Prescribed Degree Distribution Research Articles

Aligning random graphs with a sub-tree similarity message-passing algorithm

The problem of aligning Erdős–Rényi random graphs is a noisy, average-case version of the graph isomorphism problem, in which a pair of correlated random graphs is observed through a random permutation of their vertices. We study a polynomial time message-passing algorithm devised to solve the inference problem of partially recovering the hidden permutation, in the sparse regime with constant average degrees. We perform extensive numerical simulations to determine the range of parameters in which this algorithm achieves partial recovery. We also introduce a generalized ensemble of correlated random graphs with prescribed degree distributions, and extend the algorithm to this case.

Annealed inhomogeneities in random ferromagnets.

We consider spin models on complex networks frequently used to model social and technological systems. We study the annealed ferromagnetic Ising model for random networks with either independent edges (Erdős-Rényi) or prescribed degree distributions (configuration model). Contrary to many physical models, the annealed setting is poorly understood and behaves quite differently than the quenched system. In annealed networks with a fluctuating number of edges, the Ising model changes the degree distribution, an aspect previously ignored. For random networks with Poissonian degrees, this gives rise to three distinct annealed critical temperatures depending on the precise model choice, only one of which reproduces the quenched one. In particular, two of these annealed critical temperatures are finite even when the quenched one is infinite because then the annealed graph creates a giant component for all sufficiently small temperatures. We see that the critical exponents in the configuration model with deterministic degrees are the same as the quenched ones, which are the mean-field exponents if the degree distribution has finite fourth moment and power-law-dependent critical exponents otherwise. Remarkably, the annealing for the configuration model with random independent and identically distributed degrees washes away the universality class with power-law critical exponents.

Dynamical systems on large networks with predator-prey interactions are stable and exhibit oscillations.

We analyze the stability of linear dynamical systems defined on sparse, random graphs with predator-prey, competitive, and mutualistic interactions. These systems are aimed at modeling the stability of fixed points in large systems defined on complex networks, such as ecosystems consisting of a large number of species that interact through a food web. We develop an exact theory for the spectral distribution and the leading eigenvalue of the corresponding sparse Jacobian matrices. This theory reveals that the nature of local interactions has a strong influence on a system's stability. We show that, in general, linear dynamical systems defined on random graphs with a prescribed degree distribution of unbounded support are unstable if they are large enough, implying a tradeoff between stability and diversity. Remarkably, in contrast to the generic case, antagonistic systems that contain only interactions of the predator-prey type can be stable in the infinite size limit. This feature for antagonistic systems is accompanied by a peculiar oscillatory behavior of the dynamical response of the system after a perturbation, when the mean degree of the graph is small enough. Moreover, for antagonistic systems we also find that there exist a dynamical phase transition and critical mean degree above which the response becomes nonoscillatory.

Localization and Universality of Eigenvectors in Directed Random Graphs.

Although the spectral properties of random graphs have been a long-standing focus of network theory, the properties of right eigenvectors of directed graphs have so far eluded an exact analytic treatment. We present a general theory for the statistics of the right eigenvector components in directed random graphs with a prescribed degree distribution and with randomly weighted links. We obtain exact analytic expressions for the inverse participation ratio and show that right eigenvectors of directed random graphs with a small average degree are localized. Remarkably, if the fourth moment of the degree distribution is finite, then the critical mean degree of the localization transition is independent of the degree fluctuations, which is different from localization in undirected graphs that is governed by degree fluctuations. We also show that in the high connectivity limit the distribution of the right eigenvector components is solely determined by the degree distribution. For delocalized eigenvectors, we recover in this limit the universal results from standard random matrix theory that are independent of the degree distribution, while for localized eigenvectors the eigenvector distribution depends on the degree distribution.

