A system-of-systems (SoS) approach is often used for simulating disruptions to business and infrastructure system networks allowing for integration of several models into one simulation. However, the integration is frequently challenging as each system is designed individually with different characteristics, such as time granularity. Understanding the impact of time granularity on propagation of disruptions between businesses and infrastructure systems and finding the appropriate granularity for the SoS simulation remain as major challenges. To tackle these, we explore how time granularity, recovery time, and disruption size affect the propagation of disruptions between constituent systems of an SoS simulation. To address this issue, we developed a high level architecture (HLA) simulation of three networks and performed a series of simulation experiments. Our results revealed that time granularity and especially recovery time have huge impact on propagation of disruptions. Consequently, we developed a model for selecting an appropriate time granularity for an SoS simulation based on expected recovery time. Our simulation experiments show that time granularity should be less than 1.13 of expected recovery time. We identified some areas for future research centered around extending the experimental factors space. Development of new technologies results in infrastructure systems becoming more interdependent, thus introducing additional complexities. These systems require and produce inputs and outputs not only for internal use by the systems themselves, but also for other infrastructure systems and businesses. Often, those businesses also provide infrastructure resources that are then delivered over a systems network. Along with these heightened interdependencies, systems disruptions are increasing in both magnitude and frequency. This is especially visible within the context of urban settings, where various interdependent systems are vital to the survival and normal operation of a society . As a reCurrently, however, there is insufficient understanding about how such propagation of disruptions between systems occurs ,3,4,5,6.Although models have been designed to examine interdependencies among infrastructure systems ,15,16,17The objectives of our study were to develop a distributed system-of-systems model of infrastructure systems and businesses to: (1) study the effects of different disruption characteristics on propagation of disruptions between constituent systems; (2) investigate how time granularity of distributed model can affect propagation of disruptions in the model; and (3) develop a framework model for selecting the most appropriate time granularity of an SoS distributed model based on expected, estimated disruption parameters. In particular, our goal was to investigate how time granularity of a simulation, as well as the recovery time and size of a disruption to a theoretical constituent network–water supply–might propagate and affect the outcome for businesses that are networked within the simulation. In this study, we also present a general framework for performing such analysis on any SoS simulation of several constituent network systems.This study aims to improve the accuracy of SoS simulations of infrastructure systems and businesses, and so the correspondence of those simulations with the real-world. Particularly, this study addresses the issues arising from combining various infrastructure system models operating at different time granularities. Inclusion of such systems in an SoS framework poses many challenges and often presents inadequate results due to disparity of time granularities between constituent SoS systems. While the choice of the overarching SoS time granularity is vital to representing disruptions propagation in the SoS model adequately, this is not addressed adequately in the current research streams as shown in the review above. Currently, disruptions represented in an SoS simulation often do not propagate properly between SoS components due to inadequate overarching time granularity of the SoS simulation. This leads to repeated simulations, which waste simulation budgets, or results in inadequate outcomes. Results of such simulations do not bring as much value because of having significant disparities with the real-world results and not representing the actual disruption events accurately. Obtaining the suitable time granularity for such simulations would bring results of these SoS simulations closer to the real-world values, what this study attempts to achieve. Consequently, as a result of this study, practitioners, as well as decision-makers, of risk and disaster management will benefit from better simulations and decision support tools, which in turn would contribute to better responses to disruptions. In particular, the gap addressed in this study is the accurate selection of the time granularity parameter in an SoS simulation. Closing this gap will lead to more accurate simulations, and thus better decisions by the users of these simulations such as infrastructure planners, policy makers, or risk managers.Our expectation in the analysis part of this study was that time granularity would have the most significant impact on the propagation of disruptions, because coarse time granularity can completely bar disruptions of very short periods from cascading to other systems in the simulation. For this study, we did not consider a variety of networks or how their topology might affect the propagation of disruptions. The cause of the initial disruption was also outside the scope of this study.The main contributions of our study were: (1) the development of a framework to select the appropriate time granularity for an SoS simulation of disruption propagation; (2) application of this framework to infrastructure and business networks resulting in a recommendation regarding the choice of the optimal time granularity, which according to our findings should not exceed 1.13 of the expected recovery time of the disrupted system. The findings can be further extended to other fields such as finance, biology, evolutionary dynamics of social economy.This study takes up the challenge of investigating the impact of time granularity on propagation of disruptions in SoS simulation of infrastructures and businesses. The rest of this paper is organized as follows. Frameworks and methodologies have been established to model individual infrastructure systems and businesses, e.g., power  or waterThe interdependencies between infrastructure systems and businesses can be modelled in an overarching framework. The SoS approach models individual systems as being autonomous in their internal operations, but at the same time connected with and affected by other systems ,27,32. TThe conceptual model shown on In this study, time granularity is defined as the frequency of performing synchronization between constituent systems–federates–of an SoS simulation, expressed as a number of timesteps between two consecutive synchronizations of all federates. The concept of time granularity is of great importance in a simulated SoS setting  because In this section, we described the specification of the model used in the simulation. In the next section, we focus on the experiment that we performed by applying the above framework.In the previous section, we described the model in detail and we described the conceptual framework that we used to derive the experimental simulation described in this section. In this section, we focus on describing the experimental setup i.e., the general framework, layout, metrics, and implementation in detail.Select number and types of individual systems to be modeled , define Identify interdependencies between selected systems and define these interdependencies .Implement individual systems’ networks into an SoS HLA simulation .Implement disruption introduction mechanism, include this mechanism in the SoS HLA simulation .Fix the time available for the simulation experiment, hence