Node Degree Distribution Research Articles

Probabilistic Heuristics for Disseminating Information in Networks

We study the problem of disseminating a piece of information through all the nodes of a network, given that it is known originally only to a single node. In the absence of any structural knowledge on the network other than the nodes' neighborhoods, this problem is traditionally solved by flooding all the network's edges. We analyze a recently introduced probabilistic algorithm for flooding and give an alternative probabilistic heuristic that can lead to some cost-effective improvements, like better trade-offs between the message and time complexities involved. We analyze the two algorithms both mathematically and by means of simulations, always within a random-graph framework and considering relevant node-degree distributions.

Organizations of rich nodes in complex networks

Hidden regularities of complex networks may be different from the properties they show to us externally. In this paper, organization of rich nodes in real world and model networks are investigated. It is found that in scale-free networks, when the rich level is relatively low, the degree distribution of rich nodes follows a power law. On the transitivity and mixing pattern, the clustering coefficient C and assortativity coefficient r of rich nodes fluctuate with the increment of rich level ρ in World Wide Web, while in protein–protein Interaction network, a scaling law can be observed in describing the clustering coefficient C of rich nodes as a function of the rich level and an extremum can be expected in the assortativity of rich nodes at a specified rich level. Furthermore, both positive and negative assortativity coefficient can be observed in different rich levels in the studied real world networks, which dedicates that the rich nodes may be organized different from the networks of their own.

Discrete scale invariance in scale free graphs

In this work, we introduce an energy function in order to study finite scale free graphs generated with different models. The energy distribution has a fractal pattern and presents log periodic oscillations for high energies. These oscillations are related to a discrete scale invariance of certain graphs, that is, there are preferred scaling ratios suggesting a hierarchical distribution of node degrees. On the other hand, small energies correspond to graphs with evenly distributed degrees.

On the emergence of scaling in weighted networks

General conditions for the appearance of the power-law distribution of total weights concentrated in vertices of complex network systems are established. By use of the rate equation approach for networks evolving by connectivity-governed attachment of every new node to p ⩾ 1 exiting nodes and by ascription to every new link a weight taken from algebraic distributions, independent of network topologies, it is shown that the distribution of the total weight w asymptotically follows the power law, P ( w ) ∼ w - α with the exponent α ∈ ( 0 , 2 ] . The power-law dependence of the weight distribution is also proved to hold, for asymptotically large w , in the case of networks in which a link between nodes i and j carries a load w ij , determined by node degrees k i and k j at the final stage of the network growth, according to the relation w ij = ( k i k j ) θ with θ ∈ ( - 1 , 0 ] . For this class of networks, the scaling exponent σ describing the weight distribution is found to satisfy the relationship σ = ( λ + θ ) / ( 1 + θ ) , where λ is the scaling index characterizing the distribution of node degrees, n ( k ) ∼ k - λ .

Tomography of scale-free networks and shortest path trees

In this paper we model the tomography of scale-free networks by studying the structure of layers around an arbitrary network node. We find, both analytically and empirically, that the distance distribution of all nodes from a specific network node consists of two regimes. The first is characterized by rapid growth, and the second decays exponentially. We also show analytically that the nodes degree distribution at each layer exhibits a power-law tail with an exponential cutoff. We obtain similar empirical results for the layers surrounding the root of shortest path trees cut from such networks, as well as the Internet.

NODE DEGREE DISTRIBUTION IN COMPLEX MICROVASCULAR NETWORKS: A POTENTIAL NEW DIAGNOSTIC TOOL FOR EXTRACELLULAR MATRIX-RELATED DISEASES

Recent research indicates the presence of increased vascular density and irregularity on oral mucosal vascular networks in extracellular matrix (ECM)-related illness or conditions. Here, we estimated the frequency of occurrence of nodes of various degrees (K3, K4 and K5, where Kn designates a node with n connections) in patients with proven or suspected ECM-related conditions and in controls. Subjects with ECM-related conditions exhibited lower K3 and higher K4 frequency than controls (p < 0.0001) in their vascular networks. Inverse statistical correlations between the local fractal dimension and L-Z values and percentage of K3 (Pearson's r values range: -0.91 to -0.81; p values range: 0.0013 to < 0.0001), together with a positive relationship with K4 were observed (r values range: 0.81 to 0.86; p values range: 0.0015 to 0.0003). A positive correlation coefficient between D(1–46) and K5 frequency was also found (r = 0.6334, p = 0.027). K3 ≥ 52% or K4 <28% discriminated ECM patients from controls with 100% sensitivity (true positive cases to true positive + false negative ratio) and specificity (true negative cases to true negative + false positive ratio). These findings suggest that node degree distribution in oral vascular networks could be a helpful new marker of pathological conditions associated with proven or suspected ECM abnormalities.

