Neighbor network in a polydisperse hard-disk fluid: degree distribution and assortativity

The neighbor network in a two-dimensional polydisperse hard-disk fluid with diameter distribution p(sigma) approximately sigma(-4) is examined using constant-pressure Monte Carlo simulations. Graphs are constructed from vertices (disks) with edges (links) connecting each vertex to k neighboring vertices defined by a radical tessellation. At packing fractions in the range 0.24< or =eta< or =0.36, the decay of the network degree distribution is observed to be consistent with the power law k(-gamma) where the exponent lies in the range 5.6< or =gamma< or =6.0 . Comparisons with the predictions of a maximum-entropy theory suggest that this apparent power-law behavior is not the asymptotic one and that p(k) approximately k(-4) in the limit k-->infinity. This is consistent with the simple idea that for large disks, the number of neighbors is proportional to the disk diameter. A power-law decay of the network degree distribution is one of the characteristics of a scale-free network. The assortativity of the network is measured and is found to be positive, meaning that vertices of equal degree are connected more often than in a random network. Finally, the equation of state is determined and compared with the prediction from a scaled-particle theory. Very good agreement between simulation and theory is demonstrated.

Preferential attachment during the evolution of a potential energy landscape

It has previously been shown that the network of connected minima on a potential energy landscape is scale-free, and that this reflects a power-law distribution for the areas of the basins of attraction surrounding the minima. Here, the aim is to understand more about the physical origins of these puzzling properties by examining how the potential energy landscape of a 13-atom cluster evolves with the range of the potential. In particular, on decreasing the range of the potential the number of stationary points increases and thus the landscape becomes rougher and the network gets larger. Thus, the evolution of the potential energy landscape can be followed from one with just a single minimum to a complex landscape with many minima and a scale-free pattern of connections. It is found that during this growth process, new edges in the network of connected minima preferentially attach to more highly connected minima, thus leading to the scale-free character. Furthermore, minima that appear when the range of the potential is shorter and the network is larger have smaller basins of attraction. As there are many of these smaller basins because the network grows exponentially, the observed growth process thus also gives rise to a power-law distribution for the hyperareas of the basins.

Analytical approach to directed sandpile models on the Apollonian network

We investigate a set of directed sandpile models on the Apollonian network, which are inspired by the work of Dhar and Ramaswamy [Phys. Rev. Lett. 63, 1659 (1989)] on Euclidian lattices. They are characterized by a single parameter q , which restricts the number of neighbors receiving grains from a toppling node. Due to the geometry of the network, two- and three-point correlation functions are amenable to exact treatment, leading to analytical results for avalanche distributions in the limit of an infinite system for q=1,2 . The exact recurrence expressions for the correlation functions are numerically iterated to obtain results for finite-size systems when larger values of q are considered. Finally, a detailed description of the local flux properties is provided by a multifractal scaling analysis.

Griffiths singularities and algebraic order in the exact solution of an Ising model on a fractal modular network

We use an exact renormalization-group transformation to study the Ising model on a complex network composed of tightly knit communities nested hierarchically with the fractal scaling recently discovered in a variety of real-world networks. Varying the ratio KJ of intercommunity to intracommunity couplings, we obtain an unusual phase diagram: at high temperatures or small KJ we have a disordered phase with a Griffiths singularity in the free energy, due to the presence of rare large clusters, which we analyze through the Yang-Lee zeros in the complex magnetic field plane. As the temperature is lowered, true long-range order is not seen, but there is a transition to algebraic order, where pair correlations have power-law decay with distance, reminiscent of the XY model. The transition is infinite order at small KJ and becomes second order above a threshold value (KJ)_{m} . The existence of such slowly decaying correlations is unexpected in a fat-tailed scale-free network, where correlations longer than nearest neighbor are typically suppressed.

