Showing papers on "Degree distribution published in 2004"

Detection of topological patterns in complex networks: correlation profile of the internet

Abstract: A general scheme for detecting and analyzing topological patterns in large complex networks is presented. In this scheme the network in question is compared with its properly randomized version that preserves some of its low-level topological properties. Statistically significant deviation of any topological property of a network from this null model likely reflects its design principles and/or evolutionary history. We illustrate this basic scheme using the example of the correlation profile of the Internet quantifying correlations between degrees of its neighboring nodes. This profile distinguishes the Internet from previously studied molecular networks with a similar scale-free degree distribution. We finally demonstrate that the clustering in a network is very sensitive to both the degree distribution and its correlation profile and compare the clustering in the Internet to the appropriate null model.

Fitness-Dependent Topological Properties of the World Trade Web

Abstract: Among the proposed network models, the hidden variable (or good get richer) one is particularly interesting, even if an explicit empirical test of its hypotheses has not yet been performed on a real network. Here we provide the first empirical test of this mechanism on the world trade web, the network defined by the trade relationships between world countries. We find that the power-law distributed gross domestic product can be successfully identified with the hidden variable (or fitness) determining the topology of the world trade web: all previously studied properties up to third-order correlation structure (degree distribution, degree correlations, and hierarchy) are found to be in excellent agreement with the predictions of the model. The choice of the connection probability is such that all realizations of the network with the same degree sequence are equiprobable.

Cut-offs and finite size effects in scale-free networks

Abstract: We analyze the degree distribution’s cut-off in finite size scale-free networks. We show that the cut-off behavior with the number of vertices N is ruled by the topological constraints induced by the connectivity structure of the network. Even in the simple case of uncorrelated networks, we obtain an expression of the structural cut-off that is smaller than the natural cut-off obtained by means of extremal theory arguments. The obtained results are explicitly applied in the case of the configuration model to recover the size scaling of tadpoles and multiple edges.

Self-organization of collaboration networks.

Abstract: We study collaboration networks in terms of evolving, self-organizing bipartite graph models. We propose a model of a growing network, which combines preferential edge attachment with the bipartite structure, generic for collaboration networks. The model depends exclusively on basic properties of the network, such as the total number of collaborators and acts of collaboration, the mean size of collaborations, etc. The simplest model defined within this framework already allows us to describe many of the main topological characteristics (degree distribution, clustering coefficient, etc.) of one-mode projections of several real collaboration networks, without parameter fitting. We explain the observed dependence of the local clustering on degree and the degree‐degree correlations in terms of the “aging” of collaborators and their physical impossibility to participate in an unlimited number of collaborations.

Factors that predict better synchronizability on complex networks.

Abstract: While shorter characteristic path length has in general been believed to enhance synchronizability of a coupled oscillator system on a complex network, the suppressing tendency of the heterogeneity of the degree distribution, even for shorter characteristic path length, has also been reported. To see this, we investigate the effects of various factors such as the degree, characteristic path length, heterogeneity, and betweenness centrality on synchronization, and find a consistent trend between the synchronization and the betweenness centrality. The betweenness centrality is thus proposed as a good indicator for synchronizability.

Spatial Growth of Real-world Networks

Abstract: Many real-world networks have properties of small-world networks, with clustered local neighborhoods and low average-shortest path. They may also show a scale-free degree distribution, which can be generated by growth and preferential attachment to highly connected nodes, or hubs. However, many real-world networks consist of multiple, interconnected clusters not normally seen in systems grown by preferential attachment, and there also exist real-world networks with a scale-free degree distribution that do not contain highly connected hubs. We describe spatial-growth mechanisms, not using preferential attachment, that address both aspects.

Accurately modeling the internet topology.

