Despite some improvements towards reducing hunger, malnutrition remains to be a crucial challenge in the developing world. The objective of this paper is to analyze the interplay between production diversity and dietary diversity across different seasons in rural Nigeria. The paper also investigates the relationship across different income quantiles.The study uses the Living Standards Measurement Study – Integrated Surveys on Agriculture (LSMS-ISA) dataset of the World Bank. We use two rounds of survey data (2010 and 2012) from Nigeria. Data were collected in two visits: at post-planting (from September to November), and at post-harvesting (from February to April). We analyze the relationship between production diversity and dietary diversity using different panel data regression tools.In post-harvest season, an increase in farm production diversification is associated with an increase with dietary diversity. On the other hand, production diversification does not have a significant contribution to the dietary diversity at post-planting. The analysis reveals that production diversification leads to better diet diversity for households in the second and third income quantiles.Seasonal variation on the contribution of production diversification on dietary diversity in rural Nigeria calls for the role of seasonally targeted policies. A higher propensity of households in the poorest quantile for malnutrition irrespective of the season suggests the need for targeted and continuous public health and nutrition interventions. Despite some improvements towards reducing hunger in the last few decades, malnutrition remains to be a crucial challenge in the developing world –4. GlobaSustainable agriculture captures a special attention in the SDG 2 with the aim to contribute to reduce malnutrition. In addition to the role of agriculture as a source of food, it is a major source of livelihood for the rural poor in the developing world –9. IncreAgriculture can be linked to food security and improved nutritional status in multiple and bi-directional pathways. First, agriculture and food production can directly contribute to the nutrition condition of the rural poor. Second, agricultural productivity gain can improve farm income, and income gain in turn can improve nutrition condition. Third, the types and quality of the food items produced in the farm can enhance nutritional quality. A good example could be the effect of farm production diversification on the dietary diversity and nutritional quality.Understanding the interplay between farm production diversification and dietary diversity is especially relevant in smallholder agriculture and in sub-Saharan Africa (SSA). First, SSA has a hunger prevalence of over 30% and it is a region where the number of malnourished people is increasing with time –21. SecoVulnerability to hunger can be seasonal in character –28. SmalIt is of special interest in this paper to explore the interplay between farm production diversity, dietary diversity and seasonality using an unbalanced panel data from Nigeria. In this paper, we analyze the implication of farm production diversity on the dietary diversity of the household in two seasons. We separately do the analysis in post-harvest and post planting seasons, and across different income quantiles.There are multiple approaches to quantify the level of farm diversification (and sometimes biodiversity) and dietary diversity. The following section illustrates the approaches that are used mostly for the calculation of farm production diversification and dietary diversity of households in the sample.Dietary diversity is often used as an indicator of the food quality, and is constructed from the sum of unique foodstuffs consumed in a specified period of time . There aThere are also variety of approaches based on the level of aggregation in the calculation of the index , 33. TheIn this paper, we employ the household dietary diversity score (HDDS) as it better captures if the diet is composed of various dimensions of micro-nutrients. For instance, while wheat or barley count as two different types of foods in the food variety count index, they belong to the cereal category in the food diversity index. As the dataset lacks information on the 24 h format, we use the 7 day food balance sheet for the calculation of the dietary diversity of the household. Accordingly, the household dietary diversity score is calculated from the consumption of 12 food categories  consumed in a period of 1 week. The index has a continuous score that ranges from 1 to 12. This information is collected twice, at post-harvest and post-planting. This provides the possibility to evaluate the seasonality of dietary diversity of the household. It is often the case to prepare meal together and share within the family in most SSA countries. Furthermore, the food balance sheet data are collected from rural families.Farm diversification is another important variable for our empirical analysis. Nutrition and development literature measures farm diversification in a number of ways based on the availability of the foodstuff, the required level of aggregation, the purpose of the index, presence of data, etc. , 33, 34.This study uses a large an unbalanced panel data from the Living Standards Measurement Study- Integrated Surveys on Agriculture (LSMS-ISA) of the World Bank from Nigeria. We use the two rounds (of the 2010 and 2012 years) for the analysis. Data were collected in two visits: post-harvesting (from February to April), and post-planting (from September to November). This dataset is a rich panel consisting of a wide range of information including household demographic and socio-economic conditions, production and nutrition related aspects etc. together with geographical and environmental characteristics in the area.We estimate the relationship between production diversification and dietary diversity of households using unbalanced panel data using Random Effects model (RE), Mundlak random effect model (Mundlak RE) and Fixed Effects (FE) model. We estimate the relationship without controls and by controlling for other factors, and across different seasons. We further estimate the relationship using quantile regression. Unless indicated otherwise , discussions and conclusions are based on the estimation results of fixed effects (FE) model.Table Furthermore, the dietary diversity of households varies in magnitude across the harvesting and planting seasons. The dietary diversity is higher at the time of harvest. The sample comprises of a higher percentage of male headed (87.8%) households.The kernel density graph Fig.  show thaTable Table As shown in Table Among the control variables, distance to the nearest market, age of the household head, sex of the household head, family size and education level of the household head do significantly influence the dietary diversity at post-harvest in RE and Mundlak RE models. Nonetheless, most of this relationship does not exist anymore when we do the estimation using FE model. The only variable that appears to be significant is livestock holding, and it contributes positively for dietary diversity.In Table The relationship between production diversification and dietary diversity at post-planting is significant only when we estimate them without controls in RE model , and the effect will be absorbed by the parameter that we use to control the unobserved fixed effect.Estimation of the relationship between farm production diversity and dietary diversity across income quantiles reveal the following results  for the empirical analysis. Using more time points for the analysis could be more informative. Despite these limitations, this study contributes to the existing literature on production diversification, dietary diversity and seasonality in rural Nigeria.Overall, we show that production diversification in the farm can contribute to improve the dietary diversity of the household at post-harvest, and this is not the case at post-planting. In addition, unlike rural households in the second and third quantile, farm production diversification has no effect on diet diversity even at post-harvest for households in the first quantile. This finding provides some key insights for policy that aim to improve dietary diversity and food security in rural Nigeria. First, farm production diversification can play a vital role to improve nutritional quality, and it is vital to focus on nutritionally sensitive food production interventions. This finding can also support policies aimed at increasing the diversity of farm as one key strategy to improve diet quality. Such efforts are particularly relevant especially if targeted towards nutritionally rich crop and livestock products. Second, the result confirms that food availability and diet diversity are seasonal in rural Nigeria. This suggests for the role of seasonally targeted nutrition interventions. In addition, such seasonal decline in dietary diversity calls the need for interventions to improve nutrition and market infrastructures in rural Nigeria. Third, the effect of production diversity on diet diversity varies across different income quantiles, and is not significant for the households in the first income quantiles (the poorest ones). Such a higher propensity of the poor to stay malnourished regardless of the season of food production in rural Nigeria suggests the need for targeted and continuous public health and nutrition interventions.


