Degree Distribution Research Articles

Cover and hitting times of hyperbolic random graphs

AbstractWe study random walks on the giant component of Hyperbolic Random Graphs (HRGs), in the regime when the degree distribution obeys a power law with exponent in the range . In particular, we first focus on the expected time for a random walk to hit a given vertex or visit, that is, cover, all vertices. We show that, a.a.s. (with respect to the HRG), and up to multiplicative constants: the cover time is , the maximum hitting time is , and the average hitting time is . We then determine the expected time to commute between two given vertices a.a.s., up to a small factor polylogarithmic in , and under some mild hypothesis on the pair of vertices involved. Our results are proved by controlling effective resistances using the energy dissipated by carefully designed network flows associated to a tiling of the hyperbolic plane, on which we overlay a forest‐like structure.

Integrate-and-fire model of disease transmission.

We create an epidemiological susceptible-infected-susceptible model of disease transmission using integrate-and-fire nodes on a network, allowing memory of previous interactions and infections. Agents in the network sum infectious matter from their nearest neighbors at every time step, until they exceed their infection threshold, at which point they "fire" and become infected for as long as the recovery time. The model has memory of previous interactions by tracking the amount of infectious matter carried by agents as well as just binary infected or susceptible states, and the model has memory of previous infections by modeling immunity as increasing the infection threshold after recovery. Creating a simulation of the model on networks with a power-law degree distribution and homogeneous agent parameters, we find a single strain version of the model matches well with the England COVID-19 case data, with a root-mean-squared error of 0.014%. A simulation of a multistrain version of the model (where there is cross-strain immunity) matches well with the influenza strain A and strain B case numbers in Canada, with a root-mean-squared error of 0.002% and 0.0012%, respectively, though due to the coupling in the model, both strains peak in phase. Since the dynamics of the model successfully capture real-life transmission dynamics, we test interventions to study their effect on case numbers, with both quarantining and social gathering restrictions lowering the peak. Since the model has memory, the stricter the intervention, the higher the secondary peak when the restriction is removed, showing that interventions change only the shape of the curves and not the overall number infected in the population.

Degree distributions in networks: Beyond the power law

AbstractThe power law is useful in describing count phenomena such as network degrees and word frequencies. With a single parameter, it captures the main feature that the frequencies are linear on the log‐log scale. Nevertheless, there have been criticisms of the power law, for example, that a threshold needs to be preselected without its uncertainty quantified, that the power law is simply inadequate, and that subsequent hypothesis tests are required to determine whether the data could have come from the power law. We propose a modeling framework that combines two different generalizations of the power law, namely the generalized Pareto distribution and the Zipf‐polylog distribution, to resolve these issues. The proposed mixture distributions are shown to fit the data well and quantify the threshold uncertainty in a natural way. A model selection step embedded in the Bayesian inference algorithm further answers the question whether the power law is adequate.

Three-dimensional shape and connectivity of physical networks

Data describing the three-dimensional structure of physical networks is increasingly available, leading to a surge of interest in network science to explore the relationship between the shape and connectivity of physical networks. We contribute to this effort by standardizing and analyzing 15 data sets from different domains. Each network is made of tube-like objects bound together at junction points, which we treat as nodes, with the connections between them considered as links. We divide these networks into three categories: lattice-like networks, trees, and linked trees. The degree distribution of these physical networks is bounded, with most nodes having degrees one or three. Characterizing the physical properties of links, we show that links have an elongated shape and tend to follow a nearly straight trajectory, while a small fraction of links follow a winding path. These typical node and link properties must be reflected by physical network models. We also measure how confined a link is in space by comparing its trajectory to a randomized null model, showing that links that are central in the abstract network tend to be physically confined by their neighbors. The fact that the shape and connectivity of the physical networks are intertwined highlights that their three-dimensional layout must be taken into account to understand the evolution and function of physical networks.

