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DOE OSTI2023

Fast approximation to supershell partition functions: Explicit forms of the coefficients

In a previous work, we derived a formula for supershell partition functions, which are the cornerstone of the Super-Transition-Array approach to radiative-opacity calculations. The new expression takes the form of a functional of the distribution of energies within the supershell and allows for fast and accurate computations, truncating the number of terms in the expansion. The latter involves coefficients (denoted Γ $k$ ) for which we obtained a recursion relation. In the present short paper, as a complement of the previous one, we give an explicit formula for the coefficients Γ $k$ . Another recursion relation is also provided, as well as an alternative expression involving Bell polynomials. In conclusion, the connections with cycle indexes of permutation groups and with elementary symmetric polynomials are outlined.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

Active Suspension Parameters Identification: An Algebraic Approach and its Application to Suspension Travel Control

The parameters of an active suspension need to be identified online, such that the suspension control system can be adapted to mechanical wear and load change. Recursive least squares and observer-based methods are frequently utilized to fulfill this purpose. However, they can yield slow parameter identification due to their asymptotic nature. We propose an algebraic identifier to estimate the parameters of an active suspension online, which does not maintain an asymptotic convergence phase. Simulation results demonstrate the effectiveness of the proposed algebraic approach.

Wang, Zejiang↗
DOE OSTI2012

Impact of Triple-Negative Phenotype on Prognosis of Patients With Breast Cancer Brain Metastases

Purpose: To elucidate survival times and identify potential prognostic factors in patients with triple-negative (TN) phenotype who harbored brain metastases arising from breast cancer and who underwent stereotactic radiosurgery (SRS). Methods and Materials: A total of 103 breast cancer patients with brain metastases were treated with SRS and then studied retrospectively. Twenty-four patients (23.3%) were TN. Survival times were estimated using the Kaplan-Meier method, with a log-rank test computing the survival time difference between groups. Univariate and multivariate analyses to predict potential prognostic factors were performed using a Cox proportional hazard regression model. Results: The presence of TN phenotype was associated with worse survival times, including overall survival after the diagnosis of primary breast cancer (43 months vs. 82 months), neurologic survival after the diagnosis of intracranial metastases, and radiosurgical survival after SRS, with median survival times being 13 months vs. 25 months and 6 months vs. 16 months, respectively (p < 0.002 in all three comparisons). On multivariate analysis, radiosurgical survival benefit was associated with non-TN status and lower recursive partitioning analysis class at the initial SRS. Conclusion: The TN phenotype represents a significant adverse prognostic factor with respect to overall survival, neurologic survival, and radiosurgical survival in breast cancer patients with intracranial metastasis. Recursive partitioning analysis class also served as an important and independent prognostic factor.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2012

A Phase I Study of Short-Course Accelerated Whole Brain Radiation Therapy for Multiple Brain Metastases

Purpose: To define the maximum tolerated dose (MTD) of a SHort-course Accelerated whole brain RadiatiON therapy (SHARON) in the treatment of patients with multiple brain metastases. Methods and Materials: A phase 1 trial in 4 dose-escalation steps was designed: 12 Gy (3 Gy per fraction), 14 Gy (3.5 Gy per fraction), 16 Gy (4 Gy per fraction), and 18 Gy (4.5 Gy per fraction). Eligibility criteria included patients with unfavorable recursive partitioning analysis (RPA) class > or =2 with at least 3 brain metastases or metastatic disease in more than 3 organ systems, and Eastern Cooperative Oncology Group (ECOG) performance status {<=}3. Treatment was delivered in 2 days with twice-daily fractionation. Patients were treated in cohorts of 6-12 to define the MTD. The dose-limiting toxicity (DLT) was defined as any acute toxicity {>=}grade 3, according to the Radiation Therapy Oncology Group scale. Information on the status of the main neurologic symptoms and quality of life were recorded. Results: Characteristics of the 49 enrolled patients were as follows: male/female, 30/19; median age, 66 years (range, 23-83 years). ECOG performance status was <3 in 46 patients (94%). Fourteen patients (29%) were considered to be in recursive partitioning analysis (RPA) class 3. Grade 1-2 acute neurologic (26.4%) and skin (18.3%) toxicities were recorded. Only 1 patient experienced DLT (neurologic grade 3 acute toxicity). With a median follow-up time of 5 months (range, 1-23 months), no late toxicities have been observed. Three weeks after treatment, 16 of 21 symptomatic patients showed an improvement or resolution of presenting symptoms (overall symptom response rate, 76.2%; confidence interval 0.95: 60.3-95.9%). Conclusions: Short-course accelerated radiation therapy in twice-daily fractions for 2 consecutive days is tolerated up to a total dose of 18 Gy. A phase 2 study has been planned to evaluate the efficacy on overall survival, symptom control, and quality of life indices.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2013