Abstract: Based on measurements of the internet topology data, we found that there are two mechanisms which are necessary for the correct modeling of the internet topology at the autonomous systems (AS) level: the interactive growth of new nodes and new internal links, and a nonlinear preferential attachment, where the preference probability is described by a positive-feedback mechanism. Based on the above mechanisms, we introduce the positive-feedback preference (PFP) model which accurately reproduces many topological properties of the AS-level internet, including degree distribution, rich-club connectivity, the maximum degree, shortest path length, short cycles, disassortative mixing, and betweenness centrality. The PFP model is a phenomenological model which provides an insight into the evolutionary dynamics of real complex networks.

Stations,trains and small-world networks

Abstract: The clustering coefficient, path length and average vertex degree of two urban train line networks have been calculated. The results are compared with theoretical predictions for appropriate random bipartite graphs. They have also been compared with one another to investigate the effect of architecture on the small-world properties.

Network dynamics: Jamming is limited in scale-free systems

Abstract: A large number of complex networks are scale-free--that is, they follow a power-law degree distribution. Here we propose that the emergence of many scale-free networks is tied to the efficiency of transport and flow processing across these structures. In particular, we show that for large networks on which flows are influenced or generated by gradients of a scalar distributed on the nodes, scale-free structures will ensure efficient processing, whereas structures that are not scale-free, such as random graphs, will become congested.

Optimization of robustness of complex networks

Abstract: Networks with a given degree distribution may be very resilient to one type of failure or attack but not to another. The goal of this work is to determine network design guidelines which maximize the robustness of networks to both random failure and intentional attack while keeping the cost of the network (which we take to be the average number of links per node) constant. We find optimal parameters for: (i) scale free networks having degree distributions with a single power-law regime, (ii) networks having degree distributions with two power-law regimes, and (iii) networks described by degree distributions containing two peaks. Of these various kinds of distributions we find that the optimal network design is one in which all but one of the nodes have the same degree, k 1 (close to the average number of links per node), and one node is of very large degree, $k_2 \sim N^{2/3}$ , where N is the number of nodes in the network.

Analyzing Kleinberg's (and other) small-world Models

Abstract: We analyze the properties of Small-World networks, where links are much more likely to connect "neighbor nodes" than distant nodes. In particular, our analysis provides new results for Kleinberg's Small-World model and its extensions. Kleinberg adds a number of directed long-range random links to an nxn lattice network (vertices as nodes of a grid, undirected edges between any two adjacent nodes). Links have a non-uniform distribution that favors arcs to close nodes over more distant ones. He shows that the following phenomenon occurs: between any two nodes a path with expected length O(log2n) can be found using a simple greedy algorithm which has no global knowledge of long-range links.We show that Kleinberg's analysis is tight: his algorithm achieves θ(log2n) delivery time. Moreover, we show that the expected diameter of the graph is θlog n), a log n factor smaller. We also extend our results to the general k-dimensional model. Our diameter results extend traditional work on the diameter of random graphs which largely focuses on uniformly distributed arcs. Using a little additional knowledge of the graph, we show that we can find shorter paths: with expected length O(log3/2n) in the basic 2-dimensional model and O(log1+1/k) in the general k-dimensional model (fork≥1).Finally, we suggest a general approach to analyzing a broader class of random graphs with non-uniform edge probabilities. Thus we show expected θ(log n) diameter results for higher dimensional grids, as well as settings with less uniform base structures: where links can be missing, where the probability can vary at different nodes, or where grid-related factors (e.g. the use of lattice distance) has a weaker role or is dismissed, and constraints (such as the uniformness of degree distribution) are relaxed.

Structure of a large social network.

Abstract: We study a social network consisting of over 10(4) individuals, with a degree distribution exhibiting two power scaling regimes separated by a critical degree k(crit), and a power law relation between degree and local clustering. We introduce a growing random model based on a local interaction mechanism that reproduces the observed scaling features and their exponents. We suggest that the double power law originates from two very different kinds of networks that are simultaneously present in the human social network.