Gender Distribution of Coronary Artery Calcium Score and Degree of Stenosis Assessed by Computed Tomography Angiography in Iraqi Patients with Chest Pain

Gender Distribution of Coronary Artery Calcium Score and Degree of Stenosis Assessed by Computed Tomography Angiography in Iraqi Patients with Chest Pain

1H pulsed NMR measurement of reactivity of silane coupling agents for conservation of cultural heritage sites

AbstractThe network structure and reactivity of suitable restoration agent consisting of silane coupling agents (SCAs) of the brittle walls of a cultural heritage site located underground in a desert area in Egypt were investigated using 1H pulsed nuclear magnetic resonance (pulse NMR). It should possess both high coagulation strength and ductility to ensure that it can withstand an earthquake; it should also contain ethoxy rather than methoxy groups, which generate toxic methanol. The solidification rate for the SCA or oligomer with ethoxy groups was lower than those with methoxy groups; however, the final polycondensates showed the same relaxation time (hardness). The difference spectrum was developed newly to analyze the network structure combined higher strength and ductility. The distribution of crosslinking degree of such polycondensates was narrow, and the amount of lower crosslinking degree was small. On the other hand, the distribution was wide in the polycondensates with ductility but lower strength. From the compression tests of model sand solidified with the restoration agent for both methoxy and ethoxy types showed the superior properties combined higher strength and ductility. The pulse NMR experiment was useful for estimating the solidification rate and the network structure.

Accuracy of discrete- and continuous-time mean-field theories for epidemic processes on complex networks.

Discrete- and continuous-time approaches are frequently used to model the role of heterogeneity on dynamical interacting agents on the top of complex networks. While, on the one hand, one does not expect drastic differences between these approaches, and the choice is usually based on one's expertise or methodological convenience, on the other hand, a detailed analysis of the differences is necessary to guide the proper choice of one or another approach. We tackle this problem by investigating both discrete- and continuous-time mean-field theories for the susceptible-infected-susceptible (SIS) epidemic model on random networks with power-law degree distributions. We compare the discrete epidemic link equations(ELE) and continuous pair quenched mean-field (PQMF) theories with the corresponding stochastic simulations, both theories that reckon pairwise interactions explicitly. We show that ELE converges to the PQMF theory when the time step goes to zero. We performed an epidemic localization analysis considering the inverse participation ratio (IPR). Both theories present the same localization dependence on network degree exponent γ: for γ<5/2 the epidemics are localized on the maximum k-core of networks with a vanishing IPR in the infinite-size limit while, for γ>5/2, the localization happens on hubs that do not form a densely connected set and leads to a finite value of the IPR. However, the IPR and epidemic threshold of ELE depend on the time-step discretization such that a larger time step leads to more localized epidemics. A remarkable difference between discrete- and continuous-time approaches is revealed in the epidemic prevalence near the epidemic threshold, in which the discrete-time stochastic simulations indicate a mean-field critical exponent θ=1 instead of the value θ=1/(3-γ) obtained rigorously and verified numerically for the continuous-time SIS on the same networks.

Trade Network Resilience of the Global Photovoltaic Industry Chain

Based on the trade data of the global photovoltaic (PV) industry chain from 2005 to 2021, this paper constructs a global PV industry chain trade network model and analyzes its static resilience and dynamic resilience characteristics, reaching the following conclusions. The static resilience of the trade network is analyzed through aggregation, transmission, hierarchy and matching. The upstream polysilicon trade network is relatively loose, while the midstream and downstream trade networks appear to aggregate significantly. The average path length is short and the average clustering coefficient is high, indicating that the global PV industry chain trade network has high transmission efficiency and small-world characteristics. The global PV industry chain trade network is hierarchical. The degree distribution slope of the upstream polysilicon trade network is the largest and has a high level of hierarchy. Meanwhile, the global PV industry chain trade network is a heterogeneous network, which strengthens the connection between the core countries and edge countries, making the global PV industry chain trade network more resilient. Regarding the dynamic resilience of the trade network, the deterministic disturbance strategies have a greater impact on network performance than the random disturbance strategies, indicating that targeted impacts exert greater impact on the resilience of the global PV industry chain trade network. For the deterministic disturbance strategy, the interference based on intermediate centrality has a higher impact on the overall resilience of the network than the interference based on degree centrality, thus the influence of the country with strong trade radiating ability on the network elasticity is greater than that of the country with strong transit ability.