A novel mesoscale modelling approach for the investigation of mechanical properties of alginate aerogels is proposed. This method is based on the discrete element method and bonded-particle model. The nanostructure of aerogel is not directly considered, instead the highly porous structure of aerogels is represented on the mesoscale as a set of solid particles connected by solid bonds. To describe the rheological material behavior, a new elastic-plastic functional model for the solids bonds has been developed. This model has been derived based on the self-similarity principle for the material behavior on the macro and mesoscales. To analyze the effectiveness of the proposed method, the behavior of alginate aerogels with different crosslinking degrees  was analyzed. The comparison between experimental and numerical results has shown that the proposed approach can be effectively used to predict the mechanical behavior of aerogels on the macroscale. In an effort to tackle pollution in seas, fields and waterways, regulatory authorities support a wide-ranging ban on single-use plastics. The trend towards minimizing our dependency on synthetic crude-oil derived plastics is also echoed by the chemical industry, both in Europe and worldwide. One common solution currently in research and development is the use of natural and renewable sources of biopolymers. Examples are marine biopolymers  and plant-derived biopolymers such as cellulose, starch, lignin pectin and proteins. The biopolymers are generally recognized as the most sustainable starting molecules for beeing converted into functional material alternatives to plastics.Although some difficulties in their use, such as variable molecular weight, impurities and lability against microorganisms, can successfully be overcome , the areIn 2014–2015 it was reported that cellulose and alginate can be converted into highly porous nanostructured materials—aerogels—with exceptionally low thermal conductivities below that of air 26 mW/m·K), see Rudaz et al. , Kobayas mW/m·K, Many biopolymer aerogels have qualitatively similar fibrillar structures, making them very different from classical oxide-based aerogels with interconnected nanoparticles. For classical oxide-based aerogels a substantial body of experimental data have been reported and a wide range of modelling methods, including all-atom molecular dynamics, coarse-grained and macroscopic models, have been suggested ,12,13,14Another modelling approach is the bonded-particle method (BPM). It has been developed as an extension of the discrete element method (DEM) for modelling of granular materials . A materIn recent years BPM has been widely applied for modelling of different types of materials  ranging eveloped . From a eveloped  has deveeveloped  have proeveloped  have appeveloped  proposedHowever, until now the BPM was not applied for the modelling of highly porous plastically deformable materials such as biopolymer aerogels. In this contribution we extend the BPM with a new functional model for solid bonds and apply it for the modelling of the mechanical behavior of alginate aerogels.To characterize the mechanical behavior of the well-shaped cylindrical biopolymer aerogels samples see , quasi-sDuring compression of the aerogels two main regions can be generally distinguished: elastic deformation and plastic yielding. The initial region of elastic deformation is followed by the plastic yielding which is started for strains above 4%.From analysis of the unloading phase it can be observed that samples with high CaClE) is plotted versus the material density Considering the influence of the crosslinking degree on the mechanical behavior, To analyze the material behavior under different type of mechanical stresses, additionally to the uni-axial compression test, three-point bending tests have been carried out. The Texture Analyzer three-point bend rig HDP/M3PB was been used for the measurements. Detailed description of the performed experiments and obtained results is given in Significant buckling effects were observed during material compression. To characterize material deformation, samples before and after compression have been compared visually. In To analyze the possible formation of cracks or formation of zones with large density gradients the compressed samples were analyzed using micro computer tomography . All otThe resulting stress-strain curves obtained from the DEM simulations are shown in Furthermore, the simulation results show a deformation of the body similar to the experimentally observed buckling. In To analyze applicability of the model for different type of mechanical stresses, the material behavior during three-point bending tests has been experimentally and numerically analyzed. This has been done for aerogels with a high calcium chloride concentration of 5 g/L. In the upper part of In the lower part of Significant deviation between the results is related to the material breakage and initiation and propagation of cracks. Despite the fact, that the model is generally applicable for simulation of material breakage, in the scope of this work no breakage was modelled and the tensile strength was not adjusted. Thus, only in the experimental results crack formation can be observed. In a part of the samples, cracks propagated completely through the material and the sample was destroyed in two or more parts.In In the work at hand, a DEM-based model for the investigation of the mechanical behavior of aerogels has been proposed. The model consists of three main subparts: the structural model to represent the internal material structure, the functional model to describe the mechanical behavior of single components and the model parameters which have to be adjusted to the experimental data. The examined biopolymer aerogels consist of pores in the nanometer or sub-micrometer range. However, this was simplified for the modelling. Instead, the porous material structure was represented on the mesoscale as a set of primary particles connected with cylindrical solid bonds with diameters around 100 To analyze the capabilities of the model, it was applied to cylindrical alginate aerogels samples with varied crosslinking degree controlled by the calcium chloride concentration. The aerogel samples were produced with the following method: cylindrical gels were prepared in dialysis bags immersed in aqueous solution of calcium chloride followed by solvent exchange, polishing and supercritical drying. Uni-axial compression tests were performed to obtain stress-strain curves of the samples. Two different regimes are identifiable: an elastic and a plastic range. The results are in good agreement with general theory of mechanical behavior of foamed structures and reported results for aerogels ,30. Howei et al. .The experimental results were used as input for the simulations. Moreover, to test applicability of the model for other type of mechanical stressing, three-point bending tests have been performed for samples with high calcium chloride concentration. Analysis of the obtained simulation results has shown that, after adjustment of the model parameters for a specific calcium chloride concentration, the elastic-plastic material behavior can be accurately predicted for all other concentrations with different elasticity moduli. Only the unloading shows discrepancies between the experiments and simulations. They result from the construction of the bond model. Since the yield criteria in the bond model depend only on the strain, the recovered strain does not change with the Young’s modulus. By including a dependency on stresses this could be solved. Nevertheless, the developed model can mimic buckling effects occurring for cylindrical samples and can describe material behavior during three-point bending tests. Overall, we can conclude that the derived elasto-plastic model is capable to reproduce the macro-scale mechanical behavior of the studied aerogel with different crosslinking degrees. Since the fitting was performed only for one crosslinking degree, the parameters of the functional model seem to be material specific. The moderate deviations between experimental and simulation results indicate that the developed DEM-based model can be used for the modelling of aerogels with high efficiency.3 wt% alginate solution was prepared using sodium alginate  the calcium chloride solution was changed three times every 24 h with a fresh solution of the same concentration to ensure ionic equilibrium.After complete gelation b, hydrogThe sliced hydrogels were subjected to the solvent exchange with ethanol. It was performed in steps with 30, 60, 90 vol.% ethanol/water mixtures and finally using pure ethanol . The solvent exchange with pure ethanol was performed thrice to ensure complete removal of water . The gels were immersed in each ethanol grade for at least 4 h. Concaved top and bottom sides were polished by sand paper to obtain flat surfaces.Finally, the polished alcogels were subjected to dynamic supercritical drying with COBefore compression tests, the diameter of each cylindrical sample was measured at different heights using a caliper. The averaged diameter was used to estimate the cross-cut surface. The envelope density of the aerogels was estimated from the sample height and the averaged diameter.In the BPM, the aerogels are modelled as agglomerates consisting of smaller ideally spherical (primary) particles connected with solid bonds. The bonds are modelled as cylindrical objects with zero mass. Hence, particles can interact directly if they are in contact and they can interact via bonds. Both interactions can occur independently and, therefore, also at the same time between the same particles.structural model: spatial material distribution, positions and radii of primary particles, connection between primary particles, bonds radii, etc.;functional model: functional dependencies to describe forces and moments acting in the bonds and between primary particles;model parameters: model parameters to describe plasticity, softening, and so forth.The BPM is a mathematical model which is applied to represent real objects and to mimic their behavior. The model can be subdivided generally into three parts:The structural model is generated in two consequent stages. In the first stage a packing of primary particles was created using a force-biased algorithm. During this process, the particles are randomly placed into a cylindrical volume. The initial number of generated particles was calculated to reach a specified packing density. Afterwards, according to the calculated forces, particles are iteratively shifted to avoid interparticle overlaps. Finally, the radius of the particles is reduced. This results in a homogenous structure, despite the quite low packing density. In