Showing papers on "Degree distribution published in 2007"

Graph evolution: Densification and shrinking diameters

Abstract: How do real graphs evolve over timeq What are normal growth patterns in social, technological, and information networksq Many studies have discovered patterns in static graphs, identifying properties in a single snapshot of a large network or in a very small number of snapshots; these include heavy tails for in- and out-degree distributions, communities, small-world phenomena, and others. However, given the lack of information about network evolution over long periods, it has been hard to convert these findings into statements about trends over time.Here we study a wide range of real graphs, and we observe some surprising phenomena. First, most of these graphs densify over time with the number of edges growing superlinearly in the number of nodes. Second, the average distance between nodes often shrinks over time in contrast to the conventional wisdom that such distance parameters should increase slowly as a function of the number of nodes (like O(log n) or O(log(log n)).Existing graph generation models do not exhibit these types of behavior even at a qualitative level. We provide a new graph generator, based on a forest fire spreading process that has a simple, intuitive justification, requires very few parameters (like the flammability of nodes), and produces graphs exhibiting the full range of properties observed both in prior work and in the present study.We also notice that the forest fire model exhibits a sharp transition between sparse graphs and graphs that are densifying. Graphs with decreasing distance between the nodes are generated around this transition point.Last, we analyze the connection between the temporal evolution of the degree distribution and densification of a graph. We find that the two are fundamentally related. We also observe that real networks exhibit this type of relation between densification and the degree distribution.

Analysis of topological characteristics of huge online social networking services

Abstract: Social networking services are a fast-growing business in the Internet. However, it is unknown if online relationships and their growth patterns are the same as in real-life social networks. In this paper, we compare the structures of three online social networking services: Cyworld, MySpace, and orkut, each with more than 10 million users, respectively. We have access to complete data of Cyworld's ilchon (friend) relationships and analyze its degree distribution, clustering property, degree correlation, and evolution over time. We also use Cyworld data to evaluate the validity of snowball sampling method, which we use to crawl and obtain partial network topologies of MySpace and orkut. Cyworld, the oldest of the three, demonstrates a changing scaling behavior over time in degree distribution. The latest Cyworld data's degree distribution exhibits a multi-scaling behavior, while those of MySpace and orkut have simple scaling behaviors with different exponents. Very interestingly, each of the two e ponents corresponds to the different segments in Cyworld's degree distribution. Certain online social networking services encourage online activities that cannot be easily copied in real life; we show that they deviate from close-knit online social networks which show a similar degree correlation pattern to real-life social networks.

Graph theoretical analysis of complex networks in the brain

Abstract: Since the discovery of small-world and scale-free networks the study of complex systems from a network perspective has taken an enormous flight. In recent years many important properties of complex networks have been delineated. In particular, significant progress has been made in understanding the relationship between the structural properties of networks and the nature of dynamics taking place on these networks. For instance, the 'synchronizability' of complex networks of coupled oscillators can be determined by graph spectral analysis. These developments in the theory of complex networks have inspired new applications in the field of neuroscience. Graph analysis has been used in the study of models of neural networks, anatomical connectivity, and functional connectivity based upon fMRI, EEG and MEG. These studies suggest that the human brain can be modelled as a complex network, and may have a small-world structure both at the level of anatomical as well as functional connectivity. This small-world structure is hypothesized to reflect an optimal situation associated with rapid synchronization and information transfer, minimal wiring costs, as well as a balance between local processing and global integration. The topological structure of functional networks is probably restrained by genetic and anatomical factors, but can be modified during tasks. There is also increasing evidence that various types of brain disease such as Alzheimer's disease, schizophrenia, brain tumours and epilepsy may be associated with deviations of the functional network topology from the optimal small-world pattern.

