Node Degree Distribution Research Articles

Towards a bipartite graph modeling of the internet topology

Modeling the properties of the Internet topology aims at generating large scale artificial IP networks that mimic properties of real ones for simulation purposes. Current models typically consider the Internet as a simple graph where edges are point-to-point connections between routers. This approach does not take into account point-to-multipoint connections that exist at lower layers in the network, e.g. layer-2 clouds, such as Ethernet switches or MPLS networks. Instead, such physical point-to-multipoint connections are modeled as several logical IP level point-to-point connections.In this paper, we rely on recent developments in topology discovery based on IGMP probing that allows for revealing part of the network’s layer-2 structure. We take advantage of this additional knowledge for proposing an Internet model based on bipartite graphs considering both point-to-point and point-to-multipoint connections. Our model remains simple: it only takes as input the node degree sequence for both layer-2 and layer-3 nodes, randomly generates a bipartite graph respecting those distributions, and then derives the corresponding layer-3 topology. We show that, despite the simplicity of our model, realistic network properties, such as high local density, emerge naturally. This is in contrast with the now common belief that such properties can only appear with more intricate models or if explicitly injected in random models. Besides, we also provide evidences of how the analysis performed at the bipartite level might shed light on important properties of the real network structure. Finally, we propose and evaluate a bipartite graph generator based on our model that only takes two synthetic node degree distributions as input.

Statistics of Leaders in Index-Driven Networks

We study the statistical properties of leaders in growing networks with age. A leader of a network is defined as the node with the largest degree and the age of the node is trivially labeled by its index, i.e., the time it joins the network. As networks evolve with the addition of new nodes connecting to old ones with the possibility that is proportional to the index of the target, we investigate both the average number and index of leaders as well as the degree distribution of nodes. The average number of leaders first increases quickly with time and then saturates to a finite value and the average index of leaders increases algebraically with time. Both features result from the degree distribution with an exponential tail. Analytical calculations based on the rate equation are verified by numerical simulations.

Graph clusterings with overlaps: Adapted quality indices and a generation model

Many real-world networks have a structure of overlapping cohesive groups. In order to uncover this structure several clustering algorithms have been developed. In this paper, we focus on the evaluation of these algorithms. Quality measures are commonly used for this purpose and provide a means to assess the quality of a derived cluster structure. Currently, there are too few measures for graph clusterings with overlaps available that would enable a meaningful evaluation, even though many well studied crisp quality measures exist. In order to expand the pool of overlapping measures we propose three methods to adapt existing crisp quality measures so that they can handle graph overlaps appropriately. We demonstrate our methods on the well known measures Density, Modularity and Conductance. We also propose an enhancement of an existing modularity measure for networks with overlapping structure. We analyse the proposed quality indices using experiments on artificial graphs that possess overlapping structure. For this evaluation, we apply a graph generation model to create clustered graphs with overlaps that are similar to real-world networks, i.e., their node degree and cluster size distribution follow a power law.

Discrete time quantum walk on the Apollonian network

This work discusses the discrete time quantum walk (DTQW) on the first five generations of the Apollonian network. This structure is constructed in a geometric recurrent way and is characterized by, among other features, a scale-free distribution of node degrees. The DTQW formalism requires a node-dependent coin operator that, for each node, has as many different output states as that node degree, so that each local coin operator is expressed by the Fourier operator. The DTQW time evolution matrix has a larger rank than that of the network adjacency matrix or the matrix representation of the continuous time quantum walk (CTQW). Results for the time evolution, return time to specific nodes and asymptotic probability of site occupancy are obtained. Such DTQW-specific features are discussed and compared to those obtained for the classical random walk and CTQW on the same lattice.

A Transmission Power Optimization with a Minimum Node Degree for Energy-Efficient Wireless Sensor Networks with Full-Reachability

Transmission power optimization is the most significant factor in prolonging the lifetime and maintaining the connection quality of wireless sensor networks. Un-optimized transmission power of nodes either interferes with or fails to link neighboring nodes. The optimization of transmission power depends on the expected node degree and node distribution. In this study, an optimization approach to an energy-efficient and full reachability wireless sensor network is proposed. In the proposed approach, an adjustment model of the transmission range with a minimum node degree is proposed that focuses on topology control and optimization of the transmission range according to node degree and node density. The model adjusts the tradeoff between energy efficiency and full reachability to obtain an ideal transmission range. In addition, connectivity and reachability are used as performance indices to evaluate the connection quality of a network. The two indices are compared to demonstrate the practicability of framework through simulation results. Furthermore, the relationship between the indices under the conditions of various node degrees is analyzed to generalize the characteristics of node densities. The research results on the reliability and feasibility of the proposed approach will benefit the future real deployments.

