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1.
Plant J ; 117(4): 1223-1238, 2024 Feb.
Artigo em Inglês | MEDLINE | ID: mdl-37991980

RESUMO

Plant species with large genomes tend to be excluded from climatically more extreme environments with a shorter growing season. Species that occupy such environments are assumed to be under natural selection for more rapid growth and smaller genome size (GS). However, evidence for this is available only for temperate organisms. Here, we study the evolution of GS in two subfamilies of the tropical family Zingiberaceae to find out whether species with larger genomes are confined to environments where the vegetative season is longer. We tested our hypothesis on 337 ginger species from regions with contrasting climates by correlating their GS with an array of plant traits and environmental variables. We revealed 16-fold variation in GS which was tightly related to shoot seasonality. Negative correlations of GS with latitude, temperature and precipitation emerged in the subfamily Zingiberoidae, demonstrating that species with larger GS are excluded from areas with a shorter growing season. In the subfamily Alpinioideae, GS turned out to be correlated with the type of stem and light requirements and its members cope with seasonality mainly by adaptation to shady and moist habitats. The Ornstein-Uhlenbeck models suggested that evolution in regions with humid climates favoured larger GS than in drier regions. Our results indicate that climate seasonality exerts an upper constraint on GS not only in temperate regions but also in the tropics, unless species with large genomes find alternative ways to escape from that constraint.


Assuntos
Zingiber officinale , Ecossistema , Clima , Estações do Ano , Plantas
2.
Phys Rev E ; 108(2-1): 024311, 2023 Aug.
Artigo em Inglês | MEDLINE | ID: mdl-37723783

RESUMO

We study core-periphery structure in networks using inference methods based on a flexible network model that allows for traditional onionlike cores within cores, but also for hierarchical treelike structures and more general non-nested types of structures. We propose an efficient Monte Carlo scheme for fitting the model to observed networks and report results for a selection of real-world data sets. Among other things, we observe an empirical distinction between networks showing traditional core-periphery structure with a dense core weakly connected to a sparse periphery, and an alternative structure in which the core is strongly connected both within itself and to the periphery. Networks vary in whether they are better represented by one type of structure or the other. We also observe structures that are a hybrid between core-periphery structure and community structure, in which networks have a set of nonoverlapping cores that roughly correspond to communities, surrounded by a single undifferentiated periphery. Computer code implementing our methods is available.

3.
Ann Rev Mar Sci ; 15: 249-275, 2023 01 16.
Artigo em Inglês | MEDLINE | ID: mdl-36112981

RESUMO

The modes of Pacific decadal-scale variability (PDV), traditionally defined as statistical patterns of variance, reflect to first order the ocean's integration (i.e., reddening) of atmospheric forcing that arises from both a shift and a change in strength of the climatological (time-mean) atmospheric circulation. While these patterns concisely describe PDV, they do not distinguish among the key dynamical processes driving the evolution of PDV anomalies, including atmospheric and ocean teleconnections and coupled feedbacks with similar spatial structures that operate on different timescales. In this review, we synthesize past analysis using an empirical dynamical model constructed from monthly ocean surface anomalies drawn from several reanalysis products, showing that the PDV modes of variance result from two fundamental low-frequency dynamical eigenmodes: the North Pacific-central Pacific (NP-CP) and Kuroshio-Oyashio Extension (KOE) modes. Both eigenmodes highlight how two-way tropical-extratropical teleconnection dynamics are the primary mechanisms energizing and synchronizing the basin-scale footprint of PDV. While the NP-CP mode captures interannual- to decadal-scale variability, the KOE mode is linked to the basin-scale expression of PDV on decadal to multidecadal timescales, including contributions from the South Pacific.

4.
Phys Rev E ; 105(1-1): 014312, 2022 Jan.
Artigo em Inglês | MEDLINE | ID: mdl-35193232

RESUMO

Statistical methods for reconstructing networks from repeated measurements typically assume that all measurements are generated from the same underlying network structure. This need not be the case, however. People's social networks might be different on weekdays and weekends, for instance. Brain networks may differ between healthy patients and those with dementia or other conditions. Here we describe a Bayesian analysis framework for such data that allows for the fact that network measurements may be reflective of multiple possible structures. We define a finite mixture model of the measurement process and derive a Gibbs sampling procedure that samples exactly from the full posterior distribution of model parameters. The end result is a clustering of the measured networks into groups with similar structure. We demonstrate the method on both real and synthetic network populations.

