Showing papers on "Degree distribution published in 2005"

Generation of uncorrelated random scale-free networks

Abstract: Uncorrelated random scale-free networks are useful null models to check the accuracy and the analytical solutions of dynamical processes defined on complex networks. We propose and analyze a model capable of generating 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.

Subnets of scale-free networks are not scale-free: sampling properties of networks.

Abstract: †§ ¶ Most studies of networks have only looked at small subsets of the true network. Here, we discuss the sampling properties of a network’s degree distribution under the most parsimonious sampling scheme. Only if the degree distributions of the network and randomly sampled subnets belong to the same family of probability distributions is it possible to extrapolate from subnet data to properties of the global network. We show that this condition is indeed satisfied for some important classes of networks, notably classical random graphs and exponential random graphs. For scale-free degree distributions, however, this is not the case. Thus, inferences about the scale-free nature of a network may have to be treated with some caution. The work presented here has important implications for the analysis of molecular networks as well as for graph theory and the theory of networks in general. complex networks protein interaction networks random graphs sampling theory

Voter model on heterogeneous graphs.

Abstract: We study the voter model on heterogeneous graphs. We exploit the nonconservation of the magnetization to characterize how consensus is reached. For a network of N nodes with an arbitrary but uncorrelated degree distribution, the mean time to reach consensus T(N) scales as Nmu(2)1/mu(2), where mu(k) is the kth moment of the degree distribution. For a power-law degree distribution n(k) approximately k(-nu), T(N) thus scales as N for nu > 3, as N/ln(N for nu = 3, as N((2nu-4)/(nu-1)) for 2 < nu < 3, as (lnN)2 for nu = 2, and as omicron(1) for nu < 2. These results agree with simulation data for networks with both uncorrelated and correlated node degrees.

Network synchronization, diffusion, and the paradox of heterogeneity.

Abstract: Many complex networks display strong heterogeneity in the degree (connectivity) distribution. Heterogeneity in the degree distribution often reduces the average distance between nodes but, paradoxically, may suppress synchronization in networks of oscillators coupled symmetrically with uniform coupling strength. Here we offer a solution to this apparent paradox. Our analysis is partially based on the identification of a diffusive process underlying the communication between oscillators and reveals a striking relation between this process and the condition for the linear stability of the synchronized states. We show that, for a given degree distribution, the maximum synchronizability is achieved when the network of couplings is weighted and directed and the overall cost involved in the couplings is minimum. This enhanced synchronizability is solely determined by the mean degree and does not depend on the degree distribution and system size. Numerical verification of the main results is provided for representative classes of small-world and scale-free networks.

Voter Model on Heterogeneous Graphs.

Abstract: We study the voter model on heterogeneous graphs. We exploit the nonconservation of the magnetization to characterize how consensus is reached. For a network of N nodes with an arbitrary but uncorrelated degree distribution, the mean time to reach consensus TN scales as N!1=!2, where !k is the kth moment of the degree distribution. For a power-law degree distribution nk ! k"", TN thus scales as N for "> 3, as N= lnN for " # 3, as N$2""4%=$""1% for 2< "< 3, as $lnN%2 for " # 2, and as O$1% for "< 2. These results agree with simulation data for networks with both uncorrelated and correlated node degrees.

Onset of traffic congestion in complex networks

Abstract: Free traffic flow on a complex network is key to its normal and efficient functioning. Recent works indicate that many realistic networks possess connecting topologies with a scale-free feature: the probability distribution of the number of links at nodes, or the degree distribution, contains a power-law component. A natural question is then how the topology influences the dynamics of traffic flow on a complex network. Here we present two models to address this question, taking into account the network topology, the information-generating rate, and the information-processing capacity of individual nodes. For each model, we study four kinds of networks: scale-free, random, and regular networks and Cayley trees. In the first model, the capacity of packet delivery of each node is proportional to its number of links, while in the second model, it is proportional to the number of shortest paths passing through the node. We find, in both models, that there is a critical rate of information generation, below which the network traffic is free but above which traffic congestion occurs. Theoretical estimates are given for the critical point. For the first model, scale-free networks and random networks are found to be more tolerant to congestion. For the second model, the congestion condition is independent of network size and topology, suggesting that this model may be practically useful for designing communication protocols.

Random graphs as models of networks

Abstract: The random graph of Erdos and Renyi is one of the oldest and best studied models of a network, and possesses the considerable advantage of being exactly solvable for many of its average properties. However, as a model of real-world networks such as the Internet, social networks or biological networks it leaves a lot to be desired. In particular, it differs from real networks in two crucial ways: it lacks network clustering or transitivity, and it has an unrealistic Poissonian degree distribution. In this paper we review some recent work on generalizations of the random graph aimed at correcting these shortcomings. We describe generalized random graph models of both directed and undirected networks that incorporate arbitrary non-Poisson degree distributions, and extensions of these models that incorporate clustering too. We also describe two recent applications of random graph models to the problems of network robustness and of epidemics spreading on contact networks.