Biological network comparison using graphlet degree distribution

Abstract: Motivation: Analogous to biological sequence comparison, comparing cellular networks is an important problem that could provide insight into biological understanding and therapeutics. For technical reasons, comparing large networks is computationally infeasible, and thus heuristics, such as the degree distribution, clustering coefficient, diameter, and relative graphlet frequency distribution have been sought. It is easy to demonstrate that two networks are different by simply showing a short list of properties in which they differ. It is much harder to show that two networks are similar, as it requires demonstrating their similarity in all of their exponentially many properties. Clearly, it is computationally prohibitive to analyze all network properties, but the larger the number of constraints we impose in determining network similarity, the more likely it is that the networks will truly be similar. Results: We introduce a new systematic measure of a network's local structure that imposes a large number of similarity constraints on networks being compared. In particular, we generalize the degree distribution, which measures the number of nodes 'touching' k edges, into distributions measuring the number of nodes 'touching' k graphlets, where graphlets are small connected non-isomorphic subgraphs of a large network. Our new measure of network local structure consists of 73 graphlet degree distributions of graphlets with 2--5 nodes, but it is easily extendible to a greater number of constraints (i.e. graphlets), if necessary, and the extensions are limited only by the available CPU. Furthermore, we show a way to combine the 73 graphlet degree distributions into a network 'agreement' measure which is a number between 0 and 1, where 1 means that networks have identical distributions and 0 means that they are far apart. Based on this new network agreement measure, we show that almost all of the 14 eukaryotic PPI networks, including human, resulting from various high-throughput experimental techniques, as well as from curated databases, are better modeled by geometric random graphs than by Erdos--Reny, random scale-free, or Barabasi--Albert scale-free networks. Availability: Software executables are available upon request. Contact: natasha@ics.uci.edu

Mapping human whole-brain structural networks with diffusion MRI.

Abstract: Understanding the large-scale structural network formed by neurons is a major challenge in system neuroscience. A detailed connectivity map covering the entire brain would therefore be of great value. Based on diffusion MRI, we propose an efficient methodology to generate large, comprehensive and individual white matter connectional datasets of the living or dead, human or animal brain. This non-invasive tool enables us to study the basic and potentially complex network properties of the entire brain. For two human subjects we find that their individual brain networks have an exponential node degree distribution and that their global organization is in the form of a small world.

Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: theory and simulations

Abstract: The spatial structure of populations is a key element in the understanding of the large scale spreading of epidemics. Motivated by the recent empirical evidence on the heterogeneous properties of transportation and commuting patterns among urban areas, we present a thorough analysis of the behavior of infectious diseases in metapopulation models characterized by heterogeneous connectivity and mobility patterns. We derive the basic reaction-diffusion equation describing the metapopulation system at the mechanistic level and derive an early stage dynamics approximation for the subpopulation invasion dynamics. The analytical description uses degree block variables that allows us to take into account arbitrary degree distribution of the metapopulation network. We show that along with the usual single population epidemic threshold the metapopulation network exhibits a global threshold for the subpopulation invasion. We find an explicit analytic expression for the invasion threshold that determines the minimum number of individuals traveling among subpopulations in order to have the infection of a macroscopic number of subpopulations. The invasion threshold is a function of factors such as the basic reproductive number, the infectious period and the mobility process and it is found to decrease for increasing network heterogeneity. We provide extensive mechanistic numerical Monte Carlo simulations that recover the analytical finding in a wide range of metapopulation network connectivity patterns. The results can be useful in the understanding of recent data driven computational approaches to disease spreading in large transportation networks and the effect of containment measures such as travel restrictions.

Topological vulnerability of the european power grid under errors and attacks

Abstract: We present an analysis of the topological structure and static tolerance to errors and attacks of the September 2003 actualization of the Union for the Coordination of Transport of Electricity (UCTE) power grid, involving thirty-three different networks. Though every power grid studied has exponential degree distribution and most of them lack typical small-world topology, they display patterns of reaction to node loss similar to those observed in scale-free networks. We have found that the node removal behavior can be logarithmically related to the power grid size. This logarithmic behavior would suggest that, though size favors fragility, growth can reduce it. We conclude that, with the ever-growing demand for power and reliability, actual planning strategies to increase transmission systems would have to take into account this relative increase in vulnerability with size, in order to facilitate and improve the power grid design and functioning.

