Edge Weight Distribution Research Articles

A Distributed Algorithm for Reaching Average Consensus in Unbalanced Tree Networks

In this paper, a distributed algorithm for reaching average consensus is proposed for multi-agent systems with tree communication graph, when the edge weight distribution is unbalanced. First, the problem is introduced as a key topic of core algorithms for several modern scenarios. Then, the relative solution is proposed as a finite-time algorithm, which can be included in any application as a preliminary setup routine, and it is well-suited to be integrated with other adaptive setup routines, thus making the proposed solution useful in several practical applications. A special focus is devoted to the integration of the proposed method with a recent Laplacian eigenvalue allocation algorithm, and the implementation of the overall approach in a wireless sensor network framework. Finally, a worked example is provided, showing the significance of this approach for reaching a more precise average consensus in uncertain scenarios.

Alpine meadow and alpine steppe plant-soil network in Qinghai-Tibet Plateau, China

Alpine meadow and alpine steppe are crucial ecosystem types in the Qinghai-Tibet Plateau. However, due to ecosystem fragility and human disturbance, the nutrient imbalance between plant and soil leads to grassland degradation. From a systematic perspective, there needs to be a comprehensive understanding of the plant and soil traits within these two ecosystem types. This study addressed this gap by constructing plant-soil trait networks for alpine meadow and alpine steppe, identifying bridge traits, and comparing the stability and distinctions of the two trait networks. The results demonstrated a significant difference in the distribution of network structure edge weights between the alpine meadow and steppe. In the network of alpine meadow, the soil organic carbon was responsible for linking plant and soil traits, exhibiting the highest impact. Conversely, the network of alpine steppe revealed an increasing negative correlation between plant and soil traits. The results of bridge traits agreed with the optimal distribution hypothesis. The bridge traits of the alpine meadow were aboveground traits (AGB or PlantC), while those of the alpine steppe were belowground traits (BGB and RootN). Based on our findings, different management measures could be taken. The alpine meadow could be primarily restored through natural processes, and climate warming would enhance its carbon sequestration capacity. The alpine steppe ecosystem exhibits low nitrogen availability, which could be augmented through fertilisation or the return of yak manure.

Network Analysis of Internet Addiction, Online Social Anxiety, Fear of Missing Out, and Interpersonal Sensitivity among Chinese University Students

Background. Despite the growing prevalence of internet usage among young people, the relationships between internet addiction, online social anxiety, fear of missing out (FoMO), and interpersonal sensitivity remain uncertain, intricate, and multifaceted. To gain insight into the underlying psychological mechanisms, we employed network analysis to explore the interconnections between them. This endeavor may provide fresh opportunities for intervention and treatment. Methods. In this study, 470 participants were assessed at age from 18 to 22 (M=20.18 years, SD=1.861) years. Network analysis was used to examine the connections between symptoms, and statistical measures were applied to assess the stability of the network model. Results. Online social anxiety and interpersonal sensitivity had the strongest associations with other symptoms in the network, with “Evaluation anxiety” having the highest expected influence centrality, followed by “Privacy concern anxiety,” “Need for approval,” “Suspicion,” and “vulnerability.” The FoMO symptom, “Fear of missing information,” had the strongest direct relation to internet addiction. “Evaluation anxiety” and “Fear of missing information” played a key role in bridging internet addiction and interpersonal sensitivity. Additionally, the structure distribution of edge weights had a significant difference between gender. Conclusions. Our findings indicated that FoMO, interpersonal sensitivity, and online social anxiety likely play a significant role in the development and continuation of internet addiction. Interpersonal sensitivity seems to contribute to increased online social anxiety, FoMO, and the development of internet addiction, indicating that targeting these symptoms may help reduce negative online behavior and psychological burden.