Universal transient behavior in large dynamical systems on networks

We analyze how the transient dynamics of large dynamical systems in the vicinity of a stationary point, modeled by a set of randomly coupled linear differential equations, depends on the network topology. We characterize the transient response of a system through the evolution in time of the squared norm of the state vector, which is averaged over different realizations of the initial perturbation. We develop a mathematical formalism that computes this quantity for graphs that are locally tree-like. We show that for unidirectional networks the theory simplifies and general analytical results can be derived. For example, we derive analytical expressions for the average squared norm for random directed graphs with a prescribed degree distribution. These analytical results reveal that unidirectional systems exhibit a high degree of universality in the sense that the average squared norm only depends on a single parameter encoding the average interaction strength between the individual constituents. In addition, we derive analytical expressions for the average squared norm for unidirectional systems with fixed diagonal disorder and with bimodal diagonal disorder. We illustrate these results with numerical experiments on large random graphs and on real-world networks.

Metastability for Glauber dynamics on random graphs

In this paper, we study metastable behaviour at low temperature of Glauber spin-flip dynamics on random graphs. We fix a large number of vertices and randomly allocate edges according to the configuration model with a prescribed degree distribution. Each vertex carries a spin that can point either up or down. Each spin interacts with a positive magnetic field, while spins at vertices that are connected by edges also interact with each other via a ferromagnetic pair potential. We start from the configuration where all spins point down, and allow spins to flip up or down according to a Metropolis dynamics at positive temperature. We are interested in the time it takes the system to reach the configuration where all spins point up. In order to achieve this transition, the system needs to create a sufficiently large droplet of up-spins, called critical droplet, which triggers the crossover. In the limit as the temperature tends to zero, and subject to a certain key hypothesis implying metastable behaviour, the average crossover time follows the classical Arrhenius law, with an exponent and a prefactor that are controlled by the energy and the entropy of the critical droplet. The crossover time divided by its average is exponentially distributed. We study the scaling behaviour of the exponent as the number of vertices tends to infinity, deriving upper and lower bounds. We also identify a regime for the magnetic field and the pair potential in which the key hypothesis is satisfied. The critical droplets, representing the saddle points for the crossover, have a size that is of the order of the number of vertices. This is because the random graphs generated by the configuration model are expander graphs.

Universality for first passage percolation on sparse random graphs

We consider rst passage percolation on sparse random graphs with prescribed degree distributions and general independent and identically distributed edge weights assumed to have a density. Assuming that the degree distribution satises a uniform X 2 logX-condition, we analyze the asymptotic distribution for the minimal weight path between a pair of typical vertices, as well the number of edges on this path or hopcount. The hopcount satises a central limit theorem where the norming constants are expressible in terms of the parameters of an associated continuous-time branching process. Centered by a multiple of logn, where the constant is the inverse of the Malthusian rate of growth of the associated branching process, the minimal weight converges in distribution. The limiting random variable equals the sum of the logarithms of the martingale limits of the branching processes that measure the relative growth of neighborhoods about the two vertices, and a Gumbel random variable, and thus shows a remarkably universal behavior. The proofs rely on a rened coupling between the shortest path problems on these graphs and continuous-time branching processes, and on a Poisson point process limit for the potential closing edges of shortest-weight paths between the source and destination. The results extend to a host of related random graph models, ranging from random rregular graphs, inhomogeneous random graphs and uniform random graphs with a prescribed degree sequence.

Porcolation: An Invasion Percolation Model for Mercury Porosimetry

Mercury porosimetry is utilized primarily in the oil industry to determine the pore size distribution of rock samples. During the process, mercury is forced into the sample with gradually increasing pressure and the volume of the injected mercury is measured vs. the applied pressure (the saturation curve). In practice, the saturation curve is assumed to be directly related the cumulative pore size distribution. However, this distribution does not coincide with the real one because of the “nonaccessibility” of pores at a given pressure. This motivates our goal to determine a more accurate cumulative pore size distribution. To achieve this, we treat the propagation of mercury as a percolation process (dubbed “porcolation” after PORosimetry perCOLATION). Porcolation is an external pressure-driven access-limited invasion percolation model where resistance values are assigned to sites/vertices. As pressure increases, the invading mercury occupies sites with smaller resistance values along paths that are connected to the “boundaries” of the network. Simulations are carried out on regular lattices, as well as on random graphs with prescribed degree distributions (representing the pore network of rock samples). An assumed pore size distribution is considered as an input/parameter of the simulations resulting in an output saturation curve. We determine the input–output mapping (homeomorphism) and utilize its inverse to correct the discrepancies between the assumed and actual pore size distributions. The results show nice agreement between experimental saturation curves and those obtained from our homeomorphism method.