End-to-end service survivability under attacks on networks

Network survivability is a capability of a networked system to provide its services despite failures or attacks. Attacks, e.g., due to acts of war, being potentially damaging events, were basically considered in the historical definitions of a survivability phenomenon. The meaning of the term: ”network survivability” evolved in the last decade. Recently, attacks replayed the important role again. Their nature, however, including intrusions, probes, denials of service, differs from the old one. Survivability is strongly related to other fields of study. In particular, quality of service depends on network survivability. We investigate these dependencies in scale-free networks. Many networks are scale-free, i.e., their node degree distribution follows the power law. Nodes of the highest degrees, called centers, are highly vulnerable to attacks. Elimination of these nodes seriously degrades the overall performance of network services. In this paper we propose a model, which, based on traffic parameters of a demand, like delay or bit rate, allows to establish the survivable and attack proof end-to-end connections. The key idea of this model is that for the significant traffic, it establishes paths, which omit centers. The important connections become more resistant to attacks. We show that in the best case, obtained for the highest class of service, the number of broken connections is reduced even by factor 3. Example results are compared to those for the standard distance metrics. Our model is applicable to many network architectures and many classes of service.

Functional integration of ecological networks through pathway proliferation

Large-scale structural patterns commonly occur in network models of complex systems including a skewed node degree distribution and small-world topology. These patterns suggest common organizational constraints and similar functional consequences. Here, we investigate a structural pattern termed pathway proliferation. Previous research enumerating pathways that link species determined that as pathway length increases, the number of pathways tends to increase without bound. We hypothesize that this pathway proliferation influences the flow of energy, matter, and information in ecosystems. In this paper, we clarify the pathway proliferation concept, introduce a measure of the node–node proliferation rate, describe factors influencing the rate, and characterize it in 17 large empirical food-webs. During this investigation, we uncovered a modular organization within these systems. Over half of the food-webs were composed of one or more subgroups that were strongly connected internally, but weakly connected to the rest of the system. Further, these modules had distinct proliferation rates. We conclude that pathway proliferation in ecological networks reveals subgroups of species that will be functionally integrated through cyclic indirect effects.

Networks with given two-point correlations: Hidden correlations from degree correlations

This paper orders certain important issues related to both uncorrelated and correlated networks with hidden variables, in which hidden variables correspond to desired node degrees. In particular, we show that networks being uncorrelated at the hidden level are also lacking in correlations between node degrees. The observation supported by the depoissonization idea allows us to extract a distribution of hidden variables from a given node degree distribution. It completes the algorithm for generating uncorrelated networks that was suggested by other authors. In this paper we also carefully analyze the interplay between hidden attributes and node degrees. We show how to extract hidden correlations from degree correlations. Our derivations provide a mathematical background for the algorithm for generating correlated networks that was proposed by Boguñá and Pastor-Satorras.

Topology-based cancer classification and related pathway mining using microarray data

Cancer classification is the critical basis for patient-tailored therapy, while pathway analysis is a promising method to discover the underlying molecular mechanisms related to cancer development by using microarray data. However, linking the molecular classification and pathway analysis with gene network approach has not been discussed yet. In this study, we developed a novel framework based on cancer class-specific gene networks for classification and pathway analysis. This framework involves a novel gene network construction, named ordering network, which exhibits the power-law node-degree distribution as seen in correlation networks. The results obtained from five public cancer datasets showed that the gene networks with ordering relationship are better than those with correlation relationship in terms of accuracy and stability of the classification performance. Furthermore, we integrated the ordering networks, classification information and pathway database to develop the topology-based pathway analysis for identifying cancer class-specific pathways, which might be essential in the biological significance of cancer. Our results suggest that the topology-based classification technology can precisely distinguish cancer subclasses and the topology-based pathway analysis can characterize the correspondent biochemical pathways even if there are subtle, but consistent, changes in gene expression, which may provide new insights into the underlying molecular mechanisms of tumorigenesis.