5.
Science ; 374(6572): 1237-1241, 2021 Dec 03.
Artigo em Inglês | MEDLINE | ID: mdl-34855491

RESUMO

The discovery of topological order has revised the understanding of quantum matter and provided the theoretical foundation for many quantum error­correcting codes. Realizing topologically ordered states has proven to be challenging in both condensed matter and synthetic quantum systems. We prepared the ground state of the toric code Hamiltonian using an efficient quantum circuit on a superconducting quantum processor. We measured a topological entanglement entropy near the expected value of ­ln2 and simulated anyon interferometry to extract the braiding statistics of the emergent excitations. Furthermore, we investigated key aspects of the surface code, including logical state injection and the decay of the nonlocal order parameter. Our results demonstrate the potential for quantum processors to provide insights into topological quantum matter and quantum error correction.

6.
Nat Commun ; 12(1): 3911, 2021 06 23.
Artigo em Inglês | MEDLINE | ID: mdl-34162855

RESUMO

Empirical measurements of ecological networks such as food webs and mutualistic networks are often rich in structure but also noisy and error-prone, particularly for rare species for which observations are sparse. Focusing on the case of plant-pollinator networks, we here describe a Bayesian statistical technique that allows us to make accurate estimates of network structure and ecological metrics from such noisy observational data. Our method yields not only estimates of these quantities, but also estimates of their statistical errors, paving the way for principled statistical analyses of ecological variables and outcomes. We demonstrate the use of the method with an application to previously published data on plant-pollinator networks in the Seychelles archipelago and Kosciusko National Park, calculating estimates of network structure, network nestedness, and other characteristics.


Assuntos
Algoritmos , Insetos/fisiologia , Modelos Biológicos , Plantas/parasitologia , Polinização/fisiologia , Animais , Conservação dos Recursos Naturais/métodos , Conservação dos Recursos Naturais/estatística & dados numéricos , Ecossistema , Interações Hospedeiro-Parasita , New South Wales , Parques Recreativos , Seicheles
7.
BJS Open ; 5(3)2021 05 07.
Artigo em Inglês | MEDLINE | ID: mdl-33960366

RESUMO

BACKGROUND: The WHO Surgical Safety Checklist has been shown to reduce perioperative morbidity and mortality worldwide. There is evidence to suggest that sign-out is the most poorly performed phase of the checklist as it coincides with a period of high workload for team members. This study aimed to see whether modification of this process might result in greater compliance. METHODS: A controlled longitudinal (before and after) study was performed to evaluate the effect of a modified checklist sign-out on compliance in a single surgical department. Checklist quality was evaluated by measurement of checklist completion, active participation, and team member presence. Workload assessment was performed to identify the optimal moment for the sign-out process. The sign-out process was modified through an iterative multidisciplinary approach, informed by results from the workload assessment. Feedback was obtained through staff surveys. RESULTS: A total of 185 operations were used, with an intervention group in vascular surgery and a control group in orthopaedics. The optimal timing for sign-out was identified as after final wound closure. The modified sign-out process improved active participation of team members (21 of 34 versus 31 of 34; P = 0.010). In the control group, complete compliance improved (48 of 76 versus 30 of 41; P = 0.041). However, active participation decreased (53 of 76 versus 19 of 41; P = 0.022). No differences were noted between groups in team member presence. Eighteen of 21 staff questioned viewed the modifications positively. CONCLUSION: The optimal sign-out timing was identified as immediately after final wound closure prior to undraping the patient.


Assuntos
Lista de Checagem , Segurança do Paciente , Humanos , Organização Mundial da Saúde
8.
Sci Adv ; 7(17)2021 Apr.
Artigo em Inglês | MEDLINE | ID: mdl-33893102

RESUMO

Belief propagation is a widely used message passing method for the solution of probabilistic models on networks such as epidemic models, spin models, and Bayesian graphical models, but it suffers from the serious shortcoming that it works poorly in the common case of networks that contain short loops. Here, we provide a solution to this long-standing problem, deriving a belief propagation method that allows for fast calculation of probability distributions in systems with short loops, potentially with high density, as well as giving expressions for the entropy and partition function, which are notoriously difficult quantities to compute. Using the Ising model as an example, we show that our approach gives excellent results on both real and synthetic networks, improving substantially on standard message passing methods. We also discuss potential applications of our method to a variety of other problems.