A strength and sparsity preserving algorithm for generating weighted, directed networks with predetermined assortativity

Degree-preserving rewiring is a widely used technique for generating unweighted networks with given assortativity, but for weighted networks, it is unclear how an analog would preserve the strengths and other critical network features such as sparsity level. This study introduces a novel approach for rewiring weighted networks to achieve desired directed assortativity. The method utilizes a mixed integer programming framework to establish a target network with predetermined assortativity coefficients, followed by an efficient rewiring algorithm termed “strength and sparsity preserving rewiring” (SSPR). SSPR retains the node strength distributions and network sparsity after rewiring. It is also possible to accommodate additional properties like edge weight distribution, albeit with extra computational cost. The optimization scheme can be used to determine feasible assortativity ranges for an initial network. The effectiveness of the proposed SSPR algorithm is demonstrated through its application to two classes of popular network models.

Empirical Measures, Geodesic Lengths, and a Variational Formula in First-Passage Percolation

This monograph resolves—in a dense class of cases—several open problems concerning geodesics in i.i.d. first-passage percolation on Z d \mathbb {Z}^d . Our primary interest is in the empirical measures of edge-weights observed along geodesics from 0 0 to n ξ n{\boldsymbol \xi } , where ξ {\boldsymbol \xi } is a fixed unit vector. For various dense families of edge-weight distributions, we prove that these empirical measures converge weakly to a deterministic limit as n → ∞ n\to \infty , answering a question of Hoffman. These families include arbitrarily small L ∞ L^\infty -perturbations of any given distribution, almost every finitely supported distribution, uncountable collections of continuous distributions, and certain discrete distributions whose atoms can have any prescribed sequence of probabilities. Moreover, the constructions are explicit enough to guarantee examples possessing certain features, for instance: both continuous and discrete distributions whose support is all of [ 0 , ∞ ) [0,\infty ) , and distributions given by a density function that is k k -times differentiable. All results also hold for ξ {\boldsymbol \xi } -directed infinite geodesics. In comparison, we show that if Z d \mathbb {Z}^d is replaced by the infinite d d -ary tree, then any distribution for the weights admits a unique limiting empirical measure along geodesics. In both the lattice and tree cases, our methodology is driven by a new variational formula for the time constant, which requires no assumptions on the edge-weight distribution. Incidentally, this variational approach also allows us to obtain new convergence results for geodesic lengths, which have been unimproved in the subcritical regime since the seminal 1965 manuscript of Hammersley and Welsh.

Grid entropy in last passage percolation — A superadditive critical exponent approach

Working in the setting of i.i.d. last-passage percolation on RD with no assumptions on the underlying edge-weight distribution, we arrive at the notion of grid entropy — a Subadditive Ergodic Theorem limit of the entropies of paths with empirical measures weakly converging to a given target, or equivalently a deterministic critical exponent of canonical order statistics associated with the Levy-Prokhorov metric. This provides a fresh approach to an entropy first developed by Rassoul-Agha and Seppäläinen as a large deviation rate function of empirical measures along paths. In their 2014 paper, variational formulas are developed for the point-to-point/point-to-level Gibbs Free Energies as the convex conjugates of this entropy. We rework these formulas in our new framework and explicitly link our descriptions of grid entropy to theirs. We also improve on a known bound for this entropy by introducing a relative entropy term in the inequality. Furthermore, we show that the set of measures with finite grid entropy coincides with the deterministic set of limit points of empirical measures studied in a 2020 paper by Bates. We partially answer a directed polymer version of a question of Hoffman which was previously tackled in the zero temperature case by Bates.

Statistical structural inference from edge weights using a mixture of gamma distributions

Abstract The inference of reliable and meaningful connectivity information from weights representing the affinity between nodes in a graph is an outstanding problem in network science. Usually, this is achieved by simply thresholding the edge weights to distinguish true links from false ones and to obtain a sparse set of connections. Tools developed in statistical mechanics have provided particularly effective ways to locate the optimal threshold so as to preserve the statistical properties of the network structure. Thermodynamic analogies together with statistical mechanical ensembles have been proven to be useful in analysing edge-weighted networks. To extend this work, in this article, we use a statistical mechanical model to describe the probability distribution for edge weights. This models the distribution of edge weights using a mixture of Gamma distributions. Using a two-component Gamma mixture model with components describing the edge and non-edge weight distributions, we use the Expectation–Maximization algorithm to estimate the corresponding Gamma distribution parameters and mixing proportions. This gives the optimal threshold to convert weighted networks to sets of binary-valued connections. Numerical analysis shows that it provides a new way to describe the edge weight probability. Furthermore, using a physical analogy in which the weights are the energies of molecules in a solid, the probability density function for nodes is identical to the degree distribution resulting from a uniform weight on edges. This provides an alternative way to study the degree distribution with the nodal probability function in unweighted networks. We observe a phase transition in the low-temperature region, corresponding to a structural transition caused by applying the threshold. Experimental results on real-world weighted and unweighted networks reveal an improved performance for inferring binary edge connections from edge weights.