Network robustness to targeted attacks. The interplay of expansibility and degree distribution

We study the property of certain complex networks of being both sparse and highly connected, which is known as “good expansion” (GE). A network has GE properties if every subset S of nodes (up to 50% of the nodes) has a neighborhood that is larger than some “expansion factor” φ multiplied by the number of nodes in S. Using a graph spectral method we introduce here a new parameter measuring the good expansion character of a network. By means of this parameter we are able to classify 51 real-world complex networks — technological, biological, informational, biological and social — as GENs or non-GENs. Combining GE properties and node degree distribution (DD) we classify these complex networks in four different groups, which have different resilience to intentional attacks against their nodes. The simultaneous existence of GE properties and uniform degree distribution contribute significantly to the robustness in complex networks. These features appear solely in 14% of the 51 real-world networks studied here. At the other extreme we find that ∼40% of all networks are very vulnerable to targeted attacks. They lack GE properties, display skewed DD — exponential or power-law — and their topologies are changed more dramatically by targeted attacks directed at bottlenecks than by the removal of network hubs.

Emergence of scale-free networks from local connectivity and communication trade-offs

We suggest a mechanism of connectivity evolution in networks to account for the emergence of scale-free behavior. The mechanism works on a fixed set of nodes and promotes growth from a minimally connected initial topology by the addition of edges. A new edge is added between two nodes depending on the trade-off between a gain and a cost function of local connectivity and communication properties. We report on simulation results that indicate the appearance of power-law distributions of node degrees for selected parameter combinations.

Examining the potential effects of species aggregation on the network structure of food webs

One of the key measures that have been used to describe the topological properties of complex networks is the "degree distribution", which is a measure that describes the frequency distribution of number of links per node. Food webs are complex ecological networks that describe the trophic relationships among species in a community, and the topological properties of empirical food webs, including degree distributions, have been examined previously. Previously, the "niche model" has been shown to accurately predict degree distributions of empirical food webs, however, the niche model-generated food webs were referenced against empirical food webs that had their species grouped together based on their taxonomic and/or trophic relationships (aggregated food webs). Here, we explore the effects of species aggregation on the ability of the niche model to predict the total- (sum of prey and predator links per node), in- (number of predator links per node), and out- (number of prey links per node) degree distributions of empirical food webs by examining two food webs that can be aggregated at different levels of resolution. The results showed that (1) the cumulative total- and out-degree distributions were consistent with the niche model predictions when the species were aggregated, (2) when the species were disaggregated (i.e., higher resolution), there were mixed conclusions with regards to the niche model's ability to predict total- and out-degree distributions, (3) the model's ability to predict the in-degree distributions of the two food webs was generally inadequate. Although it has been argued that universal functional form based on the niche model could describe the degree distribution patterns of empirical food webs, we believe there are some limitations to the model's ability to accurately predict the structural properties of food webs.

The growth and form of tunnelling networks in ants

Many biological networks grow under strong spatial constraints, where the large-scale structure emerges from the extension, the branching and intersection of growing parts of the network. One example is provided by ant tunnelling networks, which represent the most common nest architecture in ants. Our goal was to understand how these network structures emerge from the tunnel growth dynamics. We used a standardized two-dimensional set-up shaped as a disk and studied the characteristics of tunnel growth in terms of initiation, propagation and termination of new digging sites and found that they can be described with simple probabilistic laws. We show that a model based on these simple laws and for which parameters were measured from the sand disks experiments can account for the emergence of several topological properties that were observed in experimental networks. In particular, the model accurately reproduced an allometric relation between the number of edges and the number of nodes, as well as an invariance of the node degree distribution. The model was then used to make predictions about the resulting networks’ topology when the geometry of the sand substrate was shaped as a square. Experiments aimed at testing the model's predictions showed that the predictions were indeed validated. Both in the model and in the experiments, there was a similar trend for the node degree distribution tail to be steeper in the square sand patch than in the disk sand patch, while other characteristics such as the meshedness (i.e. how densely the network is internally connected) remained constant. Because network growth based on branching/fusion events is widespread in biological systems, this general model might provide useful insights for the study of other systems and, more generally, the evolution of spatial networks in biological systems.

Convergence properties of the degree distribution of some growing network models

In this article we study a class of randomly grown graphs that includes some preferential attachment and uniform attachment models, as well as some evolving graph models that have been discussed previously in the literature. The degree distribution is assumed to form a Markov chain; this gives a particularly simple form for a stochastic recursion of the degree distribution. We show that for this class of models the empirical degree distribution tends almost surely and in norm to the expected degree distribution as the size of the graph grows to infinity and we provide a simple asymptotic expression for the expected degree distribution. Convergence of the empirical degree distribution has consequences for statistical analysis of network data in that it allows the full data to be summarized by the degree distribution of the nodes without losing the ability to obtain consistent estimates of parameters describing the network.

