prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the scaling form for the alternative order parameter O?

Read the following text and answer briefly.

Context 1: \section*{Dynamical Behaviour of $O$ in Lattice Gases}

The dynamical behaviour of the anisotropic order parameter  $m$ [see Eq ~\eqref{eq:def-m} in the Letter] following a quench to the critical point is well described by
the Gaussian theory for all the three lattice gas models studied, $i e ,$ driven lattice gas with either constant (IDLG) or random (RDLG) infinite drive and equilibrium lattice gas (LG)  In other words, in the short-time regime, $m \sim t^{1/2}$ [see Eq  \eqref{eq:mt}] and the Binder cumulant $g$ of the lowest transverse mode [defined in Eq  \eqref{eq:binder}] is zero in this regime  The alternative order parameter $O,$ however, distinguishes between the driven (IDLG, RDLG) and the equilibrium (LG) lattice gases  

In order to understand  this, we first write the phenomenological scaling form for $O$,  analogous to Eq  \eqref{eq:scalingass} in  the Letter,
\begin{eqnarray}
O (t, L_{\parallel} ; S_\Delta) = L_{\parallel}^{-\beta/[\nu(1+\Delta)]} \tilde f_O (t/L_{\parallel}^{z/(1+\Delta)} ; S_\Delta) \quad
\label{eq:Oscalingass}
\end{eqnarray}
We already remarked that, in the LG, this scaling form is not compatible with the prediction $O \sim t^{1/8}  L_{\parallel}^{-1/2}$ of the Gaussian theory   However, following Ref  \cite{AS2002}, it can be argued that, at short times, the only dependence of $O$ on the system size $L_{\parallel}$ is of the form $O \sim L_\parallel^{-1/2}$ which is very well confirmed by numerical simulations  Accordingly,  the generic behaviour of $O$ can be assumed to be
\begin{eqnarray}
O \sim t^{\alpha} L_\parallel^{-1/2}, \label{eq:O}
\end{eqnarray}
where $\alpha$ is a phenomenological exponent to be determined  This, along with Eq  \eqref{eq:Oscalingass}, implies $\tilde f_O(x) \sim x^{\alpha} $ Comparing the finite-size behaviour in Eq ~\eqref{eq:O} with Eq ~\eqref{eq:Oscalingass} one actually infers,
\begin{eqnarray}
\alpha &=& \frac{1+ \Delta -2 \beta/\nu}{2 \, (4- \eta)}  \label{eq:alpha}
\end{eqnarray}
This equation, together with the hyperscaling relation $\Delta - 2 \beta/\nu= - \eta$ in two spatial dimensions, shows that the prediction $\alpha = 1/8$ of the Gaussian theory [see Eq  \eqref{eq:Ot}] can be obtained only when $\eta=0,$ which is the case for the IDLG (exactly) and the RDLG (approximately) but not for the LG  

On the other hand,  Eq ~\eqref{eq:alpha} predicts $\alpha = 1/10$ upon substituting the values of the critical exponents corresponding to the Ising  universality class (LG)  This is consistent with the numerical simulation results presented in the main text, see Fig  \ref{fig:ising}(b) therein 

\begin{figure}[th]
\vspace*{0 2 cm}
 \centering
 \includegraphics[width=10 cm]{ /compare_binder pdf}

\caption{Comparison between the temporal evolution of the Binder cumulants $g$ corresponding to the $12^{th}$ transverse mode, $i e ,$ with $n_\perp =12,$ in the LG (lowest curve), IDLG and RDLG (two upper curves) on a $32 \times 32$ lattice  \label{fig:b}}
 \label{fig:binder}
\end{figure}


The emergence of this new value $1/10$ of the exponent $\alpha$ must be traced back to the non-Gaussian nature of higher fluctuating modes in the LG  In fact, even though the lowest mode behaves identically in all the three models we considered,  characterized by the same behaviour of $m$, higher modes show a significant difference in the non-driven case   


To illustrate this, we measured the Binder cumulants of higher modes which is defined  analogously to Eq ~(11), using transverse modes other than the first, i e , with $\mu=\tilde \sigma(0,2 \pi n_\bot/L_\bot)$ and $n_\bot>1 $  
 Figure \ref{fig:b} compares the same for all the three lattice gases for the mode with $n_\perp =12$ on a $32 \times 32$ lattice  Clearly, the curve corresponding to the LG (lowest, blue) departs from Gaussian behaviour $g=0$ (in practice, $e g ,$ $|g| \lesssim 0 005,$ corresponding to the shaded gray area) much earlier than it does for the IDLG  or RDLG (two upper curves, red and green respectively) 

Accordingly, the different dynamical behaviour of $O$, which involves a sum over all modes, can be attributed to the non-Gaussian nature of the higher modes in the LG  
Such a departure is not entirely surprising  In fact, for higher modes, mesoscopic descriptions such as the ones in Eqs  \eqref{eq:L-DLG} or \eqref{eq:g_evol} are not expected to hold, while the anisotropy at the microscopic level could be the mechanism leading to the Gaussianity of higher modes in the driven models 