Structural and algorithmic aspects of massive social networks

Abstract: We study the algorithmic and structural properties of very large, realistic social contact networks. We consider the social network for the city of Portland, Oregon, USA, developed as a part of the TRANSIMS/EpiSims project at the Los Alamos National Laboratory. The most expressive social contact network is a bipartite graph, with two types of nodes: people and locations; edges represent people visiting locations on a typical day. Three types of results are presented. (i) Our empirical results show that many basic characteristics of the dataset are well-modeled by a random graph approach suggested by Fan Chung Graham and Lincoln Lu (the CL-model), with a power-law degree distribution. (ii) We obtain fast approximation algorithms for computing basic structural properties such as clustering coefficients and shortest paths distribution. We also study the dominating set problem for such networks; this problem arose in connection with optimal sensor-placement for disease-detection. We present a fast approximation algorithm for computing near-optimal dominating sets. (iii) Given the close approximations provided by the CL-model to our original dataset and the large data-volume, we investigate fast methods for generating such random graphs. We present methods that can generate such a random network in near-linear time, and show that these variants asymptotically share many key features of the CL-model, and also match the Portland social network.The structural results have been used to study the impact of policy decisions for controlling large-scale epidemics in urban environments.

A more accurate one-dimensional analysis and design of irregular LDPC codes

Abstract: We introduce a new one-dimensional (1-D) analysis of low-density parity-check (LDPC) codes on additive white Gaussian noise channels which is significantly more accurate than similar 1-D methods. Our method assumes a Gaussian distribution in message-passing decoding only for messages from variable nodes to check nodes. Compared to existing work, which makes a Gaussian assumption both for messages from check nodes and from variable nodes, our method offers a significantly more accurate estimate of convergence behavior and threshold of convergence. Similar to previous work, the problem of designing irregular LDPC codes reduces to a linear programming problem. However, our method allows irregular code design in a wider range of rates without any limit on the maximum variable-node degree. We use our method to design irregular LDPC codes with rates greater than 1/4 that perform within a few hundredths of a decibel from the Shannon limit. The designed codes perform almost as well as codes designed by density evolution.

Scale-free and stable structures in complex ad hoc networks

Abstract: Unlike the well-studied models of growing networks, where the dominant dynamics consist of insertions of new nodes and connections and rewiring of existing links, we study ad hoc networks, where one also has to contend with rapid and random deletions of existing nodes (and, hence, the associated links). We first show that dynamics based only on the well-known preferential attachments of new nodes do not lead to a sufficiently heavy-tailed degree distribution in ad hoc networks. In particular, the magnitude of the power-law exponent increases rapidly (from 3) with the deletion rate, becoming \ensuremath{\infty} in the limit of equal insertion and deletion rates. We then introduce a local and universal compensatory rewiring dynamic, and show that even in the limit of equal insertion and deletion rates true scale-free structures emerge, where the degree distributions obey a power law with a tunable exponent, which can be made arbitrarily close to 2. The dynamics reported in this paper can be used to craft protocols for designing highly dynamic peer-to-peer networks and also to account for the power-law exponents observed in existing popular services.

Minimum spanning trees on weighted scale-free networks

Abstract: A complete understanding of real networks requires us to understand the consequences of the uneven interaction strengths between a system's components. Here we use the minimum spanning tree (MST) to explore the effect of weight assignment and network topology on the organization of complex networks. We find that if the weight distribution is correlated with the network topology, the MSTs are either scale-free or exponential. In contrast, when the correlations between weights and topology are absent, the MST degree distribution is a power-law and independent of the weight distribution. These results offer a systematic way to explore the impact of weak links on the structure and integrity of complex networks.

Random networks with tunable degree distribution and clustering.

Abstract: We present an algorithm for generating random networks with arbitrary degree distribution and clustering (frequency of triadic closure). We use this algorithm to generate networks with exponential, power law, and Poisson degree distributions with variable levels of clustering. Such networks may be used as models of social networks and as a testable null hypothesis about network structure. Finally, we explore the effects of clustering on the point of the phase transition where a giant component forms in a random network, and on the size of the giant component. Some analysis of these effects is presented.