The co-evolutionary dynamics of directed network of spin market agents

The spin market model [S. Bornholdt, Int. J. Mod. Phys. C 12 (2001) 667] is generalized by employing co-evolutionary principles, where strategies of the interacting and competitive traders are represented by local and global couplings between the nodes of dynamic directed stochastic network. The co-evolutionary principles are applied in the frame of Bak–Sneppen self-organized dynamics [P. Bak, K. Sneppen, Phys. Rev. Lett. 71 (1993) 4083] that includes the processes of selection and extinction actuated by the local (node) fitness. The local fitness is related to orientation of spin agent with respect to the instant magnetization. The stationary regime is formed due to the interplay of self-organization and adaptivity effects. The fat tailed distributions of log-price returns are identified numerically. The non-trivial model consequence is the evidence of the long time market memory indicated by the power-law range of the autocorrelation function of volatility with exponent smaller than one. The simulations yield network topology with broad-scale node degree distribution characterized by the range of exponents 1.3 < γ in < 3 coinciding with social networks.

On properties of multicast routing trees

In the last several years we witnessed the proliferation of multimedia applications on the Internet. One of the unavoidable techniques to support this type of communication is multicasting. However, even a decade after its initial proposal, multicast is still not widely deployed. One of the reasons is the lack of a solid business model. If the gain and the cost of multicast could be predicted, network operators might be encouraged to deploy multicast on a larger scale. In this paper we propose analytical expressions that could be used to estimate the gain of network-layer multicast. We show that the theoretical model matches extensive simulation and Internet measurement results remarkably well. Furthermore, we examine the reliability of traceroute data and of traceroutes-based conclusions. We investigate the node degree distributions in the Internet maps obtained from CAIDA and RIPE and we show the divergency of our results with those obtained by other researchers. We further focus on the analysis of multicast trees based on traceroute data. Only few results have been available on the node degree distribution of multicast routing trees which provided contradictory conclusions. Our results seem to indicate that the node degrees follow power laws only for a large number of multicast users. Copyright © 2005 John Wiley & Sons, Ltd.

Synchronous neural activity in scale-free network models versus random network models

Synchronous firing peaks at levels greatly exceeding background activity have recently been reported in neocortical tissue. A small subset of neurons is dominant in a large fraction of the peaks. To investigate whether this striking behavior can emerge from a simple model, we constructed and studied a model neural network that uses a modified Hopfield-type dynamical rule. We find that networks having a power-law ("scale-free") node degree distribution readily generate extremely large synchronous firing peaks dominated by a small subset of nodes, whereas random (Erdös-Rényi) networks do not. This finding suggests that network topology may play an important role in determining the nature and magnitude of synchronous neural activity.

Statistical mechanical load balancer for the web

The maximum entropy principle from statistical mechanics states that a closed system attains an equilibrium distribution that maximizes its entropy. We first show that for graphs with fixed number of edges one can define a stochastic edge dynamic that can serve as an effective thermalization scheme, and hence, the underlying graphs are expected to attain their maximum-entropy states, which turn out to be Erdös-Rényi (ER) random graphs. We next show that (i) a rate-equation-based analysis of node degree distribution does indeed confirm the maximum-entropy principle, and (ii) the edge dynamic can be effectively implemented using short random walks on the underlying graphs, leading to a local algorithm for the generation of ER random graphs. The resulting statistical mechanical system can be adapted to provide a distributed and local (i.e., without any centralized monitoring) mechanism for load balancing, which can have a significant impact in increasing the efficiency and utilization of both the Internet (e.g., efficient web mirroring), and large-scale computing infrastructure (e.g., cluster and grid computing).

Preferential attachment with information filtering—node degree probability distribution properties

A network growth mechanism based on a two-step preferential rule is investigated as a model of network growth in which no global knowledge of the network is required. In the first filtering step a subset of fixed size m of existing nodes is randomly chosen. In the second step the preferential rule of attachment is applied to the chosen subset. The characteristics of thus formed networks are explored using two approaches: computer simulations of network growth and a theoretical description based on a master equation. The results of the two approaches are in excellent agreement. Special emphasis is put on the investigation of the node degree probability distribution. It is found that the tail of the distribution has the exponential form given by exp ( - k / m ) . Implications of the node degree distribution with such tail characteristics are briefly discussed.