Is Androgen Deprivation Therapy Necessary in All Intermediate-Risk Prostate Cancer Patients Treated in the Dose Escalation Era?

Purpose: The benefit of adding androgen deprivation therapy (ADT) to dose-escalated radiation therapy (RT) for men with intermediate-risk prostate cancer is unclear; therefore, we assessed the impact of adding ADT to dose-escalated RT on freedom from failure (FFF). Methods: Three groups of men treated with intensity modulated RT or 3-dimensional conformal RT (75.6-78 Gy) from 1993-2008 for prostate cancer were categorized as (1) 326 intermediate-risk patients treated with RT alone, (2) 218 intermediate-risk patients treated with RT and ≤6 months of ADT, and (3) 274 low-risk patients treated with definitive RT. Median follow-up was 58 months. Recursive partitioning analysis based on FFF using Gleason score (GS), T stage, and pretreatment PSA concentration was applied to the intermediate-risk patients treated with RT alone. The Kaplan-Meier method was used to estimate 5-year FFF. Results: Based on recursive partitioning analysis, intermediate-risk patients treated with RT alone were divided into 3 prognostic groups: (1) 188 favorable patients: GS 6, ≤T2b or GS 3+4, ≤T1c; (2) 71 marginal patients: GS 3+4, T2a-b; and (3) 68 unfavorable patients: GS 4+3 or T2c disease. Hazard ratios (HR) for recurrence in each group were 1.0, 2.1, and 4.6, respectively. When intermediate-risk patients treated with RT alone were compared to intermediate-risk patients treated with RT and ADT, the greatest benefit from ADT was seen for the unfavorable intermediate-risk patients (FFF, 74% vs 94%, respectively; P=.005). Favorable intermediate-risk patients had no significant benefit from the addition of ADT to RT (FFF, 94% vs 95%, respectively; P=.85), and FFF for favorable intermediate-risk patients treated with RT alone approached that of low-risk patients treated with RT alone (98%). Conclusions: Patients with favorable intermediate-risk prostate cancer did not benefit from the addition of ADT to dose-escalated RT, and their FFF was nearly as good as patients with low-risk disease. In patients with GS 4+3 or T2c disease, the addition of ADT to dose-escalated RT did improve FFF.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2014

Dosimetric Predictors of Duodenal Toxicity After Intensity Modulated Radiation Therapy for Treatment of the Para-aortic Nodes in Gynecologic Cancer

Purpose: To determine the incidence of duodenal toxicity in patients receiving intensity modulated radiation therapy (IMRT) for treatment of para-aortic nodes and to identify dosimetric parameters predictive of late duodenal toxicity. Methods and Materials: We identified 105 eligible patients with gynecologic malignancies who were treated with IMRT for gross metastatic disease in the para-aortic nodes from January 1, 2005, through December 31, 2009. Patients were treated to a nodal clinical target volume to 45 to 50.4 Gy with a boost to 60 to 66 Gy. The duodenum was contoured, and dosimetric data were exported for analysis. Duodenal toxicity was scored according to Radiation Therapy Oncology Group criteria. Univariate Cox proportional hazards analysis and recursive partitioning analysis were used to determine associations between dosimetric variables and time to toxicity and to identify the optimal threshold that separated patients according to risk of toxicity. Results: Nine of the 105 patients experienced grade 2 to grade 5 duodenal toxicity, confirmed by endoscopy in all cases. The 3-year actuarial rate of any duodenal toxicity was 11.7%. A larger volume of the duodenum receiving 55 Gy (V55) was associated with higher rates of duodenal toxicity. The 3-year actuarial rates of duodenal toxicity with V55 above and below 15 cm{sup 3} were 48.6% and 7.4%, respectively (P<.01). In Cox univariate analysis of dosimetric variables, V55 was associated with duodenal toxicity (P=.029). In recursive partitioning analysis, V55 less than 13.94% segregated all patients with duodenal toxicity. Conclusions: Dose-escalated IMRT can safely and effectively treat para-aortic nodal disease in gynecologic malignancies, provided that care is taken to limit the dose to the duodenum to reduce the risk of late duodenal toxicity. Limiting V55 to below 15 cm{sup 3} may reduce the risk of duodenal complications. In cases where the treatment cannot be delivered within these constraints, consideration should be given to other treatment approaches such as resection or initial chemotherapy.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2016