TopNet: a tool for comparing biological sub-networks, correlating protein properties with topological statistics

Abstract: Biological networks are a topic of great current interest, particularly with the publication of a number of large genome-wide interaction datasets. They are globally characterized by a variety of graph-theoretic statistics, such as the degree distribution, clustering coefficient, characteristic path length and diameter. Moreover, real protein networks are quite complex and can often be divided into many sub-networks through systematic selection of different nodes and edges. For instance, proteins can be sub-divided by expression level, length, amino-acid composition, solubility, secondary structure and function. A challenging research question is to compare the topologies of sub- networks, looking for global differences associated with different types of proteins. TopNet is an automated web tool designed to address this question, calculating and comparing topological characteristics for different sub-networks derived from any given protein network. It provides reasonable solutions to the calculation of network statistics for sub-networks embedded within a larger network and gives simplified views of a sub-network of interest, allowing one to navigate through it. After constructing TopNet, we applied it to the interaction networks and protein classes currently available for yeast. We were able to find a number of potential biological correlations. In particular, we found that soluble proteins had more interactions than membrane proteins. Moreover, amongst soluble proteins, those that were highly expressed, had many polar amino acids, and had many alpha helices, tended to have the most interaction partners. Interestingly, TopNet also turned up some systematic biases in the current yeast interaction network: on average, proteins with a known functional classification had many more interaction partners than those without. This phenomenon may reflect the incompleteness of the experimentally determined yeast interaction network.

Recursive graphs with small-world scale-free properties

Abstract: We discuss a category of graphs, recursive clique trees, which have small-world and scale-free properties and allow a fine tuning of the clustering and the power-law exponent of their discrete degree distribution We determine relevant characteristics of those graphs: the diameter, degree distribution, and clustering parameter The graphs have also an interesting recursive property, and generalize recent constructions with fixed degree distributions

Urban transit system as a scale-free network

Abstract: Many systems can be represented by networks as a set of nodes joined together by links indicating interaction. Recently studies have suggested that a lot of real networks are scale-free, such as the WWW, social networks, etc. In this paper, discoveries of scale-free characteristics are reported on the network constructed from the real urban transit system data in Beijing. It is shown that the connectivity distribution of the transit network decays as a power-law, and the exponent λ is about equal to 2.24 from the simulation graph. Based on the scale-free network topology structure of the transit network, if only transit "hub nodes" are controlled well, the transit network can resist random failures (such as traffic congestion, traffic accidents, etc.) successfully.

Communication and synchronization in, disconnected networks with dynamic topology: moving neighborhood networks.

Abstract: We consider systems that are well modelled as networks that evolve in time, which we call Moving Neighborhood Networks. These models are relevant in studying cooperative behavior of swarms and other phenomena where emergent interactions arise from ad hoc networks. In a natural way, the time-averaged degree distribution gives rise to a scale-free network. Simulations show that although the network may have many noncommunicating components, the recent weighted time-averaged communication is sufficient to yield robust synchronization of chaotic oscillators. In particular, we contend that such time-varying networks are important to model in the situation where each agent carries a pathogen (such as a disease) in which the pathogen's life-cycle has a natural time-scale which competes with the time-scale of movement of the agents, and thus with the networks communication channels.

Raptor codes on symmetric channels

Abstract: This paper extends the construction and analysis of Raptor codes originally designed in A. Shokrollahi (2004) for the erasure channel to general symmetric channels. We explicitly calculate the asymptotic fraction of output nodes of degree one and two for capacity-achieving Raptor codes, and discuss techniques to optimize the output degree distribution.