Understanding the mobility of public transport systems based on weighted multiplex networks

The structure and properties of urban public transport networks are important for understanding the mobility of public transport systems. However, most studies have ignored the impact of shared mobility and lacked the support of large-scale empirical data when exploring the mobility of urban public transport networks. Using multi-mode travel data, this study constructs a three-layer multiplex network and assigns weights to reflect the transport capacity of different modes (bus, metro, and shared bike) based on ridership and travel distances. The case study was conducted on the public transport system of Shenzhen, China. The results show that the degree distribution and edge weight distribution of the multiplex network exhibit distinct scale-free characteristics. The effect of the multiplex network model on single-layer networks suggests that cooperation between networks significantly improves the capacity and efficiency of urban public transport systems. Regression analysis reveals the effects of the urban population and land use on the demand for public transport services. The results show an intrinsic association between the public transport system and urban land use, which will facilitate the prediction of public transport demand and serve the sustainable development of cities.

Diverse phase transitions in optimized directed network models with distinct inward and outward node weights.

We consider growing directed network models that aim at minimizing the weighted connection expenses while at the same time favoring other important network properties such as weighted local node degrees. We employed statistical mechanics methods to study the growth of directed networks under the principle of optimizing some objective function. By mapping the system to an Ising spin model, analytic results are derived for two such models, exhibiting diverse and interesting phase transition behaviors for general edge weight, inward and outward node weight distributions. In addition, the unexplored cases of negative node weights are also investigated. Analytic results for the phase diagrams are derived showing even richer phase transition behavior, such as first-order transition due to symmetry, second-order transitions with possible reentrance, and hybrid phase transitions. We further extend previously developed zero-temperature simulation algorithm for undirected networks to the present directed case and for negative node weights, and we can obtain the minimal cost connection configuration efficiently. All the theoretical results are explicitly verified by simulations. Possible applications and implications are also discussed.

Dynamic correlations of renewable-energy companies: Evidence from a multilayer network model

Against the background of seeking to achieve carbon neutrality, relationships among renewable-energy companies around the world have become multiple and complex. In this work, the Pearson, Kendall, tail, and partial correlation coefficients were applied to 51 global companies—including solar and wind firms, independent power plants, and utilities—to explore the linear, nonlinear, extreme-risk, and direct relations between them. Sample data from 7 August 2015 to 6 August 2021 were considered, and three sub-periods were extracted from these sample data by analysis of the evolution of multiple correlations combined with event analysis. A four-layer correlation network model was then constructed. The main results are as follows. (1) The multiple relations among the selected firms underwent dramatic changes during two external shocks (the China–US trade war and the COVID-19 pandemic). (2) The extreme-risk network layer verified that the trade war mainly affected the relationships among companies in the solar industries of China and the US. (3) During the COVID-19 pandemic period, the linear and direct relationships among wind firms from Canada, Spain, and Germany were significantly increased. In this sub-period, edge-weight distributions of the four different layers were heterogeneous and varied from power-law features to Gaussian distributions. (4) During all the sub-periods, most companies had similar numbers of neighbors, while the numbers of neighbors of a few companies varied greatly in the four different layers. These findings provide a useful reference for stakeholders and may help them understand the connectedness and evolution of global renewable-energy markets.