determine the number of experiments that will be executed and the experiment layout e.g., full-factorial layout or less dense layout .Run the simulation experiment and record the data .Perform ANCOVA analysis on the data to find factors affecting disruption propagation the most, and to determine the relationship of these factors to time granularity .Draw conclusions as to how time granularity of the SoS simulation should be selected for this type of simulation with regards to simulation parameters and desired accuracy .To perform the analysis of how time granularity and other factors affect propagation of disruptions between systems within a context of an SoS simulation, and consequently to choose an appropriate time granularity, we have developed the following general framework  consistiThe above steps show a general procedure to follow in order to analyze how propagation of disruptions between constituent systems of an SoS simulation is affected by different factors. This procedure allows to gather information on how to select an appropriate time granularity for the simulation. Furthermore, interactions of factors with time granularity can be tested too to understand their impact on propagation of disruptions. As a result, valuable information on how to select crucial parameters for the simulation can be obtained. This is of tremendous value when designing large scale SoS simulations. Small-scale prototype simulations such as presented in this study can be used to establish the range of values that time granularity should take for large-scale simulations of similar networks, and what factors should affect the selection of time granularity.We designed an experimental model system to examine the change in disruption patterns for constituent systems as a function of time granularity . When coThe three networks were abstractly generated based on their unique characteristics, as well as on their interdependencies and corresponding connections to each other. This process is described in more detail in The three networks included water supply network, power grid network and businesses network (Step 1–number and types of networks). All these networks have unique characteristics. Water supply and power grid networks tend to have branch network topology. Their reaction time is different with water network reacting to disruption within minutes and power grid within milliseconds, On the other hand, a business network has a branch topology too, however, with many links that connect to others without going through the common root, i.e., with hints of mesh topology. This is complemented by a lot of interdependencies to both water supply and electric grid network. The propagation of disruption within the network happens in matter of minutes to hours, and this is reflected in the model by appropriate delays. Hence, the propagation of disruptions from other networks if these appeared could be captured in the business network.After disruptions were introduced into the water supply network, their propagation through the SoS was assessed according to the impact they had on the business network. As shown in The electric grid network was introduced into the simulation to show how independent simulation that itself does not introduce disruption to the SoS can affect the propagation of disruptions between two other networks. The electric grid showed how a system that is not disrupted initially can become affected by a disruption originating in another system, and subsequently also additionally impact the observed businesses network. The main objective of the study was to understand time granularity effects on disruption propagation characteristics. Hence, electric grid was not measured itself because it would not bring significant additional value to this study, however, the process could be similarly repeated for the electric grid network as the target network of our observations.We also determined how time granularity might affect simulation speed. Because synchronization of the system required time, increasing the frequency of synchronizations, i.e., decreasing time granularity, would decrease the speed of simulation. Our goal was to understand the trade-off between accuracy and speed so that we could choose the most appropriate time granularity parameter for the SoS simulation and, hence, estimate the runtime of the simulation under such granularity.Develop three networks with interdependencies between those: water supply, power supply, businesses; described in Introduce a disruption to water supply simulation (Step 4). Disruption and experimental configurations (Step 5) are described in more detail in Register performance metrics :The implementation of the above is described in In this study, we present an application of the above procedure to an SoS consisting of three systems and we test impact of three experimental factors on propagation of disruptions between constituent systems of an SoS simulation. The rest of this study describes this scenario in more detail and outlines the application of the above procedure to the scenario, including the analysis of results with inferences on how to select an appropriate time granularity for an SoS simulation. However, the general framework in principle can be applied to any combination of infrastructure systems and other network systems in order to determine the impact of various factors on propagation of disruptions between constituent systems of an SoS simulation.We applied a full-factorial experimental layout (Step 5) to study the impact of time granularity (TG), actual recovery time (RT), and disruption size (DS) on simulation results . Those tNetworks (Step 2) used in this simulation experiment were abstract, and were randomly generated via the Erdős-Renyi model ,37.The uIn each network, nodes corresponded to units that performed operations and interacted with other nodes in the same network as well as with corresponding nodes in other federates. Edges corresponded to transfer links between operational units within each network. Although the mechanics of each network were similar, they were also distinct and abstract. Each node had its own intrinsic, internal performance, but also took inputs from the incoming edges of its network and from its corresponding nodes in the other two networks. These internal and external performances were then combined and transformed to determine the total performance of that particular node. Performance was then propagated to the following nodes through the outgoing edges. Similarly, performance was propagated through inter-network connections and a synchronization mechanism to the corresponding nodes of the other two networks. Interaction between different federates could take place only through synchronization. In doing so, we designed a working process for each constituent federate to simulate an individual infrastructure system or business network within the SoS simulation environment. Each network had similar but slightly different mechanics for calculating the performance of its own nodes.A section of network topologies is shown in 3. Similarly, in case of a power system it can correspond to the total electricity transported in the network e.g., expressed in Wh. For businesses, the MoP can represent the total economic activity in the area such as GDP or total value of goods and services produced in that area e.g., expressed in $. Such metrics succinctly show any disturbances in the respective networks by decreased production and delivery of resources, which would be required under normal operating conditions.We collected data that described how a disruption to the water supply might influence performance of the business network. The collected data included simulated, propagated recovery time (SPRT) and simulated, propagated disruption size (SPDS) of the business network . These mDisruptions and recovery are modelled in a similar fashion. Disruption is modelled by decreasing the performance of an individual node or a set of nodes sharply