Context 2: Hospitals as critical infrastructures has been historically dependent on various types of devices and equipment that are being revolutionized with digitalized solutions  The digitalization of conventional healthcare equipment is added with the new inclusion of numerous new devices for data collection, analysis, communication, and so on  All in all, the futures digital hospitals in 5G will be exponentially more data-dependent and digital-intensive  For that, this paper looks to theorize how the security scenario in a futures digital hospital would look like, and what relevant business possibilities could emerge for cybersecurity providers in the healthcare context  In this paper, we open up discussions on business possibilities relevant to Internet of Things-mobile device management for critical infrastructures such as future digital hospital  We apply business models as a conceptual lens to analyze how cybersecurity business could evolve for 5G enabled IoT-Mobile device management providers as a cybersecurity vendor 
The healthcare sector has progressed significantly since the introduction of Internet and proliferation of network technologies   Among many issues, the use of data, availability of data, data mass, and access control of data in healthcare remain are critical for keeping healthcare services trustworthy and secure for the end users, both hospital staff and the patients  Disruptions in healthcare services would have severe effects on people’s lives  However, as hospital managers and professionals need to design their data-dependent and digital-intensive networks in a manner, which is highly secure, they also should provide the basis for uninterrupted service for business sustainability  Security is often observed as a tradeoff between risk and business gains   Investing in security is important in order to secure business-critical systems and data for meeting business goals and eventually for creating competitive advantage [3, 4] 
Innovative technologies have the power to disrupt industries and prompt business transformation   The coming of fifth generation (5G) of telecommunications networks is seen to result in this kind of disruption  As we are gradually moving towards 5G, it is worthwhile to theorize how the security scenario in a futures digital hospital would look like, and what relevant business possibilities could emerge from cybersecurity in the healthcare context  From this perspective, in this paper, we open up discussions on business possibilities relevant to Internet of Things mobile device management for critical infrastructures such as future digital hospital  We apply business models as a conceptual lens to analyze how cybersecurity business could evolve for 5G enabled IoT device management providers as a cybersecurity vendor 
A futures digital hospital facility is envisioned to consist tech-aided advanced critical medical devices, intelligent information systems, digital communication tools, hundreds of handheld mobile devices (smartphones, tablets), wireless clinical wearables, in addition to thousands of smart IoT nodes [6, 7]  These devices should be fully integrated to improve staff productivity, hospital functions, patient safety and privacy, and, overall improve patient experience through secure and reliable healthcare services  However, inclusion of these various kinds of digital devices to the hospital context make the overall device network quite complex and heterogeneous   Thus, from a critical infrastructures view point, to manage, configure, update and secure the immense fleet of digital devices besides all the high-tech medical equipment, the futures digital hospital will need to redefine device management policy and services  
Mobile device management (MDM) systems are usually referred to “support centralized control of an entire fleet of mobile devices (smartphones and tablets) and mobile applications by applying and ensuring pre-defined configuration settings” [8, 9]  In the scope of this paper, we broadly use the term IoT-MDM to refer to a device management system that is capable of managing, configuring and updating both handheld mobile devices and IoT devices in combination in a centralized manner  We will briefly open up the concept of MDM and IoT-MDM 
The purpose of this study is to identify business potential for IoT-MDM service providers as cybersecurity vendor in the context of the future digital hospital  In doing so, we apply the concept of business model in order to make sense of a ICT-oriented business environment   Among various available conceptualizations, business model is considered as a boundary-spanning unit of analysis that explain the underlying business logic and the value creation and value capturing logic of an organization [11–15] 
Traditionally, device management has been associated with management and configuration of handheld mobile devices , thus, mobile device management (MDM)  Gartner  perceives MDM software to be a policy tool to configure and manage mobile handheld devices  They also mark that MDM services need to ensure security in reference to connectivity and content that is being transmitted  Along with surge of smart mobile devices, the Internet of Things (IoT) is growing large during the last few years and promises to flood the market with billions of devices in the coming years too   Zhang et al   states scalability, transparency and reliability as important issues that differentiates IoT from the conventional Internet  To that end, there are several IoT platforms available currently in the market for managing, updating and configuring IoT nodes, e g  IBM Bluemix, Cumulocity, ARM mbed OS, etc    However, the transition raises the question about the differences and similarities between MDM and IoT device management as approaches 
Takalo  marks MDM and IoT device management to be quite close on a conceptual level: both need solution for automated management of large device fleets consisting different form factors, device models, and operating system  Additionally, such systems conceptually needs to support various communication channels like: WiFi, cellular network and Ethernet  However, on practical level, MDM is more strictly controlled by operating systems and device vendors  IoT device management, on the other hand, is characterized by multiple operating systems, multiple hardware platform variations, non-complete standards, multiple communication methods and protocols  This agile approach allows the coexistence of various types of devices and nodes in the same environment  Recently, services are appearing where MDM and IoT device management are brought together under the same platform, which in a way reduces the complexity in device management and also improves the overall security of the system 
While we are still in early phase of mass IoT deployment and 5G deployment is approaching, it is important to revisit some of the key security threats that has been identified in recent literature relevant mainly to IoT  Zhang et al  [19, 21] identified privacy preservation as a critical issue for information security in IoT ecosystems  They further state that conventional naming, identification and authentication policies need to be improved and rather needs approaching differently  Farooq et al   look at the four layer generic IoT architecture consisting perception layer, network layer, middle-ware layer & application layer  Additionally, they offer a list of security challenges for each layer  Some of the key challenges are: unauthorized access, tag cloning, eavesdropping, spoofing, RF jamming, sinkhole attack, sleep deprivation attack, denial of service attack, malicious code injection, man-in-the-middle attack, spear-phishing attack, sniffing attack  Backman et al   state, a comprehensive security solution needs to address endpoint security, management and monitoring security, and secure data distribution and storage 
Ortbach et al   enumerate the drivers for adoption of MDM in organization through a quantitative analysis, reflecting three broad drivers: organization, environment, and technology  From organizational perspective, the company size, mobile IT usage, employee innovativeness with IT and BYOD (bring your own device) culture were identified as important drivers  From environmental viewpoint, regulations and other business partner influence are significant drivers for MDM adoption  Finally, from technology point of view, perceived security benefits and perceived cost of the service seems to affect the managerial attitude towards MDM adoption 
MDM systems are today a very common tool to manage users’ devices  With MDM, all mobile device types, tablets and PCs with typical operating systems can be controlled centrally   It is often thought that MDM can manage only mobile phones, but actually the MDM framework includes also users’ identities and profiles  This makes MDM a viable tool for organizations to manage their employees identities, user profiles, all devices, all applications and security controls under same system 
From an emerging technology perspective, SDN (software-defined network)/NFV (network functions virtualization) based 5G Slicing will challenge some of the traditional MDM features   Especially end-to-end security from device to IT cloud is difficult to realize with MDM  Of course, it is possible to force the use of VPN in mobile device with MDM, but many aspects of communications security will be still unsolved  Thus, 5G slicing provides new tools to control and manage the end-to-end communications flow with network functions (VNFs)  In the advent of IoT, this is particularly important since the billions of IoT devices of the 2020s will have only a minimal processing power and memory compared to the smart phones of today  These IoT devices may connect to network only once a month and communicate only with network edge cloud servers  Therefore, managing these new IoT devices cannot be done with conventional MDM systems of today 
Fortunately, many features of MDM can be provided by dedicated 5G slices and their VNFs  If e g  a IoT device does not have the latest anti-virus updates, the network slice may still provide the isolation and security controls so that the IoT device can send the metering data  Moreover, if the IP flow from the IoT device includes other than actual metering data, eg  due to malware in IoT device, this IP flow can be analyzed and filtered by slice specific VNFs before passing it to IoT could 
As a summary, it can be argued that SDN/NFV based 5G Slicing will provide new tools for security management, and, when combined with IoT-MDM system functionalities, together these can deliver a better device management framework for different kinds of user devices of the 2020s 
Hospital organizations are considered as critical infrastructure (CI) to nation states [3, 25]  As a critical infrastructure, hospital organizations are prone to security threats that can affect health policies, public health, healthcare services, surgical procedures, electronic patient records, patient privacy, doctor-patient communication, etc  Lehto and Ahokangas  notes, new technology (e g  next generation mobile networks, smart data storage, IoT) adoption of CIs increases the cybersecurity touchpoints and hence making the CIs more vulnerable  Broadly, from cybersecurity perspective, the hospital organizations in future will be vulnerable from management perspective (e g  organizing healthcare services, managing huge amount of patient data, clinical data, medication data, communication between health professional and patients etc ), healthcare service delivery perspective (e g  in an unwelcoming case of denial of service attacks in hospital context: like wannacry), network perspective (e g  security of the overall hospital network), and last but not the least from an individual privacy perspective (e g  individual patient records, healthcare professional logs, etc ) 
A futures digital hospital will consist various advanced technologies, such as critical medical devices, intelligent information systems and digital communication tools, which are fully integrated to improve staff productivity, hospital operations, patient safety, and the overall patient experience  Among them, many will be wireless mobile devices, wireless wearables and thousands of smart IoT devices for various kinds of measurements  
Innovations help transform healthcare in forms of advanced telecommunications techonology, new drugs and treatments like biological sensor pills or implants, new medical devices, social media interation, etc  [1, 6, 7]  In future, hospitals and surgical tools will be revolutionized with intelligence and connectivity i e  providing more assistive functions by sensors, processors, data collection, software algorithms and interfaces which are, for example, embedded in the tools that will support surgeon’s decision-making in action   Robotic-assisted Surgery (RAS) can be considered as a good example of such transformation  The rapid adoption of Minimally Invasive Surgery (MIS) and today’s information revolution not only improves the surgical outcomes and patient’s life but also changes the patient-physician relationship 
‘Smart’ i e  IoT-based technologies are going to intellectualize medical devices and service systems into Smart Hospitals  Smart hospitals extrapolate from totally digitalized and automated data collection, tracking and delivery between systems, devices, patients, and health professionals and organization   In future, it is also envisioned that virtual hospitals and personalized medication will become part of patient care  Patients will be able to visit virtual hospitals or e-clinincs for clinical purposes without the presence of health professionals, but procedures and communication are managed via remote management and telemedicine solutions  
Other features that futures digital hospitals will include: use of different AR (augmented reality) and VR (virtual reality) applications for healthcare related purposes  While VR applications are quickly evolving in the gaming industry, some applications are deemed suitable in hospital environment too  Patients can be introduced to hospital environment through VR solutions, which can reduce patient stress prior to surgical procedures   Another application of VR in healthcare has been identified as rehabilitation support for temporary physical disabilities and mental trauma  Further, AR solutions can help patients navigate within a large hospital complex 
The term, Business model, has had its root in information systems (IS) and information communication technology (ICT) since the late 1970s originating from business informatics  However, it came into management and strategic management literature as a research interest from the mid-1990s [29, 30]  Although business model had its roots in IS and ICT, the amount of research work on cyber security as a context is still quite negligible harnessing the potential of business model concept 
The concept of business models lies at the intersection of entrepreneurship and strategy, it can be observed as a bridge between abstract strategies and the practical implementation of strategic decisions and actions amidst the uncertainties of the modern business context [11–14, 31]  For instance, Zott and Amit  conceptualize business model as a ‘boundary-spanning’ set of activities aimed at creating and appropriating value  Morris et al   viewed the concept of the business model as a set of decisions related to the venture strategy, architecture, and economics of firm (value creation and capture) that need addressing to create sustainable competitive advantage in the chosen markets and specific contexts 
As a boundary-spanning unit of analysis, business models , connects an organization with its business environment, other organizations, customers, individuals and society as well; with the overall ecosystem at large [34, 35]  Trying to bridge business models and cyber security under current context, there are two core issues  First, as almost all of the entities operating within the digital sphere face multifaceted cyber threats, how can business model approach help organizations to respond to such situations? Second, how can business model approach help to identify opportunities and monetize security in future 5G? In the next two sub-chapters, we present two business model approaches that are suitable for ICT contexts, and, can help find answers to the above mentioned questions 
As the mobile telecommunications industry advanced, so did business model related discussions in the literature about extending organizational boundaries through vertical and horizontal integration in industries   Around this, Wirtz et al   offered four mutually exclusive business models classifying Internet based business models to be precise  According to Wirtz et al  , these business models present in Web 2 0 are related to connection, content, context, and commerce  Building further, Yrjölä et al   interpreted these business models as chronological layers, where “lower” layer business models are pre-requisites for the “higher” layer business models to exist 
Figure 1 The layered 4C ICT business models archetypes [Adapted from 35, 37, 38] 
The first layer is concerned with connection-related business model where a stakeholder provides connection services   Connection- related business models are relevant to connectivity for all sorts of devices and nodes through various communication channels, e g  PCs, smartphones, tablets, IoT devices, etc  The second layer is the business model focusing on monetizing content  At the content layer, all sorts of online content services (e g  mobile video streaming) are classified (i e  relevant, up-to-date and interesting) and are accessible conveniently for the end user  The content might be peer-to-peer/user-oriented contents (i e  exchange of personal content), and web browsing content (audio, video, text etc ) 
The third, context, layer concerns the ability to create and monetize user, content, equipment/user device and system profiles and turn (big) data into meaningful information and knowledge through systemic virtual contextualization  The fourth layer concerns commerce, the ability to monetize any or all of the connection, content, or context specific resources, actors or activities related to the ongoing communications  At this layer, we identify business, consumer and public/government types of communications   Thus, B2B (business-to-business), B2C and B2G communications as well as C2B, C2C and C2G or G2B, G2C and G2G communications may be monetized at this layer 
Casdesus-Masanell and Llanes’s  offers different software business models based on the openness of core software and extension software offering  According to the mixed source software business model approach, services can be: open source (open core- open extension), open core (open core- closed extension), open extensions (closed core- open extension) and proprietary (closed core- closed extension)  Although this model was offered specifically suiting the software industry, we argue that similar approach can be useful in other ICT contexts to scale up value offering by adjusting cooperation and industrial partnering model 
Figure 2 Mixed source business model approach [Adapted form 40] 
Depending on individual organization’s choice of mixed source options, it should eventually translate the value creation and value capture logic of the firm  Casadesus-Masanell & Llanes  argues that purely proprietary models results in higher captured value for organizations but lessens the scope of value creation for users  In contrast, purely open source models can deliver the maximum value to customers but reduce captured value  The key for organizations is to locate the best mix of openness-closeness for an optimum level of value capture and creation  From the business model elements collection, the mixed source approach explain key partners, key activities, key resources and technologies used  From a strategic perspective, this approach to business modeling helps an organization to find a way to scale up or down its business activities 
This study concerns business possibilities of IoT and mobile device management in the context of futures digital hospital  Since future is elusive and the speed of technology advancements has been rapid during recent decades, we consider the most suitable method for conducting this research to be through qualitative approach  Qualitative methods helps with flexibility and sensitivity to the context that is less explored, and it can facilitate understanding of how things work in a particularly complex setting   We adopt a single qualitative case study approach for this research to explain underlying business potential of a scantly explored industrial context within the existing literature  
Interview 1 Pediatric Surgery Administration, teaching, hospital development, technology and innovations Pediatric surgery patient journey, concept development of futures digital hospital, digitalization in healthcare 
Workshop 1 Network specialists, operator service provider, cybersecurity expert Business opportunity creation, new market exploration, security aspects Digitalization in healthcare, security business opportunities in health domain, business modeling 
Workshop 2 Operator service provider, cybersecurity expert Business opportunity creation, new market exploration, security aspects Digitalization in healthcare, security business opportunities in health domain, business modeling 
Table 1 summarizes the data used for this research  The empirical data used for this study was collected through two streams  First, in order to build the conceptual case of future digital hospitals, five interviews were conducted with pediatric surgery specialists, a nurse, and a general physician from two hospital districts in Finland  These semi-structured interviews were organized between April and May 2016  These interviews also focused on understanding the journey path in a pediatrics surgery case to clarify suitability of digitalization in the future  However, in the scope of this paper, we do not discuss the patient journey pediatrics surgery in detail  Second, we explored the cybersecurity and business perspectives in two workshops that were organized with telecommunications and cybersecurity experts from the industry  One of the workshops were organized in November 2016  The other workshop took place in August 2017  While the futures digital hospital context from a security perspective relates more towards hospital management and network functionality, the IoT devices and mobile devices perspective is closer to end user security needs  This paper focuses on the cyber security related business opportunities through IoT-MDM systems/services that broadly covers the above mentioned domains  Findings and analysis from the interviews and workshops are presented next 
In this section, we present our analysis in four steps  First, we briefly discuss the relevance of Internet business models (4C)  to the context of the study, addressing how this approach can help cybersecurity providers identify business opportunities in IoT-MDM  Second, we present a 4-quadrant matrix reflecting four different 5G security-provisioning scenarios, which are also relevant to IoT-MDM  Then, we connect each scenario back to business models by applying a mixed-source business model approach  and identify different possible business models for each kind of cybersecurity provider  Finally, we connect the overall discussion back to the case of future digital hospital 
The 4C business model archetypes helps decoding the boundaries between multiple business models operating either in the same verticals of the ecosystem or in the same horizontals  These archetypes can provide with a basis to classify and analyze business models of suppliers, competitors, and, at the same time, business models of customers  For this case, the 4C business model archetypes are mostly relevant for cybersecurity providers and IoT-MDM system/service providers as a tool to analyze customer business models and identify customer needs, the hospital management style  These customer needs can eventually be turned around as business opportunities  While IoT-MDM systems and services are not only concerned about cybersecurity, there is room for service providers to customize and tailor services based on customer needs  These service providers either can target customers from one layer (connection/content/context/commerce) across multiple industries, or, they can also target a specific industry (e g  healthcare) create customized services for all the layers in that industry 
To make sense of the overall cybersecurity provisioning for future 5G, we created the 4-quadrant matrix (Figure 3) by identifying major security drivers for new business creation in future 5G  These scenarios were created in workshops with telecommunication & cybersecurity experts from the industry  While these service provisioning scenarios are relevant to overall 5G, they are also significantly related to the case of IoT-MDM system/services 
Figure 3 Scenarios for 5G cybersecurity provisioning [Adapted from 35] 
As a result, we find four major drivers of security, which potentially will come with new business opportunities in the 5G era  Device driven security comprises distributed and D2D security techniques  Platform driven security will focus on centralized and D2D security techniques  Whereas, network infrastructure driven security should focus on centralized and infrastructure security methods  Lastly, location driven security should harness distributed and infrastructure security techniques 
In a quest to identify potential business entities operating in each of the quadrants relevant to device management, we recognize a ‘secure device manufacturer/provider’ focuses on device-driven security  Currently, multiple device manufacturers are developing devices where security features are built-in, regardless of which network or websites the users are accessing  This built-in security can be offered to multiple kinds of devices including smartphones, tablets, pcs, wearables, and even to IoT devices with communication and computation capability  These secure devices are built in a way that it will control access to potentially harmful networks, websites, and content; even without any commercially available security applications installed  However, when a customer enterprise (e g  the future digital hospital) is buying a fleet of hundreds of such secure devices, the question raises how to manage and configure all these devices from time-to-time  In such a case, even the secure devices will need IoT-MDM services for proper management and seamless upgrading when needed 
For the platform-driven security, we observe a ‘traditional IoT-MDM system provider’ can be a good example  In many cases, IoT-MDM system providers are selling there device management systems to enterprises directly  And, in other cases, they are selling the service through MNO’s bundled with connectivity and/or infrastructure  While as network infrastructure-driven security, a ‘mobile network operator/carrier’ or a ‘network infrastructure vendor’ can build own IoT-MDM system to offer their clients as well  And, finally, a location-specific micro operator can offer location-driven security  Micro operators offer mobile connectivity combined with specific, local services  The operation of a micro operator is spatially confined to either its premises or to a defined area of operation  As a part of the location-specific services, these micro operators can also offer IoT-MDM services for the users through outsourcing 
Further, we attempt to connect the aforementioned classification and examples of different players offering IoT-MDM services with the mixed source business model approach  Table 2 summarizes our understanding on how each kind of cybersecurity provider can open and mix the core value creation logic for end users  As mentioned previously, the mixed-source business model options are: open source (open core, open extensions), open core (open core, closed extensions), open extensions (closed core, open extensions), and proprietary (closed core, closed extensions)  In relation to these mixed-source options, we analyze the plausible options for each of the four distinct cybersecurity providers in the context of this study 
Secure device manufacturer/provider Secure devices Proprietary (own device, own IoT-MDM platform), Open extensions (own device, outsourced IoT-MDM service) 
From a secure device manufacturer perspective, device business can be considered as the core operation whereas IoT-MDM services would be extended solution  A secure device manufacturer/provider can have either a proprietary model or an open extensions model  In the proprietary model, the secure device manufacturer will offer their own devices alongwith their own IoT-MDM system/service  This is a viable case in a sense that customers who are purchasing the fleet of secure devices might prefer the IoT-MDM service from the same vendor, which is ideally less risk prone  Generally, when a customer organization buys a fleet of secure devices, they would expect the device management functions to be offered by the same service provider since integration of multiple systems/interfaces might increase vulnerability and reduce the customer’s confidence on the system  However, in practice, not all secure device manufacturer will specialize in automated device management solutions, and new R&D expenses to develop the IoT-MDM systems could result to be financially unfeasible  So, the second option for a secure device manufacturer is to have an open extensions model, where they will still offer the secure devices but outsource the IoT-MDM services to other vendors 
A micro operator’s core business relates to location specific network, while they are offering additional location specific services as extensions  Ideally, a micro operator’s business model in this case can be considered to be open source  At one hand micro operators are dependent on appropriate available spectrum resources on carriers and NIVs, and on the other hand, outsourcing the IoT-MDM services to other vendors seems more economically feasible than building own system  In contrast, a traditional IoT-MDM system/ service provider can have either a direct-open extensions business model or an indirect-open extensions business model  Unlike the other three archetypes in this discussion, IoT-MDM system/services are the core business for this kind of actor, as device and network related issues can be considered as extensions  An IoT-MDM system/service provider is characteristically dependent on connectivity providers, i e  MNOs, NIVs  In a direct-open extensions business model, while they are selling the IoT-MDM service directly to the end users, they are using the operator connectivity, but delivering their core value directly  Alternatively, in an indirect-open extensions business model, these IoT-MDM vendors can sell the services through MNOs/NIVs/device vendors to the end users 
Finally, a mobile network operator can either have an open extensions business model, or less likely a proprietary business model for IoT-MDM services  In an open extensions business model, operators will offer own connectivity to end users while outsourcing the IoT-MDM service to vendors  In very few cases though, operators might have their own IoT-MDM system on offering and thus they can sell both connectivity and IoT-MDM service as a proprietary bundle 
Looking back at the case of futures digital hospital, it is deemed to comprise various advanced technology-aided devices, let it be for clinical purposes or communications purpose that support healthcare  With the presence of devices like smartphones, tablets, wearables, connected TVs, VR touchpoints, AR touchpoints, robotic assistance for surgeries, thousands of smart IoT devices, makes the overall device network of the digital hospital complex and need automated centralized management  This centralized management of numerous IoT and mobile devices can be delivered through IoT-MDM system/services  Using the IoT-MDM system/service, the hospital organization can configure, secure, and time-to-time update their device network 
While procuring IoT-MDM services, the above analysis shows that futures digital hospitals can source it either directly from IoT-MDM service providers or through MNOs  Alternatively, if a digital hospital also plays the role of a micro operator, besides offering other location-specific services, the micro operator can also offer device management services by sourcing it from other vendors  In other cases, the future digital hospital can procure fleet of secure devices for the hospital from whom they can also source the device management service as a bundle  From the 4C business models perspective, the hospital organization seems to be a single organizational entity where different activities can be categorized in the 4C layers  Cybersecurity providers can specify and address such issues to highlight and customize their service offering for the future digital hospital 
In this article, we have looked at the futures digital hospital context from a device management perspective, and attempted to portray business model options based on how to create and capture economic value from cybersecurity business  Though in this research, we adopt a high-level stance on cybersecurity from a technical perspective, the overall discussions on business potential are relevant to issues like information security, communication security, storage security, security at vulnerable touchpoints in hospital context (end user interface layer, IoT nodes, system layer, network layer) 
Futures digital hospitals will be vulnerable to cybersecurity threats because of its data-dependency and digital-intensive device network  Thus, the business opportunity for cybersecurity providers in this case can be considered as the need for automated and centralized IoT-MDM service  To that end, this paper presents four distinct players who can provide such service to a critical infrastructure like a digital hospital  The mixed-source business model options further open up multiple alternatives that each type of vendor can consider while tailoring services for the future digital hospital   This paper also connects the 4C ICT business model archetypes to cybersecurity business context which can be used as an analytical tool to identify customer needs and scope for value creation  
Academically, this work contributes by filling up the void in discussing business models for cybersecurity as an industry  In addition, in the existing literature the hospital context has also been less discussed from a cybersecurity business perspective  From an industrial point of view, the business model options discussed in this research are timely and relevant to the market context and need  As mentioned, the mixed source business model options show how cybersecurity providers can extend their offering for different kinds of need for the hospital context based on their core businesses  This study can prove to be helpful for cybersecurity business entities and at the same time hospital managers  The scope of this paper explains business potential of cybersecurity vendors to an emerging industry from a higher level  In this paper, authors do not attempt to analyze the technical aspect of cyber security provisioning in healthcare context, however that is a forthcoming research possibility of this study  Also, since this research is based on a conceptual phenomenon, thus its empirical validation, both qualitatively and quantitatively is still yet to come, which can be considered as a limitation of the study 
All in all, we consider that applying a business perspective to IoT-MDM systems can solve many challenges of a modern mobile IT environment, not only in healthcare but also in other kinds of critical infrastructures   These IoT-MDM systems can be provided by various kind of vendors through a balanced and timely business model 
This study has been supported by the DIMECC Cyber Trust – Digital cyber security program 
 Lehto, I , and Ahokangas, P  (2017)  “Mobile Security Business Models for Critical Infrastructures – An Ecosystemic Approach”, in Proceedings of the 24th Nordic Academy of Management Conference 2017, Bodo, Norway 
 Loebbecke, C , and Picot, A  (2015)  Reflections on societal and business model transformation arising from digitization and big data analytics: A research agenda  The Journal of Strategic Information Systems, 24, 149–157 
 Himidan, S , and Kim, P
Context 3: \section{Introduction}
Given a data set and a model with some unknown parameters, the inverse problem aims to find the values of the model parameters that best fit the data   
In this work, in which we focus on systems of interacting elements,
 the inverse problem concerns  the statistical inference
 of the underling interaction network and of its coupling coefficients from observed data on the dynamics  of the system  
 Versions of this problem are encountered in physics, biology (e g , \cite{Balakrishnan11,Ekeberg13,Christoph14}), social sciences and finance (e g ,\cite{Mastromatteo12,yamanaka_15}), neuroscience (e g , \cite{Schneidman06,Roudi09a,tyrcha_13}), just to cite a few, and are becoming more and more important due to the increase in the amount of data available from these fields \\
 \indent 
 A standard approach used in statistical inference is to predict the interaction couplings by maximizing the likelihood function 
 This technique, however, requires the evaluation of the  
 