the second stage, cylindrical solid bonds are generated between primary particles. If the distance between any two primary particles was less than a specified threshold, the bond is generated. More detailed description of the algorithm can be found in ,33. NoteThe functional model in the BPM consists of three submodels: the particle-particle and the particle-wall interaction model as well as a model for the solid bonds. To calculate contact forces acting between primary particles and between particles and walls the visco-elastic Hertz-Tsuji  model foFor solid bonds a new elasto-plastic model was developed. At first, we will discuss the general approach and how it was derived. In the second paragraph the details of how it is realised are explained.For deriving the model, self-similarity of material behavior on the macroscale (cm) and mesoscale (mm) was assumed. Hence, the concept of the elasto-plasic bond model on the mesoscale was derived from the observed macroscopic mechanical behavior. As shown before, in the beginning of the deformation the aerogel reacts linearly elastic. Therefore, we assumed that in this range the internal structure of the aerogel is absorbing the forces. In our model this corresponds to the bonds. Due to the approach for generating the structural model it is ensured that the bonds are dominating at beginning. The experiments showed that at a given point the behaviour transforms into plastic yielding with a lower stiffness. This could correspond to a failure of the internal structure and the rise of forces more similar to interparticle forces. This is modelled by the failure of the bonds and the particle-particle interaction starts to dominate. The measured non-linearity of the slope with increasing strain rates also fits to this assumption. During unloading the aerogel reacts again almost linearly but with a higher stiffness. This is caused by a significant densification of the highly-porous aerogel. As a first approach, this is modelled by an increase of the stiffness.The new bond model is based on the previously proposed linear elastic beam model ,21. The A is the bond cross sectional area and l is the initial length of the bond. At a given compressive yield strain For both regimes, tension and compression, the model starts as a classical linear elastic beam with the same stiffnessBreakage of the bond due to tensile force is included in the model but was not considered for the work at hand.As shown in er MUSEN . The fmier MUSEN .To construct the structural model for the BPM simulation, data obtained from In The parameters of the functional model were adjusted only for the results of the samples that had a calcium chloride concentration of 5 g/L. Application of this procedure resulted in a strain rate of


We focus on a Bayesian inference framework for finite element (FE) model updating of a long-span cable-stayed bridge using long-term monitoring data collected from a wireless sensor network (WSN). A robust Bayesian inference method is proposed which marginalizes the prediction-error precisions and applies Transitional Markov Chain Monte Carlo (TMCMC) algorithm. The proposed marginalizing error precision is compared with other two treatments of prediction-error precisions, including the constant error precisions and updating error precisions through theoretical analysis and numerical investigation based on a bridge FE model. TMCMC is employed to draw samples from the posterior probability density function (PDF) of the structural model parameters and the uncertain prediction-error precision parameters if required. It is found that the proposed Bayesian inference method with prediction-error precisions marginalized as “nuisance” parameters produces an FE model with more accurate posterior uncertainty quantification and robust modal property prediction. When applying the identified modal parameters from acceleration data collected during a one-year period from the large-scale WSN on the bridge, we choose two candidate model classes using different parameter grouping based on the clustering results from a sensitivity analysis and apply Bayes’ Theorem at the model class level. By implementing the TMCMC sampler, both the posterior distributions of the structural model parameters and the plausibility of the two model classes are characterized given the real data. Computation of the posterior probabilities over the candidate model classes provides a procedure for Bayesian model class assessment, where the computation automatically implements Bayesian Ockham razor that trades off between data-fitting and model complexity, which penalizes model classes that “over-fit” the data. The results of FE model updating and assessment based on the real data using the proposed method show that the updated FE model can successfully predict modal properties of the structural system with high accuracy. Long-span bridges are important civil infrastructure systems as they are vital links in transportation networks. Bridge structures age due to excessive loading or environmental degradation; understanding their safety and serviceability performance is therefore necessary to achieve sustainable management and maintenance. As a result, high-fidelity structural Finite Element (FE) models of bridges are often critical to effective management of civil infrastructure systems by capturing their dynamic responses for conditions evaluation. However, modeling errors always exist in FE models, even though they are developed based on the best available knowledge from the design drawings and documents. The source of these errors could be simplification and idealization of connections and boundary conditions, and uncertainties in section geometries and material properties. While there are structural assessment methods developed to accommodate certain uncertainties ,2, it isDespite a long history, development of algorithms for FE model updating continues to be an active area of research in structural engineering and structural dynamics. FE model updating is typically formulated as optimization problems with the goal of minimizing the discrepancies between measured properties from a real structure and those of the FE model. For example, Zhang et al.  updated For real-world applications of FE model updating for large-scale bridges, an inherent challenge is that there are always modeling uncertainties involved, meaning that the exact behavior of a structure cannot be modeled using the incomplete information available. As a result, it is always necessary to take these uncertainties into account in model updating applications. There are various possible sources for these model uncertainties such as simplifying assumptions in construction of structural FE models; limited number of sensors for data collection from the structure; measurement noise; and changes in physical parameters such as mass and stiffness due to environmental and operational variations. Subsequently, researchers have been motivated to tackle the challenge from a Bayesian perspective e.g., ,9,10,11),10,119,1For exploring Bayesian model updating in bridge structures, Behmanesh and Moaveni  performeIn Bayesian model updating and assessment, it is important to explicitly quantify the uncertain prediction-errors which are the difference between the response of the real system (measured data) and the predicted response of the analytical model, since no model of the structural system is expected to give perfect predictions of structural responses. The prediction-error precision parameters have been shown to play an important role for the effectiveness of Bayesian model updating ,17. To dEffective applications of model updating are also highly dependent on the selection of the class of structural models. Structural model classes are a set of predictive input–output probability models for a structural system based on different parameterization of a structure. As an example, Jang and Smyth  utilizedThis paper performs Bayesian model updating and assessment on a full-scale FE model for a long-span cable-stayed bridge based on both synthetic and real structural health monitoring (SHM) data. In We employ low amplitude vibration testing and suppose that y is the model prediction of the modal parameters. A stochastic model class As part of the definition of the structural model class ic model . It is s class M  for the prior distributionlikelihood function, The initial probability information of the model parameter vector ty model  subject k et al. , it is aevidence  function The modal data The computation of the multi-dimensional integral in (6) is non-trivial. If there is no analytical solution for (5) and the data vector θ , the stovector θ ,26.stochastic model classmodel class assessment or model class selection. If different model classes are assigned equal plausibility a priori, The likelihood function in (2) for the modal data retation  shows thIt is known that no model of a structural system is expected to give perfect predictions because of complex dynamic behavior, e.g., the traffic loading induced vibration and the temperature effect. Therefore, in Bayesian FE model updating explicitly quantifying the uncertain prediction-errors is important, which is the difference between the real system output and the model output. Note that the uncertainty of the prediction-error parameter has a close relation with the posterior uncertainty quantification of the model parameters In this paper, we investigate a new treatment that integrates out the prediction-error precisions (inverse variances) The posterior PDF  sampler , which iIt is seen that the contribution of posterior uncertainties of MCMC has received much attention for Bayesian model updating in recent years, by which samples consistent with the posterior distribution of the model parameters are generated. An advantage of MCMC methods is that they can provide a full characterization of the posterior uncertainty, no matter the data available is sufficient to constraint the updated parameters , therebyj is the stage number and jth stage, which controls the speed of this gradual transition and is automatically computed in the process to form the intermediate PDFs. It can be seen from (12) that the series start with prior PDF The fundamental basis of TMCMC is that samples are taken from a series of intermediate PDFs in an adaptive manner, expressed as: jth stage samples j + 1)th stage are obtained by a resampling approach, which can be achieved as follows [Given the  follows : the plal = dummy index. It can be seen that if Having the plausibility weights, the uncertain model parameters are then resampled based on the normalized weight such that: The expectation of Finally, the evidence More details about TMCMC sampler are referred to . Note that