Model of Multilayer Knowledge Diffusion for Competence Development in an Organization

Growing role of intellectual capital within organizations is affecting new strategies related to knowledge management and competence development. Among different aspects related to this field, knowledge diffusion has become one of interesting areas from both practitioner and researchers perspective. Several models were proposed with main goal to simulate diffusion and to explain the nature of these processes. Existing models are focused on knowledge diffusion and they assume diffusion within a single layer using knowledge representation. From the organizational perspective connecting several types of knowledge and modelling changes of competence can bring additional value. In the article we extended existing approaches by using multilayer diffusion model and focused on analysis of competence development process. The proposed model describes competence development process in a new way through horizontal and vertical knowledge diffusion in multilayer network. In the network, agents collaborate and interchange various kind of knowledge through different layers and this mutual activities affect the competences in a positive or negative way. Taking under consideration workers cognitive and social abilities and the previous level of competence the new competence level can be estimated. The model is developed to support competence management in different organizations.


Introduction
Employees' competence become the main part of organization's intellectual capital [1]. According to [2] the management and control of knowledge and skills, and more recently the management of organizations' competencies have turned out to be essential factors of industrial processes' performance. Modern companies are no longer production systems of products and services but create and sell knowledge-based products. Including competence management into production process required integrating new decision processes regarding the cognitive dimension of business, at every managerial level [2]. Moreover, the companies have to expand knowledge management to competence management. As a result the companies will be able to fulfil the following items [3]: find the right single employee for a specific task or project, retrieve and assemble flexible project teams, develop and update employees skills, explore the employees future career paths, speed up innovation management. The workers become knowledge workers [4] and continuing needs for upgrading workplace knowledge, skills and competencies is developed. Changes in work and the ways in which it is carried out bring the need for upgrading workplace knowledge, skills and competencies. In today's workplaces, and for a number of reasons, workloads are higher than ever and stress is a growing concern [5].
Competence is an observable or measurable ability of an actor to perform a necessary action(s) in a given context(s) to achieve a specific outcome(s) [6]. After analysis of various competence definitions [7,8] one thing is common, competence is made of different knowledge-based components (e.g. knowledge, skills, behaviours). Competence development process is an acquisition of a specific set of competence's components that constitutes a particular competence [9].
In our work the modelling of the competence development process is based on the knowledge diffusion model that extends current solutions. The approach is new and required special characteristics of diffusion model. We developed a multilayer diffusion model based on the multilayer graph reflecting organisation's network. In the graph each layer represents competence's component (some kind of knowledge). There is an interaction between layers defined as a vertical diffusion. The horizontal diffusion occurs on every layer's level and relates to the diffusion of one type of knowledge between knowledge workers. Moreover, every node of organisation's network represents knowledge worker with individual set of knowledge and own cognitive and social potentials for learning (self-learning) and teaching (training). The knowledge worker, in every step of simulation, is looking for best source of knowledge. In addition, depending of node's neighbourhood the knowledge can be forgotten.
The existing diffusion models from literature were not suitable for competence modelling due to their limitations. The most important drawback is the lack of simultaneous support of vertical and horizontal diffusion. Moreover, diffusion logic proposed in literature does not reflect the competence development process. In our approach the diffusion logic is set to search for best teacher (source of knowledge) in node's neighbourhood. The best teacher is a node with the highest value of knowledge and teaching ability. The diffusion result is affected by the learning/teaching abilities of nodes, initial value of knowledge, vertical diffusion form other layers (relation between knowledge) and forgetting process.
Similarly constructed diffusion model cannot be found in the literature.
The rest of the article is organized in the following way. Next chapter covers the issue of competence development in organization. We listed components associated with competence and developed a way to include them in the diffusion model. After that, the knowledge diffusion models are analysed. We concluded that all of them are lacking some proposition for competence development. Chapter four is dedicated to the description of multilayer knowledge diffusion model for competence development in an organization. The model is the focal point of the article. The last chapter is dedicated to validation of proposed model. In this chapter some case studies will be presented.