Network analysis of comorbid depression and anxiety and their associations with quality of life among clinicians in public hospitals during the late stage of the COVID-19 pandemic in China

BackgroundMental health problems are common among clinicians working in public hospitals even in the late stage of the COVID-19 pandemic. Network analysis is a novel approach to explore interactions between mental health problems at the symptom level. This study examined the network structure of comorbid depression and anxiety and their associations with quality of life (QOL) among hospital clinicians in China during the late stage of the COVID-19 pandemic. MethodsA total of 4931 participants were recruited from October 13 to 22, 2020. The nine-item Patient Health Questionnaire (PHQ-9), seven-item Generalized Anxiety Disorder Scale (GAD-7), and the World Health Organization Quality of Life Questionnaire-Brief Version (WHOQOL-BREF) were used to measure depressive and anxiety symptoms, and QOL, respectively. Central and bridge symptoms were identified with centrality and bridge centrality indices, respectively. Network stability was examined using the case-dropping procedure. ResultsThe prevalence of depression (defined as PHQ-9 total score ≥ 5) was 35.1 % [95 % confidence interval (CI) = 33.73–36.41 %)], the prevalence of anxiety (GAD-7 total score ≥ 5) was 32.5 % (95 % CI = 31.20–33.84 %), while the prevalence of comorbid depression and anxiety was 26.9 % (95 % CI = 25.7–28.2 %). “Impaired motor skills”, “Trouble relaxing” and “Uncontrollable worry” were the central symptoms in the whole depression-anxiety network. “Irritability”, “Feeling afraid” and “Sad mood” were the most key bridge symptoms linking depression and anxiety. Three symptoms (“Fatigue”, “Trouble relaxing” and “Nervousness”) were the most strongly and negatively associated with QOL. Neither gender nor the experiences of caring for COVID-19 patients was associated with network global strength, distribution of edge weights or individual edge weights. LimitationsThe causality between variables could not be established. Depressive and anxiety symptoms were assessed by self-report measures, which may result in recall bias and limitations in capturing clinical phenomena. ConclusionsBoth the central (i.e., “Impaired motor skills”, “Trouble relaxing” and “Uncontrollable worry”) and bridge symptoms (i.e., “Irritability”, “Feeling afraid” and “Sad mood”) identified in this network analysis should be targeted in specific treatment and preventive measures for comorbid depressive and anxiety symptoms among clinicians in the late stage of the pandemic. Furthermore, “Fatigue”, “Trouble relaxing” and “Nervousness” are key symptoms to address to improve clinicians' QOL.

An Exploration of Self-Reported Sleep Inertia Symptoms Using Network Analysis.

PurposeSleep inertia (SI) is the transitional state accompanied by compromised cognitive and physical performance and sleepiness. Network analysis offers a potential new framework to conceptualize a complex network of symptom–symptom interactions, and the network structure is analyzed to reveal the core characteristics. However, no previous study examined the network structure of SI symptoms. Thus, this study aimed to elucidate characteristics and compare sex differences of SI symptom networks in the general population.Materials and MethodsA total of 1491 participants from China were recruited from 30 May to 17 June, 2021. SI symptoms were assessed by using the Sleep Inertia Questionnaire (SIQ). The network structures were estimated and compared using network analytic methods in the R version 4.1.1.ResultsCentrality properties analysis of the expected influence suggested that symptoms of “Feel sleepy”, “Groggy, fuzzy or hazy mind”, and “Dread starting your day” exerted greatest influences. The weighted adjacency matrix revealed that the “Dread starting your day” and “Anxious about the upcoming day” edge showed the strongest connection (edge weight value = 0.70). The network comparison test found no significant difference in network global strength (p=0.928), distribution of edge weights (p=0.194) and individual edge weights (all p values >0.05 after Holm–Bonferroni corrections) between males and females.ConclusionSymptoms of “Feel sleepy”, “Groggy, fuzzy or hazy mind”, and “Dread starting your day” were central in the SI symptom network. Intervention, such as the artificial dawn and change in body temperature, for symptoms of “Feel sleepy”, “Groggy, fuzzy or hazy mind”, and “Dread starting your day” might be crucial to hasten the dissipation of SI in the general population who may need to perform tasks upon waking.

