Computer Networks



As business becomes knowledge dependent, organizational productivity often relies upon in-depth knowledge of business processes, technologies, and transfer knowledge across the organizational boundary [68]. Because organizations may not always have all necessary knowledge within their organizational boundaries, many have implemented inter-organizational knowledge networks as a mechanism to obtain knowledge [35,46]. Virtual knowledge organizations (VKOs) provide an efficient mechanism to search for and access knowledge in needs and facilitate the knowledge transfer process [33,43].
A VKO can be viewed as a virtual organization in which knowledge users co-create and share knowledge [39,44]. An important feature of VKOs is to bring knowledge seekers and providers into one virtual space that is equipped with knowledge databases over networks. Being free from the constraints of hierarchy and rules of real-world organizations, people benefit fromthis virtual organization by gaining access to new knowledge, expertise, and ideas, all of which may not be availablewithin the organizational boundary [77]. Our notion of VKOfocuses on a knowledge-centric virtual community where a group of knowledge workers with common topics, interests, problems, experiences and practices co-create and share valuable knowledge. A unique characteristic of VKOs, which differentiates them from real-world organizations, is that knowledge processed through VKOs becomes explicit in the way of knowledge digitalization, and turns into common assets benefiting all participating groups [40].
This paper develops on the network and organizational perspective, which defines VKO network (VKON) as networked organizations transmitting knowledge through the network. A VKON can be viewed as a market mechanism that provides knowledge resources to problems and solutions in a decentralized manner [31]. An example of VKON is “Network of Networks (N2)” (, a Canadian national initiative that integrates 29 existing virtual organizations to enhance Canada's disease research capability and capacity.2 The Network of Networks (N2) was initiated because researchers from diverse knowledge domains confront challenges in carrying out a joint research project that requires integrating knowledge from foreign domain. They need a networked system establishing a common knowledge base and integrating knowledge from various fields. In N2, a number of members who join multiple VKOs play the role of knowledge agents transferring and distributing knowledge across domains. Similarly, in an effort to facilitate knowledge creation process between academia and industry, European academic and business organizations from nine nations designed a VKON, European Corporate Academies Transnational Best Practice Network (ECUANET).
During the last two decades, many of research and development organizations that engaged in innovative tasks have begun a rapid transformation toward using virtual organization networks [39]. They, however, have learned that acquiring knowledge fromexternal sources is not a simple task, and that there are challenges in making newly acquired knowledge useful.While the evolution of individual VKOs leads to knowledge specialization by developing local knowledge and coding schemes, it also hinders the acquisition and interpretation of knowledge transferred from other VKOs [20]. In addition, developing an effective channel to transmit knowledge kept in one VKO to another is not an easy task. Then, how can we create a VKO network thatwould effectively facilitate the knowledge transfer process among VKOs?
We suggest two efficient mechanisms to copewith the challenges of cross-VKO knowledge transfer; acquiring inter-VKO common knowledge, and forming the network based on knowledge transfer agents (knowledge agents hereafter). The current study addresses two important questions; how do we efficiently distribute and acquire necessary knowledge across the network? and howdowe maximize the collateral benefits from the valuable knowledge through the entire VKON? Using a mathematicalmodeling approach, this study attempts to answer these questions based upon the theories of common knowledge and knowledge agent. We posit that the common knowledge between VKOs and the topological properties of a VKON characterized by distribution of knowledge agents are associated with superior knowledge transfer capacity.
The capacity of knowledge network is measured by network complexity, which is a generalization of the number of spanning trees in a graph. This notion of network efficiency has been well addressed in the quantification of flow of information by Stephenson and Zelen [65]. We view a VKON as an organic social network consisting of two outstanding entities: 1) a group of VKOs in which private and common knowledge are created and 2) cross-VKO knowledge agents connecting two VKOs by involving in and transferring knowledge between the two.3 We argue that the distribution of the agents is an imperative determinant of knowledge transfer capacity and accessibility of knowledge in a VKON. We investigate how a network structure formed by knowledge agents connecting multiple VKOs along with the knowledge profile of each VKO influences the dynamics of knowledge transfer in a VKON. Our objective is to assess the effects of a knowledge network structure formed by the distribution of knowledge agents and common knowledge on knowledge transfer.
The strategy of this paper is to examine interactions between two VKOs, and to illustrate how they affect knowledge diffusion, cohesion of knowledge network, and structural issues of the entire knowledge network. Our VKON model seeks to advance an understanding of knowledge transfer in virtual organization networks in two ways. First, while extant research on knowledge transfers tends to focus on sharing activities within the boundary of a VKO, we consider the dynamics of knowledge transfer at a network level. Second, one common assumption of the studies in this area is knowledge transfer based on direct communication. This study further considered knowledge transfer through indirect communication which is a common practice in VKON.
The knowledge transfer capacity of a VKON is measured by network complexity and common knowledge accumulated in each VKO. The mathematical approach in this study has been adopted in various areas including electrical engineering and biomedical sciences where the original motivation was to address the issues of non-social networks.
We next review relevant literature, focusing on the theory of common knowledge and the knowledge agent. We, then, develop a mathematical model of VKON using information centrality in Section 3. Section 4 shows an optimal condition for knowledge dissemination in a VKON and Section 5 discusses the effect of merging two VKOs on knowledge transfer in the VKON. Section 6 suggests a way to maximize knowledge transfer capacity in a VKON and Section 7 assesses the amount of knowledge transfer between two VKOs through direct and indirect communication channels. Section 8 examines the effects of common knowledge on the dynamics of knowledge transfer in a VKON. Finally, we summarize and discuss the results and conclude with contributions, implications and limitations of this research in Section 9. Detailed mathematical proofs and leads are provided in the Appendix A.