the Bayesian FE model updating framework in this paper is based on the assumption that the structural model parameters are time-invariant. For the situation with changing structural properties, Kalman filter formulation may need to be employed which can track the changing state of the system and the estimates are updated using a state transition model and measurements. In this paper, we use the identified modal parameters from acceleration data collected during a one-year period from the large-scale WSN on the bridge, and it is assumed that the structural properties are invariant during the one-year period. In fact, the variation of the structural model parameters for real applications is one of important source of the model uncertainties which can induce larger posterior uncertainties of the structural model parameters Two other treatments for the uncertain prediction-error parameters are also investigated in the paper. The first treatment is to assign constant values for prediction-error precisions k et al. , the preAnother treatment is to update prediction-error precisions as additional uncertain parameters by Bayesian inference. Accordingly, two uncertain parameters Using the probability product rule, we can rewrite the full posterior as:It is seen that the posterior uncertainties of prediction-error precisions Bayesian model updating is performed for the Jindo Bridge, which is a twin cable-stayed bridge in South Korea, as shown in The corresponding FE model of the bridge is demonstrated in The selection of uncertain structural model parameters is part of the definition of the structural model class and plays an important role on effectiveness of the Bayesian FE model updating and accuracy of model for future response predictions. Therefore, a comprehensive sensitivity analysis is performed to determine the structural model parameters to be updated. To compute sensitivities, the parameter values are perturbed by ±10% and their corresponding natural frequencies are calculated through the FE model. Subsequently, the sensitivity of each parameter for each mode is obtained as the ratio between the frequency change and the parameter range. Moreover, complex structure systems such as long-span bridges are typically composed of a large number of structural members. Ideally, one would like to treat each structural member  as an individual substructure in the FE model so that the physical properties (mass and stiffness) of each member can be inferred. However, the information from a limited number of sensors will generally be insufficient to support such a member-level resolution of parameter inference, so substructure assemblages, each consisting of multiple structural members may be necessary in order to reduce the number of model parameters in  process . x and y directions) corresponding to the beam elements of half of the bridge girder are considered, leading to a total of 260 individual physical parameters. Subsequently, the sensitivities of these 260 parameters to the natural frequencies of the structure are analyzed.Sensitivity analysis in conjunction with clustering is performed herein to generate a series of effective uncertain structural model parameters. Firstly, to perform sensitivity analysis, a set of physical parameters from the FE model is selected. The Jindo Bridge FE model contains 129 beam elements in the bridge girder. Since the bridge girder is symmetric with respect to the location of mid-span, both mass and stiffness , and yyI (moment of inertia in y-direction) and thereby 12 clusters in total are obtained. The binary tree to the three mass clusters is shown in By performing separate clustering, three clusters for each type of physical parameters including mass, Young’s moduli, xxI and yyI). Thus, the number of uncertain parameter components in Finally, two model classes In this section, Bayesian FE model updating is performed on the bridge model. Three cases are considered corresponding to the three aforementioned treatments of prediction-error precisions, respectively, in which Case 3 considers the proposed method that marginalizes over the prediction error precision parameter. The likelihood function for the structural model parameters The simulated modal data is generated from model class ® Xeon® CPU E3-1241 and 16 GB of RAM running a 64-bit version of Windows 7.To characterize the posterior PDF of the structural model parameters, TMCMC sampler is employed for each case and 1000 samples are drawn in each stage. Uniform distributions are assigned to the prior for all normalized structural model parameter components in yy versus Ixx at the last stage are even more concentrated, though at two incorrect locations. This is due to the fact that the uncertainties of the prediction-error precision parameters have been overlooked by estimating them directly, which leads to inaccuracy in the model updating results and underestimation of the posterior uncertainties of structural model parameters, as we discussed in In i-th mode are computed as:From the comparison of the Bayesian updating results for the three cases, it is shown that the performance of the third case with prediction-error precision marginalized is superior. This is due to the fact that the posterior uncertainties of the error precision are preserved properly by integrating them as “nuisance” parameters. However, the first two cases with constant and updated error precisions have limitations that the results depend on the specified values of computed or sampled error-precisions. Therefore, prescribing a robust treatment for prediction-error precisions for Bayesian FE model updating is important because of the significant effect of the prediction error precision parameters, especially for large-scale complex civil structures such as long-span bridges. The merit of the new treatment with error precision marginalized is that the concern of the sensitivity of these parameters on the model updating performance has been alleviated significantly in a rigorous manner. In this section, Bayesian FE model updating is performed for Jindo Bridge by using long-term monitoring data from a dense deployment of a WSN that include 113 wireless sensor nodes, each composed of an Imote2 (MEMSIC) with an SHM-A  or SHM-HEnvironmental and operational conditions such as temperature and excitation amplitude have been recognized to potentially impact natural frequencies. The wireless smart sensor network measured the temperature during the deployment, offering an excellent opportunity to evaluate the relationship between modal frequencies and temperature. It was observed in Asadollahi and Li  that as Model updating problems are intrinsically ill-conditioned and often ill-posed when dealing with noisy measurement data, leading non-unique solutions. Under the Bayesian framework, incorporating as much prior knowledge of the structure as possible can help alleviate the ill-conditioning and ill-posedness ,42. TherThe modal data of all fourteen natural frequencies and the first three vertical mode shapes are used to construct the likelihood function By using the prior and posterior distributions of structural model parameters, prior and posterior predictions of modal parameters In addition to the investigation of Bayesian model updating, Bayesian model class assessment is also studied to quantify the plausibility of proposed candidate model classes given the measured data. Based on the posterior samples of mplexity . AlthougIn summary, Bayesian FE model updating of the bridge using real-world data from long-term SHM leads to an accurate FE model which has vibration properties close to the real structure system and produces more reliable predictions of structural responses. In addition, the plausibility of competing model classes for the bridge system can also be assessed based on their posterior probability from Bayes’ Theorem, giving a rigorous method for structural model selection.The presented work proposed a robust Bayesian inference method which marginalizes the prediction-error precisions and uses Transitional Markov Chain Monte Carlo (TMCMC) sampler. Bayesian FE model updating and assessment on a long-span cable-stayed bridge are performed by using both synthetic and real SHM data from a Wireless Sensor Network. This work is motivated from the fact that FE model calibration or updating for real civil structures is typically ill-conditioned and ill-posed when using noisy incomplete data because of various source of modeling uncertainties. Therefore, one should not just search for single “optimal” FE model parameters but rather attempt to describe the family of all plausible FE model parameters that are consistent with both the measured data and prior information. To achieve reliable Bayesian model updating performance for complex bridge structures, an advanced treatment of prediction-error precision parameters, named marginalizing error precisions was studied where the analytical solution of posterior probability density function (PDF) of model parameters is intractable. TMCMC was applied to draw samples from the posterior PDF of the structural model parameters. In addition, it is compared with two other existing treatments of prediction-error precisions, termed constant error precisions and updating error precisions, by investigating the associated Bayesian model updating performance. From both the theoretic analysis and simulation-based investigation, the treatment with prediction-error precisions marginalized has been found to produce the best model updating performance in terms of more accurate model parameter inference and posterior uncertainty quantification, hence achieving more reliable predictions of modal properties. In order to choose an appropriate class of models to describe the bridge system, Bayesian model class assessment has also been performed. Firstly, two model classes were defined from the parameter grouping results based on sensitivity-based clustering analysis. Bayesian FE model updating was then performed by marginalizing over prediction-error precisions given the two model classes using the identified modal data from long-term monitoring of the structure. A measure of plausibility for a class of models is given by the TMCMC sampler, which allows for rational comparisons of competing model classes based on the real data. The more probable model class gives a better balance between data-fit and model complexity. Overall, the results of FE model updating and assessment of the long-span bridge using real data measured from a long-term wireless monitoring system show that the updated FE model can successfully predict modal properties of the real structural system with high accuracy and reliability, by comparing them with the identified modal properties of the structure.