Predictive Parameters of Symptomatic Hematochezia Following 5-Fraction Gantry-Based SABR in Prostate Cancer

Purpose: This study identified predictors of high-grade late hematochezia (HH) following 5-fraction gantry-based stereotactic ablative radiation therapy (SABR). Methods and Materials: Hematochezia data for 258 patients who received 35 to 40 Gy SABR in 5-fractions as part of sequential phase 2 prospective trials was retrieved. Grade 2 or higher late rectal bleeding was labeled HH. Hematochezia needing steroid suppositories, 4% formalin, or 1 to 2 sessions of argon plasma coagulation (APC) was labeled grade 2. More than 2 sessions of APC, blood transfusion, or a course of hyperbaric oxygen was grade 3 and development of visceral fistula, grade 4. Various dosimetric and clinical factors were analyzed using univariate and multivariate analyses. Receiver operating characteristic (ROC) curve analysis and recursive partitioning analysis were used to determine clinically valid cut-off points and identify risk groups, respectively. Results: HH was observed in 19.4%, grade ≥3 toxicity in 3.1%. Median follow-up was 29.7 months (interquartile range [IQR]: 20.6-61.7) Median time to develop HH was 11.7 months (IQR: 9.0-15.2) from the start of radiation. At 2 years, cumulative HH was 4.9%, 27.2%, and 42.1% in patients who received 35 Gy to prostate (4-mm planning target volume [PTV] margin), 40 Gy to prostate (5-mm PTV margin), and 40 Gy to prostate/seminal vesicles (5-mm PTV margin), respectively (P<.0001). In the ROC analysis, volume of rectum receiving radiation dose of 38 Gy (V38) was a strong predictor of HH with an area under the curve of 0.65. In multivariate analysis, rectal V38 (≥2.0 cm{sup 3}; odds ratio [OR]: 4.7); use of anticoagulants in the follow-up period (OR: 6.5) and presence of hemorrhoids (OR: 2.7) were the strongest predictors. Recursive partitioning analysis showed rectal V38 < 2.0 cm{sup 3}, and use of anticoagulants or rectal V38 ≥ 2.0 cm{sup 3} plus 1 other risk factor resulted in an HH risk of >30%. Conclusions: Rectal V38 and 2 clinical factors were strong predictors of HH following 5-fraction SABR. Planning constraints should keep rectal V38 below 2.0 cm{sup 3}.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2014

An improved fast multipole method for electrostatic potential calculations in a class of coarse-grained molecular simulations