2.The role of common knowledge and knowledge agent

VKON can be viewed as a network that consists of multiple VKOs. Thanks to the frictionless nature of VKOs [9], they proficiently operate on relational structures or networks [46]. Recently, researchers have paid attention to various phenomena of VKOs due to their tremendous potential for knowledge creation and transfer [2,3,28,39,77]. Given the importance and potential of knowledge transfer, researchers have investigated various phenomena within the boundary of a VKO. Particularly in research and development organizations, the importance of inter-organizational knowledge transfer has been highlighted due to the emerging need of cross-disciplinary research.
While firms acquire valuable knowledge assets through network, there are substantial barriers to knowledge transfer between organizations [30,76]. The identified barriers include 1) lack of absorptive capacity of the recipient entities, 2) lack of credibility on the part of the source of knowledge, 3) disconcerted relationships between the sourcing and recipient entities, and 4) causal ambiguity caused by the complexity of knowledge [66,75]. The barriers can be overcome by increasing common knowledge between the source and recipient organizations [12,17] and by acquiring capable knowledge agents [71,78].

3.Common knowledge

Knowledge is socially embedded and highly context-specific, and these characteristics make it difficult to transfer knowledge [8,51]. Von Hippel [75] argued that knowledge transfer becomes costly particularly when the recipient organizations need to obtain pre-required knowledge and skills to be able to use the newly acquired knowledge. As a more concrete example, a group ofweb designerswho launch aweb development project usingweb development softwarewill not be aided to a great extent simply by adopting the software program unless they have already obtained the pre-required knowledge related to the web server and the fundamental operating systems. Similarly, the cost of knowledge transfer would be lowered when the software vendor provides training and technical support to the web designers who have common fundamental knowledge. In other words, when both knowledge sourcing and seeking entities share a set of necessary common knowledge, transferring knowledge would be less costly and the transfer process would be more efficient. The cost of transfer can be substantially lowered by common knowledge particularly when knowledge transfer takes place in an “explicit” form [50].
Social scientists have been emphasizing the important roles of common knowledge in knowledge transfer [17,28,45,51,70]. It is because common knowledge 1) facilitates the knowledge transfer processes by increasing similarity of knowledge profiles of the two VKOs, and 2) allows knowledge sourcing and seeking parties to share rules in the form of practices. Zhao and Anand [84] further suggest that the similarity of structural and cultural attributes of the knowledge between the two entities, which becomes an important component of common knowledge, is the vehicle acquiring and assimilating newly transferred knowledge. The depth and breadth of common knowledge, therefore, may indicate the functional and technical expertise integrated across organizational boundaries.. This notion of common knowledge has been viewed as a key component of absorptive capacity of knowledge recipients [42,83].
On the contrary, a lack of knowledge commonality can be conceptualized as the impedance of knowledge transfer. The greater the mismatch in knowledge profile and cognitive orientations of the two VKOs, the greater the difficulties of transferring [81]. Transferring knowledge involves high costs and often creates greater ambiguity, particularly when knowledge is being transferred across multiple VKOs that lack common knowledge [27]. An increasing amount of common knowledge facilitates knowledge transfer between VKOs, lends itself to knowledge development, and brings forth a VKON that is more accessible to all VKOs.
Inkpen [34] further emphasized the importance of a common language protocol for knowledge transfer between knowledge sourcing and seeking VKOs. For thosewho share common communication protocols, terms, and awareness, communication is remarkably efficient. Furthermore, a positive effect of network externalities takes place when VKOs in the network increase communality in their views and behaviors to the extent that they find proximity to other VKOs in the network [54].