in one timestep to null performance in effect by removing the node from the network, and subsequently observing how the target network reacts to this. Similarly, recovery is modelled by reverting the disrupted nodes in one sharp timestep back to their original levels of performance from before the disruption had occurred, and subsequently observing how the target network rebounds and recovers from the disruption. Individual network mechanisms and within-network as well as between-network propagations are responsible for the recovery process.rec) (SPRT) enabled us to calculate the length of time needed for a system to recover after a disruption was propagated and, subsequently, retracted. Recovery was determined to be the point at which that system had returned to within 99% of its original (pre-disrupted) performance. For our simulation, we were primarily interested in the impact of actual recovery time (RT) in water supply network on the simulated propagated recovery time (SPRT) in businesses network, which would then represent the difference in recovery times due to synchronization between federates in the SoS, as defined by time granularity–our key experimental parameter. In this way we could assess the accuracy of the SoS simulation and its dependence on time granularity.As depicted in max) (SPDS), was defined and measured as the lowest point along the performance curve after a disruption was propagated to that network. Those results indicated the magnitude of such an impact by one system on another, e.g., when a disruption to the water supply interfered with operations by the business network. This approach served as an alternative metric for assessing the accuracy of a simulation based on time granularity.Another dependent variable, maximum simulated propagated disruption size  among different time granularity configurations so that we could determine how the former changes in conjunction with the latter. This was an important factor because of the trade-off found between speed and accuracy in simulations. Successful design of a framework requires selecting the most appropriate time granularity based on desired speed and some basic knowledge about the networks being simulated. All of these were goals of our study here.Finally, we devised a model of likelihood of visibility of disruption based on recovery time to time granularity ratio. Such model can help in selection of the optimal time granularity based on expected recovery time. Similarly, we developed a model that predicts estimated error for recovery time based on the relationship between simulated propagated recovery time and actual recovery time. This model can further aid with selection of an adequate time granularity. Moreover, such model could be used to estimate the actual recovery time of the original system based on the recorded recovery time in the propagated system.Planners use distributed modelling frameworks to implement the SoS approach for businesses and infrastructure systems. This involves numerous individual, autonomous systems connected with each other through inputs and outputs to other systems. One such framework is HLA ,39,40. aA particular implementation of HLA features three components: interface specification, object model template (OMT), and rules. Interface specification defines where and how constituent systems (‘federations’) communicate with RTI, a method used to join all constituent distributed systems. This component serves as the inter-federate communication and synchronization unit of the HLA. Second, OMT describes what information is exchanged between constituent federates and what updates about the federation must be communicated to those federates. Finally, rules specify what federates have to obey to subscribe to the overall HLA SoS simulation (‘federation’). When modelling infrastructure systems, the federates within an HLA simulation can include infrastructure systems, disruption generators, observers and visualization tools, patterns of user services, and businesses. All of these federates introduce dynamic changes to the systems and allow designers to observe their effects on the overall SoS.Although HLA is perfectly suited to describing and modelling the manner in which constituent systems exchange information and synchronize with each other, it does not indicate the ideal time granularity at which such synchronization and exchange of information should take place. In this context, time granularity means the frequency of exchange of information among federates , disruption generators, and other components. As such, granularity reflects the frequency of inter-system synchronization. Although specification of HLA provides some mechanisms to perform time management ,44,45, iigraph library for C++ and Python, version 0.7.1 [igraph, for graph generation, representation, and operations; and NumPy version 1.10.1 [CanvasJS library [Our simulation was developed in C++ v11  and Python 0.7.1 . Disrupton 0.7.1 , under Aon 0.7.1 . We usedn 1.10.1 , for lin library . This obThis implementation shadowed a specific scheme (Step 3). First, the infrastructure systems were developed based on the definitions established for their interconnections, number of nodes, and working mechanisms, i.e., inputs and outputs. Within each network, the nodes depended on preceding nodes and on their corresponding nodes from other federates. Once individual networks and their mechanisms were defined, the interfaces between federates were devised. This was followed by the design of HLA interfaces which considered what information and how often needs to be exchanged between federates. Finally, the overall HLA simulation was created by combining the individual constituents together to include all components of the infrastructure systems i.e., the three networks of interest.Before arriving at our final experimental design, we evaluated the systems for different individual networks, each of which was tested to assess its representation of a real-life system. Our preliminary investigation showed that the networks and HLA SoS simulation performed well individually and as a whole, adequately representing individual networks and propagating and communicating disruptions between them as required. The networks were compared with simulations of individual infrastructures as devised by domain experts. This allowed us to assess the adequacy of the networks. A small-scale testing was developed to ensure that these networks respond well to disruptions being introduced. HLA SoS simulation as a whole was similarly tested with reference to expectations described by domain experts. Small-scale easy to estimate disruptions were introduced and response of the system was recorded. This followed the expectations of the experts. Hence, the adequacy of the model was established. Furthermore, interdependencies introduced between federates in the SoS were evaluated by experts of power systems, water supply networks, and economists to ensure that these interdependencies corresponded well with the real-world and that duality of the simulation model was maintained.In this section, we presented the experimental setup, layout and implementation of the model used in the simulation experiment. In the following section, we present the results of the experiments described in this section. Moreover, we provide an analysis of this results and derive statistical models based on our simulation results, which in turn can help us in making a recommendation to the optimal time granularity for the SoS simulation described in this section.In the previous section, we described the simulation experiment. In this section, we present and analyze the results of the simulation experiment performed in accordance with descriptions from the previous sections. Consequently, we make a recommendation for to the optimal time granularity for an SoS simulation of infrastructure systems and businesses.In this Section, we evaluate the impact of various factors described in Simulated, propagated