Statistical analysis of 22 public transport networks in Poland

Abstract: Public transport systems in 22 Polish cities have been analyzed. The sizes of these networks range from N = 152 to 2881. Depending on the assumed definition of network topology, the degree distribution can follow a power law or can be described by an exponential function. Distributions of path lengths in all considered networks are given by asymmetric, unimodal functions. Clustering, assortativity, and betweenness are studied. All considered networks exhibit small-world behavior and are hierarchically organized. A transition between dissortative small networks N approximately or = 500 is observed.

Enhancing complex-network synchronization

Abstract: Heterogeneity in the degree (connectivity) distribution has been shown to suppress synchronization in networks of symmetrically coupled oscillators with uniform coupling strength (unweighted coupling). Here we uncover a condition for enhanced synchronization in weighted networks with asymmetric coupling. We show that, in the optimum regime, synchronizability is solely determined by the average degree and does not depend on the system size and the details of the degree distribution. In scale-free networks, where the average degree may increase with heterogeneity, synchronizability is drastically enhanced and may become positively correlated with heterogeneity, while the overall cost involved in the network coupling is significantly reduced as compared to the case of unweighted coupling.

Finding, counting and listing all triangles in large graphs, an experimental study

Abstract: In the past, the fundamental graph problem of triangle counting and listing has been studied intensively from a theoretical point of view Recently, triangle counting has also become a widely used tool in network analysis Due to the very large size of networks like the Internet, WWW or social networks, the efficiency of algorithms for triangle counting and listing is an important issue The main intention of this work is to evaluate the practicability of triangle counting and listing in very large graphs with various degree distributions We give a surprisingly simple enhancement of a well known algorithm that performs best, and makes triangle listing and counting in huge networks feasible This paper is a condensed presentation of [SW05]

Efficient and Simple Generation of Random Simple Connected Graphs with Prescribed Degree Sequence

Abstract: We address here the problem of generating random graphs uniformly from the set of simple connected graphs having a prescribed degree sequence. Our goal is to provide an algorithm designed for practical use both because of its ability to generate very large graphs efficiency and because it is easy to implement simplicity. We focus on a family of heuristics for which we prove optimality conditions, and show how this optimality can be reached in practice. We then propose a different approach, specifically designed for typical real-world degree distributions, which outperforms the first one. Assuming a conjecture, we finally obtain an On log n algorithm, which, in spite of being very simple, improves the best known complexity.

Voter model dynamics in complex networks: Role of dimensionality, disorder, and degree distribution.

Abstract: We analyze the ordering dynamics of the voter model in different classes of complex networks. We observe that whether the voter dynamics orders the system depends on the effective dimensionality of the interaction networks. We also find that when there is no ordering in the system, the average survival time of metastable states in finite networks decreases with network disorder and degree heterogeneity. The existence of hubs, i.e., highly connected nodes, in the network modifies the linear system size scaling law of the survival time. The size of an ordered domain is sensitive to the network disorder and the average degree, decreasing with both; however, it seems not to depend on network size and on the heterogeneity of the degree distribution.

Networks and Cities: An Information Perspective

Abstract: Traffic is constrained by the information involved in locating the receiver and the physical distance between sender and receiver. We here focus on the former, and investigate traffic in the perspective of information handling. We replot the road map of cities in terms of the information needed to locate specific addresses and create information city networks with roads mapped to nodes and intersections to links between nodes. These networks have the broad degree distribution found in many other complex networks. The mapping to an information city network makes it possible to quantify the information associated with locating specific addresses.

Epidemic spreading in community networks

Abstract: Social networks have the structure of communities To understand how the community structure affects epidemic spreading, we present a simplified model of community network, and investigate the epidemic propagation in this model We find that, compared to the random network, the community network has a broader degree distribution, a smaller threshold of epidemic outbreak, and more prevalence to keep the outbreak endemic The formulae of epidemic threshold are given and confirmed by numerical simulations

On the bias of traceroute sampling: or, power-law degree distributions in regular graphs