Generating synthetic signaling networks for in silico modeling studies

Predictive models of signaling pathways have proven to be difficult to develop. Traditional approaches to developing mechanistic models rely on collecting experimental data and fitting a single model to that data. This approach works for simple systems but has proven unreliable for complex systems such as biological signaling networks. Thus, there is a need to develop new approaches to create predictive mechanistic models of complex systems. To meet this need, we developed a method for generating artificial signaling networks that were reasonably realistic and thus could be treated as ground truth models. These synthetic models could then be used to generate synthetic data for developing and testing algorithms designed to recover the underlying network topology and associated parameters. We defined the reaction degree and reaction distance to measure the topology of reaction networks, especially to consider enzymes. To determine whether our generated signaling networks displayed meaningful behavior, we compared them with signaling networks from the BioModels Database. This comparison indicated that our generated signaling networks had high topological similarities with BioModels signaling networks with respect to the reaction degree and distance distributions. In addition, our synthetic signaling networks had similar behavioral dynamics with respect to both steady states and oscillations, suggesting that our method generated synthetic signaling networks comparable with BioModels and thus could be useful for building network evaluation tools.

Investigating stronger tolerant network against cascading failures in focusing on changing degree distributions.

Many real-world networks with Scale-Free structure are significantly vulnerable against both intentional attacks and catastrophic cascading failures. On the other hand, it has been shown that networks with narrower degree distributions have strong robustness of connectivity by enhancing loops. This paper numerically reveals that such networks are also tolerant against cascading failures. Our findings will be useful in designing stronger tolerant network infrastructures.

Analyzing the intrastate and interstate swine movement network in the United States

Identifying and restricting animal movements is a common approach used to mitigate the spread of diseases between premises in livestock systems. Therefore, it is essential to uncover between-premises movement dynamics, including shipment distances and network-based control strategies. Here, we analyzed three years of between-premises pig movements, which include 197,022 unique animal shipments, 3973 premises, and 391,625,374 pigs shipped across 20 U.S. states. We constructed unweighted, directed, temporal networks at 180-day intervals to calculate premises-to-premises movement distances, the size of connected components, network loyalty, and degree distributions, and, based on the out-going contact chains, identified network-based control actions. Our results show that the median distance between premises pig movements was 74.37 km, with median intrastate and interstate movements of 52.71 km and 328.76 km, respectively. On average, 2842 premises were connected via 6705 edges, resulting in a weak giant connected component that included 91 % of the premises. The premises-level network exhibited loyalty, with a median of 0.65 (IQR: 0.45 – 0.77). Results highlight the effectiveness of node targeting to reduce the risk of disease spread; we demonstrated that targeting 25 % of farms with the highest degree or betweenness limited spread to 1.23 % and 1.7 % of premises, respectively. While there is no complete shipment data for the entire U.S., our multi-state movement analysis demonstrated the value and the needs of such data for enhancing the design and implementation of proactive­ disease control tactics.

Node-Level Community Detection within Edge Exchangeable Models for Interaction Processes

Scientists are increasingly interested in discovering community structure from modern relational data arising on large-scale social networks. While many methods have been proposed for learning community structure, few account for the fact that these modern networks arise from processes of interactions in the population. We introduce block edge exchangeable models (BEEM) for the study of interaction networks with latent node-level community structure. The block vertex components model (B-VCM) is derived as a canonical example. Several theoretical and practical advantages over traditional vertex-centric approaches are highlighted. In particular, BEEMs allow for sparse degree structure and power-law degree distributions within communities. Our theoretical analysis bounds the misspecification rate of block assignments while supporting simulations show the properties of the network can be recovered. A computationally tractable Gibbs algorithm is derived. We demonstrate the proposed model using post-comment interaction data from Talklife, a large-scale online peer-to-peer support network, and contrast the learned communities from those using standard algorithms including degree-corrected stochastic block models, popularity-adjusted block models, and weighted stochastic block models. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

A Note on the Distribution of the Extreme Degrees of a Random Graph via the Stein-Chen Method

We offer an alternative proof, using the Stein-Chen method, of Bollobás’ theorem concerning the distribution of the extreme degrees of a random graph. Our proof also provides a rate of convergence of the extreme degree to its asymptotic distribution. The same method also applies in a more general setting where the probability of every pair of vertices being connected by edges depends on the number of vertices.