   partition function that, in the most general case, concerns a number of computations scaling exponentially with the system size 
  
  
    Boltzmann machine learning uses Monte Carlo sampling to compute the gradients of the Log-likelihood looking for stationary points \cite{Murphy12} but this method is computationally manageable only for small systems  A series of faster approximations, such as naive mean-field, independent-pair approximation \cite{Roudi09a, Roudi09b}, inversion of TAP equations \cite{Kappen98,Tanaka98}, small correlations expansion \cite{Sessak09}, adaptive TAP \cite{Opper01}, adaptive  cluster expansion \cite{Cocco12} or Bethe approximations \cite{Ricci-Tersenghi12, Nguyen12}  have, then, been developed  These techniques take as input means and correlations of observed variables and most of them assume a fully connected graph as underlying connectivity network, or expand  around it by perturbative dilution   In most cases, network reconstruction turns out to  be not accurate for small data sizes and/or when couplings are  strong or, else, if the original interaction network is sparse \\
\indent
 A further method, substantially improving performances for small data, is the so-called Pseudo-Likelyhood Method (PLM) \cite{Ravikumar10}  In Ref  \cite{Aurell12} Aurell and Ekeberg performed a comparison between PLM and some of the just mentioned mean-field-based algorithms on the pairwise interacting Ising-spin  ($\sigma = \pm 1$) model, showing how PLM performs sensitively better, especially on sparse graphs and in the high-coupling limit, i e , for low temperature 
 