Abstract: Understanding the structure of the Internet graph is a crucial step for building accurate network models and designing efficient algorithms for Internet applications. Yet, obtaining its graph structure is a surprisingly difficult task, as edges cannot be explicitly queried. Instead, empirical studies rely on traceroutes to build what are essentially single-source, all-destinations, shortest-path trees. These trees only sample a fraction of the network's edges, and a recent paper by Lakhina et al. found empirically that the resuting sample is intrinsically biased. For instance, the observed degree distribution under traceroute sampling exhibits a power law even when the underlying degree distribution is Poisson.In this paper, we study the bias of traceroute sampling systematically, and, for a very general class of underlying degree distributions, calculate the likely observed distributions explicitly. To do this, we use a continuous-time realization of the process of exposing the BFS tree of a random graph with a given degree distribution, calculate the expected degree distribution of the tree, and show that it is sharply concentrated. As example applications of our machinery, we show how traceroute sampling finds power-law degree distributions in both δ-regular and Poisson-distributed random graphs. Thus, our work puts the observations of Lakhina et al. on a rigorous footing, and extends them to nearly arbitrary degree distributions.

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

Abstract: 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.

K-core decomposition of Internet graphs: hierarchies, self-similarity and measurement biases

Abstract: We consider the $k$-core decomposition of network models and Internet graphs at the autonomous system (AS) level. The $k$-core analysis allows to characterize networks beyond the degree distribution and uncover structural properties and hierarchies due to the specific architecture of the system. We compare the $k$-core structure obtained for AS graphs with those of several network models and discuss the differences and similarities with the real Internet architecture. The presence of biases and the incompleteness of the real maps are discussed and their effect on the $k$-core analysis is assessed with numerical experiments simulating biased exploration on a wide range of network models. We find that the $k$-core analysis provides an interesting characterization of the fluctuations and incompleteness of maps as well as information helping to discriminate the original underlying structure.

Duplication-divergence model of protein interaction network.

Abstract: We investigate a very simple model describing the evolution of protein-protein interaction networks via duplication and divergence. The model exhibits a remarkably rich behavior depending on a single parameter, the probability to retain a duplicated link during divergence. When this parameter is large, the network growth is not self-averaging and an average node degree increases algebraically. The lack of self-averaging results in a great diversity of networks grown out of the same initial condition. When less than a half of links are (on average) preserved after divergence, the growth is self-averaging, the average degree increases very slowly or tends to a constant, and a degree distribution has a power-law tail. The predicted degree distributions are in a very good agreement with the distributions observed in real protein networks.

Optimization of network robustness to waves of targeted and random attacks

Abstract: We study the robustness of complex networks to multiple waves of simultaneous sid targeted attacks in which the highest degree nodes are removed and siid random attacks sor failuresd in which fractions pt and pr, respectively, of the nodes are removed until the network collapses. We find that the network design which optimizes network robustness has a bimodal degree distribution, with a fraction r of the nodes having degree k2 =skkl ˛1+ rd / r and the remainder of the nodes having degree k1 = 1, where kkl is the average degree of all the nodes. We find that the optimal value ofr is of the order of pt / pr for pt / pr! 1.

Tuning clustering in random networks with arbitrary degree distributions

Abstract: We present a generator of random networks where both the degree-dependent clustering coefficient and the degree distribution are tunable. Following the same philosophy as in the configuration model, the degree distribution and the clustering coefficient for each class of nodes of degree k are fixed ad hoc and a priori. The algorithm generates corresponding topologies by applying first a closure of triangles and second the classical closure of remaining free stubs. The procedure unveils an universal relation among clustering and degree-degree correlations for all networks, where the level of assortativity establishes an upper limit to the level of clustering. Maximum assortativity ensures no restriction on the decay of the clustering coefficient whereas disassortativity sets a stronger constraint on its behavior. Correlation measures in real networks are seen to observe this structural bound.

Tight bounds for LDPC and LDGM codes under MAP decoding

Abstract: A new method for analyzing low-density parity-check (LDPC) codes and low-density generator-matrix (LDGM) codes under bit maximum a posteriori probability (MAP) decoding is introduced. The method is based on a rigorous approach to spin glasses developed by Francesco Guerra. It allows one to construct lower bounds on the entropy of the transmitted message conditional to the received one. Based on heuristic statistical mechanics calculations, we conjecture such bounds to be tight. The result holds for standard irregular ensembles when used over binary-input output-symmetric (BIOS) channels. The method is first developed for Tanner-graph ensembles with Poisson left-degree distribution. It is then generalized to "multi-Poisson" graphs, and, by a completion procedure, to arbitrary degree distribution

Comparison of voter and Glauber ordering dynamics on networks.

Abstract: We study numerically the ordering process of two very simple dynamical models for a two-state variable on several topologies with increasing levels of heterogeneity in the degree distribution. We find that the zero-temperature Glauber dynamics for the Ising model may get trapped in sets of partially ordered metastable states even for finite system size, and this becomes more probable as the size increases. Voter dynamics instead always converges to full order on finite networks, even if this does not occur via coherent growth of domains. The time needed for order to be reached diverges with the system size. In both cases the ordering process is rather insensitive to the variation of the degreee distribution from sharply peaked to scale free.