Growing networks under geographical constraints

Inspired by the structure of technological weblike systems, we discuss network evolution mechanisms which give rise to topological properties found in real spatial networks. Thus, we suggest that the peculiar structure of transport and distribution networks is fundamentally determined by two factors. These are the dependence of the spatial interaction range of vertices on the vertex attractiveness (or importance within the network) and on the inhomogeneous distribution of vertices in space. We propose and analyze numerically a simple model based on these generating mechanisms which seems, for instance, to be able to reproduce known structural features of the Internet.

Power-law distributions for the areas of the basins of attraction on a potential energy landscape

Energy landscape approaches have become increasingly popular for analyzing a wide variety of chemical physics phenomena. Basic to many of these applications has been the inherent structure mapping, which divides up the potential energy landscape into basins of attraction surrounding the minima. Here, we probe the nature of this division by introducing a method to compute the basin area distribution and applying it to some archetypal supercooled liquids. We find that this probability distribution is a power law over a large number of decades with the lower-energy minima having larger basins of attraction. Interestingly, the exponent for this power law is approximately the same as that for a high-dimensional Apollonian packing, providing further support for the suggestion that there is a strong analogy between the way the energy landscape is divided into basins, and the way that space is packed in self-similar, space-filling hypersphere packings, such as the Apollonian packing. These results suggest that the basins of attraction provide a fractal-like tiling of the energy landscape, and that a scale-free pattern of connections between the minima is a general property of energy landscapes.

Evolving Apollonian networks with small-world scale-free topologies

We propose two types of evolving networks: evolutionary Apollonian networks (EANs) and general deterministic Apollonian networks (GDANs), established by simple iteration algorithms. We investigate the two networks by both simulation and theoretical prediction. Analytical results show that both networks follow power-law degree distributions, with distribution exponents continuously tuned from 2 to 3. The accurate expression of clustering coefficient is also given for both networks. Moreover, the investigation of the average path length of EAN and the diameter of GDAN reveals that these two types of networks possess small-world feature. In addition, we study the collective synchronization behavior on some limitations of the EAN.

Geographical effects on the path length and the robustness in complex networks

The short paths between any two nodes and the robustness of connectivity are advanced properties of scale-free (SF) networks; however, they may be affected by geographical constraints in realistic situations. We consider geographical networks with the SF structure based on planar triangulation for online routings, and suggest scaling relations between the average distance or number of hops on the optimal paths and the network size. We also show that the tolerance to random failures and attacks on hubs is weakened in geographical networks, and that even then it is possible for the extremely vulnerable ones to be improved by adding with the local exchange of links.

Degree-dependent intervertex separation in complex networks

We study the mean length (l)(k) of the shortest paths between a vertex of degree k and other vertices in growing networks, where correlations are essential. In a number of deterministic scale-free networks we observe a power-law correction to a logarithmic dependence, (l)(k) = A ln[N/k((gamma-1)/2)]-Ck(gamma-1)/N+ in a wide range of network sizes. Here N is the number of vertices in the network, gamma is the degree distribution exponent, and the coefficients A and C depend on a network. We compare this law with a corresponding (l)(k) dependence obtained for random scale-free networks growing through the preferential attachment mechanism. In stochastic and deterministic growing trees with an exponential degree distribution, we observe a linear dependence on degree, (l)(k)approximately A ln N-Ck. We compare our findings for growing networks with those for uncorrelated graphs.

Neighborhood properties of complex networks

A concept of neighborhood in complex networks is addressed based on the criterion of the minimal number of steps to reach other vertices. This amounts to, starting from a given network R1, generating a family of networks Rl, l = 2, 3,... such that, the vertices that are l steps apart in the original R1, are only 1 step apart in Rl. The higher order networks are generated using Boolean operations among the adjacency matrices Ml that represent Rl. The families originated by the well known linear and the Erdös-Renyi networks are found to be invariant, in the sense that the spectra of Ml are the same, up to finite size effects. A further family originated from small world network is identified.