9.
Oral Oncol ; 116: 105195, 2021 05.
Artigo em Inglês | MEDLINE | ID: mdl-33618103

RESUMO

INTRODUCTION: Current research is elucidating how the addition of depth of invasion (DOI) to the 8th edition of the American Joint Committee on Cancer (AJCC) TNM staging for oral cavity squamous cell carcinoma influences its prognostic accuracy. However, there is limited research on survival in pT3N0M0 oral tongue SCC (OTSCC) patients when stratifying by DOI. OBJECTIVES: Determine 5-year overall survival (OS), and cancer-specific survival (CSS) for patients with pT3N0M0 oral OTSCC based on shallow DOI (<10 mm) and deep DOI (10-20 mm). METHODS: Retrospective review involving three tertiary care cancer centers in North America. cT3N0M0 OTSCC patients receiving primary surgical treatment from 2004 to 2018 were identified. Inclusion: age > 18 years old and confirmation of pT3N0M0 OTSCC on surgical pathology. Exclusion: patients undergoing palliative treatment or previous head and neck surgery/radiotherapy. Analysis comprised two groups: shallow pT3 (tumor diameter > 4 cm, DOI < 10 mm) and deep pT3 (DOI 10 mm-20 mm). RESULTS: One hundred and four patients with pT3N0M0 OTSCC were included. Mean age was 59.1 years (range: 18-80.74). Age, gender, and Charlson Comorbidity Index were similar between the two groups (p > 0.05). Recurrence, LVI, PNI, and positive margins were more common in deep T3 tumors (P < 0.05). 5-year OS (50% vs 26%, p = 0.006) and CSS (72% vs 24%, p = 0.005) were worse in deep pT3 tumors. Deep pT3 disease was an independent predictor of OS (p = 0.004) and CSS (p = 0.01) on Cox-Regression analysis. CONCLUSION: DOI is an independent predictor of poor survival in pT3N0M0 OTSCC patients. Consideration should be given to escalating adjuvant therapy for deep pT3N0M0 OTSCC patients.


Assuntos
Carcinoma de Células Escamosas de Cabeça e Pescoço , Neoplasias da Língua , Adulto , Humanos , Pessoa de Meia-Idade , Invasividade Neoplásica , Estudos Retrospectivos , Taxa de Sobrevida , Neoplasias da Língua/cirurgia
10.
Hernia ; 25(3): 625-630, 2021 06.
Artigo em Inglês | MEDLINE | ID: mdl-32876796

RESUMO

PURPOSE: Primary midline hernias arising in the linea alba are common. While mesh repair has been shown to reduce recurrence rates even in small hernias, many surgeons still use a suture repair for defects of less than 2 cm. The recent European and Americas Hernia Societies Guidelines recommended suture repair only for hernias smaller than 1 cm. A suture repair implies edge-to-edge or overlapping fascial margins, which necessarily involves tension on the repair. A darn is a tension-free repair where, in effect, a "mesh" is hand-woven across the defect in situ. METHODS: The darn repair is a modification of the darn techniques for inguinal hernia repair. Eligible patients undergoing this repair at the Royal Infirmary of Edinburgh between 1 January 2008 and 31 December 2017 were identified from a prospective computer-based medical record system and their case notes reviewed. Inclusion criteria were adult patients with a primary midline abdominal wall defect smaller than 2 cm in the widest diameter of the hernia defect measured intra-operatively. Patients were followed up by telephone in 2019. Those who reported possible recurrence or other symptoms in the region of their hernia repair were reviewed in the outpatient clinic. RESULTS: 47 suture-darn repairs were undertaken over the 10-year period. Fifteen of the darn repair operations (32%) were performed under local anaesthesia. Forty-one patients were followed up with a mean of 80 ± 35 and median of 87 months after surgery. Six patients (13%) were lost to follow-up. Recurrence was found in two cases (5%) and one patient has since been diagnosed with a new epigastric hernia some 5 cm cranial to the previous repair. CONCLUSIONS: The darn repair for small primary midline hernias is quick and inexpensive with promising long-term results. It can be performed under local anaesthesia. It can serve as an alternative to mesh repair for defects less than 2 cm in maximum dimension.