Degree correlations in directed scale-free networks.

Scale-free networks, in which the distribution of the degrees obeys a power-law, are ubiquitous in the study of complex systems. One basic network property that relates to the structure of the links found is the degree assortativity, which is a measure of the correlation between the degrees of the nodes at the end of the links. Degree correlations are known to affect both the structure of a network and the dynamics of the processes supported thereon, including the resilience to damage, the spread of information and epidemics, and the efficiency of defence mechanisms. Nonetheless, while many studies focus on undirected scale-free networks, the interactions in real-world systems often have a directionality. Here, we investigate the dependence of the degree correlations on the power-law exponents in directed scale-free networks. To perform our study, we consider the problem of building directed networks with a prescribed degree distribution, providing a method for proper generation of power-law-distributed directed degree sequences. Applying this new method, we perform extensive numerical simulations, generating ensembles of directed scale-free networks with exponents between 2 and 3, and measuring ensemble averages of the Pearson correlation coefficients. Our results show that scale-free networks are on average uncorrelated across directed links for three of the four possible degree-degree correlations, namely in-degree to in-degree, in-degree to out-degree, and out-degree to out-degree. However, they exhibit anticorrelation between the number of outgoing connections and the number of incoming ones. The findings are consistent with an entropic origin for the observed disassortativity in biological and technological networks.

Invariant Bipartite Random Graphs on Rd

Suppose that red and blue points occur inRdaccording to two simple point processes with finite intensities λRand λB, respectively. Furthermore, let ν and μ be two probability distributions on the strictly positive integers with means ν̅ and μ̅, respectively. Assign independently a random number of stubs (half-edges) to each red (blue) point with law ν (μ). We are interested in translation-invariant schemes for matching stubs between points of different colors in order to obtain random bipartite graphs in which each point has a prescribed degree distribution with law ν or μ depending on its color. For a large class of point processes, we show that such translation-invariant schemes matching almost surely all stubs are possible if and only if λRν̅ = λBμ̅, including the case when ν̅ = μ̅ = ∞ so that both sides are infinite. Furthermore, we study a particular scheme based on the Gale-Shapley stable marriage problem. For this scheme, we give sufficient conditions on ν and μ for the presence and absence of infinite components. These results are two-color versions of those obtained by Deijfen, Holroyd and Häggström.

Invariant Bipartite Random Graphs on Rd

Suppose that red and blue points occur in Rd according to two simple point processes with finite intensities λR and λB, respectively. Furthermore, let ν and μ be two probability distributions on the strictly positive integers with means ν̅ and μ̅, respectively. Assign independently a random number of stubs (half-edges) to each red (blue) point with law ν (μ). We are interested in translation-invariant schemes for matching stubs between points of different colors in order to obtain random bipartite graphs in which each point has a prescribed degree distribution with law ν or μ depending on its color. For a large class of point processes, we show that such translation-invariant schemes matching almost surely all stubs are possible if and only if λRν̅ = λBμ̅, including the case when ν̅ = μ̅ = ∞ so that both sides are infinite. Furthermore, we study a particular scheme based on the Gale-Shapley stable marriage problem. For this scheme, we give sufficient conditions on ν and μ for the presence and absence of infinite components. These results are two-color versions of those obtained by Deijfen, Holroyd and Häggström.