Propagation characteristic of adoption thresholds heterogeneity in double-layer networks with edge weight distribution

People normally show different relation intensity and adoption behaviors, which can affect information spreading on social networks. To describe the two phenomena, we consider edge weight distribution and adoption thresholds heterogeneity on the networks separately, where the adoption thresholds of nodes obey truncated gaussian distribution. Therefore, we propose a double-layer network model with edge weight distribution to explore the impact of heterogeneous adoption thresholds of nodes on information spreading. And we also develop an edge-based compartmental theory to analyze the information spreading mechanism. Through numerical simulations and theoretical analysis, we find that increasing the values of mean can suppress information spreading, and the impact of the values of standard deviation on information spreading depends on the values of mean. Specifically, for a small value of mean, the size of information spreading will decrease with the increase of the value of standard deviation. Meanwhile, when the values of standard deviation are fixed, the size of information spreading will increase continuously with the increase of transmission probability. For a large value of mean, the size of information spreading will increase first and then decrease with the increase of the value of standard deviation. In addition, this paper shows that the theoretical method agrees with the numerical simulations in the experimental results.

Network analysis of depressive symptoms in Hong Kong residents during the COVID-19 pandemic

In network theory depression is conceptualized as a complex network of individual symptoms that influence each other, and central symptoms in the network have the greatest impact on other symptoms. Clinical features of depression are largely determined by sociocultural context. No previous study examined the network structure of depressive symptoms in Hong Kong residents. The aim of this study was to characterize the depressive symptom network structure in a community adult sample in Hong Kong during the COVID-19 pandemic. A total of 11,072 participants were recruited between 24 March and 20 April 2020. Depressive symptoms were measured using the Patient Health Questionnaire-9. The network structure of depressive symptoms was characterized, and indices of “strength”, “betweenness”, and “closeness” were used to identify symptoms central to the network. Network stability was examined using a case-dropping bootstrap procedure. Guilt, Sad Mood, and Energy symptoms had the highest centrality values. In contrast, Concentration, Suicide, and Sleep had lower centrality values. There were no significant differences in network global strength (p = 0.259), distribution of edge weights (p = 0.73) and individual edge weights (all p values > 0.05 after Holm–Bonferroni corrections) between males and females. Guilt, Sad Mood, and Energy symptoms were central in the depressive symptom network. These central symptoms may be targets for focused treatments and future psychological and neurobiological research to gain novel insight into depression.

Network analysis of anxiety and depressive symptoms among nursing students during the COVID-19 pandemic

Background: The 2019 coronavirus disease (COVID-19) pandemic has impacted the mental health and well-being of medical personnel, including nursing students. Network analysis provides a deeper characterization of symptom-symptom interactions in mental disorders. The aim of this study was to elucidate characteristics of anxiety and depressive symptom networks of Chinese nursing students during the COVID-19 pandemic.Method: A total of 932 nursing students were included. Anxiety and depressive symptom were measured using the seven-item Generalized Anxiety Disorder Scale (GAD-7) and two-item Patient Health Questionnaire (PHQ-2), respectively. Central symptoms and bridge symptoms were identified via centrality indices and bridge centrality indices, respectively. Network stability was examined using the case-dropping procedure.Results: Irritability, Uncontrollable worry, Trouble relaxing, and Depressed mood had the highest centrality values. Three bridge symptoms (Depressed mood, Nervousness, and Anhedonia) were also identified. Neither gender nor region of residence was associated with network global strength, distribution of edge weights or individual edge weights.Limitations: Data were collected in a cross-sectional study design, therefore, causal relations and dynamic changes between anxiety and depressive symptoms over time could not be inferred. Generalizability of findings may be limited to Chinese nursing students during a particular phase of the current pandemic.Conclusions: Irritability, Uncontrollable worry, Trouble relaxing, and Depressed mood constituted central symptoms maintaining the anxiety-depression network structure of Chinese nursing students during the pandemic. Timely, systemic multi-level interventions targeting central symptoms and bridge symptoms may be effective in alleviating co-occurring experiences of anxiety and depression in this population.

Sparsity of weighted networks: Measures and applications

A majority of real life networks are weighted and sparse. The present article aims at characterization of weighted networks based on sparsity, as an indicator of inherent diversity of different network parameters. The measure called sparsity index defined on ordered degree sequence of simple networks is extended and new properties of this index are derived. The range of possible values of sparsity index of any connected network, with edge-count in specific intervals, is worked out analytically in terms of node-count and a pattern is uncovered in corresponding degree sequences. Given the edge-weight frequency distribution of a network, an expression of the sparsity index of edge-weights is formulated. Its properties are analyzed under different distributions of edge-weights. For example, the upper and lower bounds of sparsity index of edge-weights of a network, with all distinct edge-weights, is determined in terms of its node-count and edge-density. The article highlights that this summary index with low computational cost, computed on different network parameters, is useful to reveal different structural and organizational aspects of networks for performing analysis. An application of this index is demonstrated through devising a new overlapping community detection method. The results validated on artificial and real-world networks show its efficacy.