Assuntos
Parede Abdominal , Hérnia Inguinal , Parede Abdominal/cirurgia , Adulto , Hérnia Inguinal/cirurgia , Herniorrafia , Humanos , Estudos Prospectivos , Recidiva , Telas Cirúrgicas
11.
Epidemiol Infect ; 148: e147, 2020 07 01.
Artigo em Inglês | MEDLINE | ID: mdl-32605670

RESUMO

Major surgery carried out in low- and middle-income countries is associated with a high risk of surgical site infections (SSI), but knowledge is limited regarding contributory factors to such infections. This study explores factors related to patients developing an SSI in a teaching hospital in Ghana. A prospective cohort study of patients undergoing abdominal surgical procedures was conducted at Korle Bu Teaching Hospital. Patient characteristics, procedures and environmental characteristics were recorded. A 30-day daily surveillance was used to diagnose SSI, and Poisson regression analysis was used to test for association of SSI and risk factors; survival was determined by proportional hazard regression methods. We included 358 patients of which 58 (16.2%; 95% CI 12.7-20.4%) developed an SSI. The median number of door openings during an operation was 79, with 81% being unnecessary. Door openings greater than 100 during an operation (P = 0.028) significantly increased a patient's risk of developing an SSI. Such patients tended to have an elevated mortality risk (hazard ratio 2.67; 95% CI 0.75-9.45, P = 0.128). We conclude that changing behaviour and practices in operating rooms is a key strategy to reduce SSI risk.


Assuntos
Abdome/cirurgia , Microbiologia do Ar , Infecção da Ferida Cirúrgica/epidemiologia , Infecção da Ferida Cirúrgica/microbiologia , Adulto , Feminino , Gana/epidemiologia , Hospitais de Ensino , Humanos , Masculino , Pessoa de Meia-Idade , Movimento , Salas Cirúrgicas , Estudos Prospectivos , Fatores de Risco
12.
Phys Rev E ; 101(5-1): 052306, 2020 May.
Artigo em Inglês | MEDLINE | ID: mdl-32575211

RESUMO

The most widely used techniques for community detection in networks, including methods based on modularity, statistical inference, and information theoretic arguments, all work by optimizing objective functions that measure the quality of network partitions. There is a good case to be made, however, that one should not look solely at the single optimal community structure under such an objective function but rather at a selection of high-scoring structures. If one does this, one typically finds that the resulting structures show considerable variation, which could be taken as evidence that these community detection methods are unreliable, since they do not appear to give consistent answers. Here we argue that, upon closer inspection, the structures found are in fact consistent in a certain way. Specifically, we show that they can all be assembled from a set of underlying "building blocks," groups of network nodes that are usually found together in the same community. Different community structures correspond to different arrangements of blocks, but the blocks themselves are largely invariant. We propose an information theoretic method for discovering the building blocks in specific networks and demonstrate it with several example applications. We conclude that traditional community detection does in fact give a significant amount of insight into network structure.

13.
Phys Rev E ; 101(4-1): 042304, 2020 Apr.
Artigo em Inglês | MEDLINE | ID: mdl-32422767

RESUMO

The information theoretic measure known as mutual information is widely used as a way to quantify the similarity of two different labelings or divisions of the same set of objects, such as arises, for instance, in clustering and classification problems in machine learning or community detection problems in network science. Here we argue that the standard mutual information, as commonly defined, omits a crucial term which can become large under real-world conditions, producing results that can be substantially in error. We derive an expression for this missing term and hence write a corrected mutual information that gives accurate results even in cases where the standard measure fails. We discuss practical implementation of the new measure and give example applications.