GeneSPIDER2: large scale GRN simulation and benchmarking with perturbed single-cell data.

Single-cell data is increasingly used for gene regulatory network (GRN) inference, and benchmarks for this have been developed based on simulated data. However, existing single-cell simulators cannot model the effects of gene perturbations. A further challenge lies in generating large-scale GRNs that often struggle with computational and stability issues. We present GeneSPIDER2, an update of the GeneSPIDER MATLAB toolbox for GRN benchmarking, inference, and analysis. Several software modules have improved capabilities and performance, and new functionalities have been added. A major improvement is the ability to generate large GRNs with biologically realistic topological properties in terms of scale-free degree distribution and modularity. Another major addition is a simulation of single-cell data, which is becoming increasingly popular as input for GRN inference. Specifically, we introduced the unique feature to generate single-cell data based on genetic perturbations. Finally, the simulated single-cell data was compared to real single-cell Perturb-seq data from two cell lines, showing that the synthetic and real data exhibit similar properties.

Mesoscale modeling of random chain scission in polyethylene melts

Polyolefins account for more than half of global primary polymer production, however only a small fraction of these polymers are currently being recycled. Fragmentation of polymer chains into shorter chains with a targeted molecular weight distribution with the goal of reusing these fragments in subsequent chemical synthesis can potentially introduce an alternative approach to polyolefins recycling. Herein we develop a mesoscale framework to model degradation of polyethylene melts at a range of high temperatures. We use the dissipative particle dynamics approach with modified segmental repulsive potential to model the process of random scission in melts of linear polymer chains. We characterize the fragmentation process by tracking the time evolution of the distribution of degrees of polymerization of chain fragments. Specifically, we track the weight average and the number average degrees of polymerization and dispersity of polymer fragments as a function of the fraction of bonds broken. Furthermore, we track the number fraction distribution and the weight fraction distribution of polymer fragments with various degrees of polymerization as functions of the fraction of bonds broken for a range of high temperatures. Our results allow one to quantify to what extent the distribution of polymer chain fragments during random scission can be captured by the respective analytical distributions for the range of conversions considered. Understanding the thermal degradation of polyolefins on the mesoscale can result in the development of alternative strategies for recycling a range of thermoplastics.

MDDBranchNet: A Deep Learning Model for Detecting Major Depressive Disorder Using ECG Signal.

Major depressive disorder (MDD) is a chronic mental illness which affects people's well-being and is often detected at a later stage of depression with a likelihood of suicidal ideation. Early detection of MDD is thus necessary to reduce the impact, however, it requires monitoring vitals in daily living conditions. EEG is generally multi-channel and due to difficulty in signal acquisition, it is unsuitable for home-based monitoring, whereas, wearable sensors can collect single-channel ECG. Classical machine-learning based MDD detection studies commonly use various heart rate variability features. Feature generation, which requires domain knowledge, is often challenging, and requires computation power, often unsuitable for real time processing, MDDBranchNet is a proposed parallel-branch deep learning model for MDD binary classification from a single channel ECG which uses additional ECG-derived signals such as R-R signal and degree distribution time series of horizontal visibility graph. The use of derived branches was able to increase the model's accuracy by around 7%. An optimal 20-second overlapped segmentation of ECG recording was found to be beneficial with a 70% prediction threshold for maximum MDD detection with a minimum false positive rate. The proposed model evaluated MDD prediction from signal excerpts, irrespective of location (first, middle or last one-third of the recording), instead of considering the entire ECG signal with minimal performance variation stressing the idea that MDD phenomena are likely to manifest uniformly throughout the recording.

Emergent scale-free networks.