SIR dynamics in random networks with heterogeneous connectivity

Abstract: Random networks with specified degree distributions have been proposed as realistic models of population structure, yet the problem of dynamically modeling SIR-type epidemics in random networks remains complex. I resolve this dilemma by showing how the SIR dynamics can be modeled with a system of three nonlinear ODE's. The method makes use of the probability generating function (PGF) formalism for representing the degree distribution of a random network and makes use of network-centric quantities such as the number of edges in a well-defined category rather than node-centric quantities such as the number of infecteds or susceptibles. The PGF provides a simple means of translating between network and node-centric variables and determining the epidemic incidence at any time. The theory also provides a simple means of tracking the evolution of the degree distribution among susceptibles or infecteds. The equations are used to demonstrate the dramatic effects that the degree distribution plays on the final size of an epidemic as well as the speed with which it spreads through the population. Power law degree distributions are observed to generate an almost immediate expansion phase yet have a smaller final size compared to homogeneous degree distributions such as the Poisson. The equations are compared to stochastic simulations, which show good agreement with the theory. Finally, the dynamic equations provide an alternative way of determining the epidemic threshold where large-scale epidemics are expected to occur, and below which epidemic behavior is limited to finite-sized outbreaks.

Scalable modeling of real graphs using Kronecker multiplication

Abstract: Given a large, real graph, how can we generate a synthetic graph that matches its properties, i.e., it has similar degree distribution, similar (small) diameter, similar spectrum, etc? We propose to use "Kronecker graphs", which naturally obey all of the above properties, and we present KronFit, a fast and scalable algorithm for fitting the Kronecker graph generation model to real networks. A naive approach to fitting would take super-exponential time. In contrast, KronFit takes linear time, by exploiting the structure of Kronecker product and by using sampling. Experiments on large real and synthetic graphs show that KronFit indeed mimics very well the patterns found in the target graphs. Once fitted, the model parameters and the resulting synthetic graphs can be used for anonymization, extrapolations, and graph summarization.

Seed size strongly affects cascades on random networks.

Abstract: The average avalanche size in the Watts model of threshold dynamics on random networks of arbitrary degree distribution is determined analytically. Existence criteria for global cascades are shown to depend sensitively on the size of the initial seed disturbance. The dependence of cascade size upon the mean degree $z$ of the network is known to exhibit several transitions---these are typically continuous at low $z$ and discontinuous at high $z$; here it is demonstrated that the low-$z$ transition may in fact be discontinuous in certain parameter regimes. Connections between these results and the zero-temperature random-field Ising model on random graphs are discussed.

Scaling and correlations in three bus-transport networks of China

Abstract: We report the statistical properties of three bus-transport networks (BTN) in three different cities of China. These networks are composed of a set of bus lines and stations serviced by these. Network properties, including the degree distribution, clustering and average path length are studied in different definitions of network topology. We explore scaling laws and correlations that may govern intrinsic features of such networks. Besides, we create a weighted network representation for BTN with lines mapped to nodes and number of common stations to weights between lines. In such a representation, the distributions of degree, strength and weight are investigated. A linear behavior between strength and degree s ( k ) ∼ k is also observed.

Uncovering fuzzy community structure in complex networks.

Abstract: There has been an increasing interest in properties of complex networks, such as small-world property, power-law degree distribution, and network transitivity which seem to be common to many real world networks. In this study, a useful community detection method based on non-negative matrix factorization (NMF) technique is presented. Based on a popular modular function, a proper feature matrix from diffusion kernel and NMF algorithm, the presented method can detect an appropriate number of fuzzy communities in which a node may belong to more than one community. The distinguished characteristic of the method is its capability of quantifying how much a node belongs to a community. The quantification provides an absolute membership degree for each node to each community which can be employed to uncover fuzzy community structure. The computational results of the method on artificial and real networks confirm its ability.