Competence-based approach to system design
In literature we can find different definitions of competence linked by three fundamental characteristics: resources, context, and objectives [10]. The competence profile is data about a competence that may be aggregated for communication among individuals, organizations, and public administrations. The competence modelling issue has been a subject of research for a long time, starting with Frederick Taylor [11]. However, in recent years studies have greatly accelerated. An interesting review of competence notion can be found in [8,12,13]. The history and background of standardization in this area and research project are covered in [7]. Some computational approach for competence profile processing are described in [14,15]. The fuzzy nature of competence description is explained in [16]. Moreover, there is a number of high quality scientific journals with special issue dedicated only to competence including: Competence Management in Industrial Processes [17], Skills Management -Managing Competencies in the Knowledge-based Economy [18], Learning Networks for Lifelong Competence Development [19], Assessment of Competencies [20], Competencies Management [21].
Competence-based approaches have proved to be a critical tool in many organizational functions, such as employment planning, recruitment, trainings, raising work efficiency, personal development, managing key competencies [22]. In addition, competence-based system can be used for different purposes such as staff development and deployment, job analysis or economic evaluation [23]. The reasons for this are made by [24] as: -Competence-based approach can provide identification of the skills, knowledge, behaviours and capabilities needed to meet current and future personnel selection needs, in alignment with the differentiations in strategies and organizational priorities.
-Competence-based approach can focus the individual and group development plans to eliminate the gap between the competencies requested by a project, job role, or enterprise strategy and those available.
The important way of competence developing is a community of practice, because growing number of people and organizations in various sectors are now focusing on communities of practice as a key to improving their performance [10]. Communities of practice are groups of people who share a concern or a passion for something they do and learn how to do it better as they interact regularly [25].
From a pragmatic point of view competence is a combination of components, usually related to knowledge, experiences and skills/abilities. It is important to notice that it is not possible to directly develop another person's competence. It is just possible to set the scene, to provide the tools and act like a catalyst [26]. As a result, the competence development is regarded as the acquisition of a specific set of competence's components (e.g. knowledge, skills) that constitutes a particular competence [9]. Moreover, overriding principle for development of competence becomes transmitting such attributes (components) to those people who do not possess them by range of activities, such as general communication, classroom teaching, on-the-job training and job rotation. [9]. The data about the competencies value/state is produced and transformed by identification, assessment and acquisition processes [27]. Competencies can be processed because there is a certain set of tools used to test competencies and estimate their levels [22,28].
There are some challenging issues with competence-based approach [29]: sharing, push/pull power balance, synergy creation).

6
The dichotomy between definitions of competence that target individual workers and definitions that target the results of their work is a complex issue [32]. On the one hand the literature has focused on individual competences and has taken the worker's attributes as a starting point for discussing competence [33]. The worker's competence value is treated as a stock that can be developed through training and validated in "objective" rating schedules [34]. On the other hand, the competence is conceptualised as a characteristic of organisations where human competences are seen as one of the resources available to organisations [33].