 In this work, we aim at performing statistical inference  on a model whose interacting variables are continuous $XY$ spins, i e , $\sigma \equiv \left(\cos \phi,\sin \phi\right)$ with $\phi \in [0, 2\pi )$  The developed tools can, actually, be also straightforward applied  to the $p$-clock model  \cite{Potts52} where the phase $\phi$ takes discretely equispaced $p$ values  in the $2 \pi$ interval, $\phi_a =  a 2 \pi/p$, with $a= 0,1,\dots,p-1$  The $p$-clock model, else called vector Potts model, gives a hierarchy of discretization of the $XY$ model as $p$ increases  For $p=2$, one recovers the Ising model, for $p=4$ the Ashkin-Teller model \cite{Ashkin43}, for $p=6$ the ice-type model \cite{Pauling35,Baxter82} and the eight-vertex model \cite{Sutherland70,Fan70,Baxter71} for $p=8$   
It turns out to be very useful also for numerical implementations of the continuous $XY$ model  
Recent analysis on the multi-body $XY$ model has shown that for a limited number of discrete phase values ($p\sim 16, 32$) the thermodynamic critical properties of the $p\to\infty$ $XY$ limit are promptly recovered \cite{Marruzzo15, Marruzzo16}   
Our main motivation to study statistical inference is that these kind of models have recently turned out to be rather useful in describing the behavior of optical systems, 
including standard mode-locking lasers \cite{Gordon02,Gat04,Angelani07,Marruzzo15} and random lasers \cite{Angelani06a,Leuzzi09a,Antenucci15a,Antenucci15b,Marruzzo16}  
In particular, the inverse problem on the pairwise XY model analyzed here might be of help in recovering images from light propagated through random media  


      This paper is organized as follows: in Sec  \ref{sec:model} we introduce the general model  and we discuss its  derivation also  as a model for light transmission through random scattering media  
    In Sec  \ref{sec:plm} we introduce the PLM with $l_2$ regularization  and with decimation, two variants of the PLM respectively introduced in Ref  \cite{Wainwright06} and \cite{Aurell12} for the inverse Ising problem  
    Here, we analyze these techniques for continuous $XY$ spins and we test them on thermalized data generated by Exchange Monte Carlo numerical simulations of the original model dynamics  In Sec  \ref{sec:res_reg} we  present the results related to the PLM-$l_2$  In Sec  \ref{sec:res_dec} the results related to the PLM with decimation are reported and its performances are compared to the PLM-$l_2$ and to a variational mean-field method analyzed in Ref  \cite{Tyagi15}  In Sec  \ref{sec:conc}, we outline conclusive remarks and perspectives 

        \section{The leading $XY$ model}
      \label{sec:model}
 The leading model we are considering is defined, for a system of $N$ angular $XY$ variables, by the Hamiltonian 
 \begin{equation}
  \mathcal{H} = - \sum_{ik}^{1,N} J_{ik} \cos{\left(\phi_i-\phi_k\right)} 
  \label{eq:HXY}
 
  \end{equation} 
  
  The $XY$ model is well known in statistical mechanics, displaying important physical
  insights, starting from the Berezinskii-Kosterlitz-Thouless
  transition in two dimensions\cite{Berezinskii70,Berezinskii71,Kosterlitz72} and moving to, e g , the
  transition of liquid helium to its superfluid state \cite{Brezin82}, the roughening transition of the interface of a crystal in equilibrium with its vapor \cite{Cardy96}  In presence of disorder and frustration \cite{Villain77,Fradkin78} the model has been adopted to describe synchronization problems as the Kuramoto model \cite{Kuramoto75} and in the theoretical modeling of Josephson junction arrays \cite{Teitel83a,Teitel83b} and arrays of coupled lasers \cite{Nixon13} 
  Besides several derivations and implementations of the model in quantum and classical physics, equilibrium or out of equilibrium, ordered or fully frustrated systems, Eq  (\ref{eq:HXY}), in its generic form,
  has found applications also in other fields  A rather fascinating example being the behavior of starlings flocks  \cite{Reynolds87,Deneubourg89,Huth90,Vicsek95, Cavagna13} 
  Our interest on the $XY$ model resides, though, in optics  Phasor and phase models with pairwise and  multi-body interaction terms can, indeed, describe the behavior of electromagnetic modes in both linear and nonlinear optical systems in the analysis of problems such as light propagation and lasing \cite{Gordon02, Antenucci15c, Antenucci15d}  As couplings are strongly frustrated, these models turn out to be especially useful to the study of optical properties in random media \cite{Antenucci15a,Antenucci15b}, as in the noticeable case of random lasers \cite{Wiersma08,Andreasen11,Antenucci15e}  and they might as well be applied to linear scattering problems, e g , propagation of waves in opaque systems or disordered fibers  
  
  
  \subsection{A propagating wave model}
  We briefly mention a derivation of the model as a proxy for the propagation of light through random linear media  
 Scattering of light is held responsible to obstruct our view and make objects opaque  Light rays, once that they enter the material, only exit  after getting scattered multiple times within the material  In such a disordered medium, both the direction and the phase of the propagating waves are random  Transmitted light 
 yields a disordered interference pattern typically  having low intensity, random phase and almost no resolution, called a speckle  Nevertheless, in recent years it has been realized that disorder is rather a blessing in disguise \cite{Vellekoop07,Vellekoop08a,Vellekoop08b}  Several experiments have made it possible to control the behavior of light and other optical processes in a given random disordered medium,  
 by exploiting, e g ,  the tools developed for wavefront shaping to control the propagation of light and to engineer the confinement of light \cite{Yilmaz13,Riboli14} 
 \\
 \indent
 In a linear dielectric medium, light propagation can be described through a part of the scattering matrix, the transmission matrix $\mathbb{T}$, linking the outgoing to the incoming fields  
 Consider the case in which there are $N_I$ incoming channels and $N_O$ outgoing ones; we can indicate with $E^{\rm in,out}_k$ the input/output electromagnetic field phasors of channel $k$  In the most general case, i e , without making any particular assumptions on the field polarizations, each light mode and its polarization polarization state can be represented by means of the $4$-dimensional Stokes vector  Each $ t_{ki}$ element of $\mathbb{T}$, thus, is a $4 \times 4$ M{\"u}ller matrix  If, on the other hand, we know that the source is polarized and the observation is made on the same polarization, one can use a scalar model and adopt Jones calculus \cite{Goodman85,Popoff10a,Akbulut11}:
   \begin{eqnarray}
 E^{\rm out}_k = \sum_{i=1}^{N_I}  t_{ki} E^{\rm in}_i \qquad \forall~ k=1,\ldots,N_O
 \label{eq:transm}
 \end{eqnarray}
  We recall that the elements of the transmission matrix are random complex coefficients\cite{Popoff10a}  For the case of completely unpolarized modes, we can also use a scalar model similar to Eq  \eqref{eq:transm}, but whose variables are  the intensities of the outgoing/incoming fields, rather than the fields themselves \\ 
In the following, for simplicity, we will consider Eq  (\ref{eq:transm}) as our starting point,
where $E^{\rm out}_k$, $E^{\rm in}_i$ and $t_{ki}$ are all complex scalars  
If Eq  \eqref{eq:transm} holds for any $k$, we can write:
  \begin{eqnarray}
  \int \prod_{k=1}^{N_O} dE^{\rm out}_k \prod_{k=1}^{N_O}\delta\left(E^{\rm out}_k - \sum_{j=1}^{N_I}  t_{kj} E^{\rm in}_j \right) = 1
  \nonumber
  \\
  \label{eq:deltas}
  \end{eqnarray}

 Observed data are a noisy representation of the true values of the fields  Therefore, in inference problems it is statistically more meaningful to take that noise into account in a probabilistic way, 
 rather than looking  at the precise solutions of the exact equations (whose parameters are unknown)  
 To this aim we can introduce Gaussian distributions whose limit for zero variance are the Dirac deltas in Eq  (\ref{eq:deltas}) 
 Moreover, we move to consider the ensemble of all possible solutions of Eq  (\ref{eq:transm}) at given $\mathbb{T}$, looking at  all configurations of input fields  We, thus, define the function:
 
   \begin{eqnarray}
  Z &\equiv &\int_{{\cal S}_{\rm in}} \prod_{j=1}^{N_I}  dE^{\rm in}_j \int_{{\cal S}_{\rm out}}\prod_{k=1}^{N_O} dE^{\rm out}_k 
  \label{def:Z}
\\
    \times
  &&\prod_{k=1}^{N_O}
   \frac{1}{\sqrt{2\pi \Delta^2}}  \exp\left\{-\frac{1}{2 \Delta^2}\left|
  E^{\rm out}_k -\sum_{j=1}^{N_I}  t_{kj} E^{\rm in}_j\right|^2
\right\} 
\nonumber
 \end{eqnarray}
   We stress that the integral of Eq  \eqref{def:Z} is not exactly a Gaussian integral  Indeed, starting from Eq  \eqref{eq:deltas}, two constraints on the electromagnetic field intensities must be taken into account  

  The space of solutions is  delimited by the total power ${\cal P}$ received by system, i e , 
  ${\cal S}_{\rm in}: \{E^{\rm in} |\sum_k I^{\rm in}_k = \mathcal{P}\}$, also implying  a constraint on the total amount of energy that is transmitted through the medium, i  e , 
  ${\cal S}_{\rm out}:\{E^{\rm out} |\sum_k I^{\rm out}_k=c\mathcal{P}\}$, where the attenuation factor  $c<1$ accounts for total losses 
  As we will see more in details in the following, being interested in inferring the transmission matrix through the PLM, we can omit to explicitly include these terms in Eq  \eqref{eq:H_J} since they do not depend on $\mathbb{T}$ not adding any information on the gradients with respect to the elements of $\mathbb{T}$ 
  