disruption size (SPDS) was measured under different experimental configurations to understand which factors had the largest impact on the simulated disruption size (SPDS). An Analysis of Covariance (ANCOVA) (Step 7) was performed in R to determine the strength of the effect of experimental factors on SPDS. The results above suggested that correctly adjusting the ratio of time granularity to the expected recovery time in a simulation would be of immense importance when simulating the disruption size in an SoS evaluation. Therefore, careful selection of the ratio of time granularity to recovery time would have to be based on the data available to us if we were to determine the optimal time granularity for a simulation and yield the most accurate simulation results. These results signify the need for an adequate time granularity selection. Adjusting the time granularity can vary the outcomes of the simulation. Hence, when designing a simulation experiment time granularity needs to be considered. In particular, we need to make sure that the results with the time granularity selected are representing the real world accurately. As our further analysis below shows, the finer time granularity would yield more accurate results, and the coarser time granularity loses accuracy of the simulation.In this Section, we evaluate the impact of various factors described in Similar to our assessment of disruption size, we analyzed the impact of experimental factors on the variation in simulated, propagated recovery time (SPRT) of the business network. Simulated recovery time (SPRT) under various configurations was measured to understand how SPRT was affected by these factors. As before, we performed ANCOVA in R, using the data obtained when measuring the SPRT. All of the findings presented above strongly suggested that time granularity (TG) was of great importance in an SoS simulation. Making proper adjustments to time granularity in reference to recovery time would be crucial if we were to obtain reliable simulation results. Furthermore, depending upon TG, simulation results could change significantly because the impact of RT and TG on the share of variation in SPRT was similar. In both cases of SPRT and SPDS, TG accounted for a larger share of the variation than did either the RT or the actual DS. Again, here we strongly see the importance of time granularity on obtaining adequate results of the simulation. The results obtained here signify that varying time granularity can result in outcomes, which do not represent real-world accurately, and in loss of propagation effects of disruptions. In particular, it is important to select time granularity which is fine enough to ensure that propagation of disruptions is noted in the simulation. The results obtained and described in this section support this view.One reason that time granularity and actual recovery time had such a great impact on the simulation outcome was that disruptions below a certain recovery time did not get propagated through the SoS simulation to other federates. If the time granularity was large enough, then when such a disruption occurred, the individual system recovered from the disruption before that propagation took place. This implied that time granularity prevented such an occurrence. As such, this phenomenon contributed to the great impact of time granularity on variation of the metrics.Hence, in this section, we derive the optimal time granularity for the simulation of infrastructure systems and businesses described in Based on these results, we then derived a model of likelihood for disruption visibility that would allow us to select time granularity of the simulation according to the (expected) recovery time of disruptions, making their propagation visible in the system. This model was used to select the maximum time granularity so that one could observe and detect an SoS disruption. Here, visibility of a disruption was understood to be a drop in performance to below 95% of the original level in the network to which disruption should propagate.glm.fit function [This logistical model  was devefunction . It depiBecause visibility of a disruption is a key parameter when simulating an SoS infrastructure network, we wished to examine whether a disruption originating in one network could propagate to another network, for how long, and with what impact. This would enable us to understand how businesses respond to a certain disruption in the water supply network. However, if an excessively large time granularity prevented such a propagation, then the simulation would be useless. Consequently, in a real-life scenario, we would not be able to determine whether the disruption would propagate. Hence, selection of appropriate time granularity for an SoS simulation is vital.Our finding here could apply to multitude of other infrastructure networks, which also exhibit similar properties. These networks also require synchronization frequencies that are below or around expected recovery times. This is because individual networks otherwise might recover before the disruption is propagated. The general rule of maintaining recovery time above time granularity would hold. However, to obtain a detailed analysis of a different set of infrastructure networks, the simulation experiment should be conducted in a similar fashion with these particular networks following our general framework presented earlier. Such approach would allow to obtain a more accurate ratio of expected recovery time to time granularity that could apply for such networks. Similarly, uncertainty of the parameters could be estimated better with approaches such as bootstrapping and rerunning the simulation multiple times in order to obtain deviation from the numbers that we had obtained in the experiment. Stochasticity within each individual network and the overall system could be captured through such approach. However, this would require significant simulation time and might not be practical given the goals of such simulation. A representative sample of results used for calculation of the metric is sufficient in order to obtain a general rule for designing SoS of infrastructure systems and businesses.Finally, the result here is of great significance, and such logistic model should be obtained when designing large simulation experiments for infrastructure systems. The time granularity needs to be fine enough to capture cascading effects of disruptions. Such model allows us to define the time granularity, which would be fine enough to ensure propagation of disruptions. This model is derived based on the previous two Sections , which sIn this Section, we evaluate the impact of time granularity, as described in Because the SoS approach often entails performing one simulation immediately after another, we want to enable performing as many simulations as possible within a limited period of time to test different scenarios. Our goal was to achieve the most rapid simulation that was also the most accurate. Since simulation speed was affected by time granularity, at any given level of accuracy, a trade-off existed between time granularity and simulation time.In our experiment, simulation time was expressed in seconds per timestep. As time granularity increased, the simulation time decreased because reduced granularities involved more synchronization between federates. Such synchronizations are time-consuming and computationally expensive, thereby lengthening the simulations. As shown in There is a clear traded-off between disruption time and time granularity of disruption. This can be seen based on results in this section. Hence, to estimate the time of running a simulation, we can use this model coupled with the minimum time granularity required obtained from the logistic model in In our SoS simulation, selecting the appropriate time granularity was critical. However, we detected a trade-off between time granularity and the speed of the simulation. The principal factors affecting our results were time granularity (TG) and recovery time (RT), both of which influenced the size of the simulated disruption to the business network (SPDS) more than did the size of the disruption to the water supply network (DS). Therefore, we concluded that the ratio of the recovery time for the original disruption to the water supply ) to the time granularity of the simulation (TG) had an even greater impact on the variation among our experimental metrics .Our simulation experiments were specifically designed to investigate how time granularity, recovery time, and size of a disruption to the water supply might propagate and affect the outcome of a business network under simulated conditions. After developing an HLA system-of-systems simulation that incorporated two infrastructure systems (water supply and power grid), a business network, and a disruption generator, we ran full-factorial simulation experiments to analyze various impacts on the propagation of disruptions. We then developed two models to assist in selecting an adequate time granularity based on expected recovery time and desired accuracy of the simulation.The results from the simulation experiment demonstrated that time granularity, recovery time from disruptions, and disruption size all had a significant impact on the outcome of our simulation. Assessments of performance by the business network indicated that the simulated propagated recovery time (SPRT) and disruption size (SPDS) were affected by all experimental factors, but especially by time granularity and recovery time. The ratio of actual recovery time to time granularity had the greatest effect on the ratio of simulated, propagated recovery time to time granularity. Our study determined that the minimum ratio of actual recovery time to time granularity at which the propagation of disruptions was visible for infrastructure system networks and businesses was 0.88 Finally, we developed a general linear model to estimate the actual recovery time based on simulated, propagated recovery time.Another important contribution of this study, besides the optimal ratio of time granularity to expected recovery time in an SoS simulation, is the development and proof-of-concept of the framework for deriving the optimal value of time granularity. This is a key contribution of this study to various fields of research that might benefit from distributed simulations of propagating disruptions including social economics, finance, sociology or biology to name a few.Our study about the effect of time granularity on propagation of disruptions in a SoS simulation of infrastructure systems and businesses networks is novel. Our simulation experiment addressed the issue of selecting an appropriate time granularity when modelling disruptions in distributed modelling SoS environments such as the Portico HLA. This has closed a major gap in current research stream. Obtaining the suitable time granularity for such simulations will bring results of these SoS simulations closer to true values. Consequently, as a result of this study, practitioners, as well as decision-makers, of risk and disaster management will benefit from better simulations and decision support tools, which in turn would contribute to better responses to disruptions. In particular, the gap addressed in this study is the accurate selection of the time granularity parameter in an SoS simulation. Closing this gap will lead to more accurate simulations, and thus better decisions by the users of these simulations.Our research findings have several further implications. Scientists can use the findings of this study to develop better models of infrastructure systems and business networks. The closing of the gap regarding selection of the appropriate time granularity will enable them to achieve better simulation representations at lower simulation costs. Scientists and researchers can select time granularity that yields the most optimal results of SoS simulations in terms of the most accurate disruption propagation representation. The experimental framework presented here can also be used to define time granularity of SoS simulations for applications other than infrastructure systems such as military systems, business systems or evolutionary dynamics of social economics. This would result in more accurate simulations in these fields thus decreasing the simulation time and budgets required, as well as yielding better results. For practitioners, policymakers, and scientists, better simulations can help them to achieve better estimates of the size and cost of actual disruptions. Thus, more accurate and faster simulations would result in better decisions regarding infrastructure development, contingency plans, and responses to ongoing disruptions.Practitioners and scientists who regularly perform such simulations can benefit from identifying the maximum time granularity required if they are still to register the propagation of disruptions. This will aid in solving the trade-off between speed and accuracy of the simulation, which in turn would allow them to perform more simulations within the same timeframe, thereby leading to savings in both cost and time. Moreover, they can benefit from the framework to perform analysis of how individual factors affect propagation of disruptions between constituent systems in an SoS simulation of infrastructure systems. Addressing the gap of selecting the optimal time granularity, through faster and more accurate simulations will result in improved speed of simulations and so decreased time and budget spent on simulations. This is advantageous to scientists, practitioners, and decision-makers alike.Our study is limited in its ability to apply these results to other networks and infrastructure systems. Although the test system was analyzed for various parameters of disruptions, the topologies and mechanics of the network were constant. In real-world scenarios, the nature of the networks, as well as their sizes, could differ. However, this can be remedied by applying our generalized framework to different situations and simulation in a small-scale, drawing conclusions from these to utilize when developing a large-scale simulation of the same systems. Similarly, the experimental space and metrics are restricted. Here, we focused on only a subset of possible factors that might influence a disruption. Another limitation was that the model of the ratio of simulated, propagated recovery time to time granularity based on the ratio of actual recovery time to time granularity may have several regimes that could be analyzed separately. Finally, our model was limited in capturing and representing stochasticity and uncertainty of results and parameters. To address these challenges, future work on this subject might include similar simulation experiments over a broader field of network topologies with different operating parameters. A real-life network could be used to investigate whether the results obtained here remain valid. Bootstrapping methods could be used to capture and represent uncertainty of results as a standard deviation of the resulting parameters. This would contribute to better representation and understanding of key results to make these more applicable. Finally, a wider range of disruption parameters could be studied, along with more experimental metrics, so that we could determine what other factors can affect the propagation of disruptions, and whether the models obtained through our study still apply.