14.
J Hosp Infect ; 104(3): 321-327, 2020 Mar.
Artigo em Inglês | MEDLINE | ID: mdl-31931045

RESUMO

BACKGROUND: Surveillance systems for surgical site infections (SSIs), as a measure of patient safety, help health institutions devise strategies to reduce or prevent them. No surveillance systems exist to monitor SSIs in Ghana. AIM: To establish a system for monitoring trends and detecting outbreaks in order to create awareness of and control SSIs. METHODS: An active 30-day surveillance was undertaken at the general surgical unit of the Korle Bu Teaching Hospital, from July 1st, 2017 to December 31st, 2018 to identify SSI. It involved a daily inpatient surveillance of patients who had had a surgical procedure, followed by post-discharge surveillance by means of a healthcare personnel-based survey and a patient-based telephone survey. We supplied quarterly feedback of results to surgeons. FINDINGS: Among the 3267 patients included, 331 were identified with an SSI, a 10% incidence risk. Patients who acquired an SSI experienced increased morbidity including nine extra days in hospital and an adjusted relative mortality risk of 2.3 (95% confidence interval: 1.3 - 4.1; P=0.006) compared to patients without SSI. Forty-nine per cent (161/331) of SSIs were diagnosed post discharge using the healthcare personnel-based survey. The patient-based telephone survey contributed 12 additional cases. SSI incidence risk decreased from 12.8% to 7.5% during the study period. CONCLUSION: Post-discharge surveillance is feasible using existing healthcare personnel, and the results highlight the high risk and burden of SSIs in Ghana. A surveillance system with feedback for monitoring SSIs may contribute to reducing SSIs; however, firm conclusions regarding the impact need longer observation time.


Assuntos
Infecção Hospitalar/epidemiologia , Segurança do Paciente , Infecção da Ferida Cirúrgica/epidemiologia , Adolescente , Adulto , Idoso , Criança , Estudos de Coortes , Feminino , Gana , Hospitais de Ensino , Humanos , Masculino , Pessoa de Meia-Idade , Vigilância da População , Estudos Prospectivos , Fatores de Risco , Infecção da Ferida Cirúrgica/prevenção & controle , Adulto Jovem
15.
Osteoporos Int ; 31(2): 277-289, 2020 Feb.
Artigo em Inglês | MEDLINE | ID: mdl-31720722

RESUMO

The trial compared three physiotherapy approaches: manual or exercise therapy compared with a single session of physiotherapy education (SSPT) for people with osteoporotic vertebral fracture(s). At 1 year, there were no statistically significant differences between the groups meaning there is inadequate evidence to support manual or exercise therapy. INTRODUCTION: To evaluate the clinical and cost-effectiveness of different physiotherapy approaches for people with osteoporotic vertebral fracture(s) (OVF). METHODS: >Prospective, multicentre, adaptive, three-arm randomised controlled trial. Six hundred fifteen adults with back pain, osteoporosis, and at least 1 OVF participated. INTERVENTIONS: 7 individual physiotherapy sessions over 12 weeks focused on either manual therapy or home exercise compared with a single session of physiotherapy education (SSPT). The co-primary outcomes were quality of life and back muscle endurance measured by the QUALEFFO-41 and timed loaded standing (TLS) test at 12 months. RESULTS: At 12 months, there were no statistically significant differences between groups. Mean QUALEFFO-41: - 1.3 (exercise), - 0.15 (manual), and - 1.2 (SSPT), a mean difference of - 0.2 (95% CI, - 3.2 to 1.6) for exercise and 1.3 (95% CI, - 1.8 to 2.9) for manual therapy. Mean TLS: 9.8 s (exercise), 13.6 s (manual), and 4.2 s (SSPT), a mean increase of 5.8 s (95% CI, - 4.8 to 20.5) for exercise and 9.7 s (95% CI, 0.1 to 24.9) for manual therapy. Exercise provided more quality-adjusted life years than SSPT but was more expensive. At 4 months, significant changes above SSPT occurred in endurance and balance in manual therapy, and in endurance for those ≤ 70 years, in balance, mobility, and walking in exercise. CONCLUSIONS: Adherence was problematic. Benefits at 4 months did not persist and at 12 months, we found no significant differences between treatments. There is inadequate evidence a short physiotherapy intervention of either manual therapy or home exercise provides long-term benefits, but arguably short-term benefits are valuable. TRIAL REGISTRATION: ISRCTN 49117867.