Competence as a union of associated components
For the propose of model design it is required to identified the structure of competence. Competencies are considered as an union of different components (see Tab. 1). -Knowledge (includes theoretical knowledge and procedural knowledge) -Skills (includes formalized know-how and empirical knowhow -Behavioural aptitudes (individual's behaviour at work) [23] After the presentation of the components of competence it is crucial to understand what the relationship between them is. A good way of understanding the relationship is use of competence ontology structures, which can be found in literature [14,24,40,41,42]. Most of them is designed to [41]: define an organization-wide role structure based on the competencies required by job functions and organizational positions; identification of the competencies required in order to perform the various activities involved in each business process and assignment of roles to process activities based on these competencies; identification of the competencies acquired in the organization and assignment of users to roles through competence matching.
Moreover, the competence ontology is the most important part of an effective competence management system [24]. The competence's ontology is important because competence management system has to collaborate with other similar systems or e-learning and human resources applications. More formal approach to competence ontology building, based on the Description Logics, can be found in [42]. treated as a part of an innovative process, is the process by which an innovation is communicated through certain channels over time among the members of a social system [43]. Research shows the importance of social network structure, which should be analysed, developed and managed for continuous innovation in organizations [44].
In the area of scientific research knowledge diffusion can be defined as the adaptations and applications of knowledge documented in scientific publications and patents [45].
Technology diffusion is a complex social communication process. According to [46] technology diffusion starts with an innovation generated by a particular source. After that, potential adopters are informed about the availability of the new technology and persuaded by contact with prior users to adopt them. It is important to understand that technology diffusion is very sectoral. Ribeiro et al. [47] shows that any given innovation related to an area X is intensely used and diffused only within its specific area and will hardly permeate into other areas.
In a learning organisation knowledge diffusion is a process of knowledge communication and learning [48]. Moreover, the close relation among members results in strong willingness of knowledge diffusion.
It is important to say that tracing knowledge diffusion is a challenging issue due to the extreme complexity of diffusion processes [45]. Morone and Taylor [49] point out that knowledge diffusion is a complex social phenomenon which consists of, among others, knowledge spillover, knowledge transfer and knowledge integration. The nature of human interactions and information flow is affected mainly by the creation of new knowledge and the process of learning at an individual level [50]. Moreover, the different organizational and 9 teamwork structure conducts different knowledge behaviours and their performance.
However, we should have in mind that for group of people learning efficiency is in most cases accelerated [51]. In addition, diffusion makes neighbouring agents tend to display similar knowledge levels [52]. Social influence theories provide an interpretation that different social proximity evoke distinguish contagion effects [51]. The best learning outcome can be determined by the best suit payoff schemes and network structure changes within a complex social network [50]. In other words the effectiveness of the diffusion is a function of the network structure and seeding strategy used in delivering the initial broadcast [53].
From the economic point of view the knowledge diffusion process is related to the transfer of intellectual capital. Knowledge diffusion takes place through worker mobility [54] and the research task is related to finding the equilibrium between the host and the mobile worker. The [55] model offers a quantitative approach to explore the dynamic relationship between knowledge value and enterprise benefits in a given period.
At cognitive level the research of knowledge diffusion is related to the problem of [56]: how do individuals perceive and cognitively represent the social networks that surround them, and how do individuals' perceptions of their social networks affect their behaviours and outcomes?
The next important issue, related to knowledge transfer, is homophily, defining as tendency of people to associate relatively more with those who are similar to them than with those who are not [57]. The Golub and Jackson [57] show that homophily and the segregation induces in networks has important consequences for processes of interest, particularly the ones of information flow.
Knowledge diffusion must be based on efficient communication channels between all actors. The importance of such efficient channels is empirically supported by McGarvie [58] who shows that technological knowledge diffusion is faster in countries which share a common official language, whose inventors communicate more frequently by phone, and are geographically closer to each other.
The process of diffusion of knowledge is based on several communication mechanisms [59]: formal way of communication through documents, databases, face-to-face meetings, e-mails, videoconferencing, and social communication (excluding commercial transaction) throughout communities-or networks-of practice. Typical artefacts are opinion, practice, know-how [60]. The knowledge benefits can be externalized from the following three knowledge sources [61]: (1) the use of original knowledge inside the organization; (2) the improvement of original knowledge due to internal investment; and (3) the integration of innovative knowledge.

From one point of view knowledge diffusion is intended by the organization.
According to Canals [59] the diffusion process takes place in a formal way through the use of documents and databases or through interaction in face-to-face meetings or by using technological means as e-mail or videoconference. Form the another point of view unintended diffusion of knowledge is performed in accordance with knowledge spillovers process. The diffusion process takes advantage of the social relationships between employees of the firms, be it of a professional type through communities-or networks-of practice or more of a personal nature [59]. The most important issue is to combine in knowledge diffusion the intended knowledge diffusion mechanisms and unintended spreading of knowledge. For such reason knowledge diffusion is not equivalent to other diffusion processes modelled in natural sciences as epidemics or in social sciences (e.g. like the spread of rumors) [59]. Nevertheless, there are some attempts to do that e.g. tacit knowledge diffusion modelled as a SIR epidemic transformation [62].

