Properties of Random Graphs with Hidden Color

We investigate in some detail a recently suggested general class of ensembles of sparse undirected random graphs based on a hidden stub-coloring, with or without the restriction to nondegenerate graphs. The calculability of local and global structural properties of graphs from the resulting ensembles is demonstrated. Cluster size statistics are derived with generating function techniques, yielding a well-defined percolation threshold. Explicit rules are derived for the enumeration of small subgraphs. Duality and redundancy is discussed, and subclasses corresponding to commonly studied models are identified.Comment: 14 pages, LaTeX, no figure

Percolation transition and distribution of connected components in generalized random network ensembles

In this work, we study the percolation transition and large deviation properties of generalized canonical network ensembles. This new type of random networks might have a very rich complex structure, including high heterogeneous degree sequences, non-trivial community structure or specific spatial dependence of the link probability for networks embedded in a metric space. We find the cluster distribution of the networks in these ensembles by mapping the problem to a fully connected Potts model with heterogeneous couplings. We show that the nature of the Potts model phase transition, linked to the birth of a giant component, has a crossover from second to first order when the number of critical colors $q_c = 2$ in all the networks under study. These results shed light on the properties of dynamical processes defined on these network ensembles.Comment: 27 pages, 15 figure

Random acyclic networks

Directed acyclic graphs are a fundamental class of networks that includes citation networks, food webs, and family trees, among others. Here we define a random graph model for directed acyclic graphs and give solutions for a number of the model's properties, including connection probabilities and component sizes, as well as a fast algorithm for simulating the model on a computer. We compare the predictions of the model to a real-world network of citations between physics papers and find surprisingly good agreement, suggesting that the structure of the real network may be quite well described by the random graph.Comment: 4 pages, 2 figure

Laplacian spectra of complex networks and random walks on them: Are scale-free architectures really important?

We study the Laplacian operator of an uncorrelated random network and, as an application, consider hopping processes (diffusion, random walks, signal propagation, etc.) on networks. We develop a strict approach to these problems. We derive an exact closed set of integral equations, which provide the averages of the Laplacian operator's resolvent. This enables us to describe the propagation of a signal and random walks on the network. We show that the determining parameter in this problem is the minimum degree $q_m$ of vertices in the network and that the high-degree part of the degree distribution is not that essential. The position of the lower edge of the Laplacian spectrum $\lambda_c$ appears to be the same as in the regular Bethe lattice with the coordination number $q_m$. Namely, $\lambda_c>0$ if $q_m>2$, and $\lambda_c=0$ if $q_m\leq2$. In both these cases the density of eigenvalues $\rho(\lambda)\to0$ as $\lambda\to\lambda_c+0$, but the limiting behaviors near $\lambda_c$ are very different. In terms of a distance from a starting vertex, the hopping propagator is a steady moving Gaussian, broadening with time. This picture qualitatively coincides with that for a regular Bethe lattice. Our analytical results include the spectral density $\rho(\lambda)$ near $\lambda_c$ and the long-time asymptotics of the autocorrelator and the propagator.Comment: 25 pages, 4 figure

Non-equilibrium mean-field theories on scale-free networks

Many non-equilibrium processes on scale-free networks present anomalous critical behavior that is not explained by standard mean-field theories. We propose a systematic method to derive stochastic equations for mean-field order parameters that implicitly account for the degree heterogeneity. The method is used to correctly predict the dynamical critical behavior of some binary spin models and reaction-diffusion processes. The validity of our non-equilibrium theory is furtherly supported by showing its relation with the generalized Landau theory of equilibrium critical phenomena on networks.Comment: 4 pages, no figures, major changes in the structure of the pape

Spin Glass Phase Transition on Scale-Free Networks

We study the Ising spin glass model on scale-free networks generated by the static model using the replica method. Based on the replica-symmetric solution, we derive the phase diagram consisting of the paramagnetic (P), ferromagnetic (F), and spin glass (SG) phases as well as the Almeida-Thouless line as functions of the degree exponent $\lambda$, the mean degree $K$, and the fraction of ferromagnetic interactions $r$. To reflect the inhomogeneity of vertices, we modify the magnetization $m$ and the spin glass order parameter $q$ with vertex-weights. The transition temperature $T_c$ ($T_g$) between the P-F (P-SG) phases and the critical behaviors of the order parameters are found analytically. When $2 < \lambda < 3$, $T_c$ and $T_g$ are infinite, and the system is in the F phase or the mixed phase for $r > 1/2$, while it is in the SG phase at $r=1/2$. $m$ and $q$ decay as power-laws with increasing temperature with different $\lambda$-dependent exponents. When $\lambda > 3$, the $T_c$ and $T_g$ are finite and related to the percolation threshold. The critical exponents associated with $m$ and $q$ depend on $\lambda$ for $3 < \lambda < 5$ ($3 < \lambda < 4$) at the P-F (P-SG) boundary.Comment: Phys. Rev. E in pres

Sampling properties of random graphs: the degree distribution

We discuss two sampling schemes for selecting random subnets from a network: Random sampling and connectivity dependent sampling, and investigate how the degree distribution of a node in the network is affected by the two types of sampling. Here we derive a necessary and sufficient condition that guarantees that the degree distribution of the subnet and the true network belong to the same family of probability distributions. For completely random sampling of nodes we find that this condition is fulfilled by classical random graphs; for the vast majority of networks this condition will, however, not be met. We furthermore discuss the case where the probability of sampling a node depends on the degree of a node and we find that even classical random graphs are no longer closed under this sampling regime. We conclude by relating the results to real {\it E.coli} protein interaction network data.Comment: accepted for publication in Phys.Rev.

Localization Transition of Biased Random Walks on Random Networks

We study random walks on large random graphs that are biased towards a randomly chosen but fixed target node. We show that a critical bias strength b_c exists such that most walks find the target within a finite time when b>b_c. For b<b_c, a finite fraction of walks drifts off to infinity before hitting the target. The phase transition at b=b_c is second order, but finite size behavior is complex and does not obey the usual finite size scaling ansatz. By extending rigorous results for biased walks on Galton-Watson trees, we give the exact analytical value for b_c and verify it by large scale simulations.Comment: 4 pages, includes 4 figure

Interfaces and the edge percolation map of random directed networks

The traditional node percolation map of directed networks is reanalyzed in terms of edges. In the percolated phase, edges can mainly organize into five distinct giant connected components, interfaces bridging the communication of nodes in the strongly connected component and those in the in- and out-components. Formal equations for the relative sizes in number of edges of these giant structures are derived for arbitrary joint degree distributions in the presence of local and two-point correlations. The uncorrelated null model is fully solved analytically and compared against simulations, finding an excellent agreement between the theoretical predictions and the edge percolation map of synthetically generated networks with exponential or scale-free in-degree distribution and exponential out-degree distribution. Interfaces, and their internal organization giving place from "hairy ball" percolation landscapes to bottleneck straits, could bring new light to the discussion of how structure is interwoven with functionality, in particular in flow networks.Comment: 20 pages, 4 figure

Generation of uncorrelated random scale-free networks

Uncorrelated random scale-free networks are useful null models to check the accuracy an the analytical solutions of dynamical processes defined on complex networks. We propose and analyze a model capable to generate random uncorrelated scale-free networks with no multiple and self-connections. The model is based on the classical configuration model, with an additional restriction on the maximum possible degree of the vertices. We check numerically that the proposed model indeed generates scale-free networks with no two and three vertex correlations, as measured by the average degree of the nearest neighbors and the clustering coefficient of the vertices of degree $k$, respectively