Complexity indices for the traveling salesman problem continued

We consider the symmetric traveling salesman problem (TSP) with instances represented by complete graphs G with distances between cities as edge weights. A complexity index is an invariant of an instance I by which we predict the execution time of an exact TSP algorithm for I. In the paper [5] we have considered some short edge subgraphs of G and defined several new invariants related to their connected components. Extensive computational experiments with instances on 50 vertices with the uniform distribution of integer edge weights in the interval [1,100] show that there exists correlation between the sequences of selected invariants and the sequence of execution times of the well-known TSP Solver Concorde. In this paper we extend these considerations for instances up to 100 vertices.

Phase transitions in optimized network models

We consider network connection growing models that aim at minimizing the wiring cost while at the same time maximizing the network connections. By mapping the system to an Ising spin model, we obtain analytic results for two such models, both of them show interesting, but different phase transition behaviors for general wiring cost distributions. The phase diagrams for these transitions are also obtained. These results are also extended for networks optimized with weighted node degree for connections. Depending on the properties of the edge and node weight distributions, the system can exhibit a variety of phase transitions, including first-order, second-order and hybrid ones, from no connection to a network of finite fraction of connections. Furthermore, mean-field theory leads to an effective algorithm for finding the fully optimized network configurations in these models. All these results are also verified by numerical simulations.

Optimizing Connectivity-Driven Brain Parcellation Using Ensemble Clustering

This work addresses the problem of constructing a unified, topologically optimal connectivity-based brain atlas. The proposed approach aggregates an ensemble partition from individual parcellations without label agreement, providing a balance between sufficiently flexible individual parcellations and intuitive representation of the average topological structure of the connectome. The methods exploit a previously proposed dense connectivity representation, first performing graph-based hierarchical parcellation of individual brains, and subsequently aggregating the individual parcellations into a consensus parcellation. The search for consensus—based on the hard ensemble (HE) algorithm—approximately minimizes the sum of cluster membership distances, effectively estimating a pseudo-Karcher mean of individual parcellations. Computational stability, graph structure preservation, and biological relevance of the simplified representation resulting from the proposed parcellation are assessed on the Human Connectome Project data set. These aspects are assessed using (1) edge weight distribution divergence with respect to the dense connectome representation, (2) interhemispheric symmetry, (3) network characteristics' stability and agreement with respect to individually and anatomically parcellated networks, and (4) performance of the simplified connectome in a biological sex classification task. Ensemble parcellation was found to be highly stable with respect to subject sampling, outperforming anatomical atlases and other connectome-based parcellations in classification as well as preserving global connectome properties. The HE-based parcellation also showed a degree of symmetry comparable with anatomical atlases and a high degree of spatial contiguity without using explicit priors.

Gene Networks with Transcriptional Bursting Recapitulate Rare Transient Coordinated High Expression States in Cancer.

Non-genetic transcriptional variability is a potential mechanism for therapy resistance in melanoma. Specifically, rare subpopulations of cells occupy a transient pre-resistant state characterized by coordinated high expression of several genes and survive therapy. How might these rare states arise and disappear within the population? It is unclear whether the canonical models of probabilistic transcriptional pulsing can explain this behavior, or if it requires special, hitherto unidentified mechanisms. We show that a minimal model of transcriptional bursting and gene interactions can give rise to rare coordinated high expression states. These states occur more frequently in networks with low connectivity and depend on three parameters. While entry into these states is initiated by a long transcriptional burst that also triggers entry of other genes, the exit occurs through independent inactivation of individual genes. Together, we demonstrate that established principles of gene regulation are sufficient to describe this behavior and argue for its more general existence. A record of this paper's transparent peer review process is included in the Supplemental Information.