This paper presents a novel algorithm to approximate the long-range electrostatic potential field in the Cartesian coordinates applicable to 3D coarse-grained simulations of biopolymers. In such models, coarse-grained clusters are formed via treating groups of atoms as rigid and/or flexible bodies connected together via kinematic joints. Therefore, multibody dynamic techniques are used to form and solve the equations of motion of such coarse-grained systems. In this article, the approximations for the potential fields due to the interaction between a highly negatively/positively charged pseudo-atom and charged particles, as well as the interaction between clusters of charged particles, are presented. These approximations are expressed in terms of physical and geometrical properties of the bodies such as the entire charge, the location of the center of charge, and the pseudo-inertia tensor about the center of charge of the clusters. Further, a novel substructuring scheme is introduced to implement the presented far-field potential evaluations in a binary tree framework as opposed to the existing quadtree and octree strategies of implementing fast multipole method. Using the presented Lagrangian grids, the electrostatic potential is recursively calculated via sweeping two passes: assembly and disassembly. In the assembly pass, adjacent charged bodies are combined together to form new clusters. Then, the potential field of each cluster due to its interaction with faraway resulting clusters is recursively calculated in the disassembly pass. The method is highly compatible with multibody dynamic schemes to model coarse-grained biopolymers. Since the proposed method takes advantage of constant physical and geometrical properties of rigid clusters, improvement in the overall computational cost is observed comparing to the tradition application of fast multipole method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

High-dimensional methods for quantum homodyne tomography

We provide optimized recursion relations for homodyne tomography. We improve previous methods by mitigating the divergences intrinsic in the calculation of the pattern functions used previously, and detail how to implement the data analysis through Monte Carlo simulations. Our refinements are necessary for the reconstruction of excited quantum states which populate a high-dimensional subspace of the electromagnetic field Hilbert space. •Stabilization of numerics for quantum homodyne tomography reconstructions.•High dimensional quantum homodyne tomography reconstruction.•Monte Carlo simulations of quantum homodyne tomography.•Quantum Homodyne tomography simulations.

Mosco, Nicola↗
DOE OSTI2016

X-ray and neutron total scattering analysis of Hy·(Bi0.2Ca0.55Sr0.25)(Ag0.25Na0.75)Nb3O10·xH2O perovskite nanosheet booklets with stacking disorder

Ion-exchanged Aurivillius materials form perovskite nanosheet booklets wherein well-defined bi-periodic sheets, with ~11.5 Å thickness, exhibit extensive stacking disorder. The perovskite layer contents were defined initially using combined synchrotron X-ray and neutron Rietveld refinement of the parent Aurivillius structure. The structure of the subsequently ion-exchanged material, which is disordered in its stacking sequence, is analyzed using both pair distribution function (PDF) analysis and recursive method simulations of the scattered intensity. Combined X-ray and neutron PDF refinement of supercell stacking models demonstrates sensitivity of the PDF to both perpendicular and transverse stacking vector components. Further, hierarchical ensembles of stacking models weighted by a standard normal distribution are demonstrated to improve PDF fit over 1–25 Å. Recursive method simulations of the X-ray scattering profile demonstrate agreement between the real space stacking analysis and more conventional reciprocal space methods. The local structure of the perovskite sheet is demonstrated to relax only slightly from the Aurivillius structure after ion exchange.

Metz, Peter↗
DOE OSTI2023

Machine Learning Prediction of the Experimental Transition Temperature of Fe(II) Spin-Crossover Complexes

Spin-crossover (SCO) complexes are materials that exhibit changes in the spin state in response to external stimuli, with potential applications in molecular electronics. It is challenging to know a priori how to design ligands to achieve the delicate balance of entropic and enthalpic contributions needed to tailor a transition temperature close to room temperature. Here, we leverage the SCO complexes from the previously curated SCO-95 data set [Vennelakanti et al. J. Chem. Phys. 159, 024120 (2023)] to train three machine learning (ML) models for transition temperature (T 1/2 ) prediction using graph-based revised autocorrelations as features. We perform feature selection using random forest-ranked recursive feature addition (RF-RFA) to identify the features essential to model transferability. Of the ML models considered, the full feature set RF and recursive feature addition RF models perform best, achieving moderate correlation to experimental T 1/2 values. We then compare ML T 1/2 predictions to those from three previously identified best-performing density functional approximations (DFAs) which accurately predict SCO behavior across SCO-95, finding that the ML models predict T 1/2 more accurately than the best-performing DFAs. In addition, we study ML model predictions for a set of 18 SCO complexes for which only estimated T 1/2 values are available. Upon excluding outliers from this set, the RF-RFA RF model shows a strong correlation to estimated T 1/2 values with a Pearson’s r of 0.82. In contrast, DFA-predicted T 1/2 values have large errors and show no correlation to estimated T 1/2 values over the same set of complexes. Overall, our study demonstrates slightly superior performance of ML models in comparison with some of the best-performing DFAs, and we expect ML models to improve further as larger data sets of SCO complexes are curated and become available for model training.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2026