We claim that networks are created according to the priority attachment mechanism. We introduce a simple model, which uses the priority attachment to generate both synthetic and close to empirical networks. Priority attachment is a mechanism, which generalizes previously proposed mechanisms, such as small world creation or preferential attachment, but we also observe its presence in a range of real-world networks. In this paper, we show that by using priority attachment we can generate networks of very diverse topologies, as well as re-create empirical ones. An additional advantage of the priority attachment mechanism is an easy interpretation of the latent processes of network formation. We substantiate our claims by performing numerical experiments on both synthetic and empirical networks. The two main contributions of the paper are: the development of the priority attachment mechanism, and the design of Priority Rank: a simple network generative model based on the priority attachment mechanism. Each of these generative network models is based on some phenomenon which (as is often claimed) explains the underlying process of network formation. For instance, in the case of the small-world model, the alleged phenomenon is the tendency of many systems to form tightly connected groups  with incidental connections between groups serving as long distance bridges. In the case of the preferential attachment model the phenomenon which purportedly fuels the network formation process is the strong preference of vertices to connect to already well-connected vertices. In the stochastic block model, which is is a probabilistic model of the interactions between pairs of vertices, each vertex belongs to one of the groups or’blocks’, and each edge exists with a certain probability depending only on the vertices’ membership to the particular group. Fluctuations in the same block are stochastically equivalent, indicating their equivalent role in generating of the network structure. This model is popular because it can generate a wide range of large-scale networking patterns and can learn social structure in unweighted and weighted networks11.Many generative models of network formation have been proposed in the scientific literature12. Recently, an interesting proposal has been formulated to model complex networks using a stochastic sequence of predefined base actions13 and turning the re-creation of an empirical network into an optimization problem.Some of the network generators do not attempt to model real-world processes directly, but they rely on some mathematical formalism, like the sequence of Kronecker products applied to a small seed set of networks14. In systems science priority queues play an important role in controlling and scheduling16 or in recommender systems for evolving networks17. In sociology there are strong clues that the stochastic queuing theory can provide valuable insights into human dynamics20. In ecology prioritization influences the inter-species interactions23.For a long time, we have suspected that these individual phenomena are specialized instances of a more general mechanism of network creation. The main reason for this forefeeling was the fact that the generative network models seemed to be narrowly defined and each of them covered only a specific class of possible network topologies. A closer inspection of the generative network models further revealed that all of them were using, sometimes inadvertently, some type of prioritization while choosing target vertices during edge formation. The mechanism of prioritization using priority queues has a very long presence in almost all disciplines of science. In psychology priority queues are believed to be the driving mechanisms of human attention24. They have shown that substituting vertex feature distributions with global rankings of these features  as the basis for preferential attachment, one always obtains a scale-free degree distribution in the resulting network. In this context a global ranking is a universal ordering of vertices common to all vertices, whereas a local ranking denotes an ordering of vertices as perceived by an individual vertex. In addition, they have proven that it is sufficient to use ranking samples in the network growth process, as long as these samples maintain the original partial order of vertices. However, the approach presented in this paper differs from24 in that we consider not only global rankings of vertices, but we allow local rankings as well. This is a fundamental difference since our model is not limited to scale-free networks, but can produce arbitrary network topologies. In addition, our model provides means for computational re-creation of empirical networks resulting in many instances of networks with nearly identical network profiles. It is a unique and promising property in the field of complex networks.In the domain of complex networks, the idea of network growth based on rankings was introduced by Fortunato, Flammini, and MenczerIn order to verify whether the priority attachment can explain processes of network formation, we have developed Priority Rank, a simple generative network model. This model uses the priority attachment phenomenon to drive the network formation. Our experiments show that Priority Rank is capable of generating a very wide spectrum of networks. We were able to successfully re-create network topologies generated by the most popular generative network models, which suggests that the priority attachment mechanism is indeed the common denominator for these models. More importantly, examining the prioritization scheme, which resulted in the best re-generation of the original network, we could provide viable interpretations of the latent network generative process. This unique feature of the Priority Rank model is best pronounced in empirical networks.8. Is it fair to conclude that this phenomenon should apply to all complex networks? Or maybe it is valid only for scale-free networks characterized by the preferential attachment mechanism of edge formation? Unfortunately, it is impossible to conduct a proper statistical inference since only one realization of each network is available. Furthermore, this problem cannot be addressed by sampling of networks, because sampling distorts the topology of networks to the point where sub-networks of a network loose their identifying properties25.Research on complex networks is constantly struggling with a serious methodological obstacle. Despite the vast availability of networks that can be analyzed, each individual network has its distinctive characteristics and presents to a researcher a single data point. Any attempt to generalize the results obtained on one network to other networks strives with inherent vulnerabilities. The choice of target networks is arbitrary and the transfer of discovered patterns has to be selective. Oftentimes, unspoken assumptions come into play, for instance, a phenomenon discovered in a few networks of a particular class is presumed to be applicable to all networks of this class. These assumptions are much more questionable than usually assumed and they significantly weaken the scientific appeal of a discovery. Let us consider the phenomenon of the shrinking diameter of a network26. As the result, statistical comparison of seemingly similar networks suggests a much greater variability and difference between networks, which makes any machine learning extremely difficult in the domain of complex networks. Without the access to diversified training, testing, and validation sets of networks, all patterns discovered in networks are bound to over-fit and not generalize well. Also, the relationships between network generation processes proposed in the literature are very unclear. Is the preferential attachment an alternative to the small world phenomenon, or is it merely a supplementation? We note that there is no common language at the level of network generating processes, making the comparisons between different networks even more challenging.Another problem addressed in this paper is the inherent incompatibility of networks. Trying to analyze different networks and to make quantitative comparisons between them is very difficult, because in the realm of complex networks even minuscule perturbations of local topology can lead to significant differences in global network properties. This is true even for synthetic networks which have been generated from the same generative model and with slightly different parameters. Distributions of centrality measures, such as degree, betweenness, closeness or clustering coefficient, vary dramatically in response to minor changes of initial configurationsThe main contribution presented in this paper is the priority attachment mechanism of network formation. We show how priority queues built on local rankings can lead to very diversified network topologies. The function, which generates these local rankings, yields itself to interpretation. As it turns out, popular generative network models can be easily mimicked by the priority attachment, and some of these models  are a special case of the priority attachment. The phenomenon of the priority attachment can be readily incorporated into a generative Priority Rank network model, which is a simple procedure of network creation. The Priority Rank model can produce networks with very different topologies using the same principle of priority attachment. Instead of using dedicated network models one can successfully force Priority Rank to produce random networks, preferential attachment networks, small world networks, and many more. The second main feature of Priority Rank is its applicability to empirical networks. Due to the priority attachment mechanism, which forms the basis of the model, it is possible to adjust Priority Rank to deliver networks with topology very similar to a given empirical network. The Priority Rank model has only to learn the distance function used in the priority attachment. We demonstrate Priority Rank’s ability to re-create empirical networks while preserving the distributions of centrality measures and other network characteristics. This ability allows for generating whole families of similar networks, thus, for conducting statistical inference on families of networks, etc.Priority Rank is a comprehensive generative network model which can produce a very wide spectrum of network topologies, including the most popular network models.Priority Rank offers insights into the generative processes behind modeled networks. Classical machine learning algorithms can be used to find the most fitting distance function for the priority attachment mechanism. In most cases such a distance function is easily interpretable and provides explanations for the latent network formation process.Priority Rank allows us to generate multiple instances of networks with the same characteristics and distributions, because the model discovers the main generative process of network formation. It should be noted that this is incomparable with network sampling which oftentimes distorts the profile of sampled networks. Instead, the Priority Rank model allows to multiply networks for the purpose of A/B testing, statistical inference, simulations, etc.a priori such as edge creation probability in the random network model or edge rewiring probability in the small world network. This feature of the Priority Rank model is very important because, contrary to popular belief, generative network models are very sensitive to the initialization of these parameters.Priority Rank does not require any hyper-parameters to be set There are four main reasons, which make the Priority Rank model valuable:position in the local ranking. One should regard this local ranking as the priority queue which orders all of the vertices of the network from the point of view of a single vertex. This means that each vertex uses its own local ranking while creating edges. In other words, the main mechanism of network formation is the attachment of vertices driven by their individual perception of priority of other vertices, hence the name “priority attachment”. The power of the priority attachment mechanism stems from the fact that local rankings can be computed by arbitrarily complex distance functions which can either model real-world phenomena, or model adjacency matrices of empirical networks.The idea behind the priority attachment mechanism is fairly simple. Consider a new vertex which joins a network. The primary issue is the selection of target vertices to which the new vertex creates edges. Previously, several mechanisms have been proposed to model this selection process. For instance, in the Erdös-Rényi model the vertex selects target vertices randomly using the uniform distribution. In the Albert-Barabási model the vertex selects target