Assuntos
Terapia por Exercício , Modalidades de Fisioterapia , Fraturas da Coluna Vertebral , Idoso , Idoso de 80 Anos ou mais , Análise Custo-Benefício , Terapia por Exercício/economia , Feminino , Humanos , Masculino , Pessoa de Meia-Idade , Osteoporose/complicações , Modalidades de Fisioterapia/economia , Estudos Prospectivos , Qualidade de Vida , Fraturas da Coluna Vertebral/economia , Fraturas da Coluna Vertebral/terapia
16.
NPJ Genom Med ; 4: 27, 2019.
Artigo em Inglês | MEDLINE | ID: mdl-31700678

RESUMO

A growing body of evidence points to a considerable and heterogeneous genetic aetiology of cerebral palsy (CP). To identify recurrently variant CP genes, we designed a custom gene panel of 112 candidate genes. We tested 366 clinically unselected singleton cases with CP, including 271 cases not previously examined using next-generation sequencing technologies. Overall, 5.2% of the naïve cases (14/271) harboured a genetic variant of clinical significance in a known disease gene, with a further 4.8% of individuals (13/271) having a variant in a candidate gene classified as intolerant to variation. In the aggregate cohort of individuals from this study and our previous genomic investigations, six recurrently hit genes contributed at least 4% of disease burden to CP: COL4A1, TUBA1A, AGAP1, L1CAM, MAOB and KIF1A. Significance of Rare VAriants (SORVA) burden analysis identified four genes with a genome-wide significant burden of variants, AGAP1, ERLIN1, ZDHHC9 and PROC, of which we functionally assessed AGAP1 using a zebrafish model. Our investigations reinforce that CP is a heterogeneous neurodevelopmental disorder with known as well as novel genetic determinants.

17.
Proc Natl Acad Sci U S A ; 116(47): 23398-23403, 2019 11 19.
Artigo em Inglês | MEDLINE | ID: mdl-31685640

RESUMO

Message passing is a fundamental technique for performing calculations on networks and graphs with applications in physics, computer science, statistics, and machine learning, including Bayesian inference, spin models, satisfiability, graph partitioning, network epidemiology, and the calculation of matrix eigenvalues. Despite its wide use, however, it has long been recognized that the method has a fundamental flaw: It works poorly on networks that contain short loops. Loops introduce correlations that can cause the method to give inaccurate answers or to fail completely in the worst cases. Unfortunately, most real-world networks contain many short loops, which limits the usefulness of the message-passing approach. In this paper we demonstrate how to rectify this shortcoming and create message-passing methods that work on any network. We give 2 example applications, one to the percolation properties of networks and the other to the calculation of the spectra of sparse matrices.

18.
Phys Rev E ; 100(1-1): 012314, 2019 Jul.
Artigo em Inglês | MEDLINE | ID: mdl-31499775

RESUMO

The spectrum of the adjacency matrix plays several important roles in the mathematical theory of networks and network data analysis, for example in percolation theory, community detection, centrality measures, and the theory of dynamical systems on networks. A number of methods have been developed for the analytic computation of network spectra, but they typically assume that networks are locally treelike, meaning that the local neighborhood of any node takes the form of a tree, free of short loops. Empirically observed networks, by contrast, often have many short loops. Here we develop an approach for calculating the spectra of networks with short loops using a message passing method. We give example applications to some previously studied classes of networks and find that the presence of loops induces substantial qualitative changes in the shape of network spectra, generating asymmetries, multiple spectral bands, and other features.

19.
Sociol Sci ; 6: 219-234, 2019.
Artigo em Inglês | MEDLINE | ID: mdl-31363485

RESUMO

We study the structure of heterosexual dating markets in the United States through an analysis of the interactions of several million users of a large online dating website, applying recently developed network analysis methods to the pattern of messages exchanged among users. Our analysis shows that the strongest driver of romantic interaction at the national level is simple geographic proximity, but at the local level, other demographic factors come into play. We find that dating markets in each city are partitioned into submarkets along lines of age and ethnicity. Sex ratio varies widely between submarkets, with younger submarkets having more men and fewer women than older ones. There is also a noticeable tendency for minorities, especially women, to be younger than the average in older submarkets, and our analysis reveals how this kind of racial stratification arises through the messaging decisions of both men and women. Our study illustrates how network techniques applied to online interactions can reveal the aggregate effects of individual behavior on social structure.

20.
Phys Rev E ; 99(4-1): 042309, 2019 Apr.
Artigo em Inglês | MEDLINE | ID: mdl-31108596

RESUMO

We derive a message-passing method for computing the spectra of locally treelike networks and an approximation to it that allows us to compute closed-form expressions or fast numerical approximates for the spectral density of random graphs with arbitrary node degrees-the so-called configuration model. We find that the latter approximation works well for all but the sparsest of networks. We also derive bounds on the position of the band edges of the spectrum, which are important for identifying structural phase transitions in networks.

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