Empirical analysis of a scale-free railway network in China

Abstract: We present a detailed, empirical analysis of the statistical properties of the China Railway Network (CRN) consisting of 3915 nodes (train stations) and 22 259 edges (railways). Based on this, CRN displays two explicit features already observed in numerous real-world and artificial networks. One feature, the small-world property, has the fingerprint of a small characteristic shortest-path length, 3.5, accompanied by a high degree of clustering, 0.835. Another feature is characterized by the scale-free distributions of both degrees and weighted degrees, namely strengths. Correlations between strength and degree, degree and degree, and clustering coefficient and degree have been studied and the forms of such behaviors have been identified. In addition, we investigate distributions of clustering coefficients, topological distances, and spatial distances.

Random trees and general branching processes

Abstract: We consider a tree that grows randomly in time. Each time a new vertex appears, it chooses exactly one of the existing vertices and attaches to it. The probability that the new vertex chooses vertex x is proportional to w(deg(x)), a weight function of the actual degree of x. The weight function w : ℕ ➝ ℝ+ is the parameter of the model. In [4] and [11] the authors derive the asymptotic degree distribution for a model that is equivalent to the special case, when the weight function is linear. The proof therein strongly relies on the linear choice of w. Using well-established results from the theory of general branching processes we give the asymptotical degree distribution for a wide range of weight functions. Moreover, we provide the asymptotic distribution of the tree itself as seen from a randomly selected vertex. The latter approach gives greater insight to the limiting structure of the tree. Our proof is robust and we believe that the method may be used to answer several other questions related to the model. It relies on the fact that considering the evolution of the random tree in continuous time, the process may be viewed as a general branching process, this way classical results can be applied. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007

Relating Network Structure to Diffusion Properties through Stochastic Dominance

Abstract: We examine the spread of a disease or behavior through a social network. In particular, we analyze how infection rates depend on the distribution of degrees (numbers of links) among the nodes in the network. We introduce new techniques using first- and second order stochastic dominance relationships of the degree distribution in order to compare infection rates across different social networks.

Complex network synchronizability: analysis and control.

Abstract: In this paper, the investigation is first motivated by showing two examples of simple regular symmetrical graphs, which have the same structural parameters, such as average distance, degree distribution, and node betweenness centrality, but have very different synchronizabilities. For a given network with identical node dynamics, it is further shown that two key factors influencing the network synchronizability are the network inner linking matrix and the eigenvalues of the network topological matrix. Several examples are then provided to show that adding new edges to a network can either increase or decrease the network synchronizability. In searching for conditions under which the network synchronizability may be increased by adding edges, it is found that for networks with disconnected complementary graphs, adding edges never decreases their synchronizability. Moreover, it is found that an unbounded synchronized region is always easier to analyze than a bounded synchronized region. Therefore to effectively enhance the network synchronizability, a design method is finally presented for the inner linking matrix of rank 1 such that the resultant network has an unbounded synchronized region, for the case where the synchronous state is an equilibrium point of the network.