 Taking the same number of incoming and outcoming channels, $N_I=N_O=N/2$, and  ordering the input fields in the first $N/2$ mode indices and the output fields in the last $N/2$ indices, we can drop the ``in'' and ``out'' superscripts and formally write $Z$  as a partition function
    \begin{eqnarray}
        \label{eq:z}
 && Z =\int_{\mathcal S} \prod_{j=1}^{N} dE_j \left(   \frac{1}{\sqrt{2\pi \Delta^2}} \right)^{N/2} 
 \hspace*{- 4cm} \exp\left\{
  -\frac{ {\cal H} [\{E\};\mathbb{T}] }{2\Delta^2}
  \right\}
  \\
&&{\cal H} [\{E\};\mathbb{T}] =
-  \sum_{k=1}^{N/2}\sum_{j=N/2+1}^{N} \left[E^*_j t_{jk} E_k + E_j t^*_{kj} E_k^* 
\right]
 \nonumber
\\
&&\qquad\qquad \qquad + \sum_{j=N/2+1}^{N} |E_j|^2+ \sum_{k,l}^{1,N/2}E_k
U_{kl} E_l^*
 \nonumber
 \\
 \label{eq:H_J}
 &&\hspace*{1 88cm } = - \sum_{nm}^{1,N} E_n J_{nm} E_m^*
 \end{eqnarray}
 where ${\cal H}$ is a real-valued function by construction, we have introduced the effective input-input coupling matrix
\begin{equation}
U_{kl} \equiv \sum_{j=N/2+1}^{N}t^*_{lj} t_{jk} 
 \label{def:U}
 \end{equation}
 and the whole interaction matrix reads (here $\mathbb{T} \equiv \{ t_{jk} \}$)
 \begin{equation}
 \label{def:J}
 \mathbb J\equiv \left(\begin{array}{ccc|ccc}
 \phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}\\
 \phantom{()}&-\mathbb{U} \phantom{()}&\phantom{()}&\phantom{()}&{\mathbb{T}}&\phantom{()}\\
\phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}\\
 \hline
\phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}\\
 \phantom{()}& \mathbb   T^\dagger&\phantom{()}&\phantom{()}& - \mathbb{I} &\phantom{()}\\
\phantom{a}&\phantom{a}&\phantom{a}&\phantom{a}&\phantom{a}&\phantom{a}\\
 \end{array}\right)
 \end{equation}
 
 Determining the electromagnetic complex amplitude configurations that minimize the {\em cost function} ${\cal H}$, Eq   (\ref{eq:H_J}),  means to maximize the overall distribution peaked around the solutions of the transmission Eqs  (\ref{eq:transm})  As the variance $\Delta^2\to 0$, eventually, the initial set of Eqs  (\ref{eq:transm}) are recovered  The ${\cal H}$ function, thus, plays the role of an Hamiltonian and  $\Delta^2$ the role of a noise-inducing temperature  The exact numerical problem corresponds to the zero temperature limit of the statistical mechanical problem  Working with real data, though, which are noisy, a finite ``temperature''
  allows for a better representation of the ensemble of solutions to the sets of equations of continuous variables  
  

  
  
  Now, we can express every phasor in Eq  \eqref{eq:z}  as $E_k = A_k e^{\imath \phi_k}$  As a working hypothesis we will consider the intensities $A_k^2$ as either homogeneous or as \textit{quenched} with respect to phases 
The first condition occurs, for instance, to the input intensities $|E^{\rm in}_k|$ produced by a phase-only spatial light modulator (SLM) with homogeneous illumination \cite{Popoff11} 
With \textit{quenched} here we mean, instead, that the intensity of each mode is the same for every solution of Eq  \eqref{eq:transm} at fixed $\mathbb T$ 
We stress that, including intensities in the model does not preclude the inference analysis but it is out of the focus of the present work and will be considered elsewhere  

If all intensities are uniform in input and in output, this amount to a constant rescaling for each one of the four sectors of matrix $\mathbb J$ in Eq  (\ref{def:J}) that will not change the properties of the matrices 
For instance, if the original transmission matrix is unitary, so it will be the rescaled one and the matrix $\mathbb U$ will be  diagonal 
Otherwise, if intensities are \textit{quenched}, i e , they can be considered as constants in Eq  (\ref{eq:transm}),
they are inhomogeneous with respect to phases  The generic Hamiltonian element will, therefore, rescale as 
  \begin{eqnarray}
  E^*_n J_{nm} E_m = J_{nm} A_n A_m e^{\imath (\phi_n-\phi_m)} \to J_{nm} e^{\imath (\phi_n-\phi_m)}
  \nonumber
  \end{eqnarray}
  and the properties of the original  $J_{nm}$ components are not conserved  in the rescaled one  In particular, we have no argument, anymore, to possibly set the rescaled $U_{nm}\propto \delta_{nm}$ 
  Eventually, we end up with the complex couplings $XY$ model, whose real-valued Hamiltonian is written as
 \begin{eqnarray}
  \mathcal{H}& = &  - \frac{1}{2} \sum_{nm} J_{nm} e^{-\imath (\phi_n - \phi_m)}  + \mbox{c c } 
    \label{eq:h_im}
\\    &=&  - \frac{1}{2} \sum_{nm} \left[J^R_{nm} \cos(\phi_n - \phi_m)+
  J^I_{nm}\sin (\phi_n - \phi_m)\right] 
  \nonumber
 \end{eqnarray}
where $J_{nm}^R$ and $J_{nm}^I$ are the real and imaginary parts of $J_{nm}$  Being $\mathbb J$  Hermitian, $J^R_{nm}=J^R_{mn}$ is symmetric and $J_{nm}^I=-J_{mn}^I$ is skew-symmetric 

\begin{comment}
\textcolor{red}{
F: comment about quenched:
I think that to obtain the XY model, it is not necessary that the intensities are strictly quenched (that is also a quite unfeasible situation, I guess) 
Indeed eq (2) does not deal with the dynamics of the modes, but just connect the in and out ones 
For this, what it is necessary to have the XY model, it is that the intensities are always the same on the different samples
(so that the matrix $t_{ij}$ is the same for different phase data)  If the intensities are fixed, then they can be incorporated in $t_{ij}$ and eq (2) can be written just for phases as described  \\
}
\end{comment}


  \section{Pseudolikelihood Maximization}
  \label{sec:plm}
The inverse problem consists in the reconstruction of the parameters $J_{nm}$ of the Hamiltonian, Eq  (\ref{eq:h_im})   
Given a set of $M$ data configurations of $N$ spins
 $\bm\sigma = \{ \cos \phi_i^{(\mu)},\sin \phi_i^{(\mu)} \}$, $i = 1,\dots,N$ and $\mu=1,\dots,M$, we want to \emph{infer} the couplings:
 \begin{eqnarray}
\bm \sigma  \rightarrow  \mathbb{J} 
\nonumber
 \end{eqnarray}
 With this purpose in mind,
 in the rest of this section we implement the working equations for the techniques used  
 In order to test our methods, we generate the input data, i e , the configurations, by Monte-Carlo simulations of the model 
 The joint probability distribution of the $N$ variables $\bm{\phi}\equiv\{\phi_1,\dots,\phi_N\}$, follows the Gibbs-Boltzmann distribution:
 \begin{equation}\label{eq:p_xy}
 P(\bm{\phi}) = \frac{1}{Z} e^{-\beta \mathcal{H\left(\bm{\phi}\right)}} \quad \mbox{ where } \quad Z = \int \prod_{k=1}^N d\phi_k  e^{-\beta \mathcal{H\left(\bm{\phi}\right)}}  
 \end{equation}
 and where we denote $\beta=\left( 2\Delta^2 \right)^{-1}$ with respect to Eq  (\ref{def:Z}) formalism 
 In order to stick to usual statistical inference notation, in the following we will rescale the couplings by a factor $\beta / 2$: $\beta J_{ij}/2 \rightarrow J_{ij}$  
 The main idea of the PLM is to work with the conditional probability distribution of one variable $\phi_i$ given all other variables, 
 $\bm{\phi}_{\backslash i}$:
 
  \begin{eqnarray}
	\nonumber
   P(\phi_i | \bm{\phi}_{\backslash i}) &=& \frac{1}{Z_i} \exp \left \{ {H_i^x (\bm{\phi}_{\backslash i})
  	\cos \phi_i + H_i^y (\bm{\phi}_{\backslash i}) \sin \phi_i } \right \}
	\\
 \label{eq:marginal_xy}
	&=&\frac{e^{H_i(\bm{\phi}_{\backslash i}) \cos{\left(\phi_i-\alpha_i(\bm{\phi}_{\backslash i})\right)}}}{2 \pi I_0(H_i)}
  \end{eqnarray}
  where $H_i^x$ and $H_i^y$ are defined as
   \begin{eqnarray}
   H_i^x (\bm{\phi}_{\backslash i})  &=& \sum_{j (\neq i)} J^R_{ij} \cos \phi_j  - \sum_{j  (\neq i) } J_{ij}^{I} \sin \phi_j \phantom{+  h^R_i} \label{eq:26} \\
   H_i^y  (\bm{\phi}_{\backslash i})  &=&  \sum_{j  (\neq i)} J^R_{ij}  \sin \phi_j  + \sum_{j  (\neq i) } J_{ij}^{I}   \cos \phi_j \phantom{ + h_i^{I} }\label{eq:27}
   \end{eqnarray}
and $H_i= \sqrt{(H_i^x)^2 + (H_i^y)^2}$, $\alpha_i = \arctan H_i^y/H_i^x$   and we introduced the modified Bessel function of the first kind:
  \begin{equation}
  \nonumber
   I_k(x) = \frac{1}{2 \pi}\int_{0}^{2 \pi} d \phi e^{x \cos{ \phi}}\cos{k \phi}
  \end{equation}
  
   Given $M$ observation samples $\bm{\phi}^{(\mu)}=\{\phi^\mu_1,\ldots,\phi^\mu_N\}$, $\mu = 1,\dots, M$, the
   pseudo-loglikelihood for the variable $i$ is given by the logarithm of Eq  (\ref{eq:marginal_xy}),
   \begin{eqnarray}
   \label{eq:L_i}
   L_i &=& \frac{1}{M}  \sum_{\mu = 1}^M  \ln P(\phi_i^{(\mu)}|\bm{\phi}^{(\mu)}_{\backslash i})
   \\
   \nonumber
   & =&  \frac{1}{M}  \sum_{\mu = 1}^M  \left[ H_i^{(\mu)} \cos( \phi_i^{(\mu)} - \alpha_i^{(\mu)}) - \ln  2 \pi I_0\left(H_i^{(\mu)}\right)\right] \,  
    \end{eqnarray}
The underlying idea of PLM is that an approximation of the true parameters of the model is obtained for values that maximize the functions $L_i$ 
The specific maximization scheme differentiates the different techniques 


   
   
   \subsection{PLM with $l_2$ regularization}
   Especially for the case of sparse graphs, it is useful to add a regularizer, which prevents the maximization routine to move towards high values of 
   $J_{ij}$ and $h_i$ without converging  We will adopt an $l_2$ regularization so that the Pseudolikelihood function (PLF) at site $i$  reads:
   \begin{equation}\label{eq:plf_i}
  {\cal L}_i = L_i
  - \lambda \sum_{i \neq j} \left(J_{ij}^R\right)^2 - \lambda \sum_{i \neq j} \left(J_{ij}^I\right)^2  
   \end{equation}
   with $\lambda>0$ 
   Note that the values of $\lambda$ have to be chosen arbitrarily, but not too large, in order not to overcome $L_i$ 
   The standard implementation of the PLM consists in maximizing each ${\cal L}_i$, for $i=1\dots N$, separately  The expected values of the couplings are then:
   \begin{equation}
   \{ J_{i j}^*\}_{j\in \partial i} :=  \mbox{arg max}_{ \{ J_{ij} \}}
    \left[{\cal L}_i\right]
   \end{equation}
   In this way, we obtain two estimates for the coupling $J_{ij}$, one from maximization of ${\cal L}_i$, $J_{ij}^{(i)}$, and another one from ${\cal L}_j$, say $J_{ij}^{(j)}$ 
    Since the original Hamiltonian of the $XY$ model is Hermitian, we know that the real part of the couplings is symmetric while the imaginary part is skew-symmetric  
   