The modeling of scale-free networks

Abstract: In order to explore further the mechanism responsible for scale-free networks, we introduce two extended models of the BA model. The model A, where the system incorporates the addition of new links between existing nodes, a new node with new links and the rewiring of some links at every time step, all sites are born with some initial attractiveness. We calculate analytically the degree distribution. The system self-organizes into a scale-free network, the scaling exponent γ >2. The model B is a new model; we consider that some old links are deleted with the anti-preferential probability. The result indicates that the system evolves itself into a scale-free network, the scaling exponent γ varies from 2 to 3.

Generation models for scale-free networks

Abstract: In the last few years it has been established that the connectivity distribution of the large real-world networks often follows the power-law, i.e., they are scale-free networks. In this article stochastic models leading to scale-free network are considered and a model close to them is proposed. Deterministic models for creating scale-free networks with given nodes (static model) are demonstrated. A characteristic of graphs, which could be used for determining the scale-free topology of networks, is suggested.

Potts model on complex networks

Abstract: We consider the general p-state Potts model on random networks with a given degree distribution (random Bethe lattices). We find the effect of the suppression of a first order phase transition in this model when the degree distribution of the network is fat-tailed, that is, in more precise terms, when the second moment of the distribution diverges. In this situation the transition is continuous and of infinite order, and size effect is anomalously strong. In particular, in the case of p = 1, we arrive at the exact solution, which coincides with the known solution of the percolation problem on these networks.

Phenix: supporting resilient low-diameter peer-to-peer topologies

Abstract: Peer-to-peer networks are mainly unstructured, where no specific topology is imposed on the network during its operations. These networks offer a high degree of resilience against network dynamics disrupting the network's operation. Unstructured networks, based on random connections are limited, however, in the performance and node reachability they can offer to applications. In contrast, structured networks impose predetermined connectivity relationships between nodes in order to offer a guarantee on the diameter between requesting nodes and the requested objects. We observe that neither structured nor unstructured networks can simultaneously offer both good performance and resilience in a single algorithm. To address this challenge, we propose Phenix, a peer-to-peer algorithm that can construct low-diameter resilient topologies. Phenix supports low diameter operations by creating a topology of nodes whose degree distribution follows a power-law, while the implementation of the underlying algorithm is fully distributed requiring no central server, thus, eliminating the possibility of a single point of failure in the system. We present the design and evaluation of the algorithm and show through analysis, simulation, and experimental results obtained from an implementation on the PlanetLab testbed that Phenix is robust to network dynamics such as joins/leaves, node failure and large-scale network attacks, while maintaining low overhead when implemented in an experimental network.

Spectral analysis of random graphs with skewed degree distributions

Abstract: We extend spectral methods to random graphs with skewed degree distributions through a degree based normalization closely connected to the normalized Laplacian. The normalization is based on intuition drawn from perturbation theory of random matrices, and has the effect of boosting the expectation of the random adjacency matrix without increasing the variances of its entries, leading to better perturbation bounds. The primary implication of this result lies in the realm of spectral analysis of random graphs with skewed degree distributions, such as the ubiquitous "power law graphs". Mihail and Papadimitriou (2002) argued that for randomly generated graphs satisfying a power law degree distribution, spectral analysis of the adjacency matrix simply produces the neighborhoods of the high degree nodes as its eigenvectors, and thus miss any embedded structure. We present a generalization of their model, incorporating latent structure, and prove that after applying our transformation, spectral analysis succeeds in recovering the latent structure with high probability.

Network of econophysicists: a weighted network to investigate the development of econophysics

Abstract: The development of Econophysics is studied from the perspective of scientific communication networks. Papers in Econophysics published from 1992 to 2003 are collected. Then a weighted and directed network of scientific communication, including collaboration, citation and personal discussion, is constructed. Its static geometrical properties, including degree distribution, weight distribution, weight per degree, and betweenness centrality, give a nice overall description of the research works. The way we introduced here to measure the weight of connections can be used as a general one to construct weighted network.