From the perspective of inter-community knowledge transfer, knowledge owned by a VKO can be classified into two types: 1) private knowledge that is exclusively owned by a knowledge sourcing VKO and can be transferred to other VKOs, and 2) common knowledge that is commonly owned by the two interacting communities (Fig. 1). By definition, once knowledge is transferred from one VKO to another, transferred knowledge becomes common for the two, and in turn the amount of common knowledge increases. Our goal is to understand the dynamic effects of common knowledge and its facilitating mechanism for knowledge transfer (c.f. Section 8).

4.The roles of inter-VKO knowledge agents

Researchers suggest that one way to efficiently deal with the costly knowledge transfer process is through knowledge agentswho are capable of understanding and translating contrasting knowledge of both knowledge sourcing and seeking VKOs [71,78]. We define Inter-VKO knowledge agents (knowledge agents afterwards) as thosewho actively engage multiple VKOs and connect them by transferring knowledge. Knowledge agents communicate with knowledge seeking VKO to identify the need of external knowledge, coordinate and align an access to a knowledge source, understand the knowledge froma knowledge sourcing VKO, and localize and disseminate it to a knowledge seeking VKO.
According to Katz and Lazarsfeld'smodel of “two-step flow” of communication [37], a small group of knowledge agents act as useful intermediary channels between the source of information and information users. Inkpen [34] emphasized the important role of business managers as the knowledge agents transferring between organizations. Tushman

Figure 1 Decomposition of knowledge profiles: private vs common knowledge
and Katz's early work in R&D identified the organizational benefits of having gatekeepers as knowledge agents who gather relevant knowledge from the external environment and enable the flow of knowledge into the group [71]. Hargadon and Sutton [32] highlighted the role of these knowledge brokers and translators who coordinate the transfer of knowledge through external network. Social researchers called these key individuals stars, defined them as the most important actors in a social network [62], and attempted to quantify the effects of their existence in the network by “centrality” [78]. In a similar vein, Cohen and Levinthal [13] viewed the aggregation of the prior relevant knowledge or experiences of the key individuals as an important source of absorptive capacity.
Knowledge agents in a VKON are important in that they are the links and play the role of transfer channels between VKOs they are associated with, and they are the ones who understand common knowledge of both VKOs. With these knowledge agents, external knowledge can be transferred into a knowledge seeking VKO from a knowledge sourcing one by means of two-step process as shown in Fig. 2: gathering and internalizing knowledge from a knowledge sourcing VKO, and translating and externalizing it in local language of the knowledge seeking VKO [35,49]. Because transferring knowledge requires an understanding of the fundamental knowledge of business processes, culture, or norms of both VKOs, it is critical to have knowledge agents who understand the fundamental common knowledge of both VKOs. Inter-community knowledge transferwithout a knowledge agent may lead to misperceptions and an incomplete understanding of the transferred knowledge.
Another important role of these key individuals is formation of knowledge transfer network. In a social network theory, a network structure can be viewed as an arrangement of the network elements that patternize the flow of communication in a network [47]. Knowledge agents formulate the network structure of a VKONby continuously engaging, acquiring and communicating with multiple VKOs. Because knowledge agents are important channels of knowledge flow among VKOs, the capacity of knowledge transfer in a network created by knowledge agents becomes a meaningful determinant of knowledge transfer channel between VKOs.