High-resolution esophageal manometry is used for the study of esophageal motility disorders, with the help of catheters with up to 36 sensors. Color pressure topography plots are generated and analyzed and using the Chicago algorithm a final diagnosis is established. One of the main parameters in this algorithm is integrated relaxation pressure (IRP). The procedure is time consuming. Our aim was to firstly develop a machine learning based solution to detect probe positioning failure and to create a classifier to automatically determine whether the IRP is in the normal range or higher than the cut-off, based solely on the raw images. The first step was the preprocessing of the images, by finding the region of interest—the exact moment of swallowing. Afterwards, the images were resized and rescaled, so they could be used as input for deep learning models. We used the InceptionV3 deep learning model to classify the images as correct or failure in catheter positioning and to determine the exact class of the IRP. The accuracy of the trained convolutional neural networks was above 90% for both problems. This work is just the first step in fully automating the Chicago Classification, reducing human intervention. The esophagus is a muscular tube that extends from the bottom part of the throat  to the stomach. The main function of the esophagus is to transport solids and liquids into the stomach. The complex synchronization of esophageal striated and smooth muscles allows food propagation. Problems emerge when patients have difficulty swallowing or in cases of gastric reflux. These disorders can be investigated using techniques such as an upper gastrointestinal swallow study, esophagogastroduodenoscopy, pH monitoring, and esophageal manometry. In this work we will focus on esophageal motility disorders (EMD) and their diagnosis using high-resolution esophageal manometry (HRM). Esophageal HRM uses solid state or water perfused catheters, with as many as 36 circumferential sensors which generate color pressure topography plots, representing the pressures generated in the esophageal body secondary to muscle contractions during a swallow ,2,3,4.Efficient esophageal transport necessitates a coordinated, sequential motility pattern that pushes food from above while clearing acid and bile reflux from below. Disruption of this highly integrated muscular action limits food and fluid delivery while also causing an unpleasant sensation of dysphagia and chest discomfort. Peristalsis is a movement pattern produced by the synchronization of these concurrently contracting muscle layers. Peristalsis is a synchronized, consecutive contraction wave that spans the length of the esophagus, pushing food to the stomach. In gastroenterology, esophageal motility problems are not rare. The spectrum of these disorders ranges from achalasia and esophago-gastric junction outflow obstruction to minor disorders of peristalsis. Motility patterns during swallows give important information regarding esophageal contractility and sphincter relaxation in response to food administration. Esophageal manometry is the measurement of esophageal movement and pressure. Conventional esophageal manometry measures contraction and pressure using probes positioned at every 5 cm in the esophagus. This technique has recently progressed, and HRM has replaced conventional esophageal manometry as the gold standard. HRM transmits intraluminal pressure data through a high-resolution catheter, which is then translated into dynamic esophageal pressure topography graphs . The catheter was inserted trans-nasally and positioned with at least 3 sensors in the stomach. The protocol for the examination consisted of a baseline recording of 2 min, followed by 10 wet 5 mL swallows, spaced at more than 30 s, with the patient in the supine position and the thorax angulated at 30°. Every wet swallow was marked during the exam, by the performing nurse or physician, with a white vertical line from the software. In this way, it was ascertained that only wet swallows  would be analyzed, while all other swallows  would be ignored, based on current recommendations. In our study, the upper normal limit of IRP, using an Unisensor® probe, was set at 28 mmHg , and the resulted matrix was fed to the feature extraction part. For the feature extraction step, we used the InceptionV3 CNN model, without the final classification layer, and pre-trained on the ImageNet dataset, in this way leveraging the power of transfer learning and overcoming the small data problem. We built two separate models, one for IRP classification and one for probe positioning failure detection. Both models used InceptionV3 as a feature extractor. In the final part of the models, we added a custom classification layer in which a global average layer  with a dAccuracy: The number of correct classifications compared to the total number of examples.Precision: The ratio of the correctly classified positives to the total number of positive classifications.Recall: The ratio of the correctly classified positives to the total number of positives from the dataset.F1-Score: The harmonic mean of Precision and Recall.Confusion Matrix: A confusion matrix is a summary of prediction results on a classification problem. The number of correct and incorrect predictions are summarized with count values and broken down by each class. The confusion matrix shows how the classification model is confused when it makes predictions.We employed several assessment criteria to conduct a thorough review of the solution:To correctly calculate these metrics, it is important to mention that in the case of the probe position failure detection, the positive class was represented by the correct positioning of the catheter, and in the case of the IRP classification problem, the positive class was represented by the normal IRP class.After preprocessing the whole dataset of images and finding the region of interest for IRP in each image, we trained our CNN model to classify normal or high IRP images. The results of the trained neural network were very promising, and the evaluation scores are presented in The confusion matrix that was achieved on the validation set can be seen in After passing the images through the pipeline presented in the previous section and after training our CNN model, the confusion matrix presented in Medicine has already benefited from the use of Artificial Intelligence (AI) and machine learning. Two recent reviews analyzed the applications of AI and machine learning in gastroenterology ,13. AI wSearching the literature, only a few articles tackle the problem of esophageal motility diagnosis or the automation of the Chicago Classification algorithm. Most of the studies analyzed only pharyngeal changes and swallowing patterns ,22,23,24Mielens et al.  comparedTo our knowledge, only three studies ,26,27 trKou et al.  proposedJell et al. evaluated the feasibility of autonomous analysis of ambulatory long-term HRM using an AI-based system . This teIn contrast to the solutions presented above, our final goal is to fully automate the Chicago Classification algorithm, meaning that the final solution would be capable of classifying EMD based on raw images, with no input from physicians. This work is the first part of the final algorithm, and is focused on the two most important steps before applying the Chicago algorithm, namely probe positioning failure detection, which would make the interpretation of HRM recording impossible, and IRP classification. IRP is one of the main parameters in the Chicago Classification and in the algorithm. Therefore, our first objective was to determine whether the IRP is normal or higher than the cut-off, based solely on raw images. The second objective was to identify if the catheter was positioned correctly, allowing swallow interpretation. The innovation of our approach is supported by both the few references quoted and two recent reviews ,13 on apFor probe positioning failure detection, our dataset was very unbalanced, which is normal, because the specialists always try to position the catheter correctly. The recall results were smaller compared with the accuracy. This shows the effects of not having enough examples for the wrong positioning situation. We cannot apply traditional data augmentation techniques to increase the number of wrong positioning examples, because the position, the scale, and the angles are important, and obtaining these images manually would be hard because it would require positioning the catheter wrongly in a conscious