     The final estimate for $J_{ij}$ can then be obtained  averaging the two results:
  
  
  
   \begin{equation}\label{eq:symm}
   J_{ij}^{\rm inferred} = \frac{J_{ij}^{(i)} + \bar{J}_{ij}^{(j)}}{2} 
   \end{equation}
   where with $\bar{J}$ we indicate the complex conjugate 
   It is worth noting that the pseudolikelihood $L_i$, Eq  \eqref{eq:L_i}, is characterized by the
   following properties: (i) the normalization term of Eq \eqref{eq:marginal_xy} can be
   computed analytically at odd with the {\em full} likelihood case that
   in general require a computational time which scales exponentially
   with the size of the systems; (ii) the $\ell_2$-regularized pseudolikelihood
   defined in Eq \eqref{eq:plf_i} is strictly concave (i e  it has a single
   maximizer)\cite{Ravikumar10}; (iii) it is consistent, i e  if $M$ samples are
   generated by a model $P(\phi | J*)$ the maximizer tends to $J*$
   for $M\rightarrow\infty$\cite{besag1975}  Note also that (iii) guarantees that  
   $|J^{(i)}_{ij}-J^{(j)}_{ij}| \rightarrow 0$ for $M\rightarrow \infty$ 
   In Secs  \ref{sec:res_reg}, \ref{sec:res_dec} 
   we report the results obtained and we analyze the performances of the PLM having taken the configurations from Monte-Carlo simulations of models whose details are known 
   

   
   \subsection{PLM with decimation}
   Even though the PLM with $l_2$-regularization allows to dwell the inference towards the low temperature region and in the low sampling case with better performances that mean-field methods, in some situations some couplings are overestimated and not at all symmetric  Moreover, in the technique there is the bias of the $l_2$ regularizer 
   Trying to overcome these problems, Decelle and Ricci-Tersenghi introduced a new method \cite{Decelle14}, known as PLM + decimation: the algorithm maximizes the sum of the $L_i$,
   \begin{eqnarray}
    {\cal L}\equiv \frac{1}{N}\sum_{i=1}^N \mbox{L}_i
    \end{eqnarray}  
    and, then, it recursively set to zero couplings which are estimated very small  We expect that as long as we are setting to zero couplings that are unnecessary to fit the data, there should be not much changing on ${\cal L}$  Keeping on with decimation, a point is reached where ${\cal L}$ decreases abruptly indicating  that relevant couplings are being decimated and under-fitting is taking place 
   Let us define  by $x$  the fraction of non-decimated couplings  To have a quantitative measure for the halt criterion of the decimation process, a tilted ${\cal L}$ is defined as,
   \begin{eqnarray}
  \mathcal{L}_t &\equiv& \mathcal{L}  - x \mathcal{L}_{\textup{max}} - (1-x) \mathcal{L}_{\textup{min}} \label{$t$PLF} 
   \end{eqnarray}
   where 
   \begin{itemize}
   \item $\mathcal{L}_{\textup{min}}$ is the pseudolikelyhood of a model with independent variables  In the XY case: $\mathcal{L}_{\textup{min}}=-\ln{2 \pi}$ 
   \item
   $\mathcal{L}_{\textup{max}}$ is the pseudolikelyhood in the fully-connected model and it is maximized over all the $N(N-1)/2$ possible couplings  
   \end{itemize}
   At the first step, when $x=1$, $\mathcal{L}$ takes value $\mathcal{L}_{\rm max}$ and  $\mathcal{L}_t=0$  On the last step, for an empty graph, i e , $x=0$, $\mathcal{L}$ takes the value $\mathcal{L}_{\rm min}$ and, hence, again $\mathcal{L}_t =0$  
   In the intermediate steps, during the decimation procedure, as $x$ is decreasing from $1$ to $0$, one observes firstly that $\mathcal{L}_t$ increases linearly and, then, it displays an abrupt decrease indicating that from this point on relevant couplings are being decimated\cite{Decelle14}  In Fig  \ref{Jor1-$t$PLF} we give an instance of this behavior for the 2D short-range XY model with ordered couplings  We notice that the maximum point of $\mathcal{L}_t$ coincides with the minimum point of the reconstruction error, the latter defined as 
   \begin{eqnarray}\label{eq:errj}
   \mbox{err}_J \equiv \sqrt{\frac{\sum_{i<j} (J^{\rm inferred}_{ij} -J^{\rm true}_{ij})^2}{N(N-1)/2}} \label{err}
   \end{eqnarray}
  We stress that the ${\cal L}_t$ maximum is obtained ignoring the underlying graph, while the err$_J$ minimum can be evaluated once the true graph has been reconstructed  
   
     \begin{figure}[t!]
   	\centering
   	\includegraphics[width=1\linewidth]{Jor1_dec_tPLF_new eps}
   	\caption{The tilted likelyhood ${\cal L}_t$ curve and the reconstruction error vs the number of decimated couplings  for an ordered, real-valued J on 2D XY model with $N=64$ spins  The peak of ${\cal L}_t$ coincides with the dip of the error }  
   	\label{Jor1-$t$PLF}
   \end{figure} 
       
   
   In the next sections we will show the results obtained on the  $XY$ model analyzing the performances of the two methods and comparing them also with a mean-field method \cite{Tyagi15} 
   
  
    
   \section{Inferred couplings with PLM-$l_2$}
    \label{sec:res_reg}
    \subsection{$XY$ model with real-valued couplings}
    
    In order to obtain the vector of couplings, $J_{ij}^{\rm inferred}$ the function $-\mathcal{L}_i$ is minimized through the vector of derivatives ${\partial \mathcal{L}_i}/\partial J_{ij}$



ground_truth: O(t, L_{\parallel}; S_\Delta) = L_{\parallel}^{-\beta/[\nu(1+\Delta)]} \tilde f_O(t/L_{\parallel}^{z/(1+\Delta)}; S_\Delta).

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the definition of mobile device management (MDM)?

Read the following text and answer briefly.

Context 2:  (2015)  The evolving identity, capacity, and capability of the future surgeon  In Seminars in pediatric surgery, 24, 145–149, WB Saunders 
 Beimborn, D , and Palitza, M  (2013)  “Enterprise app stores for mobile applications-development of a benefits framework”, in Proceedings of the Nineteenth Americas Conference on Information Systems, Chicago, Illinois 
 Ortbach, K , Brockmann, T , and Stieglitz, S  (2014)  “Drivers for the adoption of mobile device management in organizations”, in proceedings of the Twenty Second European Conference on Information Systems, Tel Aviv 2014 
 Wirtz, B  W , Schilke, O , and Ullrich, S  (2010)  Strategic development of business models: implications of the Web 2 0 for creating value on the internet  Long range planning, 43, 272–290 
 Afuah, A  (2004)  Business Models: A Strategic Management Approach  McGraw-Hill/Irwin 
 Alt, R , and Zimmermann, H  D  (2001)  Preface: introduction to special section–business models  Electronic markets, 11, 3–9 
 Chesbrough, H , and Rosenbloom, R  S  (2002)  The role of the business model in capturing value from innovation: evidence from Xerox Corporation’s technology spin-off companies  Industrial and corporate change, 11, 529–555 
 Shafer, S  M , Smith, H  J , and Linder, J  C  (2005)  The power of business models  Business horizons, 48, 199–207 
 Zott, C , and Amit, R  (2010)  Business model design: an activity system perspective  Long range planning, 43, 216–226 
 Xu, T , Wendt, J  B , and Potkonjak, M  (2014)  Security of IoT systems: Design challenges and opportunities  In Proceedings of the 2014 IEEE/ACM International Conference on Computer-Aided Design, 417–423  IEEE Press 
 Zhang, Z  K , Cho, M  C  Y , Wang, C  W , Hsu, C  W , Chen, C  K , and Shieh, S  (2014)  IoT security: ongoing challenges and research opportunities  In Service-Oriented Computing and Applications (SOCA), 2014 IEEE 7th International Conference, 230–234  IEEE 
 Farooq, M  U , Waseem, M , Khairi, A , and Mazhar, S  (2015)  A critical analysis on the security concerns of internet of things (IoT)  Int  J  Computer Appl  111 
 Backman, J , Väre, J , Främling, K , Madhikermi, M , and Nykänen, O  (2016)  IoT-based interoperability framework for asset and fleet management  In Emerging Technologies and Factory Automation (ETFA), 2016 IEEE 21st International Conference, 1–4  IEEE 
 Isotalo, L  (2017)  “5G Slicing as a Tool to Test User Equipment Against Advanced Persistent Threats”, in International Conference on Network and System Security, 595–603  Springer, Cham 
 Alcaraz, C , and Zeadally, S  (2015)  “Critical infrastructure protection: requirements and challenges for the 21st century”, in International Journal of Critical Infrastructure Protection, 8, 53–66 
 Sutherland, J , and van den Heuvel, W  J  A  M  (2006)  Towards an intelligent hospital environment: adaptive workflow in the OR of the future  In System Sciences, 2006  HICSS’06, in Proceedings of the 39th Annual Hawaii International Conference, 5, 100b–100b  IEEE 
 Haq, M  M  (2008)  U S  Patent No  7,412,396  Washington, DC: U S  Patent and Trademark Office 
 Spector, N  D , Cull, W , Daniels, S  R , Gilhooly, J , Hall, J , Horn, I , and Stanton, B  F  (2014)  Gender and generational influences on the pediatric workforce and practice  Pediatrics, 133, 1112–1121 
 Wirtz, B  W  (2011)  Business model management  Design–Instrumente–Erfolgsfaktoren von Geschäftsmodellen, 2 
 Gomes, J  F , Ahokangas, P , and Owusu, K  A  (2016)  Business modeling facilitated cyber preparedness  Int  J  Business & Cyber Security, 1, 54–67 
 Richardson, J  (2008)  The business model: an integrative framework for strategy execution  Strategic change, 17, 133–144 
 Morris, M , Schindehutte, M , and Allen, J  (2005)  The entrepreneur’s business model: toward a unified perspective  J  Business Res  58, 726–735 
 Zott, C , Amit, R , and Massa, L  (2011)  The business model: recent developments and future research  J  Manage  37, 1019–1042 
 Gomes, J F , Iivari, M , Ahokangas, P , Isotalo, L , Sahlin, B , and Melen, J  (Forthcoming, in press), Cyber security business models in 5G  Comprehensive guide to 5G security, Wiley Publishers, London 
 Ballon, P  (2007)  Business modelling revisited: the configuration of control and value  info, 9, 6–19 
 Yrjölä, S , Ahokangas, P , and Matinmikko, M  (2015)  Evaluation of recent spectrum sharing concepts from business model scalability point of view  In Dynamic Spectrum Access Networks (DySPAN, in 2015 IEEE International Symposiu, 241–250  IEEE 
 Ahokangas, P , Moqaddamerad, S , Matinmikko, M , Abouzeid, A , Atkova, I , Gomes, J  F , and Iivari, M  (2016)  Future micro operators business models in 5G  The Business & Management Review, 7, 143 
 Mitola, J , Guerci, J , Reed, J , Yao, Y  D , Chen, Y , Clancy, T , and Guo, Y  (2014)  Accelerating 5G QoE via public-private spectrum sharing  IEEE Communications Magazine, 52, 77–85 
 Casadesus-Masanell, R , and Llanes, G  (2011)  Mixed source  Manage  Sci  57, 1212–1230 
 Mason, J  (2002)  Qualitative researching  Sage 
 Miles, M  B , and Huberman, A  M  (1994)  Qualitative data analysis: An expanded sourcebook  Sage 
Julius Francis Gomes is pursuing his Ph D  in international business from the University of Oulu  He currently works at the Oulu Business School as a Doctoral Student to research the futuristic business models for digital intensive industries  His research focuses on using business models as a mean to look in to future industries  He is interested to research business ecosystems in different contexts like cyber security, healthcare, future’s network etc  with a business model perspective  He received his M Sc  (2015) in international business from the University of Oulu  Prior to that he acquired MBA (2011) specializing in managing information systems in business applications  Previously, he has also enjoyed about three years in a top tier bank in Bangladesh as a channel innovator 
Marika Iivari is a postdoctoral researcher at the Martti Ahtisaari Institute within Oulu Business School  She defended her doctoral dissertation on business models in ecosystemic contexts  She holds M Sc  in International Business from the Ulster University, Northern Ireland  Her research interests are in the areas of open innovation, business models and strategy in the context of innovation ecosystems and smart cities, digital and ICT business ecosystems  She has been involved in several research projects around 5G and the Internet of Things, most recently in the health care sector  She is also an active member of the Business Model Community, the Open Innovation Community and the Society for Collaborative Networks 
Petri Ahokangas received his M Sc  (1992) and D Sc  (1998) degrees from the University Vaasa, Finland  He is currently Adjunct Professor (International software entrepreneurship) and Senior research fellow at Martti Ahtisaari Institute, Oulu Business School, University of Oulu, Finland  His research interests are in how innovation and technological change affect international business creation, transformation, and strategies in highly technology – intensive or software – intensive business domains  He has over 100 publications in scientific journals, books, conference proceedings, and other reports  He is actively working in several ICT-focused research consortia leading the business-related research streams 
Lauri Isotalo has received his M Sc  from Helsinki University of Technology (currently Aalto University) in 1992  He has also a postgraduate Diploma in Business Administration  At first Lauri worked in Nokia Corp  in Mobile Technology & System Marketing unit specializing in Intelligent Networks  In 1992 he joined Elisa Corp  where he has held several managerial positions in value added services business, system and process security and mobile network development  Since 2005 Lauri has also led Elisa SME teams in various international collaboration projects and acquired a deep knowledge of the cyber security of legacy telecommunication networks, ip core, access networks, user terminals and modern virtualized data center IT platforms/cloud systems  From 2014 Lauri has headed SDN&NFV development in Elisa 
Riikka Niemelä, M Sc , MHSc , is a cross-disciplinary health-tech professional who obtained her degrees in Electrical Engineering (Master of Science) and Medical technology (Master of Health Sciences) from the University of Oulu, Finland  After R&D engineering, she worked as a research assistant to research the national eHealth development of the Finnish healthcare  Thereafter, she has been researching and promoting the adaptation of connected health technologies in surgical processes of a Nordic future hospital, resulting in scientific publications  Currently, she works as a project manager in Tuttlingen, Germany – the world center of Medical Technology – to generate R&D and innovation projects in cooperation with the ICT and LifeScience companies from Oulu, Finland  Her aim is to promote the digitalization and internationalization of the both high-tech regions and to generate innovations serving the increasing demands for healthcare 
Journal of ICT, Vol  5_1, 107–128 