Knowledge diffusion modelling
The problem of knowledge diffusion is an important element of complex network theory application. Based on the literature analysis we can recognise two approaches to problem modelling [63]. The first one is focuses on knowledge exchange behavioural patterns between a pair of individuals. The cognitive and social psychology and economics investigated absorptive capability, effectiveness and stimulation of knowledge share [64]. The second one used computer simulation to discuss the influence of the topology of social networks [63]. The simulation results show in what way the network structural characteristics influence knowledge diffusion.
When discussing the knowledge diffusion modelling we should keep in mind two dimensions of this problem: network topology and design of interaction rules driving knowledge transmission. Many studies show that the effectiveness of the diffusion mainly depends on the network structure and the seeding strategy used [65]. The problem of network topology analysis is solved by the utilisation of an existing network models which support real-world phenomena such as power-law ("scale-free") degree distributions, high clustering, short network diameter. In addition some authors make a debate about how accurately present models and corresponding analytic solutions or simulations render real-world network [66].
The main concept of knowledge transmission mechanism, according to [66], are: payoff based models [67], opinions vectors, continuous or discrete, one-dimensional [68] or ndimensional [69]. The other issue is a mechanisms of knowledge diffusion, most of models is aiming to maximizing the spread of influence in a network and they are based on assumed rather than measured influence effects [70].
The complex nature of knowledge diffusion problem is difficult to conceptualise and formalize. However, there are some propositions in literature. In [71] formal approach to create integrated ontology, which covers a number of learning activities, is proposed. Due to utilization of OpenCyc framework the way to computational semantics is clear. The real application of complex ontology, which is formalised for Computer-Aided Control Engineering, can be found in [72]. However, before we begin to think of the formalization we should try to define the based learning process in order to recognise the objectives outside the individual, and the transformation of these activities into measurable, efficient behaviour [73].
The presented literature gives some idea how to formalized different parts of knowledge diffusion model.

Related work in the areas of knowledge diffusion models
A number of papers studied a model of a population of agents whose interaction network co-evolves with knowledge diffusion and accumulation. General idea of the proposed model is based on the Cowan and Jonard model (CJ) [74]. Similar to CJ model the proposed model is designed to capture effects of incremental innovation and their diffusion over a network of heterogeneous agents. The CJ model assumes [75]: agents are arranged in one-dimensional space; each agent occupies one vertex and may interact with their k nearest neighbours on either side; the population of individuals is endowed with different levels of initial knowledge; a small number of agents are 'expert' and are endowed with a high level of knowledge in at least one value of the vector; all individuals interact among themselves, exchanging information; knowledge is a non-rival good and can be transferred without decreasing the level of knowledge possessed by each trader.
In our work we extended classical CJ model to multidimensional vertical and horizontal diffusion scheme. Moreover, new mechanism of knowledge processing was introduced includes self-learning and forgetting process. Some authors noticed the importance of this factors (e.g. dissemination ability and knowledge forgetting in Xiaoqing and Runqing work [76]). However, this phenomenon is not regularly analysed because of the complexity.
In [77] two processes on the network are proposed: knowledge diffusion refers to the distribution of existing knowledge in the network, while knowledge upgrade means the discovery of new knowledge. Additionally, authors took into account the agent's knowledge absorptivity and forgetting factors represents some cognitive ability of agent. However, the proposed model works only for one type of knowledge, absorptivity and forgetting factors are constant and not associated with agent's network localisation.
The paper [78] focuses on knowledge diffusion as an economic process of different types of knowledge exchanging. Similar to previous work the paper covers the knowledge diffusion process (agent broadcasts his knowledge to the agents to whom he is directly connected) and knowledge creation process (agents receive new knowledge which is combined with their existing knowledge stocks). However, this paper only examines the relationship between network architecture and aggregate knowledge levels.
The key factors that affect the speed and the distribution of knowledge diffusion are identified in Morone et al. [79]. Theory where content of the messages has no meaning. However, there is an opinion in the competence literature that the process of competence computing should be understood as enabling the use of competence databases for inference and combination of competencies for different functions and processes, not as a reductionist account of competencies to numeric models [14]. In our case we focused on knowledge flow and linked it to the diffusion process, which is based on mathematical transformations.
Proposition of 3th class' layers model for diffusion model for competence development [29]: -Class 1: know-how-practical, hands-on forms of knowledge gained through incremental improvements to products and processes.
-Class 2: know-why-theoretical forms of understanding that enable the creation of new kinds of products and processes.
-Class 3: know-what-a strategic form of understanding about the value creating purposes to which available know-how and know-why forms of knowledge may be applied We assume that one layer in our model is dedicated to one kind of knowledge, which belongs to one class of competence's components . The question arise: Can each component of competence be called knowledge? The works [10,25] prove that it is correct and knowledge can be expressed in various forms: -Knowledge can be explicitly formalizedtexts, documents multimedia.
-Knowledge can be a practiceit rests on the accumulation of experiments.
-Knowledge can be tacitall cannot be formalized. Its transmission requires suitable means: conversation, training, joint work, etc.
-Knowledge can be socialthe technical know-how of a company does not rest on an individual but on the interaction of all the members of its technical community. It is while collaborating, by confronting their points of view, that these technicians create and finally hold new knowledge.
-Knowledge is dynamic, and evolves/moves in time.
In our approach the term knowledge will be used for description of all components of competence. Based on the term knowledge all competence's components like instances of class: knowledge, skills, behaviour can be modelled in a multilayer graph as a single layer.