5.Modeling virtual knowledge organization network (VKON)

5.1 Graph theory

One useful method in analyzing and representing a network is the graph theory. The graph theory has been considered a useful tool in various areas including computer network research (i.e. [25,67]), electrical engineering (i.e. [82]) and medical sciences (i.e. [36]). This connection topology has been adopted in social network analysis for many reasons [23]. One, graph theory offers practical terms and labels representing various properties of a social network structure. For example, we consider a VKON a network that is formed with a group of VKOs connected by agents transferring knowledge among them. In this study, VKOs respectively. These terms provide us with a set of fundamental tools for analyzing the dynamics created by various network properties. Two, graph theory and knowledge agents are represented by nodes and arcs,
Figure 2 Decomposition of knowledge transfer
offers useful mathematical operations and notions with which social network properties and their effects can be quantified. Finally, the theory allows us to prove theorems regarding network and representations of a social structure. In addition to its mathematical utility, the graph theory offers a vehicle reconceptualizing a VKON as a model of a knowledge transfer system. This model can be viewed as a simplified representation of a VKON characterized by distribution of knowledge agents and common knowledge.

5.2 Measuring social network capacity: information centrality

We use Information Centrality in Social Network [78] as themethod of analysis of a virtual knowledge organization network (VKON). The term “Centrality” as applied to the entire network relates the compactness of the network as well [22]. A network can be regarded compact when the average length of knowledge transfer paths (in terms of the number of necessary connections) between any two VKOs is short. It is reasonable to assume that knowledge can be transferred more accurately and rapidly, when there are fewer connections involved between two parties
The score of centrality is determined by the distribution of nodes (or VKOs) and arcs (or knowledge agents). Since the idea of network centrality applied to human communication was originally introduced by Bavelas [5], the concept has been used to examine various phenomena of social networks such as political integration [14], design of organization [6], diffusion of technology innovation [1,16], inter-organization relationships [59], group stability [11], network structure in virtual organizations [2], and individual centrality in virtual groups [3].
Among various measures of centrality, Information Centrality [78] measures the importance of a node (as a hub) in a network. The measure focuses on the flow of knowledge originating from a VKO, and reveals its influence on the whole VKON.

5.3 Virtual knowledge organization network (VKON)

A VKON can be viewed as a collection of VKOs which are connected by knowledge agents. Members in each VKO create and share knowledge within the community boundary, developing their own set of knowledge focusing on a domain-specific knowledge space. Among members, there are thosewho acquire and transfer knowledge between two VKOs. A model of a VKON consists of nodes and a set of arcs connecting pairs of nodes, where each node and arc represent a VKO and a knowledge agent connecting two related VKOs respectively. The central idea of VKON is that a VKO takes advantage of autonomy among nodes, but uses arcs (agents) to transmit valuable knowledge between them. Clearly, this formof network precludes the idea of global interaction in the sense that a VKOcan interactwith selective other [15]. Indeed, VKOs tend to interact only with a relatively circumscribed subset of the entire VKON.
Fig. 3 depicts a simplified example of a VKON. Knowledge can be transferred between two VKOs when there is at least one knowledge agent. The distribution of knowledge agents, therefore, forms the network structure of a VKON. An intuitive notion of the knowledge agent in the context of inter-community knowledge transfer is that the distribution of knowledge agents affects the probability that knowledge can be transferred from one VKO to another. Knowledge transfer wouldn't be enabled without knowledge agents. We investigate the effects of knowledge agents on the dynamics of knowledge transfer in a VKON. The weight4 assigned to each arc indicates the number of agents assigned to the VKOs. Knowledge agents involve knowledge sourcing and seeking VKOs, and transfer knowledge in two ways. We, therefore, assume that a VKON N is non-directed
Figure 3 Simple virtual knowledge organization network (VKON)