vertices with the probability proportional to the current degree of each vertex. According to the priority attachment mechanism each vertex has a local ranking which arranges all possible target vertices by their “importance” from the point of view of the new vertex. The new vertex selects target vertices with the probability proportional to their D which is used to generate local rankings. Figure a(v), resulting in a strong preference for vertices in the local neighborhood  is the Euclidean distance between vertices vi, vj and DH is the hierarchical distance between these two vertices . When applied to the Priority Rank model, such distance function produces complex hierarchical structures  would be a sound choice for a simple model of disease spreading. In addition, if one would like to differentiate the probability of edge formation based on additional factors , the incorporation of this factor into the distance function would be trivial. Similarly, when trying to model semantic relationships between words embedded in multidimensional space (a standard tool in the contemporary NLP), it is reasonable to assume that the proximity of word embeddings is an indication of some semantic relatedness of the words. A simple way to model these relationships would be to use cosine distance to define the distance function n vertices vi and vj, such that D does not have to be symmetrical.V\{vi} representing the local ranking of vertices for the vertex viLet us now formally define the priority attachment mechanism and present the Priority Rank model. Further, we will use the following notation.vj as the target vertex for an edge originating from the vertex vi is inversely proportional to the position of the vertex vj in the ranking of vertices for vi. The probability mass function of selecting the ith element of the ranking is given byHn is the nth harmonic number, serving as the normalizing constant so that Equation nth harmonic number as γ is the Euler-Mascheroni constant, k of vertices, following examples of well-established generative models such as Watts-Strogatz small-world model3 or the Albert-Barabási preferential attachment model5. In principle, however, the number of edges each vertex creates can vary from vertex to vertex. When using the Priority Rank model to re-create empirical networks, one can sample the out-degree distribution of the empirical network to obtain a particular value of k for a given vertex.According to the priority attachment mechanism, the probability of selecting a vertex name, age, and sex  random rewiring of a single edge can drastically change network characteristics. Likewise, a rewiring of a single edge can leave the network topologically identical to the original network, not allowing to test a given algorithm on a diversified population of networks. Our model enables us, to a certain degree, to discover the process underlying the network formation, and repeat this process many times, producing “similar” network instances generated using the same generative principle. We are, in fact, not interested in the exact re-creating of empirical networks, but rather in generating many similar networks which could be treated as instantiations of a single process. In the remainder of the paper, we use the term “to re-create a network” in this less strict sense, i.e. equivalent to “to generate a similar instance of the network”.Given an empirical network 28. However, these measures tend to be computationally exhaustive and difficult to apply to really large networks. For this reason, we have decided to use a simple and well-understood network similarity measure. In order to measure the degree of network similarity we compare the distributions of centrality measures using the Kolmogorov-Smirnov (K-S) non-parametric two-sample test of the equality of continuous one-dimensional distributions. The K-S test computes the maximal distance between cumulative distribution functions and provides rejection thresholds for the null hypothesis that the compared samples are drawn from the same distribution. The question remains, how to find the distance function D which produces networks that minimize the K-S statistic for centrality measure distributions.The problem of defining a robust and flexible measure of network similarity has been studied for many years and several network similarity measures have been proposed in the literatureδ be the set of shortest paths between vertices vi and vj in the network G, and let δk be the set of shortest paths between vertices vi and vj which pass through vertex vk. Finally, let Δ denote the length of the shortest path between vertices vi and vj. A centrality measure is a function G. Four the most popular centrality measures include29:degree centralityvi. The assumption here is that a vertex is important if it is directly connected to many vertices in the network.betweenness centralityvi. This interpretation of importance highlights the influence of a vertex on communication pathways through the network.closeness centralityvi to all other vertices in the network. According to this definition, a vertex is important if it can quickly communicate with all remaining vertices in the network.page rank centralityvi. According to this definition, a vertex is important if it connects to other important vertices in the network.Let D given the description of the empirical network. Table vi, vj, and then applies weights wk to distances computed on each attribute.Sometimes it is possible to “guess” the distance function D from the existing network, which, if applied to the model, would generate the network most similar to the original one. The learning task can be defined as follows. Consider a network D to be minimized, and at the same time, the distance D should be maximized G provides the training data for a machine learning algorithm under the form of the adjacency matrix, which can be interpreted as a function:However, in most cases it is impossible to approximate the generative process of a network using a single, simple distance function. Given an empirical network, it is often precisely the aim of a researcher to deduce the guiding generative principle of a network. The main advantage of the Priority Rank model is its ability to derive the proper distance function vi, vj), which are adjacent in G, and the negative cases consist of tuples vi, vk) which are not adjacent in G. The training set can be fed into a classification algorithm, such as logistic regression, naive Bayes classifier, or SVM, to find patterns in vertex attribute co-occurrences which influence the probability of edge’s presence or absence. The model resulting from a classification algorithm can be interpreted as a condensed representation of the underlying principle of network formation. The last two distance functions presented in Table The positive cases in the training set consist of tuples G, we have applied all distance functions presented in Table The main result reported in this paper is the development of the attachment priority mechanism. In this section, we present the outcomes of conducted experiments. Their aim was to verify whether the priority attachment mechanism is capable of explaining the underlying network formation process. We have used four popular generative network models to produce synthetic networks, and we have collected 18 empirical networks from various domains to test the ability of the priority attachment to re-create these networks. The experimental protocol was as follows. Since we cannot guess which distance function will be able to best reproduce a given X, Y drawn randomly from the uniform distribution in the range Before presenting the results obtained on various synthetic and empirical networks, we want to make the following note. Priority Rank model assumes that there is an underlying process of network formation, which can be discovered using the priority attachment mechanism. Obviously, if there is no such process, or if the process is too “messy” or too complex to produce a clear signal for the formation of the network topology, Priority Rank will not be able to re-create the network. Our conjecture is that Priority Rank can be regarded as a testing mechanism for the presence, strength and consistency of the network formation process. Consider three networks N1: each vertex selects its m-nearest neighbors and creates edges to them,N2: each vertex selects its m of them to which it creates edges,N3: each vertex selects its All three networks are generated using very simple processes which are based on the Euclidean distance between vertices:N1 and N2, but it should be inherently incapable of recreating network N3. Experimental evaluation fully confirms our suspicions. As can be seen in Table N3, Priority Rank fails to find a suitable ranking of vertices for obvious reasons. In this network vertices explicitly choose their targets simultaneously from the head and tail of the distance-based ranking. The results are exacerbated when the greater network (N3 large) is considered. The Priority Rank tries to use the Euclidean distance, producing a small-world network as the result, which can be seen in the much larger values of the diameter and average path length. For networks N1 and N2, the appropriate measures do not change with the increasing size of the network by an order of magnitude. Far from being a definitive proof, above results suggest that if a network has an underlying well-defined generative process, our model combined with the priority attachment mechanism is likely to discover this process. However, if the original process cannot be represented by means of considered simple distance functions, priority attachment approach fails.The first network is generated by a straightforward deterministic process of nearest neighbors selection. In the case of the second network, the process remains basically the same, but the two-tier selection introduces some non-trivial noise to the process output. Finally, although the third network is also generated based on vertex distances, the non-linearity introduced by the process (selecting at the same time the nearest and the farthest vertices) obfuscates the output of the process. We hypothesize that the Priority Rank model should successfully re-generate networks Erdös-Rényi random model2: an empty network with Watts-Strogatz small world model3: initially, Albert-Barabási scale free model5: initial topology of the network consists of n0 vertices forming a complete graph Leskovec forest fire model8: vertices are added sequentially to the network, each out of k edges to uniformly selected targets, and then adds more edges to direct neighbors of selected targets with the burning probability Dorogovtsev-Goltsev-Mendes hierarchical model30: initially, the network consists of a single edge connecting two vertices, and in each step of the network growth an edge produces a new vertex which is immediately attached to both ends of the edge. Although the resulting network is not strictly a fractal, it has a strong hierarchical structure with power law distributions of vertex degree and local clustering coefficient. In addition, the network contains multiple loops and is far from a trivial tree-like structure.Disassortative model: an empty network with We have tested the ability of the Priority Rank model to re-create networks using both synthetic networks obtained from the theoretical network models, and empirical networks representing various domains. Generative network models used in our experiments include the following:Synthetic network statistics and the corresponding statistics of networks generated by the Priority Rank model are shown in Table The Priority Rank model is able to re-create Erdös-Rényi random network very well. As can be seen in Table The initial topology of the ring is formed in one dimension. None of the considered distance functions was able to re-create the degree distribution, although the euclidean distance function managed to generate networks with similar betweenness and closeness distributions. The standard deviation of the K-S statistic is quite large, which weakens the result. The euclidean distance function used two synthetic attributes generated from exponential and normal distributions. The failure to re-generate the degree distribution is caused by the fact that Priority Rank does not have anything akin to random edge re-wiring, which is essential for the small-world model. The ring structure can be very easily reproduced. However, the presence of a few randomly re-wired edges, which reduce average path lengths in the network, is very difficult to mimic using any distance function. Apart from this deficiency, the Priority Rank model generates networks, which are very similar to the original small world network in terms of network diameter, density, and average shortest path