A Weighted Configuration Model and Inhomogeneous Epidemics

A random graph model with prescribed degree distribution and degree dependent edge weights is introduced. Each vertex is independently equipped with a random number of half-edges and each half-edge is assigned an integer valued weight according to a distribution that is allowed to depend on the degree of its vertex. Half-edges with the same weight are then paired randomly to create edges. An expression for the threshold for the appearance of a giant component in the resulting graph is derived using results on multi-type branching processes. The same technique also gives an expression for the basic reproduction number for an epidemic on the graph where the probability that a certain edge is used for transmission is a function of the edge weight (reflecting how closely ‘connected’ the corresponding vertices are). It is demonstrated that, if vertices with large degree tend to have large (small) weights on their edges and if the transmission probability increases with the edge weight, then it is easier (harder) for the epidemic to take off compared to a randomized epidemic with the same degree and weight distribution. A recipe for calculating the probability of a large outbreak in the epidemic and the size of such an outbreak is also given. Finally, the model is fitted to three empirical weighted networks of importance for the spread of contagious diseases and it is shown that R 0 can be substantially over- or underestimated if the correlation between degree and weight is not taken into account.

Epidemics and vaccination on weighted graphs

A Reed–Frost epidemic with inhomogeneous infection probabilities on a graph with prescribed degree distribution is studied. Each edge ( u, v) in the graph is equipped with two weights W ( u, v) and W ( v, u) that represent the (subjective) strength of the connection and determine the probability that u infects v in case u is infected and vice versa. Expressions for the epidemic threshold are derived for i.i.d. weights and for weights that are functions of the degrees. For i.i.d. weights, a variation of the so called acquaintance vaccination strategy is analyzed where vertices are chosen randomly and neighbors of these vertices with large edge weights are vaccinated. This strategy is shown to outperform the strategy where the neighbors are chosen randomly in the sense that the basic reproduction number is smaller for a given vaccination coverage.

Rewiring dynamical networks with prescribed degree distribution for enhancing synchronizability

In this paper, we present an algorithm for enhancing synchronizability of dynamical networks with prescribed degree distribution. The algorithm takes an unweighted and undirected network as input and outputs a network with the same node-degree distribution and enhanced synchronization properties. The rewirings are based on the properties of the Laplacian of the connection graph, i.e., the eigenvectors corresponding to the second smallest and the largest eigenvalues of the Laplacian. A term proportional to the eigenvectors is adopted to choose potential edges for rewiring, provided that the node-degree distribution is preserved. The algorithm can be implemented on networks of any sizes as long as their eigenvalues and eigenvectors can be calculated with standard algorithms. The effectiveness of the proposed algorithm in enhancing the network synchronizability is revealed by numerical simulation on a number of sample networks including scale-free, Watts-Strogatz, and Erdős-Rényi graphs. Furthermore, a number of network's structural parameters such as node betweenness centrality, edge betweenness centrality, average path length, clustering coefficient, and degree assortativity are tracked as a function of optimization steps.

Tailored graph ensembles as proxies or null models for real networks I: tools for quantifying structure

We study the tailoring of structured random graph ensembles to real networks, with the objective of generating precise and practical mathematical tools for quantifying and comparing network topologies macroscopically, beyond the level of degree statistics. Our family of ensembles can produce graphs with any prescribed degree distribution and any degree–degree correlation function; its control parameters can be calculated fully analytically, and as a result we can calculate (asymptotically) formulae for entropies and complexities and for information-theoretic distances between networks, expressed directly and explicitly in terms of their measured degree distribution and degree correlations.

Designing threshold networks with given structural and dynamical properties

The threshold model can be used to generate random networks of arbitrary size with given local properties such as the degree distribution, clustering, and degree correlation. We summarize the properties of networks created using the threshold model and present an alternative deterministic construction. These networks are threshold graphs and therefore contain a highly compressible layered structure and allow computation of important network properties in linear time. We show how to construct arbitrarily large, sparse, threshold networks with (approximately) any prescribed degree distribution or Laplacian spectrum. Control of the spectrum allows careful study of the synchronization properties of threshold networks including the relationship between heterogeneous degrees and resistance to synchrony.

Generating Simple Random Graphs with Prescribed Degree Distribution

Let $F$ be a probability distribution with support on the non-negative integers. Four methods for generating a simple undirected graph with (approximate) degree distribution $F$ are described and compared. Two methods are based on the so called configuration model with modifications ensuring a simple graph, one method is an extension of the classical Erd\H{o}s-R\'{e}nyi graph where the edge probabilities are random variables, and the last method starts with a directed random graph which is then modified to a simple undirected graph. All methods are shown to give the correct distribution in the limit of large graph size, but under different assumptions on the degree distribution $F$ and also using different order of operations.

