Degree Exponent Research Articles

Sea Bed Logging Applications: Predicting Hydrocarbon Depth Using Mathematical Equations

The aim of this research is to develop 1-D mathematical equation that relates Normalized mean Square Error (NMSE) to depth of hydrocarbon (HC) for data processed using cubic spline interpolation in sea bed logging (SBL) application. Simulations were conducted using CST software that replicated real SBL environment to generate synthetic data. Frequency of 0.25Hz was used and sediment thicknesses were varied from 1000m to 3000m and incremented at every 250m. Data collected were interpolated using cubic spline and NMSE were calculated between original and interpolated data. Mathematical equation that relates NMSE (y) and depth of hydrocarbon (x) was constructed in terms of linear, exponential, logarithm and polynomial degree 2, 3, 4 and R2 were calculated for each equation. Mathematical equation using exponential function was adopted because it has reasonably high R2 and can be inversed easily. Average percentage error was calculated between calculated and measured data to achieve 5 % or lower values. If achievable, this model was accepted as forward mathematical model for SBL. Otherwise, the coefficients were adjusted and the processes repeated until minimum average error was achieved.

Discovery of Power-Law Growth in the Self-Renewal of Heterogeneous Glioma Stem Cell Populations.

BackgroundAccumulating evidence indicates that cancer stem cells (CSCs) drive tumorigenesis. This suggests that CSCs should make ideal therapeutic targets. However, because CSC populations in tumors appear heterogeneous, it remains unclear how CSCs might be effectively targeted. To investigate the mechanisms by which CSC populations maintain heterogeneity during self-renewal, we established a glioma sphere (GS) forming model, to generate a population in which glioma stem cells (GSCs) become enriched. We hypothesized, based on the clonal evolution concept, that with each passage in culture, heterogeneous clonal sublines of GSs are generated that progressively show increased proliferative ability.Methodology/Principal FindingsTo test this hypothesis, we determined whether, with each passage, glioma neurosphere culture generated from four different glioma cell lines become progressively proliferative (i.e., enriched in large spheres). Rather than monitoring self-renewal, we measured heterogeneity based on neurosphere clone sizes (#cells/clone). Log-log plots of distributions of clone sizes yielded a good fit (r>0.90) to a straight line (log(% total clones) = k*log(#cells/clone)) indicating that the system follows a power-law (y = xk) with a specific degree exponent (k = −1.42). Repeated passaging of the total GS population showed that the same power-law was maintained over six passages (CV = −1.01 to −1.17). Surprisingly, passage of either isolated small or large subclones generated fully heterogeneous populations that retained the original power-law-dependent heterogeneity. The anti-GSC agent Temozolomide, which is well known as a standard therapy for glioblastoma multiforme (GBM), suppressed the self-renewal of clones, but it never disrupted the power-law behavior of a GS population.Conclusions/SignificanceAlthough the data above did not support the stated hypothesis, they did strongly suggest a novel mechanism that underlies CSC heterogeneity. They indicate that power-law growth governs the self-renewal of heterogeneous glioma stem cell populations. That the data always fit a power-law suggests that: (i) clone sizes follow continuous, non-random, and scale-free hierarchy; (ii) precise biologic rules that reflect self-organizing emergent behaviors govern the generation of neurospheres. That the power-law behavior and the original GS heterogeneity are maintained over multiple passages indicates that these rules are invariant. These self-organizing mechanisms very likely underlie tumor heterogeneity during tumor growth. Discovery of this power-law behavior provides a mechanism that could be targeted in the development of new, more effective, anti-cancer agents.

Complex networks from space-filling bearings.

Two-dimensional space-filling bearings are dense packings of disks that can rotate without slip. We consider the entire first family of bearings for loops of four disks and propose a hierarchical construction of their contact network. We provide analytic expressions for the clustering coefficient and degree distribution, revealing bipartite scale-free behavior with a tunable degree exponent depending on the bearing parameters. We also analyze their average shortest path and percolation properties.

Evolving Scale-Free Networks by Poisson Process: Modeling and Degree Distribution.