Models and Algorithms for Equilibrium Analysis of Mixed-Material Nucleic Acid Systems

Dynamic programming algorithms within the NUPACK software suite enable analysis of equilibrium base-pairing properties for complex and test tube ensembles containing arbitrary numbers of interacting nucleic acid strands. Currently, calculations are limited to single-material systems that are either all-RNA or all-DNA. Here, to enable analysis of mixed-material systems that are critical for modern applications in vitro, in situ, and in vivo, we develop physical models and dynamic programming algorithms that allow the material of the system to be specified at nucleotide resolution. Free energy parameter sets are constructed for both RNA/DNA and RNA/2'OMe-RNA mixed-material systems by combining available empirical mixed-material parameters with single-material parameter sets to enable treatment of the full complex and test tube ensembles. New dynamic programming recursions account for the material of each nucleotide throughout the recursive process. For a complex with N nucleotides, the mixed-material dynamic programming algorithms maintain the O(N 3 ) time complexity of the single-material algorithms, enabling efficient calculation of diverse physical quantities over complex and test tube ensembles (e.g., complex partition function, equilibrium complex concentrations, equilibrium base-pairing probabilities, minimum free energy secondary structure(s), and Boltzmann-sampled secondary structures) at a cost increase of roughly 2.0-3.5×. The results of existing single-material algorithms are exactly reproduced when applying the new mixed-material algorithms to single-material systems. Accuracy is significantly enhanced using mixed-material models and algorithms to predict RNA/DNA and RNA/2'OMe-RNA duplex melting temperatures from the experimental literature as well as RNA/DNA melt profiles from new experiments. In conclusion, mixed-material analyses can be performed online using the NUPACK web app (www.nupack.org) or locally using the NUPACK Python module.

2′OMe-RNA↗
DOE OSTI2012

Many-body calculations of low energy eigenstates in magnetic and periodic systems with self healing diffusion Monte Carlo: steps beyond the fixed-phase

The self-healing diffusion Monte Carlo algorithm (SHDMC) [Reboredo, Hood and Kent, Phys. Rev. B {\bf 79}, 195117 (2009), Reboredo, {\it ibid.} {\bf 80}, 125110 (2009)] is extended to study the ground and excited states of magnetic and periodic systems. A recursive optimization algorithm is derived from the time evolution of the mixed probability density. The mixed probability density is given by an ensemble of electronic configurations (walkers) with complex weight. This complex weigh allows the amplitude of the fix-node wave function to move away from the trial wave function phase. This novel approach is both a generalization of SHDMC and the fixed-phase approximation [Ortiz, Ceperley and Martin Phys Rev. Lett. {\bf 71}, 2777 (1993)]. When used recursively it improves simultaneously the node and phase. The algorithm is demonstrated to converge to the nearly exact solutions of model systems with periodic boundary conditions or applied magnetic fields. The method is also applied to obtain low energy excitations with magnetic field or periodic boundary conditions. The potential applications of this new method to study periodic, magnetic, and complex Hamiltonians are discussed.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2012

Exponential renormalisation. II. Bogoliubov's R-operation and momentum subtraction schemes

This article aims at advancing the recently introduced exponential method for renormalisation in perturbative quantum field theory. It is shown that this new procedure provides a meaningful recursive scheme in the context of the algebraic and group theoretical approach to renormalisation. In particular, we describe in detail a Hopf algebraic formulation of Bogoliubov's classical R-operation and counterterm recursion in the context of momentum subtraction schemes. This approach allows us to propose an algebraic classification of different subtraction schemes. Our results shed light on the peculiar algebraic role played by the degrees of Taylor jet expansions, especially the notion of minimal subtraction and oversubtractions.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2015

Multi-layer Lanczos iteration approach to calculations of vibrational energies and dipole transition intensities for polyatomic molecules