manner, which would be unethical and could hurt the patient. In the future, we will try to obtain more images from other hospitals, in this way improving the performance of the probe positioning failure classifier.In this article we presented a solution for detecting the catheter positioning failure in esophageal HRM and we also trained a model to classify IRP as normal or high. In the first part of the article, we defined HRM and presented the steps of the procedure, the resulted images, and the disorders that can be detected with it. In the second part we described the preprocessing steps that were necessary to prepare the input data for the CNN models and to find the region of interest for the IRP classification problem. Then we presented the solution pipeline which was used to setup and train the models using the preprocessed images. We used two InceptionV3 deep learning models to classify the images as correct or failure in catheter positioning and to determine the exact class of the IRP. To overcome the problem of small training data, as a starting point for feature extraction we used pretrained InceptionV3 models (pretrained on the ImageNet dataset) and changed the last fully connected part to match the size of our specific problem.In the last part of the article, we presented the experimental results. These results were quite impressive, achieving an accuracy over 90% and an F1-score over 84% in both cases. Compared to other articles/attempts to automatically detect IRP higher than the cut-off, the advantage of our solution is that it is fully automatic, the features used for classification were automatically obtained by the InceptionV3 CNN model, and the classification was made based on these features with no human intervention. Furthermore, using the pre-trained InceptionV3 model we obtained even higher evaluation scores than in other articles, where the features and attributes were prepared manually by human experts.This work is just the first step in fully automating the Chicago Classification (version 3.0) algorithm, in this way assisting hospitals and doctors in their daily workplace and reducing costs and time wasted on repetitive tasks.


The aging of the population, coupled with increasing divorce and remarriage rates, are changing the structure of potential non-financial support for older parents. The purpose of this study was to examine support provided to parents aged 50+ in stepfamilies and to determine if the difference existed between help provided by natural children and stepchildren. The primary objective was to investigate whether blood ties were a significant determinant of the support if the quality of the relationship between the parent and a natural child or a stepchild was taken into account. The secondary objective was to answer the question to what extent the reciprocal exchange motive of support was observed in stepfamilies. The probability of non-financial support from children and stepchildren was estimated based on the sixth wave of the SHARE  database for European countries. Children in stepfamilies provided less non-financial help to parents than those in intact families. Stepchildren were less likely to be in stepparents’ social networks, and stepparents provided less help with childcare for grandchildren than they did to their biological children. Relationship closeness and looking after grandchildren increased the probability of non-financial support to older parents, regardless of whether the donor was a natural child or a stepchild. Rising divorce rates and falling marriage rates in Europe are changing the structure of the family ,2,3. SinThe process of population aging increased the number of people requiring support in their old age. The current developments regarding both divorce and remarriage rates had an impact on the family structure of older people and, consequently, the availability of potential donors of support ,12. The The two basic theories explaining intergenerational transfers are kinship selection and the reciprocal exchange theory. The kinship selection theory  assumes In both theories, relationship quality and emotional closeness between family members played an important role in intergenerational support. Children provided more support when relationships were close, compared to when they were distant ,26,27. RThe aim of this study was to identify the determinants of receiving support by Europeans over the age of 50 in stepfamilies while assessing the differences between non-financial help given by natural children and stepchildren. The secondary objective was to determine whether factors other than blood ties were significant in stepfamilies in the context of providing non-financial support. In particular, the quality of the relationship between the parent and the natural child or stepchild and the reciprocal exchange motive of support might have an influence. The research questions are:Q1: Do stepchildren provide less non-financial help to parents than natural children in stepfamilies?Q2: Is the likelihood of providing non-financial help to parents in stepfamilies greater for children with close relationships, regardless of whether it is a natural child or stepchild?Q3: Is the probability of receiving non-financial help by parents in stepfamilies greater from children whose children are looked after by the grandparents, regardless of whether they are natural grandchildren or step grandchildren?Understanding to what extent stepchildren could be donors of informal, non-financial help for older stepparents in the light of the ongoing demographic changes was crucial for both the point of view of older individuals as well as for future changes in social policies of countries. Prior research on support among European stepfamilies’ members was scarce, and neither provided such a broad overview of multiple European countries nor did it include all the above-mentioned factors and comparisons.In the analysis, the SHARE  database was used. The SHARE is a biennial panel study conducted on a representative sample of people aged 50 years or older and their partners across European countries . The stunorthern countries: Denmark, Swedenwestern countries: Austria, Germany, France, Switzerland, Belgium, Luxemburgsouthern countries: Spain, Italy, Greece, Slovenia, Croatia, Portugaleastern countries: Poland, Czech Republic, Estonia.The empirical analysis included seventeen European countries (Israel was excluded) grouped into four geographical regions that differed with regard to intergenerational relations and welfare systems ,33. The The sample covered 58,242 respondents after excluding those who did not have children or did not provide information about them. The data was complemented with information provided by respondents in the previous waves of the survey (for example regarding children). Respondents with missing data, after completing, were excluded from the analysis. For each respondent, the information on their family composition was available, including the relation to the child, whether it was a natural child of the respondent and the partner, a natural child of the respondent but not the partner, a natural child of the partner but not the respondent. In the sample, 3330 respondents declared to have at least one stepchild. The econometric analysis was conducted at the level of a parent and child dyad. The database included 138,490 such dyads, of which 5952 were between a stepparent and a stepchild. In order to focus on the factors influencing the probability of receiving help from children in stepfamilies, the sample of the econometric analysis was limited only to parent and child dyads in stepfamilies. The first model included all children in stepfamilies. Then, to see the exact differences between stepchildren and natural children in stepfamilies, a second model including only stepparent and stepchild dyads and a third with parent and natural child dyads in stepfamilies were estimated.For each natural child and stepchild, the parents indicated whether they received informal help from that child. Information about informal care was based on responses to these two questions:“Thinking about the last twelve months, has any family member from outside the household, any friend or neighbour given you any personal care or practical household help?” .