Knowledge domain
Every node in the network represents single knowledge worker. According to [4] the knowledge employee's main tasks related to knowledge are capture/extract, analyse/organise, find, create/synthetize and distribute/share. In the organisation's network different types of knowledge is propagated in order to acquire competence by employee. Moreover, the productivity of knowledge workers is enhanced through competence enhancement and learning which take place directly at workers' workplaces [82].

Definition: Organization members
The organization X composed of knowledge worker determined by index i . The methods of knowledge modelling mainly focus on the formally representation of relationship between different areas/element/types of knowledge. The best way to do is to use the ontological approach. Ontology is a formal, explicit specification of a shared conceptualization [83]. The main components of an ontology are concepts, relations, instances and axioms [84]. The relations describe the interactions between concepts or a concept's properties. In the problem of knowledge diffusion is essential to show the existence of the relationship between areas/elements of knowledge. Because defining the types of relationship between is not very useful for numerical processing and difficult to determine, we focused on causality and mutual order of knowledge areas/elements. Therefore, we choose the method of presentation of the knowledge domain based on the Knowledge Space Theory [85].
In our approach the competence acquisition is a result of combination of different competence's components [86]. In other words the proper combinations of different knowledge elements, which reflect competence's components, result in an efficient competence acquisition by a knowledge worker. Competence is an ability to find an effective way to theoretical knowledge usage in order to solve a practical problem and the ability to verify the solutions. We use the Knowledge Space Theory to describe the relation between knowledge element in domain K .
Based on the Knowledge Space Theory let ) , (  K be a partial ordered set. In this theory the prerequisite relationship is cover by the surmise relation [87] and function  represents prerequisite relationship. Two knowledge items a and b are in surmise relation b a  if, whenever a person has solved/maintained item b correctly, we can surmise that this person is also able to solve item a correctly [88]. According to [89] we say that for Moreover, the graph ) , (  K is a Hasse digram for K . The important assumption for future discussion is that due to cognitive nature of problem and mutual relation between them, not all potential knowledge states have to be observed [90]. :

Knowledge network
In a knowledge network the node actively processes knowledge and edges represents channels for knowledge relocation [91].

Structure of competence use in the model
Competence can get gradually stronger, in a situation where surroundings affect and stimulate its components. For example, we acquire new skills in a training session or while working (e.g. software developers programming everyday). Competence (its level) can also degrade. The most common reason for it, is not using the given competence in everyday work. The other is thanks to technology progress which makes the components of competence outdated. We can distinguish different relations between competencies which affect the interaction between them. Increasing competence in a certain competence group (e.g. communication) can affect the increase of other competencies (e.g. sales of products). Next issues regarding competence processing in an organisation start to show up when we take a look from a company's perspective. From the company's point of view, certain competencies are created only by combining the competencies of a greater number of employees. The complexity of these combined competencies is too great for a single person to obtain this kind of competence.
In this approach we do not analyse the content of knowledge included in competence.
In our case we are interested in the competence's level, which allow us to analyse the knowledge and competencies growth and dynamics in the organization. In presented method the competence value will be normalized to range   1 , 0 in order to be compatible with scale in literature. The level of competence is related to the expertise of an employee (see Tab. 2) [93]. According to cognitive science employees with more competence (expert) within their domains are skilled, competent and think in qualitatively different ways than novices do [94].