Since the great mathematician Leonhard Euler initiated the study of graph theory, the network has been one of the most significant research subject in multidisciplinary. In recent years, the proposition of the small-world and scale-free properties of complex networks in statistical physics made the network science intriguing again for many researchers. One of the challenges of the network science is to propose rational models for complex networks. In this paper, in order to reveal the influence of the vertex generating mechanism of complex networks, we propose three novel models based on the homogeneous Poisson, nonhomogeneous Poisson and birth death process, respectively, which can be regarded as typical scale-free networks and utilized to simulate practical networks. The degree distribution and exponent are analyzed and explained in mathematics by different approaches. In the simulation, we display the modeling process, the degree distribution of empirical data by statistical methods, and reliability of proposed networks, results show our models follow the features of typical complex networks. Finally, some future challenges for complex systems are discussed.

Growing optimal scale-free networks via likelihood.

Preferential attachment, by which new nodes attach to existing nodes with probability proportional to the existing nodes' degree, has become the standard growth model for scale-free networks, where the asymptotic probability of a node having degree k is proportional to k^{-γ}. However, the motivation for this model is entirely ad hoc. We use exact likelihood arguments and show that the optimal way to build a scale-free network is to attach most new links to nodes of low degree. Curiously, this leads to a scale-free network with a single dominant hub: a starlike structure we call a superstar network. Asymptotically, the optimal strategy is to attach each new node to one of the nodes of degree k with probability proportional to 1/N+ζ(γ)(k+1)(γ) (in a N node network): a stronger bias toward high degree nodes than exhibited by standard preferential attachment. Our algorithm generates optimally scale-free networks (the superstar networks) as well as randomly sampling the space of all scale-free networks with a given degree exponent γ. We generate viable realization with finite N for 1≪γ<2 as well as γ>2. We observe an apparently discontinuous transition at γ≈2 between so-called superstar networks and more treelike realizations. Gradually increasing γ further leads to reemergence of a superstar hub. To quantify these structural features, we derive a new analytic expression for the expected degree exponent of a pure preferential attachment process and introduce alternative measures of network entropy. Our approach is generic and can also be applied to an arbitrary degree distribution.

A Distributed and Local-World Topology Evolution Model for Wireless Sensor Network

In this paper, a new evolution model based on complex network among the cluster heads in wireless sensor network is proposed. The evolution model considered distributed and local-world mechanism during the evolving process. The theoretical analysis of this model exhibits a power-law degree distribution with mean-field theory, which provides good fault-tolerance. The degree exponent is not a fixed number, which changes with the distribution of the cluster heads and the energy as well as the communication radius. Furthermore, the degree exponent can lead to an upper limit -2 when the distribution of the cluster heads and the energy are both uniform distribution. Analysis and simulation show that the network exhibits well robustness and a power-law degree distribution.

A thermostatistical approach to scale-free networks

We describe an ensemble of growing scale-free networks in an equilibrium framework, providing insight into why the exponent of empirical scale-free networks in nature is typically robust. In an analogy to thermostatistics, to describe the canonical and microcanonical ensembles, we introduce a functional, whose maximum corresponds to a scale-free configuration. We then identify the equivalents to energy, Zeroth-law, entropy and heat capacity for scale-free networks. Discussing the merging of scale-free networks, we also establish an exact relation to predict their final "equilibrium" degree exponent. All analytic results are complemented with Monte Carlo simulations. Our approach illustrates the possibility to apply the tools of equilibrium statistical physics to study the properties of growing networks, and it also supports the recent arguments on the complementarity between equilibrium and nonequilibrium systems.

Quantum phase transition of the transverse-field quantum Ising model on scale-free networks.

I investigate the quantum phase transition of the transverse-field quantum Ising model in which nearest neighbors are defined according to the connectivity of scale-free networks. Using a continuous-time quantum Monte Carlo simulation method and the finite-size scaling analysis, I identify the quantum critical point and study its scaling characteristics. For the degree exponent λ=6, I obtain results that are consistent with the mean-field theory. For λ=4.5 and 4, however, the results suggest that the quantum critical point belongs to a non-mean-field universality class. Further simulations indicate that the quantum critical point remains mean-field-like if λ>5, but it continuously deviates from the mean-field theory as λ becomes smaller.