Assortativity and act degree distribution of some collaboration networks

Abstract: Empirical investigation results on weighted and un-weighted assortativity, act degree distribution, degree distribution and node strength distribution of nine real world collaboration networks have been presented. The investigations propose that act degree distribution, degree distribution and node strength distribution usually show so-called “shifted power law” (SPL) function forms, which can continuously vary from an ideal power law to an ideal exponential decay. Two parameters, α and η , can be used for description of the distribution functions. Another conclusion is that assortativity coefficient and the parameter, α or η , is monotonously dependent on each other. By the collaboration network evolution model introduced in a reference [P. Zhang et al., Physica A 360 (2006) 599], we analytically derived the SPL distributions, which typically appeared in general situations where nodes are selected partially randomly, with a probability p , and partially by linear preferential principle, with the probability 1 - p . The analytic discussion gives an explicit expression on the relationship between the random selection proportion p and the parameters α and η . The numerical simulation results by the model show a monotonic dependence of assortativity on the random selection proportion p . The empirically obtained assortativity versus α or η curve for the four collaboration networks with small maximal act size, T max , shows a good agreement with the model prediction. According to the curves, one can qualitatively judge the random selection proportion of the real world network in its evolution process.

Empirical analysis of the evolution of a scientific collaboration network

Abstract: We present an analysis of the temporal evolution of a scientific coauthorship network, the genetic programming network. We find evidence that the network grows according to preferential attachment, with a slightly sublinear rate. We empirically find how a giant component forms and develops, and we characterize the network by several other time-varying quantities: the mean degree, the clustering coefficient, the average path length, and the degree distribution. We find that the first three statistics increase over time in the growing network; the degree distribution tends to stabilize toward an exponentially truncated power-law. We finally suggest an effective network interpretation that takes into account the aging of collaboration relationships.

Small-world network topology of hippocampal neuronal network is lost, in an in vitro glutamate injury model of epilepsy.

Abstract: Neuronal network topologies and connectivity patterns were explored in control and glutamate-injured hippocampal neuronal networks, cultured on planar multielectrode arrays. Spontaneous activity was characterized by brief episodes of synchronous firing at many sites in the array (network bursts). During such assembly activity, maximum numbers of neurons are known to interact in the network. After brief glutamate exposure followed by recovery, neuronal networks became hypersynchronous and fired network bursts at higher frequency. Connectivity maps were constructed to understand how neurons communicate during a network burst. These maps were obtained by analysing the spike trains using cross-covariance analysis and graph theory methods. Analysis of degree distribution, which is a measure of direct connections between electrodes in a neuronal network, showed exponential and Gaussian distributions in control and glutamate-injured networks, respectively. Although both the networks showed random features, small-world properties in these networks were different. These results suggest that functional two-dimensional neuronal networks in vitro are not scale-free. After brief exposure to glutamate, normal hippocampal neuronal networks became hyperexcitable and fired a larger number of network bursts with altered network topology. The small-world network property was lost and this was accompanied by a change from an exponential to a Gaussian network.

Partitioning and modularity of graphs with arbitrary degree distribution.

Abstract: We solve the graph bipartitioning problem in dense graphs with arbitrary degree distribution using the replica method. We find the cut size to scale universally with k. In contrast, earlier results studying the problem in graphs with a Poissonian degree distribution had found a scaling with kFu and Anderson, J. Phys. A 19, 1605 1986. Our results also generalize to the problem of q partitioning. They can be used to find the expected modularity Q Newman and Girvan, Phys. Rev. E 69, 026113 2004 of random graphs and allow for the assessment of the statistical significance of the output of community detection algorithms. Given a graph or network GN ,M of N nodes and M edges, the problem of graph partitioning is finding a partition of the nodes into q equal sized parts, such that the number of edges connecting different parts, the cut size, is minimal. The solution of this problem has many important practical applications in multiprocessor scheduling for parallel computing, very large-scale integrated chip design VLSI, and data mining 1. Due to the problem being NP-complete 2, i.e., no algorithm is known which is guaranteed to find an optimal solution in a number of steps that grows only polynomially with the size of the graph, only heuristics exist to approximate a solution. Furthermore, one cannot determine in polynomial time whether a solution found is indeed optimal. The only way to assess the quality of a heuristic algorithm or a particular result is to compare with rigorous bounds on the cut size.