lengths, these characteristics are re-created almost flawlessly.As a matter of fact, Priority Rank is capable of producing small world networks. The main issue is with reproducing the topology of a particular implementation of the Watts-Strogatz model. The original Watts-Strogatz model places all vertices on the circumference of a circle and attaches neighboring nodes. Thus, the proximity of vertices is defined by the Euclidean distance in two dimensions, with vertices artificially placed on a circle. This is a highly contrived scenario which bares little resemblance to any practical application of the model. On the other hand, we have assumed in our experiment that each vertex has a single numerical attribute. The proximity of vertices is defined by the one dimensional distance in the space of this attribute. For instance, one can interpret this attribute as yearly income. The distance function reflects the tendency of people to socialize within levels of financial hierarchy. Another example could be the tendency of people to form social ties with people of similar age, i.e., age is the attribute. Any scenario when the social network is driven by strong homophily can be modeled using our distance function, provided that the driver of homophily can be expressed by a single numerical attribute. For Priority Rank successfully re-creates degree and betweenness distributions, while struggling to preserve the distribution of the closeness centrality (the linear regression distance function barely manages to pass the K-S test). The Priority Rank model also builds networks with larger diameters than the original network, but the density and the centralization of the degree distribution are very close to original values. We also note that the networks generated based on Priority Rank, in which average shortest paths are slightly longer than in the original Albert-Barabási model. Nevertheless, we conclude that synthetic scale-free networks can be mimicked by the Priority Rank model sufficiently.The Priority Rank model can re-create forest fire networks very precisely. The best results are obtained when using euclidean distance based on one attribute with values drawn randomly from the normal distribution. Such distance function delivers networks with similar centrality measure distributions , and with very similar diameter, density, average shortest path lengths and degree distribution.The Priority Rank model is capable of successful generating hierarchical network structures, such as pseudo-fractal topologies of the Dorogovtsev-Goltsev-Mendes network. The only centrality measure that has not been reproduced is the closeness. Network density is reproduced exactly whereas the remaining descriptors are rather close to their original values.As can be expected, the degree distance function was the best choice for re-creating disassortative networks. Distributions of degree and betweenness are re-created very well together with density and the average shortest path length. The diameter of the network is almost reproduced, and the only network descriptor which significantly differs from the original is Freeman’s degree centralization.Overall, the Priority Rank model can easily mimic synthetic networks produced by popular network generative models. A simple substitution of the distance function allows the Priority Rank model to produce instances of random networks, small world, scale-free, forest fire, hierarchical, and disassortative networks. In addition, one can easily introduce different variations of these generative models by modifying the distance function used to compute local priority rankings. This flexibility and ability to generalize multiple models is a unique and practical feature of the Priority Rank model.31 and The Network Repository32. Metrics used to compare empirical networks and networks generated by the Priority Rank model are presented in Table American bisons: The Priority Rank model correctly captures the underlying network structure and can re-create the network to a sufficient extent. All centrality measure distributions are retained in the generated networks, and these networks exhibit densities, and average shortest path lengths similar to the original network. The Priority Rank model slightly underestimates the diameter and the centralization of the degree distribution. Since the original network resembles the Erdös-Rényi random network, no wonder that the random distance best matches recreation process. Since Priority Rank is capable of re-creating the American bison network, this means that one could generate multiple instances of this network and belief that the generated instances reflect the same principle guiding the formation of the original network. These new networks could be interpreted as observations of another herd of bisons, or snapshots of the same group for different periods. This may be very useful for simulation purposes while analyzing transmission of a disease between individual animals living in the wild. Network instances generated with Priority Rank could be used as multiple alternative scenarios.Bighorn sheep: The Priority Rank model very convincingly re-creates the topology of the original network maintaining distributions of all centrality measures, density and the average shortest path lengths. The generated networks have slightly smaller diameters than the original one, and the reciprocity is over-estimated. This experiment, however, supports our general claim that the Priority Rank model can discover the latent generative principle of an empirical network. The page rank distance aggregates the importance of all animals in the herd. Intuitively page rank reflects the true importance of animals given the history of dominance relationships. For the sake of brevity, we report only on the best fitting distance function for each network. Nevertheless, we have analyzed several different distance functions for each network. In this case, very similar results were obtained for two distance functions, which both made use of the age attribute. Naive Bayes classifier distance and linear regression distance functions are machine learning algorithms which map the relationship between two animals’ ages onto the probability of the existence of the dominance relationship between these animals. These two distance functions were close competitors of the page rank distance function. Thus, the interpretation of these distance functions allows us to extract a general principle of this particular network formation.C. elegans: The Priority Rank model generates networks which are very similar in terms of degree and betweenness distributions to the C. elegans network, but we were unable to re-create a similar distribution of the closeness centrality measure (this could be due to a large number of vertices in the network). Also the distance, the density and the average path lengths are faithfully re-created with Priority Rank. One missing characteristic of the original network is the reciprocity. Our model cannot capture this feature using the closeness distance function. We note, however, that adding reciprocity to the distance function is straightforward. It is sufficient to include a component that would diminish the estimated distance between vertices vi and vj, if an edge  is already present in the network.Mouse visual cortex: This small network is very difficult to re-create, most probably due to the fact that its generative process may be complex. Unfortunately, the source network does not contain any additional attributes and we have to rely only on topological features of the network. In this case, the best result is obtained for the page rank distance which creates networks similar to the scale free model. The Priority Rank model is able to produce networks with very similar degree and betweenness distributions.Enzyme 108: The Priority Rank model using euclidean distance can re-generate all centrality measure distributions and obtain very similar values of diameter, density, and the average shortest paths length. The euclidean distance measure uses two synthetic attributes drawn from the uniform distribution. A distance function tends to produce networks similar to the small world model, and the enzyme 108 network definitely belongs to this network family. This small-worldliness of enzyme 108 is best manifested in the density and the average shortest path length, which are the largest among all analyzed networks. Interestingly, although Priority Rank had problems re-creating the synthetic small-world network, it manages to approximate the real world example of a small-world network very faithfully. Look at Illinois high school to see the same Priority Rank behavior. This might indicate that the original small-world model proposed by Watts and Strogatz over-simplifies the reality and the priority attachment based on the similarity of attributes is a better representation of real network formation phenomena.Cage5: The degree distance function reproduces distributions of degree and betweenness, but cannot re-create the original distribution of the closeness centrality. As for the topology of the generated network, the Priority Rank model generates very similar networks in terms of diameter, density, average shortest path length and degree centralization. Our model underestimates the reciprocity of the network, most probably due to the fact that the degree distance function alone cannot account for the increased probability of reciprocal relationships.Political books: This network is easily re-created with the Priority Rank model using the euclidean distance function based on a single discrete attribute, which reduces the function to a simple binary flag comparison. This result agrees very well with our intuition. One can interpret the synthetic discrete attribute as an indicator of a broad book category, and two books are purchased together if they belong to the same category. The Priority Rank model re-creates degree and betweenness distributions, but fails to retain the original distribution of closeness. Also the diameter, the density, the average shortest path length, and the centralization of degree are very similar to the original network. The difference in the reciprocity estimation results from lack of direction of edges in the original network. The Priority Rank model produces inherently directed networks. This example illustrates well the ability of the Priority Rank model to discover the latent process of network formation and provide simple, interpretable explanations of the underlying network structure.Primary school: The main challenge that this network poses is the density of social interactions and very short average shortest path lengths. The Priority Rank model re-creates this network reasonably well using the euclidean distance function on a single discrete attribute. Similarly to the political books network, a simple comparison of a discrete attribute is sufficient to produce a good approximation of the original network. Although the Priority Rank model cannot reproduce original distributions of betweenness and centrality, the degree distribution is very well preserved. The remaining topological characteristics of the network are also retained, except for the reciprocity — for the same reasons as for political books network. The interpretation of the distance function is similar to the political books network, namely, the discrete attribute serves as a label of a coherent social group and students have a strong preference to create relationships within the social group they belong to.Vickers 7th graders: The Priority Rank model re-creates this network perfectly using the closeness distance, where the closeness of the vertex is computed on incoming edges. With the exception of the reciprocity (slightly underestimated), all the remaining network profile is re-created precisely. One reason for this result is the fact that multiple types of edges have been flattened to a single layer in the original network, making the guessing of edge existence a bit easier. Incoming closeness seems to be a very reasonable explanation for edge presence. Popular students who have been nominated by many peers have indeed high value of inbound closeness centrality. Using this measure to generate ranking lists in the Priority Rank model yields good approximations of the original network. Again, we see how the Priority Rank model provides a simple and interpretable explanation of the latent network generation process.Freeman researchers: An interesting feature of this network is the fact that it represents communication patterns in the pre-internet era. As expected, the network is characterized by very high reciprocity, low diameter and average shortest path length. The best distance function uses betweenness centrality. Our model not only re-generates identical