The multiplicative coalescent, inhomogeneous continuum random trees, and new universality classes for critical random graphs

One major open conjecture in the area of critical random graphs, formulated by statistical physicists, and supported by a large amount of numerical evidence over the last decade [23, 24, 28, 63] is as follows: for a wide array of random graph models with degree exponent $\tau\in (3,4)$, distances between typical points both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like $n^{(\tau-3)/(\tau-1)}$. In this paper we study the metric space structure of maximal components of the multiplicative coalescent, in the regime where the sizes converge to excursions of L\'evy processes without replacement [10], yielding a completely new class of limiting random metric spaces. A by-product of the analysis yields the continuum scaling limit of one fundamental class of random graph models with degree exponent $\tau\in (3,4)$ where edges are rescaled by $n^{-(\tau-3)/(\tau-1)}$ yielding the first rigorous proof of the above conjecture. The limits in this case are compact tree-like random fractals with finite fractal dimensions and with a dense collection of hubs (infinite degree vertices) a finite number of which are identified with leaves to form shortcuts. It is generally believed that dynamic versions of a number of fundamental random graph models, as one moves from the barely subcritical to the critical regime can be approximated by the multiplicative coalescent. In work in progress, the general theory developed in this paper is used to prove analogous limit results for other random graph models with degree exponent $\tau\in (3,4)$. Our proof makes crucial use of inhomogeneous continuum random trees (ICRT), which have previously arisen in the study of the entrance boundary of the additive coalescent. We show that tilted versions of the same objects using the associated mass measure, describe connectivity properties of the multiplicative coalescent. Since convergence of height processes of corresponding approximating p-trees is not known, we use general methodology in [14] and develop novel techniques relying on first showing convergence in the Gromov-weak topology and then extending this to Gromov-Hausdorff-Prokhorov convergence by proving a global lower mass-bound. Keywords: Multiplicative coalescent, p-trees, inhomogeneous continuum random trees, critical random graphs, Gromov-Hausdorff distance, Gromov-weak topology

Power-Law Property of High Clustering Network Induced by Deactivation Mechanism

We study the high clustering model of K. Klemm and V.M. Eguiliuz [1]. The basic idea is the growing networks based on a finite memory of the nodes. Our main contributions are as follows: based on the high clustering model, we investigate analytically that it does generate a power-law degree distribution and give the exact solution from the perspective of Markov chain. And the degree exponent varying is depend on the parameters α and m.

Growing networks with superjoiners.

We study the Krapivsky-Redner (KR) network growth model, but where new nodes can connect to any number of existing nodes, m, picked from a power-law distribution p(m)∼m^{-α}. Each of the m new connections is still carried out as in the KR model with probability redirection r (corresponding to degree exponent γ_{KR}=1+1/r in the original KR model). The possibility to connect to any number of nodes resembles a more realistic type of growth in several settings, such as social networks, routers networks, and networks of citations. Here we focus on the in-, out-, and total-degree distributions and on the potential tension between the degree exponent α, characterizing new connections (outgoing links), and the degree exponent γ_{KR}(r) dictated by the redirection mechanism.

A Network Theory View on the Thalamo-Cortical Loop

We used a network theory approach, based on the dynamic core hypothesis (DCH), to study the thalamo-cortical loop (TCL) and its subsets regarding their role in consciousness. We used the Collation of Connectivity Data on the Macaque Brain (CoCoMac) and calculated the degree distributions, transmission coefficients, connection density, clustering coefficients, path lengths, and modularity. Our results showed that the TCL and cortex exhibit exponential degree distributions, and the ratio of efferent/afferent connections in the thalamus is smaller than 1.0 This may support the notion that the connections received by the thalamus from the cortex play a key role in improving information processing in the conscious states. The average values of transmission coefficients for the cortex and TCL were found to be equal to 1.49 and 1.28, respectively. This indicates that: (i) the cortex is a system that mainly transmits information outward rather than receives it; (ii) the TCL is a cooperative system that performs this in a give-and-take manner; (iii) connections of the cortex are denser than those in the TCL, showing that the cortex might be advantageous for processing of complicated information during consciousness; (iv) both the TCL and cortex are small-world systems; (v) the scaled value of the characteristic path length in the TCL is smaller than that in the cortex, which implies a higher speed potential for information processing in the TCL than in the cortex; (vi) the scaled value of the clustering coefficient is nearly the same in the cortex and TCL, and (vii) the number of modules is 5 in the cortex and 6 in the TCL.