Peer-to-peer information retrieval using shared-content clustering

Peer-to-peer (p2p) networks are used by millions for searching and downloading content. Recently, clustering algorithms were shown to be useful for helping users find content in large networks. Yet, many of these algorithms overlook the fact that p2p networks follow graph models with a power-law node degree distribution. This paper studies the obtained clusters when applying clustering algorithms on power-law graphs and their applicability for finding content. Driven by the observed deficiencies, a simple yet efficient clustering algorithm is proposed, which targets a relaxed optimization of a minimal distance distribution of each cluster with a size balancing scheme. A comparative analysis using a song-similarity graph collected from 1.2 million Gnutella users reveals that commonly used efficiency measures often overlook search and recommendation applicability issues and provide the wrong impression that the resulting clusters are well suited for these tasks. We show that the proposed algorithm performs well on various measures that are well suited for the domain.

Analysis on Refinery System as a Complex Task-resource Network

Refinery system, a typical example of process systems, is presented as complex network in this paper. The topology of this system is described by task-resource network and modeled as directed and weighted graph, in which nodes represent various tasks and edges denote the resources exchanged among tasks. Using the properties of node degree distribution, strength distribution and other weighted quantities, we demonstrate the heterogeneity of the network and point out the relation between structural characters of vertices and the functionality of corresponding tasks. The above phenomena indicate that the design requirements and principles of production process contribute to the heterogeneous features of the network. Besides, betweenness centrality of nodes can be used as an importance indicator to provide additional information for decision making. The correlations between structure and weighted properties are investigated to further address the influence brought by production schemes in system connectivity patterns. Cascading failures model is employed to analyze the robustness of the network when targeted attack happens. Two capacity assignment strategies are compared in order to improve the robustness of the network at certain cost. The refinery system displays more reliable behavior when the protecting strategy considers heterogeneous properties. This phenomenon further implies the structure-activity relationship of the refinery system and provides insightful suggestions for process system design. The results also indicate that robustness analysis is a promising application of methodologies from complex networks to process system engineering.

A kinetic model of the evolution of a protein interaction network

BackgroundKnown protein interaction networks have very particular properties. Old proteins tend to have more interactions than new ones. One of the best statistical representatives of this property is the node degree distribution (distribution of proteins having a given number of interactions). It has previously been shown that this distribution is very close to the sum of two distinct exponential components. In this paper, we asked: What are the possible mechanisms of evolution for such types of networks? To answer this question, we tested a kinetic model for simplified evolution of a protein interactome. Our proposed model considers the emergence of new genes and interactions and the loss of old ones. We assumed that there are generally two coexisting classes of proteins. Proteins constituting the first class are essential only for ecological adaptations and are easily lost when ecological conditions change. Proteins of the second class are essential for basic life processes and, hence, are always effectively protected against deletion. All proteins can transit between the above classes in both directions. We also assumed that the phenomenon of gene duplication is always related to ecological adaptation and that a new copy of a duplicated gene is not essential. According to this model, all proteins gain new interactions with a rate that preferentially increases with the number of interactions (the rich get richer). Proteins can also gain interactions because of duplication. Proteins lose their interactions both with and without the loss of partner genes.ResultsThe proposed model reproduces the main properties of protein-protein interaction networks very well. The connectivity of the oldest part of the interaction network is densest, and the node degree distribution follows the sum of two shifted power-law functions, which is a theoretical generalization of the previous finding. The above distribution covers the wide range of values of node degrees very well, much better than a power law or generalized power law supplemented with an exponential cut-off. The presented model also relates the total number of interactome links to the total number of interacting proteins. The theoretical results were for the interactomes of A. thaliana, B. taurus, C. elegans, D. melanogaster, E. coli, H. pylori, H. sapiens, M. musculus, R. norvegicus and S. cerevisiae.ConclusionsUsing these approaches, the kinetic parameters could be estimated. Finally, the model revealed the evolutionary kinetics of proteome formation, the phenomenon of protein differentiation and the process of gaining new interactions.

