The two-star model: exact solution in the sparse regime and condensation transition

The $2$-star model is the simplest exponential random graph model that displays complex behavior, such as degeneracy and phase transition. Despite its importance, this model has been solved only in the regime of dense connectivity. In this work we solve the model in the finite connectivity regime, far more prevalent in real world networks. We show that the model undergoes a condensation transition from a liquid to a condensate phase along the critical line corresponding, in the ensemble parameters space, to the Erd\"os-R\'enyi graphs. In the fluid phase the model can produce graphs with a narrow degree statistics, ranging from regular to Erd\"os-R\'enyi graphs, while in the condensed phase, the "excess" degree heterogeneity condenses on a single site with degree $\sim\sqrt{N}$. This shows the unsuitability of the two-star model, in its standard definition, to produce arbitrary finitely connected graphs with degree heterogeneity higher than Erd\"os-R\'enyi graphs and suggests that non-pathological variants of this model may be attained by softening the global constraint on the two-stars, while keeping the number of links hardly constrained.Comment: 20 pages, 3 figure

The statistical geometry of scale-free random trees

The properties of scale-free random trees are investigated using both preconditioning on non-extinction and fixed size averages, in order to study the thermodynamic limit. The scaling form of volume probability is found, the connectivity dimensions are determined and compared with other exponents which describe the growth. The (local) spectral dimension is also determined, through the study of the massless limit of the Gaussian model on such trees.Comment: 21 pages, 2 figures, revtex4, minor changes (published version

The Internet AS-Level Topology: Three Data Sources and One Definitive Metric

We calculate an extensive set of characteristics for Internet AS topologies extracted from the three data sources most frequently used by the research community: traceroutes, BGP, and WHOIS. We discover that traceroute and BGP topologies are similar to one another but differ substantially from the WHOIS topology. Among the widely considered metrics, we find that the joint degree distribution appears to fundamentally characterize Internet AS topologies as well as narrowly define values for other important metrics. We discuss the interplay between the specifics of the three data collection mechanisms and the resulting topology views. In particular, we show how the data collection peculiarities explain differences in the resulting joint degree distributions of the respective topologies. Finally, we release to the community the input topology datasets, along with the scripts and output of our calculations. This supplement should enable researchers to validate their models against real data and to make more informed selection of topology data sources for their specific needs.Comment: This paper is a revised journal version of cs.NI/050803

Trapped surfaces and emergent curved space in the Bose-Hubbard model

A Bose-Hubbard model on a dynamical lattice was introduced in previous work as a spin system analogue of emergent geometry and gravity. Graphs with regions of high connectivity in the lattice were identified as candidate analogues of spacetime geometries that contain trapped surfaces. We carry out a detailed study of these systems and show explicitly that the highly connected subgraphs trap matter. We do this by solving the model in the limit of no back-reaction of the matter on the lattice, and for states with certain symmetries that are natural for our problem. We find that in this case the problem reduces to a one-dimensional Hubbard model on a lattice with variable vertex degree and multiple edges between the same two vertices. In addition, we obtain a (discrete) differential equation for the evolution of the probability density of particles which is closed in the classical regime. This is a wave equation in which the vertex degree is related to the local speed of propagation of probability. This allows an interpretation of the probability density of particles similar to that in analogue gravity systems: matter inside this analogue system sees a curved spacetime. We verify our analytic results by numerical simulations. Finally, we analyze the dependence of localization on a gradual, rather than abrupt, fall-off of the vertex degree on the boundary of the highly connected region and find that matter is localized in and around that region.Comment: 16 pages two columns, 12 figures; references added, typos correcte

Epidemic spreading in complex networks with degree correlations

We review the behavior of epidemic spreading on complex networks in which there are explicit correlations among the degrees of connected vertices.Comment: Contribution to the Proceedings of the XVIII Sitges Conference "Statistical Mechanics of Complex Networks", eds. J.M. Rubi et, al (Springer Verlag, Berlin, 2003

Spin models on random graphs with controlled topologies beyond degree constraints

We study Ising spin models on finitely connected random interaction graphs which are drawn from an ensemble in which not only the degree distribution $p(k)$ can be chosen arbitrarily, but which allows for further fine-tuning of the topology via preferential attachment of edges on the basis of an arbitrary function Q(k,k') of the degrees of the vertices involved. We solve these models using finite connectivity equilibrium replica theory, within the replica symmetric ansatz. In our ensemble of graphs, phase diagrams of the spin system are found to depend no longer only on the chosen degree distribution, but also on the choice made for Q(k,k'). The increased ability to control interaction topology in solvable models beyond prescribing only the degree distribution of the interaction graph enables a more accurate modeling of real-world interacting particle systems by spin systems on suitably defined random graphs.Comment: 21 pages, 4 figures, submitted to J Phys

Kinetic growth walks on complex networks

Kinetically grown self-avoiding walks on various types of generalized random networks have been studied. Networks with short- and long-tailed degree distributions $P(k)$ were considered ($k$, degree or connectivity), including scale-free networks with $P(k) \sim k^{-\gamma}$. The long-range behaviour of self-avoiding walks on random networks is found to be determined by finite-size effects. The mean self-intersection length of non-reversal random walks, , scales as a power of the system size $N$: $ \sim N^{\beta}$, with an exponent $\beta = 0.5$ for short-tailed degree distributions and $\beta < 0.5$ for scale-free networks with $\gamma < 3$. The mean attrition length of kinetic growth walks, , scales as $\sim N^{\alpha}$, with an exponent $\alpha$ which depends on the lowest degree in the network. Results of approximate probabilistic calculations are supported by those derived from simulations of various kinds of networks. The efficiency of kinetic growth walks to explore networks is largely reduced by inhomogeneity in the degree distribution, as happens for scale-free networks.Comment: 10 pages, 8 figure