ground_truth: Centralized control of mobile devices and applications.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What are some fields in which the inverse problem is encountered?

Read the following text and answer briefly.

Context 3:  The process is repeated for all the couplings obtaining then a fully connected adjacency matrix  The results here presented are obtained with $\lambda = 0 01$ 
   For the minimization we have used the MATLAB routine \emph{minFunc\_2012}\cite{min_func}  
      
      \begin{figure}[t!]
    	\centering
    	\includegraphics[width=1\linewidth]{Jor11_2D_l2_JR_soJR_TPJR}
    	\caption{Top panels: instances of single site coupling reconstruction for the case of $N=64$ XY spins on a 2D lattice with ordered $J$ (left column) and bimodal distributed $J$ (right column)  
    	
    	Bottom panels: sorted couplings }
    	\label{PL-Jor1}
    \end{figure}

   
To produce the data by means of numerical Monte Carlo simulations a system with $N=64$  spin variables  is considered on a deterministic 2D lattice with periodic boundary conditions  
Each spin has then connectivity $4$, i e , we expect to infer an adjacency matrix with $N c = 256$ couplings different from zero  
The dynamics of the simulated model is based on the Metropolis algorithm and parallel tempering\cite{earl05} is used to speed up the thermalization of the system 
The thermalization is tested looking at the average energy over logarithmic time windows and
the acquisition of independent configurations
starts only after the system is well thermalized 

   For the values of the couplings we considered two cases: an ordered case, indicated in the figure as  $J$ ordered (e g , left column of Fig  \ref{PL-Jor1}) where the couplings can take values $J_{ij}=0,J$, with $J=1$, 
   and a quenched disordered case, indicated in the figures as  $J$ disordered (e g , right column of Fig  \ref{PL-Jor1})
   where the couplings can take also  negative values, i e , 
    $J_{ij}=0,J,-J$, with a certain probability   The results here presented were obtained with bimodal distributed $J$s: 
   
    $P(J_{ij}=J)=P(J_{ij}=-J)=1/2$   The performances of the PLM have shown not to depend on $P(J)$  
   
    We recall that in Sec  \ref{sec:plm} we used the temperature-rescaled notation, i e , $J_{ij}$ stands for $J_{ij}/T$  
   
    To analyze the performances of the PLM, in Fig  \ref{PL-Jor1} the inferred couplings, $\mathbb{J}^R_{\rm inf}$, are shown on top of the original couplings,  $\mathbb{J}^R_{\rm true}$ 
     The first figure (from top) in the left column shows  the $\mathbb{J}^R_{\rm inf}$ (black) and the $\mathbb{J}^R_{\rm tru}$ (green) for a given spin
     at temperature $T/J=0 7$ and number of samples  $M=1024$  PLM appears to reconstruct the correct couplings, though zero couplings are always given a small inferred non-zero value  
     In the left column of Fig   \ref{PL-Jor1},  both the $\mathbb{J}^R_{\rm{inf}}$ and the $\mathbb{J}^R_{\rm{tru}}$ are sorted in decreasing order and plotted on top of each other  
     We can clearly see that $\mathbb{J}^R_{\rm inf}$ reproduces the expected step function  Even though the jump is smeared, the difference between inferred couplings corresponding to the set of non-zero couplings 
     and to the set of zero couplings can be clearly appreciated 
     Similarly, the plots in the right column of Fig  \ref{PL-Jor1} show the results obtained for the case with  bimodal disordered couplings, for the same working temperature and number of samples  
     In particular, note that the algorithm infers half positive and half negative couplings, as expected 
     
     
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{Jor11_2D_l2_errJ_varT_varM}
\caption{Reconstruction error $\mbox{err}_J$, cf  Eq  (\ref{eq:errj}), plotted as a function of temperature (left) for three values of the number of samples $M$ and  as a function $M$ (right) for three values of temperature in the ordered system, i e , $J_{ij}=0,1$  
The system size is $N=64$ }
\label{PL-err-Jor1}
\end{figure}

In order to analyze the effects of the number of samples and of the temperature regimes, we plot in Fig  \ref{PL-err-Jor1} the reconstruction error, Eq  (\ref{err}), as a function of temperature for three different sample sizes $M=64,128$ and $512$  
The error is seen to sharply rise al low temperature, incidentally, in the ordered case, for  $T<T_c \sim 0 893$, which is the Kosterlitz-Thouless transition temperature of the 2XY model\cite{Olsson92}  
 However, we can see that if only $M=64$ samples are considered, $\mbox{err}_J$ remains high independently on the working temperature  
 In the right plot of Fig  \ref{PL-err-Jor1},  $\mbox{err}_J$ is plotted as a function of $M$ for three different working temperatures $T/J=0 4,0 7$ and $1 3$  As we expect, 
 $\mbox{err}_J$  decreases as $M$ increases  This effect was observed also with mean-field inference techniques on the same model\cite{Tyagi15} 

To better understand the performances of the algorithms, in Fig  \ref{PL-varTP-Jor1} we show several True Positive (TP) curves obtained for various values of $M$ at three different temperatures $T$  As $M$ is large and/or temperature is not too small,  we are able to reconstruct correctly all the couplings present in the system (see bottom plots) 
The True Positive curve displays how many times the inference method finds a true link of the original network  as a function of the index of the vector of sorted absolute value of reconstructed couplings $J_{ij}^{\rm inf}$  
The index $n_{(ij)}$ represents the related spin couples $(ij)$  The TP curve is obtained as follows: 
first  the values $|J^{\rm inf}_{ij}|$ are sorted in descending order and  the spin pairs $(ij)$ are ordered according to the sorting position of $|J^{\rm inf}_{ij}|$  Then,
     	a cycle over the ordered set of pairs $(ij)$, indexed by $n_{(ij)}$, is performed, comparing with the original network coupling $J^{\rm true}_{ij}$ and verifying whether it is zero or not  The true positive curve is computed as
\begin{equation}
\mbox{TP}[n_{(ij)}]= \frac{\mbox{TP}\left[n_{(ij)}-1\right] (n_{ij}-1)+ 1 -\delta_{J^{\rm true}_{ij},0}}{n_{(ij)}}
\end{equation}
As far as $J^{\rm true}_{ij} \neq 0$, TP$=1$  As soon as the true coupling of a given $(ij)$ couple in the sorted list is zero, the TP curve departs from one  
In our case, where the connectivity per spin of the original system is  $c=4$ and there are $N=64$ spins, we know that we will have  $256$ non-zero couplings   
     	If the inverse problem is successful, hence, we expect a steep decrease of the TP curve when  $n_{ij}=256$ is overcome 

In Fig  \ref{PL-varTP-Jor1}
it is  shown that,  almost independently of $T/J$, the TP score improves as $M$ increases  Results are plotted for three different temperatures, $T=0 4,1$ and $2 2$, with increasing number of samples $M = 64, 128,512$ and $1024$ (clockwise)  
We can clearly appreciate the improvement in temperature if the size of the data-set is not very large: for small $M$, $T=0 4$ performs better  
When $M$ is high enough (e g , $M=1024$), instead, the TP curves do not appear to be strongly influenced by the temperature 

\begin{figure}[t!]
	\centering
	\includegraphics[width=1\linewidth]{Jor11_2D_l2_TPJR_varT_varM}
	\caption{TP  curves  for 2D short-range ordered $XY$ model with $N=64$ spins at three different values of $T/J$ with increasing - clockwise from top - $M$ }
	\label{PL-varTP-Jor1} 
\end{figure} 