Rank-based deactivation model for networks with age

We study the impact of age on network evolution which couples addition of new nodes and deactivation of old ones. During evolution, each node experiences two stages: active and inactive. The transition from the active state to the inactive one is based on the rank of the node. In this paper, we adopt age as a criterion of ranking, and propose two deactivation models that generalize previous research. In model A, the older active node possesses the higher rank, whereas in model B, the younger active node takes the higher rank. We make a comparative study between the two models through the node-degree distribution.

Phase transitions in the Potts model on complex networks

The Potts model is one of the most popular spin models of statistical physics. The prevailing majority of work done so far corresponds to the lattice version of the model. However, many natural or man-made systems are much better described by the topology of a network. We consider the q-state Potts model on an uncorrelated scale-free network for which the node-degree distribution manifests a power-law decay governed by the exponent \lambda. We work within the mean-field approximation, since for systems on random uncorrelated scale-free networks this method is known to often give asymptotically exact results. Depending on particular values of q and \lambda one observes either a first-order or a second-order phase transition or the system is ordered at any finite temperature. In a case study, we consider the limit q=1 (percolation) and find a correspondence between the magnetic exponents and those describing percolation on a scale-free network. Interestingly, logarithmic corrections to scaling appear at \lambda=4 in this case.

Topological characterization and modeling of dynamic evolving power distribution networks

Existing complex network models are either with unvaried network size or based on simple growth mechanisms which cannot accurately describe the operational dynamics and characteristics of realistic networks. In this paper, we exploit the dynamic evolving phenomenon of power distribution networks covering its growth, reconnection and shrinking characteristics from the network topology perspective, and attempt to produce a novel dynamic evolving model through introducing the locating probability and shrinking mechanism. The proposed modeling approach is assessed and validated through extensive numerical simulation experiments for a range of standard IEEE power test systems. The statistical results reveal that the node degree distribution of the power network follows the power-law distribution and the node removal probability of the network dynamic has a significant impact on the network evolvement and robustness. Such macroscopic topological findings can greatly benefit the power distribution network operators (DNOs) from many aspects, including network planning, vulnerability analysis, fault prediction and cost-effective reinforcement.

On optimal node and polynomial degree distribution in one-dimensional $$hp$$ -FEM

We are concerned with the task of constructing an optimal higher-order finite element mesh under a constraint on the total number of degrees of freedom. The motivation for this work is to obtain a truly optimal higher-order finite element mesh that can be used to compare the quality of automatic adaptive algorithms. Minimized is the approximation error in a global norm. Optimization variables include the number of elements, positions of nodes, and polynomial degrees of elements. Optimization methods and software that we use are described, and numerical results are presented.

Motifs insights from engineering systems architectures

“Network Motifs” is a research area in “Network Science” which has captured the attention of many researchers. Complex biological and social networks have displayed presence of some triad motifs far in excess (or short) of their expected values. Some of these over(under)-represented motifs have explained the basic functionality of systems, for example, in the sensory transcription networks of biology, over-represented motifs are shown to perform signal processing tasks. This suggests purposeful, selective retention of these motifs in the studied biological systems. Another interesting feature is the high correlation of triad motif significance profiles (MSPs) of all systems that belong to a family of naturally grouped systems, thereby suggesting that all the systems in a family have the same function to perform and hence the correlation. Engineering systems also display over(under)-represented motifs. The MSPs of a family of naturally grouped engineering systems show high correlation. Unlike biological and social networks, engineering systems are designed by humans and offer an opportunity for investigation based on known design rules. We show that over(under)-represented motifs in engineering systems are not purposefully retained/avoided to perform functions but are a natural consequence of design by decomposition. We also discover that naturally grouped systems have remarkably correlated in(out) degree distribution across nodes resulting in a high correlation in MSP. Therefore, we argue that the idea of network motifs has no significance in engineering systems (unlike biological and other evolutionary systems), and we caution the engineering research community to be careful while drawing conclusions based on network motifs. We report a remarkable correlation of in(out) degree distribution of systems within a family of engineering systems. We further show that biological and social networks also display signs of decomposition. They also show a high correlation of in–out degree distribution of nodes for systems within a family. This opens up an interesting opportunity to investigate these systems through their observed decomposition. © 2012 Wiley Periodicals, Inc. Complexity, 2012 © 2012 Wiley Periodicals, Inc.