5.4 Complexity of network

Knowledge becomes more valuable when it is swiftly distributed to and consumed by more knowledge users [26]. The network complexity is a key measurement computing the probability of knowledge flows froma sourcing VKOto a recipient VKOin a network [19]. Fromthisperspective, network complexity that assesses the degree of knowledge dissemination in a network is a meaningful indicator of knowledge transfer capacity of a network [10,65]. The complexity, c(N), of a VKON N quantifies the number of possible ways connecting all VKOs without a recursive loop. It is a generalization of the number of spanning trees in a graph and general indicator of cohesiveness of a network, measuring how evenly knowledge agents are dispersed in a network [65]. The implication of network complexity is that the greater c(N) of a VKON is, the more likely knowledge is disseminated throughout the network. When the weights of all arcs in N are constant, c(N) counts the number of possible loop-free spanning trees each of which forms a set of arcs connecting all nodes in the graph (c.f.West [80] for details). That is, we can develop a more efficient network using the same number of knowledge agents.
Consider a simple VKON N formed by three VKOs C1, C2, and C3 with knowledge agents connecting them. The connection created by knowledge agents between Ci and Cj will be represented by an arc eij. Let wij denote a weight assigned to eij indicating the number of knowledge agents between Ci and Cj. Then, the complexity of N is defined as
1                                                                                  (1)
where wij = wji because N is non-directed. wij = 0 when there is no agent placed between Ci and Cj. Each term counts different ways connecting all VKOs without looping. For example, the first term, w12•w23, computes the number of possible knowledge transfer paths between C1 and C3 via C2, and thus c(N) counts the total number of paths from each and all VKOs to the rest of VKON.5 If N has n VKOs, there are possible n(n − 1) / 2 connections in N. We measure the dynamics of knowledge transfer in a VKON using the Laplacian matrix (also called Kirchhoff matrix) which is a widely adopted mathematical tool for examining various properties of a graph. We define the Laplacian matrix L(N) of the VKON N as a n × n matrix, whose n2 entries are denoted by Lij (1 ≤ i, j ≤ n) such that each diagonal entry Lij equals the sum of the weights of all arcs that are connected to the node Ci and each Lij (i ≠ j) equals −wij. Now, let L0(N) be the (n − 1) × (n − 1) matrix obtained by deleting one arbitrary row and one arbitrary column of L(N). Then we define c(N) to be the absolute value of the determinant of L0(N). When the weights of the arcs in N are non-negative integers and represent the multiplicities of the arcs, c(N) is the number of spanning trees in N [72].
Alternately, c(N) for a weighted VKON can be described as follows. Let T be a spanning tree in N. The complexity c(T) of T is defined as the product of all weights in T. Note that c(T) is a square-free monomial of degree (n − 1) in wij's as variables. Then, c(N) is the sum of c(T) over all spanning trees T:
1                                                                                                                                   (2)
where β(N) denotes the set of all spanning trees in N. Clearly, c(N) is a square-free homogeneous symmetric polynomial of degree (n − 1) in wij's. c(N) is symmetric in the sense that it remains the same after permutations of nodes.

6. Optimizing knowledge transfer capacity of VKON

Communication network literature suggests that short length of communication path leads to an efficient information transfer [73,78]. Knowledge dissemination in a VKONmainly depends on the distribution of knowledge agents in the network, which plays a role of communication path. Then, how do we create an efficient knowledge transferring network with a certain number of knowledge agents? In this section, we discuss a general guiding principle to optimize c(N)with varying distribution of knowledge agents in a VKON. We obtain the total number of knowledge agents by summing all weights of the arcs.
Theorem 1. If the total weight of N is bounded, network complexity c(N) will be maximized when the weights are uniformly distributed among all arcs in N. That is, with the constant number of knowledge agents, the degree of knowledge dissemination throughout an entire VKON is maximized when knowledge agents are uniformly dispersed across the network.
Proof. This theorem is a weighted version of the fact that a complete graph has more spanning trees than any other graph of the same size [38].
The implication of Theorem 1 is that, assuming that the number of knowledge agents in a VKON remains constant, the network complexity that indicates the degree of knowledge dissemination in a VKON can be maximized by placing knowledge agents evenly throughout the network. This is consistent with the implications of the diffusion literatures that evenly generated networks tend to be more effective [73,79].