“Is there someone living in this household who has helped you regularly during the last twelve months with personal care, such as washing, getting out of bed, or dressing?”Respondents indicated, for outside and inside the household care, three people who provided help, and if it was a child, they indicated exactly which child it was. The dependent variable was support from a child, which was a binary variable with the value one if the given child was providing help and zero otherwise. The regression models were fitted with logistic regressions. The control variables were the respondents’ sex, age, education (in years), household size, and place of residence , and country group variable. In order to control people, who required more help due to their physical condition, the model took into account the number of ADL (Activity of Daily Living) difficulties and the number of IADL  difficulties. When analysing help for older parents provided by children in stepfamilies, it was important to take into account the characteristics of the child. From the data available in the SHARE database, the child’s gender, occupation status (working or not), the geographical distance between the child and the parent (distance up to 5 km (including co-residence), 5–25 km, 25–100 km, 100 km, and over), and whether the parent looks after the grandchildren of a given natural child or stepchild were the selected variables. Apart from child characteristics, it was important to understand how the presence of other children and help received from them influenced children’s decisions to provide help. Therefore, the number of natural children and a variable indicating whether the parent received help from them were included in the second model. In the first and third models, the total number of children was included.Including the information on whether the child was in the parents’ social network was essential for testing the research questions. In the SHARE database, respondents were asked the following question:“Most people discuss with others the good or bad things that happen to them, problems they are having, or important concerns they may have. Looking back over the last 12 months, who are the people with whom you most often discussed important things? These people may include your family members, friends, neighbours, or other acquaintances. Please refer to these people by their first names.”Each respondent reported up to seven such people who were in their social network, and it was possible to identify if a particular child was in this network. A variable indicating whether a given child was in the parents’ social network was included in the second and third models. In the first model, an interaction between the variable indicating whether the child was in the parents’ social network and the variable determining whether the child was natural or stepchild was included.Due to the fact that each parent could have several children, the data in logistic regressions were clustered. This clustering method took into account nonindependence of each dyad of the same parent.Parents with natural children were the majority of the SHARE sample (94%), and parents with at least one stepchild accounted for only about 6 percent. The differences were visible between the groups of countries. In Western and Eastern Europe, the percentages were similar . Significant differences could be seen in northern countries, where this percentage was higher, 12.7%, and in southern countries, where the percentage of stepfamilies was the lowest at 1.9%. The average number of children in stepfamilies was 1.6, and it was higher than in families without stepchildren, where the average was 1.1.In stepfamilies, the share of stepchildren was higher than natural children and equalled 64 percent. Stepchildren were much less often included in stepparents’ social networks than natural children , which confirmed the results obtained by Wimer et al.  or PezziAlthough the share of stepchildren was higher than natural children, natural children accounted for 58% of all support providers in stepfamilies. Members of parents’ social network contributed most to the support in stepfamilies . Parents’ education and place of living were not statistically significant. Mothers received help more frequently from natural children in stepfamilies, and being an older stepparent increased the probability of receiving support from a stepchild. The total number of children did not have a statistically significant influence on the probability of receiving help by a parent. However, a higher number of natural children statistically significantly reduced the probability that a stepchild would provide non-financial help. On the other hand, non-financial support from other children in the family increased the probability that a given child would help. This might be explained by the prevailing values and the way of upbringing children in a given family. Children from homes where helping other family members was highly valued tended to help their parents or stepparents alike. Country effects were significant for Eastern European countries, where the probability of support from natural children and stepchildren in stepfamilies was higher than in Northern European countries. Additionally, with regards to helping received from stepchildren, the probability was higher also in Northern European countries. Due to the cultural background in the Eastern countries, informal non-financial help provided mainly by children was much more frequent than in the Northern and Western European countries.The study had some limitations, mostly related to the SHARE database itself. First, we only analysed the fact of receiving support without information about the amount of help. The help as such, defined as zero-one-variable, did not provide full information about the involvement in support. It was possible that both natural children and stepchildren helped their parents in stepfamilies, but the former spend much more time doing this. Another limitation concerned the sample size of parents in stepfamilies. The SHARE database included people over 50 years old. The average age of the parents was around 68, while the stepparents were younger. Stepchildren were rare among the oldest respondents. Due to the growing number of divorces and remarriages, the percentage of stepfamilies will increase in the following years, which would allow the study to be repeated on a larger sample. This could lead to the identification of a greater number of statistically significant determinants of receiving help.The presented study showed that children in stepfamilies provided less help than those in intact families. Non-financial support within stepfamilies was more often provided by natural children. It was also worth noting that stepchildren were less likely to be in stepparents’ social networks and received less help from them in looking after the grandchildren, as compared to biological children. However, both these factors, i.e., the closeness of the relationship and caring for grandchildren, motivated natural children and stepchildren alike to provide non-financial assistance to older parents or stepparents. Investing in a good and close relationship between a stepchild and a stepparent was less common and probably more difficult due to the lack of blood ties. However, it might result in better mutual non-financial support. In the face of demographic changes leading to an increase in the number of elderly people with stepchildren in the family, further research into the relationship and exchange of non-financial support between stepparents and stepchildren will help determine the scale and type of support that older people can receive.