Localization transition, Lifschitz tails, and rare-region effects in network models.

Effects of heterogeneity in the suspected-infected-susceptible model on networks are investigated using quenched mean-field theory. The emergence of localization is described by the distributions of the inverse participation ratio and compared with the rare-region effects appearing in simulations and in the Lifschitz tails. The latter, in the linear approximation, is related to the spectral density of the Laplacian matrix and to the time dependent order parameter. I show that these approximations indicate correctly Griffiths phases both on regular one-dimensional lattices and on small-world networks exhibiting purely topological disorder. I discuss the localization transition that occurs on scale-free networks at γ=3 degree exponent.

Fractal and complex network analyses of protein molecular dynamics

Based on protein molecular dynamics, we investigate the fractal properties of energy, pressure and volume time series using the multifractal detrended fluctuation analysis (MF-DFA) and the topological and fractal properties of their converted horizontal visibility graphs (HVGs). The energy parameters of protein dynamics we considered are bonded potential, angle potential, dihedral potential, improper potential, kinetic energy, Van der Waals potential, electrostatic potential, total energy and potential energy. The shape of the h(q) curves from MF-DFA indicates that these time series are multifractal. The numerical values of the exponent h(2) of MF-DFA show that the series of total energy and potential energy are non-stationary and anti-persistent; the other time series are stationary and persistent apart from series of pressure (with H≈0.5 indicating the absence of long-range correlation). The degree distributions of their converted HVGs show that these networks are exponential. The results of fractal analysis show that fractality exists in these converted HVGs. For each energy, pressure or volume parameter, it is found that the values of h(2) of MF-DFA on the time series, exponent λ of the exponential degree distribution and fractal dimension dB of their converted HVGs do not change much for different proteins (indicating some universality). We also found that after taking average over all proteins, there is a linear relationship between 〈h(2)〉 (from MF-DFA on time series) and 〈dB〉 of the converted HVGs for different energy, pressure and volume.

Local rewiring rules for evolving complex networks

The effects of link rewiring are considered for the class of directed networks where each node has the same fixed out-degree. We model a network generated by three mechanisms that are present in various networked systems; growth, global rewiring and local rewiring. During a rewiring phase a node is randomly selected, one of its out-going edges is detached from its destination then re-attached to the network in one of two possible ways; either globally to a randomly selected node, or locally to a descendant of a descendant of the originally selected node. Although the probability of attachment to a node increases with its connectivity, the probability of detachment also increases, the result is an exponential degree distribution with a small number of outlying nodes that have extremely large degree. We explain these outliers by identifying the circumstances for which a set of nodes can grow to very high degree.

Degree-Degree Dependencies in Random Graphs with Heavy-Tailed Degrees

We investigate degree-degree correlations for scale-free graph sequences. The main conclusion of this paper is that the assortativity coefficient is not the appropriate way to describe degree-dependences in scale-free random graphs. Indeed, we study the infinite volume limit of the assortativity coefficient, and show that this limit is always non-negative when the degrees have finite first but infinite third moment, i.e., when the degree exponent $\gamma + 1$ of the density satisfies $\gamma \in (1,3)$. More generally, our results show that the correlation coefficient is inappropriate to describe dependencies between random variables having infinite variance. We start with a simple model of the sample correlation of random variables $X$ and $Y$, which are linear combinations with non-negative coefficients of the same infinite variance random variables. In this case, the correlation coefficient of $X$ and $Y$ is not defined, and the sample covariance converges to a proper random variable with support that is a subinterval of $(-1,1)$. Further, for any joint distribution $(X,Y)$ with equal marginals being non-negative power-law distributions with infinite variance (as in the case of degree-degree correlations), we show that the limit is non-negative. We next adapt these results to the assortativity in networks as described by the degree-degree correlation coefficient, and show that it is non-negative in the large graph limit when the degree distribution has an infinite third moment. We illustrate these results with several examples where the assortativity behaves in a non-sensible way. We further discuss alternatives for describing assortativity in networks based on rank correlations that are appropriate for infinite variance variables. We support these mathematical results by simulations.