7.Improving knowledge transfer capacity of VKON(c(N)) by merging two VKOs

Based on Theorem 1, two natural questions are 1) what is the most efficient way to achieve uniform distribution of agents (discussed in Section 6)? and 2) among those, which agent would be more important than others in enhancing the degree of knowledge dissemination (discussed in Section 7)? In this section, we deploy a two-step approach as a preliminary work leading to answers for the questions. First, we use a measure to assess the capacity of knowledge transfer between two VKOs (Ci and Cj), Iij, from the notion of knowledge centrality suggested by the seminal work of Stephenson and Zelen [65]. Knowledge centrality is mathematically defined as
where cij denotes ij entry in (J + L(N))−1 and J is an n × n matrix with all entries equal to 1. Iij indicates the degree of ease of knowledge flow between two VKOs in a network. An analogous application of Iij is found in the area of the electrical engineering inwhich Iij is known as effective conductance (a measurement of how easily electrical current flows between nodes in a network) [19,82]. In addition, according to the theory of random walks, Iij is an expected value of escape probability [19]. That is, Iij can be viewed as a probabilistic capacity that the network transfers knowledge from one place to another.
The measure allows us to observe the marginal impact on the entire network complexity caused by changing a relationship between any two VKOs. Second, we consider merging VKOs under certain conditions as a way to increase the degree of knowledge dissemination of the entire VKON. The structure of virtual organization networks is to be a more amorphous web of connections and to continuously evolve in response to the need of knowledge [2]. Researchers argue thatmerge and split of virtual organizations may cause only a marginal cost, while offering substantial benefits [55,57]. Due to the frictionless nature of virtual organization [9], merging VKOs would be beneficial to the process of improving knowledge transfer. In this section, we devise a measure of knowledge transfer capacity between two VKOs, and then use it to determine when to merge the two. Let N / eij denote a new VKON with n − 1 VKOs aftermerging Ci and Cj. Aftermerging two VKOs, the knowledge agents between the two, eij, are no longer present. Let Iij be the knowledge transfer capacity between two VKOs, Ci and Cj, then we can derive Theorem 2 as follows.
Theorem 2. The knowledge transfer capacity Iij between Ci and Cj is the ratio of the complexity of N and that of N / eij, That is,
Proof. See the Appendix A.
By reformulating Eq. (4), we can obtain an optimal point at which it is better for two VKOs to merge, providing benefits to the entire network.
As such, merging two VKOs reduces the network complexity when Iij N 1, but escalates when Iij b 1. Theorem 2 implies that merging two VKOs with a high degree of knowledge transfer capacity (a high score of Iij) reduces the complexity of the resulting VKON, while merging two with a low degree of knowledge transfer capacity (a low score of Iij) raises the complexity. In other words, when there are relatively a small number of knowledge transfer activities taking place between two VKOs (that is, the two have a low degree of knowledge transfer capacity (Iij)), merging the two lowers the cost of knowledge transfer by avoiding unnecessary detours in a network.