Many complex systems-from the Internet to social, biological, and communication networks-are thought to exhibit scale-free structure. However, prevailing explanations require that networks grow over time, an assumption that fails in some real-world settings. Here, we explain how scale-free structure can emerge without growth through network self-organization. Beginning with an arbitrary network, we allow connections to detach from random nodes and then reconnect under a mixture of preferential and random attachment. While the numbers of nodes and edges remain fixed, the degree distribution evolves toward a power-law with an exponent that depends only on the proportion p of preferential (rather than random) attachment. Applying our model to several real networks, we infer p directly from data and predict the relationship between network size and degree heterogeneity. Together, these results establish how scale-free structure can arise in networks of constant size and density, with broad implications for the structure and function of complex systems.

Host-virus coevolution drives soil microbial function succession along a millennium land reclamation chronosequence

IntroductionGene exchange between viruses and hosts plays an important role in driving virus-host coevolution, enabling adaptation of both viruses and hosts to environmental changes. However, the mechanisms and functional significance of virus-host gene exchanges over long-term scales remain largely unexplored. ObjectiveThe present study aimed to gain insights into the role of viruses in virus-host interactions and coevolution by monitoring virome dynamics along a millennium-long land reclamation chronosequence. MethodsWe collected 24 soil samples from 8 stages of a millennium-long land reclamation chronosequence, including non-reclamation, and reclamation periods of 10, 50, 100, 300, 500, 700, and 1000 years. We characterized their metagenomes, and identified DNA viruses within these metagenomes. ResultsOur findings reveal a significant shift in viral community composition after 50 years of land reclamation, but soil viral diversity reached a stable phase approximately 300 years after the initial reclamation. Analysis of the virus-host network showed a scale-free degree distribution and a reduction in complexity over time, with generalist viruses emerging as key facilitators of horizontal gene transfer. ConclusionThese findings highlight the integral role of viruses, especially generalist types, in mediating gene exchanges between viruses and hosts, thereby influencing the coevolutionary dynamics in soil ecosystems over significant timescales. This study offers novel insights into long-term virus-host interactions, showing how the virome responds to environmental changes, driving shifts in various microbial functions in reclaimed land.

Information propagation characteristic by individual hesitant-common trend on weighted network

Within the context of contemporary society, the propagation of information is often subject to the influence of inter-individual connectivity, and individuals may exhibit divergent receptive attitudes towards identical information, a phenomenon denoted as the Hesitant-Common (HECO) trait. In light of this, the present study initially constructs a propagation network model devoid of correlation configurations to investigate the HECO characteristics within weighted social networks. Subsequently, the study employs a theoretical framework for edge partitioning, predicated on edge weights and HECO traits, to quantitatively analyze the mechanisms of individual information dissemination. Theoretical analyses and simulation outcomes consistently demonstrate that an augmentation in the proportion of common individuals facilitates both the diffusion and adoption of information. Concurrently, a phase transition crossover is observed, wherein the growth pattern of the ultimate adoption range, denoted as R(∞), transitions from a first-order discontinuous phase transition to a second-order continuous phase transition as the proportion of common individuals increases. An escalation in the weight distribution exponent is found to enhance information propagation. Furthermore, a reduction in the heterogeneity of degree distribution is conducive to the spread of information. Conversely, an increase in degree distribution heterogeneity and a diminution in the collective decision-making capacity can both exert inhibitory effects on the propagation of information.

Networks and their degree distribution, leading to a new concept of small worlds

The degree distribution, referred to as the delta-sequence of a network is studied. Using the non-normalized Lorenz curve, we apply a generalized form of the classical majorization partial order.Next, we introduce a new class of small worlds, namely those based on the degrees of nodes in a network. Similar to a previous study, small worlds are defined as sequences of networks with certain limiting properties. We distinguish between three types of small worlds: those based on the highest degree, those based on the average degree, and those based on the median degree. We show that these new classes of small worlds are different from those introduced previously based on the diameter of the network or the average and median distance between nodes. However, there exist sequences of networks that qualify as small worlds in both senses of the word, with stars being an example. Our approach enables the comparison of two networks with an equal number of nodes in terms of their “small-worldliness”.Finally, we introduced neighboring arrays based on the degrees of the zeroth and first-order neighbors.