Intermolecular β-Strand Networks Avoid Hub Residues and Favor Low Interconnectedness: A Potential Protection Mechanism against Chain Dissociation upon Mutation

Altogether few protein oligomers undergo a conformational transition to a state that impairs their function and leads to diseases. But when it happens, the consequences are not harmless and the so-called conformational diseases pose serious public health problems. Notorious examples are the Alzheimer's disease and some cancers associated with a conformational change of the amyloid precursor protein (APP) and of the p53 tumor suppressor, respectively. The transition is linked with the propensity of β-strands to aggregate into amyloid fibers. Nevertheless, a huge number of protein oligomers associate chains via β-strand interactions (intermolecular β-strand interface) without ever evolving into fibers. We analyzed the layout of 1048 intermolecular β-strand interfaces looking for features that could provide the β-strands resistance to conformational transitions. The interfaces were reconstructed as networks with the residues as the nodes and the interactions between residues as the links. The networks followed an exponential decay degree distribution, implying an absence of hubs and nodes with few links. Such layout provides robustness to changes. Few links per nodes do not restrict the choices of amino acids capable of making an interface and maintain high sequence plasticity. Few links reduce the “bonding” cost of making an interface. Finally, few links moderate the vulnerability to amino acid mutation because it entails limited communication between the nodes. This confines the effects of a mutation to few residues instead of propagating them to many residues via hubs. We propose that intermolecular β-strand interfaces are organized in networks that tolerate amino acid mutation to avoid chain dissociation, the first step towards fiber formation. This is tested by looking at the intermolecular β-strand network of the p53 tetramer.

Topological properties and fractal analysis of a recurrence network constructed from fractional Brownian motions.

Many studies have shown that we can gain additional information on time series by investigating their accompanying complex networks. In this work, we investigate the fundamental topological and fractal properties of recurrence networks constructed from fractional Brownian motions (FBMs). First, our results indicate that the constructed recurrence networks have exponential degree distributions; the average degree exponent 〈λ〉 increases first and then decreases with the increase of Hurst index H of the associated FBMs; the relationship between H and 〈λ〉 can be represented by a cubic polynomial function. We next focus on the motif rank distribution of recurrence networks, so that we can better understand networks at the local structure level. We find the interesting superfamily phenomenon, i.e., the recurrence networks with the same motif rank pattern being grouped into two superfamilies. Last, we numerically analyze the fractal and multifractal properties of recurrence networks. We find that the average fractal dimension 〈dB〉 of recurrence networks decreases with the Hurst index H of the associated FBMs, and their dependence approximately satisfies the linear formula 〈dB〉≈2-H, which means that the fractal dimension of the associated recurrence network is close to that of the graph of the FBM. Moreover, our numerical results of multifractal analysis show that the multifractality exists in these recurrence networks, and the multifractality of these networks becomes stronger at first and then weaker when the Hurst index of the associated time series becomes larger from 0.4 to 0.95. In particular, the recurrence network with the Hurst index H=0.5 possesses the strongest multifractality. In addition, the dependence relationships of the average information dimension 〈D(1)〉 and the average correlation dimension 〈D(2)〉 on the Hurst index H can also be fitted well with linear functions. Our results strongly suggest that the recurrence network inherits the basic characteristic and the fractal nature of the associated FBM series.

Topology and vulnerability of the Iranian power grid

In this paper we investigated the structural properties of the ultra high voltage power transmission network of Iran. We modeled the power grid as a network with 105 nodes and 142 connection links. We found that the Iranian power grid displays a relatively moderate clustering coefficient–much larger than that of corresponding random networks–and small characteristics path length comparable to that of corresponding random networks; i.e. the power grid is a small-world network with exponential degree distribution. Global efficiency was considered as an indicator of grid’s performance and the influence of random and intentional nodal failures on the efficiency was investigated. We also studied the influence of cascaded failures on the largest connected component of the network. The power grid was vulnerable against cascaded failures, which should be considered serious in redesigning the network topology.

On relations among the entropic chaos degree, the Kolmogorov-Sinai entropy and the Lyapunov exponent

There exist several criteria to describe the chaotic behaviour of a dynamical system. In this paper, we discuss the relations among three criteria: Entropic Chaos Degree, Kolmogorov-Shinai entropy, and Lyapunov exponent. Moreover, the problems of their computation are discussed.