Degree distribution in plant–animal mutualistic networks: forbidden links or random interactions?

Abstract: Recent studies show that ecological interaction networks depart from the ‘‘scale-free’’ topologies observed in many other real world networks. Such a departure has been hypothesized to result from non-matching biological attributes of species, such as phenology or morphology, that prevent the occurrence of certain interactions (‘‘forbidden links’’). Here I compare the topology of 17 plant � /animal mutualistic networks with that predicted by a simple null model that assumes that a species’ degree (number of interspecific interactions) is a function of its frequency of interaction. The topology predicted by this null model is strikingly close to that observed in the real networks. Thus, this null model provides a simple alternative interpretation of patterns observed in ecological interaction networks that does not require the existence of non-matching species traits.

Modular synchronization in complex networks.

Abstract: We study the synchronization transition (ST) of a modified Kuramoto model on two different types of modular complex networks. It is found that the ST depends on the type of intermodular connections. For the network with decentralized (centralized) intermodular connections, the ST occurs at finite coupling constant (behaves abnormally). Such distinct features are found in the yeast protein interaction network and the Internet, respectively. Moreover, by applying the finite-size scaling analysis to an artificial network with decentralized intermodular connections, we obtain the exponent associated with the order parameter of the ST to be beta approximately 1 different from beta(MF) approximately 1/2 obtained from the scale-free network with the same degree distribution but the absence of modular structure, corresponding to the mean field value.

Graph Theoretic and Spectral Analysis of Enron Email Data

Abstract: Analysis of social networks to identify communities and model their evolution has been an active area of recent research. This paper analyzes the Enron email data set to discover structures within the organization. The analysis is based on constructing an email graph and studying its properties with both graph theoretical and spectral analysis techniques. The graph theoretical analysis includes the computation of several graph metrics such as degree distribution, average distance ratio, clustering coefficient and compactness over the email graph. The spectral analysis shows that the email adjacency matrix has a rank-2 approximation. It is shown that preprocessing of data has significant impact on the results, thus a standard form is needed for establishing a benchmark data.

Sampling properties of random graphs: the degree distribution.

Abstract: 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 distributions 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 satisfied 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 Eschericia coli protein interaction network data.

Clustering and the synchronization of oscillator networks.

Abstract: By manipulating the clustering coefficient of a network without changing its degree distribution, we examine the effect of clustering on the synchronization of phase oscillators on networks with Poisson and scale-free degree distributions. For both types of networks, increased clustering hinders global synchronization as the network splits into dynamical clusters that oscillate at different frequencies. Surprisingly, in scale-free networks, clustering promotes the synchronization of the most connected nodes (hubs) even though it inhibits global synchronization. As a result, they show an additional, advanced transition instead of a single synchronization threshold. This cluster-enhanced synchronization of hubs may be relevant to the brain that is scale-free and highly clustered.

Characterizing the network topology of the energy landscapes of atomic clusters.

Abstract: By dividing potential energy landscapes into basins of attractions surrounding minima and linking those basins that are connected by transition state valleys, a network description of energy landscapes naturally arises. These networks are characterized in detail for a series of small Lennard-Jones clusters and show behavior characteristic of small-world and scale-free networks. However, unlike many such networks, this topology cannot reflect the rules governing the dynamics of network growth, because they are static spatial networks. Instead, the heterogeneity in the networks stems from differences in the potential energy of the minima, and hence the hyperareas of their associated basins of attraction. The low-energy minima with large basins of attraction act as hubs in the network. Comparisons to randomized networks with the same degree distribution reveals structuring in the networks that reflects their spatial embedding.

Lessons from Three Views of the Internet Topology

Abstract: Network topology plays a vital role in understanding the performance of network applications and protocols. Thus, recently there has been tremendous interest in generating realistic network topologies. Such work must begin with an understanding of existing network topologies, which today typically consists of a relatively small number of data sources. In this paper, we calculate an extensive set of important characteristics of Internet AS-level topologies extracted from the three data sources most frequently used by the research community: traceroutes, BGP, and WHOIS. We find that traceroute and BGP topologies are similar to one another but differ substantially from the WHOIS topology. We discuss the interplay between the properties of the data sources that result from specific data collection mechanisms and the resulting topology views. We find that, among metrics widely considered, the joint degree distribution appears to fundamentally characterize Internet AS-topologies: it narrowly defines values for other important metrics. We also introduce an evaluation criteria for the accuracy of topology generators and verify previous observations that generators solely reproducing degree distributions cannot capture the full spectrum of critical topological characteristics of any of the three 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.