distributions of centrality measures, but also captures all the remaining network statistics. Again, only the reciprocity has not been matched, due to the fact that the betweenness distance function is not capable of taking the existence of an edge into consideration. Since the network represents the flow of communication between people, it is not surprising that the best distance function uses the betweenness centrality. It ranks network vertices according to their impact on the communication pathways through the network. We find this result to be a strong indication that the Priority Rank model really can provide a viable explanation for the phenomenon driving the network formation.9/11 terrorists: Similarly to the network of researchers discussed above, the Priority Rank model re-creates the 9/11 terrorists network precisely. Although the distribution of closeness centrality is not preserved, the remaining network characteristics are almost identical to the original network. The choice of the distance function is also obvious. The network has been gathered post hoc in a way to outline the associations of a selected group of terrorists, and, as the result, these terrorists possess prominent degrees in the network. Again, the interpretation of the distance function is straightforward and the result supports our claim about the explanatory power of the Priority Rank model.Karate club: Priority Rank is capable of re-creating the structures of the original network, with the notable exception of the closeness distribution and the overestimated average shortest path length. The best distance function and perfect network explanation is the closeness distance. Recall that the network represents social contacts within the karate club community at a university. The community has split in half due to a conflict between the two principal members, the administrator and instructor. Almost all of the members of the original club have links to one of the leaders of the club. Thus, one may conclude that the members had a strong preference to prioritize social contacts to these leaders, who are the two vertices with the shortest average paths to all the remaining vertices.Illinois high school: The most distinguishing feature of this network is its very large diameter and relatively high average shortest path length. Theoretically, this network tends to follow the small world network model of Watts and Strogatz. The Priority Rank model re-generates this network almost perfectly, retaining the distributions of centrality measures and all other network descriptors. As expected, the best fitting is obtained for the aggregate distance function. This function compares the values of two attributes randomly drawn from the normal distribution. As a result, vertices with many similar attribute values tend to form communities , with very few edges interconnecting these communities, hence the large network diameter. One may conclude that the evolution of these types of small world networks is primarily driven by the homogeneity of vertices constituting communities. We find it surprising that the Priority Rank model struggled to re-create the synthetic small world network, whereas it re-creates real world examples of small world networks very convincingly. In our opinion, this indicates that the assumptions behind the small world model of Watts and Strogatz are not fully supported by empirical networks.Marseille high school: Surprisingly, the Priority Rank model has struggled to reconstruct this network, not being able to re-create distributions of betweenness and closeness. However, despite the problem with retaining the original centrality measure distributions, the networks generated by Priority Rank are very similar w.r.t. the remaining network characteristics. The best distance function is linear regression, which indicates that the attributes of vertices  are crucial for inferring the existence of relationships. This is even more encouraging than previous examples where networks have been re-created using synthetic attributes. The linear regression distance function can be easily interpreted. It even provides quantitative estimations of the latent network generative process by means of regression coefficients.CAG: The main focus of this experiment was to check whether Priority Rank is capable of capturing the generative process of highly atypical network induced by integer matrices used in computations of characteristic polynomials. Our model re-creates degree and betweenness distributions, as well as most of the descriptors of the original network, except reciprocity. Since the best fitting function is the degree distance, we are tempted to conclude that the CAG network is generated primarily by the preferential attachment mechanism.Power network: This dataset representing the structure of power stations connections is challenging due to its very low density, large diameter and lack of degree centralization. Despite this, the Priority Rank model successfully reproduces the network, preserving all the distributions and measures. The best results are obtained for the random distance function, suggesting a haphazard setup of the network.Football: Priority Rank produces networks very similar to the original network, with slightly over-estimated average shortest path length and degree centralization. The former is probably caused by the greater diameters than the original network. As expected, the best fitting function is the degree distance. It reflects the socio-economic reality of the process underlying the network formation: poorer countries are exporting best players to a few countries where very wealthy football clubs reside. Here we can see that the Priority Rank model provides an interpretation of the latent network formation mechanism, even if the mechanism involves complex socio-economical processes.St. Marks ecosystem: In this experiment, we were particularly interested in discovering the underlying generative process of network formation. We can see that the Priority Rank model re-generates this network very accurately, preserving distributions of degree and betweenness and producing very similar diameters, densities, and average shortest path lengths. The betweenness distance turned out to be the best function. It is not surprising given the fact that the network focuses on modeling flows within the ecosystem. Again, we believe that the Priority Rank model discovers the main generative process of this network formation.Apart from popular generative network models, the Priority Rank model can generate networks with topologies not available through traditional models if provided with a suitable custom distance function. However, the most interesting and valuable property of the Priority Rank model is its ability to learn the generative processes of empirical networks and to generate multiple instances of these networks. The experimental evaluation of this feature is carried out on networks listed in Table In this paper, we have developed the priority attachment, a plausible principle of network formation. We have shown that priority attachment can generalize previously proposed mechanisms of network formation, such as the small world phenomenon or preferential attachment. The second major contribution of this paper is the introduction of the Priority Rank model, a comprehensive generative network model which utilizes the priority attachment principle. The Priority Rank model is capable of mimicking both synthetic networks produced by popular generative network models, as well as re-creating empirical networks. The priority attachment mechanism requires a distance function to compute local rankings for each vertex. The interpretation of the distance function allows us to infer the properties of generated networks in case of synthetic networks. More importantly, it captures the nature of the generative process imprinting formation of empirical networks. In our opinion this is the most valuable and handy feature of Priority Rank. The ability to discover the guiding principle of network formation facilitates generation of multiple similar but slightly different realizations for a given network, resulting in the whole population of networks. The stochastic nature of the priority attachment mechanism provides a small degree of randomization required to introduce variance into such population of networks, while preserving distributions of centrality measures and network profile at the same time.Enzyme 108 and Illinois high school networks are quite typical examples of small-world networks that are better represented by the Priority Rank model than by the Watts-Strogatz model3.Additionally, the Priority Rank model and a suitable distance function can provide a network generation principle that better matches real networks than the other well-known generative models. The Many different applications of the Priority Rank model can be enumerated. Apart from the obvious substitution of multiple generative network models by a single model, the Priority Rank model can be used for A/B testing of networks. Using our model, series of networks with a given profile may be generated. As a result, statistical inference on networks is possible owing to presence of many network instances belonging to the same family as the original one, which represents only a single data point. Priority Rank facilitates the simulation of various scenarios for network formation and growth. Given a model of network formation principle under the form of the distance function, we can predict evolution of networks.Priority Rank is an early implementation of the priority attachment principle, which is capable of creating or re-creating networks of varying topologies, but it should not be regarded as a final model. In fact, in the next section we outline several limitations of the current model and we sketch possible solutions. But despite some deficiencies of the Priority Rank model we see strong evidence in favor of our main claim, namely, that priority attachment is a generalization of previously proposed network formation principles and can successfully be used to explain the emergence of many network topologies.The main focus of this paper is the introduction of the priority attachment, an important and useful mechanism of network formation. Priority Rank, on the other hand, is a generative model which uses the priority attachment when generating networks. As such, Priority Rank has certain limitations which are unavoidable, since every network model has to make a compromise between accuracy and simplicity. Firstly, we note that the Priority Rank model does not always re-create empirical networks fully. It seems that the closeness distribution is notoriously difficult to reproduce. Also, we note the lack of a mechanism which would allow us to easily account for the reciprocity. Of course, this can be introduced into the model by modifying the distance function to put more weight to existing edges.Another shortcoming of the current model is the fact, that while trying to re-create an empirical network, the search for the suitable distance function is performed via brute force approach. As of now, we are searching through the parameter grid of a fixed set of template distance functions and finding the distance function which best reproduces the input network. This is both time consuming and ineffective, as the search space is limited. Possible solution to this shortcoming would be to replace currently used distance functions by a single universal distance function discovery module. We believe that for larger networks, where adjacency matrices contain sufficient amount of information, the application of neural networks to form distance functions is a viable direction of future research. As of now, our experimental evaluation is limited to fairly small empirical networks due to an excessive cost of brute force searching for a matching distance function. We have not put additional effort into optimizing this brute force approach, because we are currently working on a universal distance function discovery by a machine learning model.Finally, we note that the amount of randomness in the Priority Rank model is limited, which is most pronounced in the inability of the model to re-create small world networks (at least the networks produced by the Watts-Strogatz model). There are two places where randomness can be introduced into the Priority Rank model: either the distance function can produce random results for a subset of vertices, or one can modify the rank probabilities used by the model to select target vertices from local vertex rankings. The latter can be achieved, e.g., by flattening the distribution of probabilities of picking vertices at given positions in the ranking. This issue, will be another subject of future research.Supplementary File