Tab. 2: Linking competence value with expertise level
The distinguished or brilliant journeyman, highly regarded by peers, whose judgments are uncommonly accurate and reliable, whose performance shows consummate skill and economy of effort, and who can deal effectively with rare or ''tough'' cases. Also and expert is one who has special skills or knowledge derived from extensive experience with subdomains.
1 Master Master is any journeyman or expert who is also qualified to teach those at a lower level. Traditionally, a master is one of an elite group of experts whose judgments set the regulations, standards or ideals.

Definition: Competence
The competence set for organization X is defined in the following way: There relationships between competence and knowledge in organization X are represent by the matrix:

Process definition
In order to analyse the competence development in an organization in addition to the structure of the network, which represents the relationships that exist between staff, we also need to describe the processes associated with the competence development.

Horizontal knowledge diffusion
Horizontal process of knowledge diffusion is related to knowledge diffusion between knowledge network nodes (representing employees) on a selected layer j . In fact, it involves 20 simulating a situation in which the relationship will be created at the level of tacit knowledge.
The relationship affects the knowledge of involved employees with regard to their ability to teach and learn. In the classic Nonaka's model this process is called socialization [96].
According to [97] for knowledge sharing to be most effective, it should take place between people who have a common knowledge and can work together effectively. The mutual relationship should be strong. Thus tacit knowledge sharing is connected to ideas of communities and collaboration.
Horizontal knowledge diffusion occurs only between active nodes. There are two possible methods of knowledge diffusion between nodes: -Broadcast: the node transfers knowledge to all connected nodes; -Multicast: the node transfers knowledge to a selected set of nodes. The set of receiving nodes may be selected random or based on some strategies.
In our approach we focus on the multicast scheme for horizontal knowledge diffusion.
Every node on selected layer of multilayer graph X G is looking for most effective source of knowledge in its neighbourhood on this layer. In this context effective means with best combination of knowledge and social abilities. As a result the horizontal knowledge diffusion occurs only when the node is able to locate node with a greater potential for knowledge transfer in his neighbourhood. In other words weak looks for strong.

Definition: Horizontal knowledge diffusion
Horizontal knowledge diffusion is calculated for node i on layer j based on the function  defined in following way:  Let us propose some form of function  : where max d is a maximal node degree on layer j and node z has ) max( The A is a fixed value. In function (2)

Vertical knowledge diffusion
Vertical knowledge diffusion takes place in a single node and occurs between its knowledge layers. Generally speaking the knowledge value increasing on layer j may increase the knowledge value on other layers. The relationship between layers can be deducted from ) , (  K or described by organisation members and saved in an dedicated matrix. Moreover, the vertical knowledge diffusion process is an internal process in contrast to the horizontal knowledge diffusion which is an external process for a knowledge worker.
Let's define the vertical diffusion matrix for worker i : Let us propose some form function for vertical knowledge diffusion: where J n  is a layer's index. The function (3) is one of the possible linear relationships. For the real processes the matrix i M has to be defined by the expert from organisation with some cognitive competencies. As a result the relationship between different layers can be non-linear, nested and with feedback.