Network Characteristics Analysis of the Xinjiang Regional Logistic Based on Complex Networks

Reasonable logistics system plays a fundamental role in the process of regional economic development. With the development of Xinjiang's energy resources conversion strategy is ceaseless and thorough, as the country imported energy and scarce mineral resources international channel, Xinjiang internal and external logistics hub action are highlighted with each passing day. In this paper the complex network theory is applied to the Xinjiang regional logistics network, important cities, ports, warehousing point for a node to connect these nodes highway straight-line distance of the railway side, constructed the Xinjiang regional characteristics of the logistics network model. Select the node degree distribution, betweenness, average path length, clustering coefficient as a measure, static statistical properties of the network are analyzed. Provide basic data for the Xinjiang regional logistics network.

A PEG Construction of Finite-Length LDPC Codes with Low Error Floor

The progressive-edge-growth (PEG) algorithm of Hu et al. is modified to improve the error floor performance of the constructed low-density parity-check (LDPC) codes. To improve the error floor, the original PEG algorithm is equipped with an efficient algorithm to find the dominant elementary trapping sets (ETS's) that are added to the Tanner graph of the under-construction code by the addition of each variable node and its adjacent edges. The aim is to select the edges, among the candidates available at each step of the original PEG algorithm, that prevent the creation of dominant ETS's. The proposed method is applicable to both regular and irregular variable node degree distributions. Simulation results are presented to demonstrate the superior ETS distribution and error floor performance of the constructed codes compared to similar codes constructed by the original and other modifications of the PEG algorithm.

Creation and growth of online social network

Social networks are an example of complex systems consisting of nodes that can interact with each other and based on these activities the social relations are defined. The dynamics and evolution of social networks are very interesting but at the same time very challenging areas of research. In this paper the formation and growth of one of such structures extracted from data about human activities within online social networking system is investigated. Dynamics of both local and global characteristics are studied. Analysis of the dynamics of the network growth showed that it changes over time—from random process to power-law growth. The phase transition between those two is clearly visible. In general, node degree distribution can be described as the scale-free but it does not emerge straight from the beginning. Social networks are known to feature high clustering coefficient and friend-of-a-friend phenomenon. This research has revealed that in online social network, although the clustering coefficient grows over time, it is lower than expected. Also the friend-of-a-friend phenomenon is missing. On the other hand, the length of the shortest paths is small starting from the beginning of the network existence so the small-world phenomenon is present. The unique element of the presented study is that the data, from which the online social network was extracted, represents interactions between users from the beginning of the social networking site existence. The system, from which the data was obtained, enables users to interact using different communication channels and it gives additional opportunity to investigate multi-relational character of human relations.

Localized geometric topologies with bounded node degree for three-dimensional wireless sensor networks

Abstract Three-dimensional (3D) wireless sensor networks have attracted a lot of attention due to their great potential usages in both commercial and civilian applications, such as environmental data collection, pollution monitoring, space exploration, disaster prevention, and tactical surveillance. Topology control in 3D sensor networks has been studied recently, and different 3D geometric topologies were proposed to be the underlying network topologies to achieve the sparseness of the communication networks. However, most of these proposed 3D topologies cannot bound the maximum node degree, i.e., some nodes may need to maintain a large number of neighbors in the constructed topologies, which is not energy efficient and may lead to large contention. In this article, we extend several existing 3D geometric topologies to a set of new 3D topologies with bounded node degree. We provide both theoretical analysis and simulation evaluation on their power efficiency and node degree distributions. Our simulation results over random 3D sensor networks confirm nice performances of these proposed 3D topologies.