Topological structural classes of complex networks

Abstract: We use theoretical principles to study how complex networks are topologically organized at large scale. Using spectral graph theory we predict the existence of four different topological structural classes of networks. These classes correspond, respectively, to highly homogenous networks lacking structural bottlenecks, networks organized into highly interconnected modules with low inter-community connectivity, networks with a highly connected central core surrounded by a sparser periphery, and networks displaying a combination of highly connected groups (quasicliques) and groups of nodes partitioned into disjoint subsets (quasibipartites). Here we show by means of the spectral scaling method that these classes really exist in real-world ecological, biological, informational, technological, and social networks. We show that neither of three network growth mechanisms---random with uniform distribution, preferential attachment, and random with the same degree sequence as real network---is able to reproduce the four structural classes of complex networks. These models reproduce two of the network classes as a function of the average degree but completely fail in reproducing the other two classes of networks.

Maximal planar scale-free Sierpinski networks with small-world effect and power law strength-degree correlation

Abstract: Many real networks share three generic properties: they are scale-free, display a small-world effect, and show a power law strength-degree correlation. In this paper, we propose a type of deterministically growing networks called Sierpinski networks, which are induced by the famous Sierpinski fractals and constructed in a simple iterative way. We derive analytical expressions for degree distribution, strength distribution, clustering coefficient, and strength-degree correlation, which agree well with the characterizations of various real-life networks. Moreover, we show that the introduced Sierpinski networks are maximal planar graphs.

A study on some urban bus transport networks

Abstract: In this paper, we present the empirical investigation results on the urban bus transport networks (BTNs) of four major cities in China. In BTN, nodes are bus stops. Two nodes are connected by an edge when the stops are serviced by a common bus route. The empirical results show that the degree distributions of BTNs take exponential function forms. Other two statistical properties of BTNs are also considered, and they are suggested as the distributions of so-called “the number of stops in a bus route” (represented by S ) and “the number of bus routes a stop joins” (by R ). The distributions of R also show exponential function forms, while the distributions of S follow asymmetric, unimodal functions. To explain these empirical results and attempt to simulate a possible evolution process of BTN, we introduce a model by which the analytic and numerical result obtained agrees well with the empirical facts. Finally, we also discuss some other possible evolution cases, where the degree distribution shows a power law or an interpolation between the power law and the exponential decay.

Fractality and self-similarity in scale-free networks

Abstract: Fractal scaling and self-similar connectivity behaviour of scale-free (SF) networks are reviewed and investigated in diverse aspects. We first recall an algorithm of box-covering that is useful and easy to implement in SF networks, the so-called random sequential box-covering. Next, to understand the origin of the fractal scaling, fractal networks are viewed as comprising of a skeleton and shortcuts. The skeleton, embedded underneath the original network, is a spanning tree specifically based on the edge-betweenness centrality or load. We show that the skeleton is a non-causal tree, either critical or supercritical. We also study the fractal scaling property of the k-core of a fractal network and find that as k increases, not only does the fractal dimension of the k-core change but also eventually the fractality no longer holds for large enough k. Finally, we study the self-similarity, manifested as the scale-invariance of the degree distribution under coarse-graining of vertices by the box-covering method. We obtain the condition for self-similarity, which turns out to be independent of the fractality, and find that some non-fractal networks are self-similar. Therefore, fractality and self-similarity are disparate notions in SF networks.

Optimal Degree Distribution for LT Codes with Small Message Length

Abstract: Fountain codes provide an efficient way to transfer information over erasure channels. We give an exact performance analysis of a specific type of fountain codes, called LT codes, when the message length N is small. Two different approaches are developed. In a Markov chain approach the state space explosion, even with reduction based on permutation isomorphism, limits the analysis to very short messages, N < 4. An alternative combinatorial method allows recursive calculation of the probability of decoding after N received packets. The recursion can be solved symbolically for values of N < 10 and numerically up to N ap30. Examples of optimization results give insight into the nature of the problem. In particular, we argue that a few conditions are sufficient to define an almost optimal LT encoding.