Knowledge deterioration (forgetting)
Over time employee competencies (knowledge) are reduced if they are not stimulated by the workers form surrounding and the work itself. From the formal point of view knowledge forgetting model can be found in [98]. The main concept is to incorporated in knowledge model fact that agents didn't always remember their previous knowledge (i.e. agents have perfect recall). Sometime we want to model the fact that certain knowledge is discarded. From the business point of view organizations must forget old habits in order to learn new and more appropriate ways of doing things [99]. Organisation may be forgetting knowledge intentionally (avoiding bad habits, unlearning) and accidentally (failure to capture, memory decay) [100]. From the cognitive science point of view, man develop their skills in an environment that stimulates them [94]. In this case, when my co-workers are less competent, with time my capacity will decrease (equal to their average level). A lot of works in psychology show that the environment impact on our activity. In the case of knowledge processes uninspiring surroundings causes progressively lose our knowledge.
Let us introduced a formula for average knowledge transfer potential for node neighbourhood calculation: where ) ( j i card  is a number of nodes in i neighbourhood. If the value of (4) is less than worker knowledge acquisition potential (product of worker's knowledge level and his/her cognitive abilities) then the worker's knowledge starts to deteriorate.

Definition: Knowledge deterioration (forgetting) process
If for node i on layer j following condition occurs , for node i on layer j the forgetting factor is related to node neighbourhood j i  and knowledge worker's cognitive ability . In general, worker with high cognitive ability forgets slower and the worker is forced to start forgetting by the weakness of his neighbourhood.

Assumption: non-zero knowledge condition
If the knowledge value for node i on layer j is greater than zero in the next steps of the knowledge value has to be always greater than 0   : The minimal value of knowledge value is represent by the variable: 0   One of the forgetting formula proposition is following:

Self-learning
Due to the rapid obsolescence of knowledge and the requirements of increasingly complex processes there is a need to continuously acquire new knowledge by employees.
Lifelong learning philosophy [101] assumes that any worker maintains ongoing, voluntary, and self-motivated pursuit of knowledge. In the proposed model this movement is described as a self-learning process.

Definition: Self-learning process
If for node i on layer j following condition occurs The function  incorporated node's surrounding and cognitive ability into selflearning process. If the average knowledge level of j i  is higher than node's knowledge then knowledge worker has to invest some time in order not to stand out from the rest and be valuable for communication. Moreover, the high clustering coefficient reflects larger environment that may have more pressure due to high cliqueness. In the article we propose following self-learning formula: where C and D are fixed values.

Procedure of knowledge diffusion
Competence development based on the knowledge diffusion involves various processes: horizontal knowledge diffusion, vertical knowledge diffusion, knowledge forgetting, self-learning. In addition, knowledge diffusion is a two dimensional process. In this section we will develop the main points of procedure for knowledge diffusion calculation in multilayer networks.
Layer selection is based on layer's ranking layyers  , calculated based on vertical diffusion matrix for variable w kˆ :

5
Areas of applications

Competence management paradigm
The model of knowledge diffusion is used to analyze the development of competencies in an organization. However, its final application is competence management.
Keeping in mind main management axioms [103] we are going to discuss whether competence management is possible based on the proposed model: -Axioms of management 1: The object is suitable for observations and measurements.
The presented model gives ability to observe and measure every component of competence. Moreover, all knowledge flow between actors can be tracked and analysed.
-Axioms of management 2: At the interval of observation object can change its state.
The dynamics of knowledge flows on different levels of network is noticeable. The process of knowledge diffusion is based on the continuously changing value of actors (workers) knowledge.
-Axioms of management 3: The predetermined target defined expected object's state.
In the organization the target is set on the strategic level and concerns for expected values of competencies.
-Axioms of management 4: There are alternative ways to influence the behaviour of an object Any types of knowledge, components of competence, can be changed by training (increase of knowledge level ), expert's mentoring (direct diffusion of knowledge) or team building (network reconfiguration).
-Axioms of management 5: There is a pre-defined criterion of management efficiency.
The criterion determines the degree of matching acquired competences to market or company requirements.
-Axioms of management 6: There are resources for the execution of the decision.
The network consists of nodes which represents knowledge workers (actors).
Moreover, in the discussed context, competence management is a process of tracking changes in the content of knowledge related to the competences.