Evidence for synchronicity of lightning activity in networks of spatially remote thunderstorms

Visual observations by space shuttle astronauts have described a phenomenon in which spatially distant thunderstorm cells seem to reciprocally “ignite” lightning flashes in a semi-cyclic sequence. Lightning occurring in one cell is immediately followed by lightning in other cells, separated by tens or hundreds of kilometers. We present quantitative analysis of lightning observations conducted within the framework of the MEIDEX-sprite campaign on board the space shuttle Columbia in January 2003 [Yair, Y., Israelevich, P., Devir, A., Moalem, M., Price, C., Joseph, J., Levin, Z., Ziv, B., Teller, A., 2004. New sprites observations from the space shuttle. Journal of Geophysical Research 109, D15201/10.1029/2003JD004497]. Video footage of 6 storm systems with varying flash rates, which occurred over Africa, South America, Australia and the Pacific Ocean were analyzed. It is found that when the storm flash rate was high, lightning activity in horizontally remote electrically active cells became clustered, with bursts of nearly simultaneous activity separated by quiet periods. The recurrence time was ∼2.5 s, close to the previously reported time delay between consecutive ELF transient signals in the Schumann resonance range [Füllekrug, M., 1995. Schumann resonances in magnetic filed components. Journal of Atmospheric and Terresterial Physics 57, 479–484]. We propose that this behavior is similar to the collective dynamics of a network of weakly coupled limit-cycle oscillators [Strogatz, S.H., 2000, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica, D, 1–20]. Thunderstorm cells embedded within a mesoscale convective system (MCS) constitute such a network, and their lightning frequency is best described in terms of phase-locking of a globally coupled array [Kourtchatov, S.Y., Yu, V.V., Likhanskii, V.V., Napartovitch, A.P., Arecchi, F.T., Lapucci, A., 1995 Theory of phase locking of globally coupled laser arrays. Phys. Rev. A 52, 4089–4094]. Comparison of basic parameters of the lightning networks with predictions of random-graph models reveals that the networks cannot be described by the classical random-graph model [Erdos, P., Renyi, A., 1960. On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci., 5, 17–61], but are more compatible with generalized random-graphs with a prescribed degree distribution [Newman, M.E.J., Strogatz, S.H., Watts, D.J., 2001. Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64, 026118] that exhibit a high clustering coefficient and small average path lengths. Such networks are capable of supporting fast response, synchronization and coherent oscillations [Lago-Fernandez, L.F., Huerta, R., Corbacho, F., Siguenza, J.A., 2000. Fast response and temporal coherent oscillations in small-world networks. Physical Review Letters 84, 2758–2761]. Several physical mechanisms are suggested to explain the observed phenomenon.

Bipartite graphs as models of complex networks

It appeared recently that the classical random graph model used to represent real-world complex networks does not capture their main properties. Since then, various attempts have been made to provide accurate models. We study here a model which achieves the following challenges: it produces graphs which have the three main wanted properties (clustering, degree distribution, average distance), it is based on some real-world observations, and it is sufficiently simple to make it possible to prove its main properties. This model consists in sampling a random bipartite graph with prescribed degree distribution. Indeed, we show that any complex network may be viewed as a bipartite graph with some specific characteristics, and that its main properties may be viewed as consequences of this underlying structure. We also propose a growing model based on this observation.

Synchronization of networks with prescribed degree distributions

We show that the degree distributions of graphs do not suffice to characterize the synchronization of systems evolving on them. We prove that, for any given degree sequence satisfying certain conditions, there exists a connected graph having that degree sequence for which the first nontrivial eigenvalue of the graph Laplacian is arbitrarily close to zero. Consequently, complex dynamical systems defined on such graphs have poor synchronization properties. The result holds under quite mild assumptions, and shows that there exists classes of random, scale-free, regular, small-world, and other common network architectures which impede synchronization. The proof is based on a construction that also serves as an algorithm for building nonsynchronizing networks having a prescribed degree distribution.