We report a rigorous full dimensional quantum dynamics algorithm, the multi-layer Lanczos method, for computing vibrational energies and dipole transition intensities of polyatomic molecules without any dynamics approximation. The multi-layer Lanczos method is developed by using a few advanced techniques including the guided spectral transform Lanczos method, multi-layer Lanczos iteration approach, recursive residue generation method, and dipole-wavefunction contraction. The quantum molecular Hamiltonian at the total angular momentum J = 0 is represented in a set of orthogonal polyspherical coordinates so that the large amplitude motions of vibrations are naturally described. In particular, the algorithm is general and problem-independent. An application is illustrated by calculating the infrared vibrational dipole transition spectrum of CH{sub 4} based on the ab initio T8 potential energy surface of Schwenke and Partridge [Spectrochimica Acta, Part A 57, 887 (2001)] and the low-order truncated ab initio dipole moment surfaces of Yurchenko et al. [J. Mol. Spectrosc. 291, 69 (2013)]. A comparison with experiments is made. The algorithm is also applicable for Raman polarizability active spectra.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2015

An algebraic function operator expectation value based eigenstate determinations for quantum systems with one degree of freedom

This proceedings paper aims to show the efficiency of an expectation value identity for a given algebraic function operator which is assumed to be depending pn only position operator. We show that this expectation value formula becomes enabled to determine the eigenstates of the quantum system Hamiltonian as long as it is autonomous and an appropriate basis set in position operator is used. This approach produces a denumerable infinite recursion which may be considered as revisited but at the same time generalized form of the recursions over the natural number powers of the position operator. The content of this short paper is devoted not only to the formulation of the new method but also to show that this novel approach is capable of catching the eigenvalues and eigenfunctions for Hydrogen-like systems, beyond that, it can give a hand to us to reveal the wavefunction structure. So it has also somehow a confirmative nature.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2019

Non-iterative characteristics analysis for high-pressure ramp loading

Here in the canonical ramp compression experiment, a smoothly increasing load is applied to the surface of the sample, and the particle velocity history is measured at two or more different distances into the sample, at interfaces where the surface of the sample can be probed. The velocity histories are used to deduce a stress-density relation, usually using iterative Lagrangian analysis to account for the perturbing effect of the impedance mismatch at the interface. In that technique, a stress-density relation is assumed in order to correct for the perturbation and is adjusted until it becomes consistent with the deduced stress-density relation. This process is subject to the usual difficulties of nonlinear optimization, such as the existence of local minima (sensitivity to the initial guess), possible failure to converge, and relatively large computational effort. We show that, by considering the interaction of successive characteristics reaching a free surface, the stress-density relation can be deduced directly by recursion rather than iteration. This calculation is orders of magnitude faster than iterative analysis and does not require an initial guess. Direct recursion may be less suitable for very noisy data, but it was robust when applied to trial data. The stress-density relation deduced was identical to the result from iterative Lagrangian analysis.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Distribution of centrality measures on undirected random networks via the cavity method

The Katz centrality of a node in a complex network is a measure of the node’s importance as far as the flow of information across the network is concerned. For ensembles of locally tree-like undirected random graphs, this observable is a random variable. Its full probability distribution is of interest but difficult to handle analytically because of its “global” character and its definition in terms of a matrix inverse. Leveraging a fast Gaussian Belief Propagation-Cavity algorithm to solve linear systems on tree-like structures, we show that i) the Katz centrality of a single instance can be computed recursively in a very fast way, and ii) the probability P ( K ) that a random node in the ensemble of undirected random graphs has centrality K satisfies a set of recursive distributional equations, which can be analytically characterized and efficiently solved using a population dynamics algorithm. We test our solution on ensembles of Erdős-Rényi and Scale Free networks in the locally tree-like regime, with excellent agreement. The analytical distribution of centrality for the configuration model conditioned on the degree of each node can be employed as a benchmark to identify nodes of empirical networks with over- and underexpressed centrality relative to a null baseline. We also provide an approximate formula based on a rank- 1 projection that works well if the network is not too sparse, and we argue that an extension of our method could be efficiently extended to tackle analytical distributions of other centrality measures such as PageRank for directed networks in a transparent and user-friendly way.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