Simulation model
All the concepts of knowledge diffusion models require validation. In real conditions only few models can be checked due to the limitation of data. As a result, a number of simulation network models is used. The description of models can be found in [104]. The literature review shows that the diffusion models are validated based on the Watts-Strogatz model [105]. The Watts-Strogatz model reflected the "small-world" characteristic of complex network. According to Cowan and Jonard [74,78] "small world" networks generate the fastest knowledge growth. Moreover, Cowan and Jonard found that the steady-state level of average knowledge is maximal when the network structure is a small world, It means that most connections are local, but roughly 10% of them are long distance. The relatively big clustering coefficient is beneficial for knowledge diffusion in the agents' network [48]. vector for worker will be set randomly. The workers are divided into two groups, 'normal' workers and experts with knowledge level significantly greater than other workers.

Applications
Due to high stochastic nature of competence development process and multidimensional of the proposed model the deep simulation analysis is very difficult to maintain. In order to illustrate the different aspects of the proposed model, in the context of competence management, we will discuss a number of case studies.

Modelling the competence development based on multilayer diffusion
The  Combining within model knowledge components and building metric for competences makes possible to track different competence development over time. In Fig. 9 competence development for each competence that can be modelled using different relations between knowledge components and each competence is shown.

Fig. 9: Development of competences based on four layers
In the next step simulation is based on the asymmetric settings for vertical diffusion and results for knowledge in each layer are shown in Fig. 10. Results for incoming knowledge in each layer are visible in Fig. 11. showed diffusion processes between different layers. Dynamics of processes was simulated using both vertical and horizontal diffusion. Effect of deterioration was simulated as well as self-learning which results in changes over the time.

Modelling the role of experts
In the next step the role of experts within the network was modelled using asymmetric relations between knowledge components. Using the proposed model it is possible to simulate changes after adding experts with assigned knowledge higher than all network members. In the first stage of the simulation showed in Fig. 12   Knowledge transferred with vertical diffusion to other layer resulted a stable increase.
In the next stage, 5 experts were added with maximal knowledge at layer one at the level of 25 and it was repeated after step 300. Adding continuously experts with smaller knowledge delivered better results than one time action based on experts with knowledge much higher than average knowledge within network. Activity for incoming and outgoing knowledge is showed in Fig. 13 and Fig. 14 respectively. Using this approach it is possible to evaluate a better strategy to add a smaller number of experts with high knowledge or add higher number of experts with smaller knowledge. In the simulated conditions adding experts with high knowledge delivered worse results because observed deterioration process.

Modelling the changes in employment
Proposed model can be used for simulating situations of reduction of employment or job quitting. It was simulated in the next step and results are showed in Fig. 15. After 200-th step of the simulation 50 random employees were removed and 40% knowledge drop was observed. Improving this situation was possible After 300-th step where 10 experts with knowledge value at 50 were added at layer one and it helped to recover average knowledge.

Fig. 15: Multilayer knowledge diffusion with changes of employment
Even though experts were added at single layer vertical diffusion helped to recover average knowledge at layer number two. Changes in employment are resulting different activity within incoming and outgoing knowledge at each layer what is illustrated in Fig. 16 and Fig. 17. -In future approach to the modelling of the discussed issue it will be possible to change the relationship between the layers based on time-dependent function, or semantic relations reflecting business rules. In the presented approach, there are linear relationships between the layers described by vertical diffusion matrix.
-When the network of knowledge, competence and links is large the complexity of proposed approach is growing. The computational complexity depends on formulas for horizontal and vertical diffusion and self-learning/forgetting processes (2)(3)(4)(5)(6)(7)(8)(9)(10). If these formulas are nonlinear and mutually nested then the resources needed for calculations are significantly higher. The number of objects in the knowledge domain is not crucial due to formal nature of the Knowledge Space Theory (KST). In contrast to the semantic-rooted language (like OWL) in KST all relationships can be explicit interpreted and handled based on mathematical mechanisms. Moreover, the number of connection between workers (nodes) actually does not affect the whole approach, because the worker collaborated only with one other worker all the time.
-The notation of upper and lower shadow for worker's knowledge set gives opportunity to develop a cost estimation method for commence development. The cost estimation algorithm in the form of a group competences expansion algorithm is proposed in [109].
In this approach we have to recognise the acquired and required competence set and then based on the Competence Set Theory the cost of competence expansion is calculated.

Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments
The work was partially supported by Fellowship co-Financed by European Union