\subsection{$XY$ model with complex-valued couplings}
For the complex $XY$ we have to contemporary  infer $2$ apart coupling matrices,  $J^R_{i j}$ and $J^I_{i j}$   As before, a system of $N=64$ spins is considered on a 2D lattice 
For the couplings we have considered both ordered and bimodal disordered cases 
In Fig  \ref{PL-Jor3}, a single row of the matrix $J$ (top) and the whole sorted couplings (bottom) are displayed for the ordered model (same legend as in Fig  \ref{PL-Jor1}) for the real, $J^R$ (left column), and the imaginary part, $J^I$   

\begin{figure}[t!]
	\centering
\includegraphics[width=1\linewidth]{Jor3_l2_JRJI_soJRJI_TPJRJI}
	\caption{Results related to the ordered complex XY model with $N=64$ spins on a  2D lattice  Top: instances of  single site reconstruction for the real, JR (left column), and
		the imaginary, JI (right column), part of $J_{ij}$  Bottom: sorted values of JR (left) and JI (right) }
		
	\label{PL-Jor3}
\end{figure}
 
 
  \section{PLM with Decimation}
 \label{sec:res_dec}
 

\begin{figure}[t!]
   	\centering
   	\includegraphics[width=1\linewidth]{Jor1_dec_tPLF_varT_varM}
    	\caption{Tilted Pseudolikelyhood, ${\cal L}_t$, plotted as a function of decimated couplings  Top: Different ${\cal L}_t$ curves obtained for different values of $M$ plotted on top of each other  Here $T=1 3$  The black line indicates the expected number of decimated couplings, $x^*=(N (N-1) - N c)/2=1888$  As we can see, as $M$ increases, the maximum point of ${\cal L}_t$ approaches $x^*$  Bottom: Different ${\cal L}_t$ curves obtained for different values of T with $M=2048$  We can see that, with this value of $M$, no differences can be appreciated on the maximum points of the different  ${\cal L}_t$ curves }
   	\label{var-$t$PLF}
   \end{figure}

\begin{figure}[t!]
    \centering
    \includegraphics[width=1\linewidth]{Jor1_dec_tPLF_peak_statistics_varM_prob eps}
    	\caption{Number of most likely decimated couplings, estimated by the maximum point of $\mathcal{L}_t$, as a function of the number of samples $M$  We can clearly see that the maximum point of $\mathcal{L}_t$ tends toward $x^*$, which is the right expected number of zero couplings in the system }  
    	\label{PLF_peak_statistics}
    \end{figure}
      
      For the ordered real-valued XY model we show in Fig  \ref{var-$t$PLF}, top panel, the outcome on the tilted pseudolikelyhood, $\mathcal{L}_t$ Eq  \eqref{$t$PLF}, of the progressive decimation: from a fully connected lattice  down to an empty lattice  The figure shows the behaviour of $\mathcal{L}_t$ for three different data sizes $M$  A clear data size dependence of the maximum point of  $\mathcal{L}_t$, signalling the most likely value for decimation, is shown  For small $M$ the most likely number of  couplings is overestimated and for increasing $M$ it tends to the true value, as displayed in Fig  \ref{PLF_peak_statistics}   In the bottom panel of Fig  \ref{var-$t$PLF} we display instead different 
 $\mathcal{L}_t$ curves obtained for three different values of $T$ 
  Even though the values of $\mathcal{L}_t$ decrease with increasing temperature, the value of the most likely number of decimated couplings appears to be quite independent on $T$ with $M=2048$ number of samples 
In Fig  \ref{fig:Lt_complex} we eventually display the tilted pseudolikelyhood for a 2D network with complex valued ordered couplings, where the decimation of the real and imaginary coupling matrices proceeds in parallel, that is, 
when a real coupling is small enough to be decimated its imaginary part is also decimated, and vice versa 
One can see that though the apart errors for the real and imaginary parts are different in absolute values, they display the same dip, to be compared with the maximum point of $\mathcal{L}_t$ 
     
       \begin{figure}[t!]
    	\centering
      	\includegraphics[width=1\linewidth]{Jor3_dec_tPLF_new}
      	\caption{Tilted Pseudolikelyhood, ${\cal L}_t$, plotted with the reconstruction errors for the XY model with $N=64$ spins on a 2D lattice  These results refer to the case of  ordered and complex valued couplings  The full (red) line indicates ${\cal L}_t$  The dashed (green) 
      		and the dotted (blue) lines show the reconstruction errors (Eq  \eqref{eq:errj}) obtained for the real and the imaginary couplings respectively  We can see that both ${\rm err_{JR}}$ and ${\rm err_{JI}}$ have a minimum at $x^*$ }
          	\label{fig:Lt_complex}
    \end{figure}

\begin{figure}[t!]
   	\centering
   	\includegraphics[width=1\linewidth]{Jor1_dec_JR_soJR_TPJR}
   	\caption{XY model on a 2D lattice with $N=64$ sites and real valued couplings  The graphs show the inferred (dashed black lines) and true couplings (full green lines) plotted on top of each other  The left and right columns refer to the
   		 cases of ordered and bimodal disordered couplings, respectively  Top figures: single site reconstruction, i e , one row of the matrix $J$  Bottom figures: couplings are plotted sorted in descending order }  
   	\label{Jor1_dec}
   \end{figure}
   
\begin{figure}[t!]
    	\centering
    	\includegraphics[width=1\linewidth]{Jor3_dec_JRJI_soJRJI_TPJRJI}
    	\caption{XY model on a 2D lattice with $N=64$ sites and ordered complex-valued couplings 
    		The inferred and true couplings are plotted on top of each other  The left and right columns show the real and imaginary parts, respectively, of the couplings  Top figures refer to a single site reconstruction, i e , one row of the matrix $J$  Bottom figures report the couplings sorted in descending order }
    	\label{Jor3_dec}
    \end{figure}
    
       







       \begin{figure}[t!]
     	\centering
     	\includegraphics[width=1\linewidth]{MF_PL_Jor1_2D_TPJR_varT}
     	\caption{True Positive curves obtained with the three techniques: PLM with decimation, (blue) dotted line,  PLM with $l_2$ regularization, (greed) dashed line, and mean-field, (red) full line   These results refer to real valued ordered couplings with $N=64$ spins on a 2D lattice  The temperature is here $T=0 7$ while the four graphs refer to different sample sizes: $M$ increases clockwise }
     	\label{MF_PL_TP}
     \end{figure}
    
    \begin{figure}[t!]
    	\centering
    	\includegraphics[width=1\linewidth]{MF_PL_Jor1_2D_errJ_varT_varM}
    	\caption{Variation of reconstruction error, ${\rm err_J}$, with respect to temperature as obtained with the three different techniques, see Fig  \ref{MF_PL_TP}, for four different sample size:  clockwise from top   $M=512,1024, 2048$ and $4096$ } 
    	\label{MF_PL_err}
    \end{figure}
    
     Once the most likely network has been identified through the decimation procedure, we perform the same analysis displayed in Fig  \ref{Jor1_dec}  for ordered and then quenched disordered real-valued couplings
and in Fig  \ref{Jor3_dec} for  complex-valued ordered couplings   In comparison to the results shown in Sec  \ref{sec:res_reg},
  the PLM with decimation leads to rather cleaner results  In Figs  \ref{MF_PL_err} and \ref{MF_PL_TP} we compare the performances of the PLM with decimation in respect to ones of the PLM with $l_2$-regularization  These two techniques are also analysed in respect to a mean-field technique previously implemented on the same XY systems\cite{Tyagi15} 
  
    For what concerns the network of connecting links, in Fig  \ref{MF_PL_TP} we compare the TP curves obtained with the three techniques  The results refer to the case of ordered and real valued couplings, but similar behaviours were obtained for the other cases analysed  
  The four graphs are related to different sample sizes, with $M$ increasing clockwise  When $M$ is high enough, all techniques reproduce the true network  
  However, for lower values of $M$ the performances of the PLM with $l_2$ regularization and with decimation drastically overcome those ones of the previous mean field technique  
  In particular, for $M=256$ the PLM techniques still reproduce the original network while the mean-field method fails to find more than half of the couplings  
  When $M=128$, the network is clearly reconstructed only through the PLM with decimation while the PLM with $l_2$ regularization underestimates the couplings  
  Furthermore, we notice that the PLM method with decimation is able to clearly infer the network of interaction even when $M=N$ signalling that it could be considered also in the under-sampling regime $M<N$   
 
 
In Fig  \ref{MF_PL_err} we compare the temperature behaviour of the reconstruction error 
In can be observed that for all temperatures and for all sample sizes  the reconstruction error, ${\rm err_J}$, (plotted here in log-scale) obtained with the PLM+decimation is always smaller than 
that one obtained with the other techniques  The temperature behaviour of ${\rm err_J}$ agrees with the one already observed for Ising spins in \cite{Nguyen12b} and for XY spins  in \cite{Tyagi15} with a mean-field approach:  ${\rm err_J}$ displays a minimum around $T\simeq 1$ and then it increases for very lower $T$; however,
 the error obtained with the PLM with decimation is several times smaller  than the error estimated by the other methods 



 
 

     
     \section{Conclusions}
     \label{sec:conc}


Different statistical inference methods have been applied to the inverse problem of the XY model 
After a short review of techniques based on pseudo-likelihood and their formal generalization to the model we have tested their performances against data generated by means of Monte Carlo numerical simulations of known instances
with diluted, sparse, interactions 

The main outcome is that the best performances are obtained by means of the  pseudo-likelihood method combined with decimation  Putting to zero (i e , decimating) very weak bonds, this technique turns out to be very precise for  problems whose real underlying interaction network is sparse, i e , the number of couplings per variable does not scale with number of variables 
The PLM + decimation method is compared to the PLM + regularization method, with $\ell_2$ regularization and to a mean-field-based method  The behavior of the quality of the network reconstruction is analyzed by looking at the overall sorted couplings and at the single site couplings, comparing them with the real network, and at the true positive curves in all three approaches  In the PLM +decimation method, moreover, the identification of the number of decimated bonds at which the tilted pseudo-likelihood is maximum allows for a precise estimate of the total number of bonds  Concerning this technique, it is also shown that the network with the most likely number of bonds is also the one of least reconstruction error, where not only the prediction of the presence of a bond is estimated but also its value 

The behavior of the inference quality in temperature and in the size of data samples is also investigated, basically confirming the low $T$ behavior hinted by Nguyen and Berg \cite{Nguyen12b} for the Ising model  In temperature, in particular, the reconstruction error curve displays a minimum at a low temperature, close to the critical point in those cases in which a critical behavior occurs, and a sharp increase as temperature goes to zero  The decimation method, once again, appears to enhance this minimum of the reconstruction error of almost an order of magnitude with respect to other methods 
 
The techniques displayed and the results obtained in this work can be of use in any of the many systems whose theoretical representation is given by Eq  \eqref{eq:HXY} or Eq  \eqref{eq:h_im}, some of which are recalled in Sec  \ref{sec:model}  In particular, a possible application can be the field of light waves propagation through random media and the corresponding problem of the  reconstruction of an object seen through an opaque medium or a disordered optical fiber \cite{Vellekoop07,Vellekoop08a,Vellekoop08b, Popoff10a,Akbulut11,Popoff11,Yilmaz13,Riboli14} 

 	



ground_truth: Physics, biology, social sciences, finance, and neuroscience are some fields in which the inverse problem is encountered.