Creating and analyzing pathway and protein interaction compendia for modelling signal transduction networks

BackgroundUnderstanding the information-processing capabilities of signal transduction networks, how those networks are disrupted in disease, and rationally designing therapies to manipulate diseased states require systematic and accurate reconstruction of network topology. Data on networks central to human physiology, such as the inflammatory signalling networks analyzed here, are found in a multiplicity of on-line resources of pathway and interactome databases (Cancer CellMap, GeneGo, KEGG, NCI-Pathway Interactome Database (NCI-PID), PANTHER, Reactome, I2D, and STRING). We sought to determine whether these databases contain overlapping information and whether they can be used to construct high reliability prior knowledge networks for subsequent modeling of experimental data.ResultsWe have assembled an ensemble network from multiple on-line sources representing a significant portion of all machine-readable and reconcilable human knowledge on proteins and protein interactions involved in inflammation. This ensemble network has many features expected of complex signalling networks assembled from high-throughput data: a power law distribution of both node degree and edge annotations, and topological features of a “bow tie” architecture in which diverse pathways converge on a highly conserved set of enzymatic cascades focused around PI3K/AKT, MAPK/ERK, JAK/STAT, NFκB, and apoptotic signaling. Individual pathways exhibit “fuzzy” modularity that is statistically significant but still involving a majority of “cross-talk” interactions. However, we find that the most widely used pathway databases are highly inconsistent with respect to the actual constituents and interactions in this network. Using a set of growth factor signalling networks as examples (epidermal growth factor, transforming growth factor-beta, tumor necrosis factor, and wingless), we find a multiplicity of network topologies in which receptors couple to downstream components through myriad alternate paths. Many of these paths are inconsistent with well-established mechanistic features of signalling networks, such as a requirement for a transmembrane receptor in sensing extracellular ligands.ConclusionsWide inconsistencies among interaction databases, pathway annotations, and the numbers and identities of nodes associated with a given pathway pose a major challenge for deriving causal and mechanistic insight from network graphs. We speculate that these inconsistencies are at least partially attributable to cell, and context-specificity of cellular signal transduction, which is largely unaccounted for in available databases, but the absence of standardized vocabularies is an additional confounding factor. As a result of discrepant annotations, it is very difficult to identify biologically meaningful pathways from interactome networks a priori. However, by incorporating prior knowledge, it is possible to successively build out network complexity with high confidence from a simple linear signal transduction scaffold. Such reduced complexity networks appear suitable for use in mechanistic models while being richer and better justified than the simple linear pathways usually depicted in diagrams of signal transduction.

A Dimensionality Reduction Framework for Detection of Multiscale Structure in Heterogeneous Networks

Graph clustering has been widely applied in exploring regularities emerging in relational data. Recently, the rapid development of network theory correlates graph clustering with the detection of community structure, a common and important topological characteristic of networks. Most existing methods investigate the community structure at a single topological scale. However, as shown by empirical studies, the community structure of real world networks often exhibits multiple topological descriptions, corresponding to the clustering at different resolutions. Furthermore, the detection of multiscale community structure is heavily affected by the heterogeneous distribution of node degree. It is very challenging to detect multiscale community structure in heterogeneous networks. In this paper, we propose a novel, unified framework for detecting community structure from the perspective of dimensionality reduction. Based on the framework, we first prove that the well-known Laplacian matrix for network partition and the widely-used modularity matrix for community detection are two kinds of covariance matrices used in dimensionality reduction. We then propose a novel method to detect communities at multiple topological scales within our framework. We further show that existing algorithms fail to deal with heterogeneous node degrees. We develop a novel method to handle heterogeneity of networks by introducing a rescaling transformation into the covariance matrices in our framework. Extensive tests on real world and artificial networks demonstrate that the proposed correlation matrices significantly outperform Laplacian and modularity matrices in terms of their ability to identify multiscale community structure in heterogeneous networks.

On the evolution of Linux kernels: a complex network perspective

This paper presents a novel method to study the evolution of Linux kernel components using complex networks to understand how Linux kernel components evolve over time. After analyzing the node degree distribution, clustering coefficient, and average path length of the call graphs corresponding to the kernel components of 130 development versions and 94 stable versions (V1.0 to V2.4.35), we found that the call graphs of the file system, driver, kernel, memory management, and net components are scale-free, small-world complex networks. In addition, all of the five components exhibit very strong preferential attachment tendency. With such in-depth understanding of the features of the Linux kernel components, we propose a generic method that could be used to find major structural changes that occur during the evolution of software systems. Copyright © 2012 John Wiley & Sons, Ltd.