From simple to complex networks: inherent structures, barriers, and valleys in the context of spin glasses

Given discrete degrees of freedom (spins) on a graph interacting via an energy function, what can be said about the energy local minima and associated inherent structures? Using the lid algorithm in the context of a spin glass energy function, we investigate the properties of the energy landscape for a variety of graph topologies. First, we find that the multiplicity N(s) of the inherent structures generically has a log-normal distribution. In addition, the large volume limit of ln <N(s)>/<ln N(s)> differs from unity, except for the Sherrington-Kirkpatrick model. Second, we find simple scaling laws for the growth of the height of the energy barrier between the two degenerate ground states and the size of the associated valleys. For finite connectivity models, changing the topology of the underlying graph does not modify qualitatively the energy landscape, but at the quantitative level the models can differ substantially.

Synchronous and asynchronous recursive random scale-free nets

We investigate the differences between scale-free recursive nets constructed by a synchronous, deterministic updating rule (e.g., Apollonian nets), versus an asynchronous, random sequential updating rule (e.g., random Apollonian nets). We show that the dramatic discrepancies observed recently for the degree exponent in these two cases result from a biased choice of the units to be updated sequentially in the asynchronous version.

Magnetic models on Apollonian networks

Thermodynamic and magnetic properties of Ising models defined on the triangular Apollonian network are investigated. This and other similar networks are inspired by the problem of covering a Euclidian domain with circles of maximal radii. Maps for the thermodynamic functions in two subsequent generations of the construction of the network are obtained by formulating the problem in terms of transfer matrices. Numerical iteration of this set of maps leads to very precise values for the thermodynamic properties of the model. Different choices for the coupling constants between only nearest neighbors along the lattice are taken into account. For both ferromagnetic and antiferromagnetic constants, long-range magnetic ordering is obtained. With exception of a size-dependent effective critical behavior of the correlation length, no evidence of asymptotic criticality was detected.

Maximal planar networks with large clustering coefficient and power-law degree distribution

In this article, we propose a simple rule that generates scale-free networks with very large clustering coefficient and very small average distance. These networks are called random Apollonian networks (RANs) as they can be considered as a variation of Apollonian networks. We obtain the analytic results of power-law exponent gamma=3 and clustering coefficient C= (46/3)-36 ln 3/2 approximately 0.74, which agree with the simulation results very well. We prove that the increasing tendency of average distance of RANs is a little slower than the logarithm of the number of nodes in RANs. Since most real-life networks are both scale-free and small-world networks, RANs may perform well in mimicking the reality. The RANs possess hierarchical structure as C(k) approximately k(-1) that are in accord with the observations of many real-life networks. In addition, we prove that RANs are maximal planar networks, which are of particular practicability for layout of printed circuits and so on. The percolation and epidemic spreading process are also studied and the comparisons between RANs and Barabási-Albert (BA) as well as Newman-Watts (NW) networks are shown. We find that, when the network order N (the total number of nodes) is relatively small (as N approximately 10(4)), the performance of RANs under intentional attack is not sensitive to N , while that of BA networks is much affected by N. And the diseases spread slower in RANs than BA networks in the early stage of the susceptible-infected process, indicating that the large clustering coefficient may slow the spreading velocity, especially in the outbreaks.

Identifying communities within energy landscapes

Potential energy landscapes can be represented as a network of minima linked by transition states. The community structure of such networks has been obtained for a series of small Lennard-Jones (LJ) clusters. This community structure is compared to the concept of funnels in the potential energy landscape. Two existing algorithms have been used to find community structure, one involving removing edges with high betweenness, the other involving optimization of the modularity. The definition of the modularity has been refined, making it more appropriate for networks such as these where multiple edges and self-connections are not included. The optimization algorithm has also been improved, using Monte Carlo methods with simulated annealing and basin hopping, both often used successfully in other optimization problems. In addition to the small clusters, two examples with known heterogeneous landscapes, the 13-atom cluster (LJ13) with one labeled atom and the 38-atom cluster (LJ38) , were studied with this approach. The network methods found communities that are comparable to those expected from landscape analyses. This is particularly interesting since the network model does not take any barrier heights or energies of minima into account. For comparison, the network associated with a two-dimensional hexagonal lattice is also studied and is found to have high modularity, thus raising some questions about the interpretation of the community structure associated with such partitions.