8.Improving knowledge transfer in VKON

We learn from Theorem 1 that the degree of knowledge dissemination through a VKON is maximized when knowledge agents are uniformly distributed across the network. Then, if there are knowledge agents newly joining a network, what is the most efficient way to achieve a uniform distribution of agents?, and where should the agents be placed in order to optimize the network complexity? Note that researchers in a joint project between two separate knowledge domains are good examples of a knowledge agent. Then, those are often questions of officers in such organizations as National Science Foundation (NSF) and National Institutes of Health (NIH),who seek for an opportunity of a cross-disciplinary project linking two research domains. By extending Theorem 2, we offer a new theorem providing an answer to the question. The proof is based on a new interpretation of Iij as the inverse of the relative growth rate of c(N) with respect to wij.
Theorem3. Amarginal increase of VKON complexity is maximized when a  new knowledge agent is placed between two VKOs with the minimum knowledge transfer capacity (Iij).
Proof. See the Appendix A
Via numerical simulation (Fig. 4),we attempt to find the optimal positioning of a new knowledge agent in a VKON, and to show how the appearance of this newagent is conducive to the network complexity c(N) and knowledge transfer capacity Iij between the Ci and Cj. Suppose there is a VKON of five VKOs. Among these five VKOs, one knowledge agent can connect to any two between which the number of knowledge agents are w12 = 3, w23 = 2, w25 = 1, w15 = 1, and w45 = 3 while others 0 (c.f. VKON (i) in Fig. 4). Information flows, Iij, are calculated by using Eq. (A.5) in the Appendix A. Note that an increase in network complexity is largest (c(N) = 101) when an additional agent is positioned between C3 and C4 (VKON (ii) in Fig. 4) where information flow is minimal (I34 = 0.71). In general, the smaller the information flow between two VKOs, the greater the marginal increase in network complexity by adding a new agent. It is worthwhile to note that the mean of Iij which indicates the knowledge transfer capacity of the network is greatest (Mean(Iij) = 2.22) and the asymmetry of knowledge transfer capacity in the network is lowest (Std(Iij) = 0.76), when the network complexity is the greatest (VKON (ii) in Fig. 4).
Theorem 3 suggests that optimal positioning of a knowledge agent can improve the capacity of knowledge transfer of a VKON. Among numerous innovations, the costs are lowered and benefits increase as they aremorewidely accepted. A VKON, like other information technologies, can generate positive network-based economies through optimal distribution of knowledge agents which, in turn, results in enhanced accessibility of knowledge [24,58]. This can be achieved by knowledge agentswho grab a better understanding of culture, norms, and common knowledge of the two VKOs, which eventually leads to improved knowledge transfer activities. The cost of placing a new knowledge agent between two VKOs will be also lowered as the number of agents between the two increases thanks to the externalities among the agents and related communities [4,52].

Figure 4 Optimal positioning of a new knowledge agent in VKON

9.Direct vs. indirect transfer

The knowledge transfer capacity Iij between two VKOs, Ci and Cj, relates direct and indirect communication. Indirect transfer is when there are intermediating VKOs between Ci and Cj,while there is none in direct transfer. For example, the path C1 → C2 in Fig. 3 is a direct transferwhile C1 → C3 → C2 is an indirect transfer. The information transfer capacity through direct transfer between Ci and Cj is represented by theweight wij of the arc eij. We can represent the indirect knowledge transfer by
The significance of the indirect transfer is that it represents the effect of externality on knowledge transfer at a network level. Thus, it is reasonable to expect that ew ij should be computable without any reference to the direct flow wij. The following proposition gives an expression of 1ijthat is independent of Iij or wij. Now, we define a new network N \ eij which is created by deleting eij from N.
Proposition 4. The knowledge transfer capacity of indirect transfer between Ci and Cj in a VKON can be calculated by the ratio of the complexity of N \ eij and that of N / eij, That is,

Proof. The following equation (Eq.8) is called the Deletion– Contraction Recursion8 [53] for c(N).

Now the result follows from 1ij 1Note that if all VKOs in N are reachable, then c(N / eij) ≠ 0, because N / eij remains connected and the complexity of a connected network is positive. Proposition 4 implies that 1ij will be zero iff c(N \ eij) = 0, which happens only when N \ eij is no longer a connected network. That is, knowledge agents between two VKOs will be called essential to VKON if their disappearance severs the network into two. As such, disappearance of essential knowledge agents will result in two disconnected networks (a technical term for an essential arc representing the essential agent is an isthmus).

10.Common knowledge and network complexity

The literature on social capital emphasizes the importance of collective production of network-based capital [56]. Knowledge transfer, however, becomes costly when it takes place between two with little common knowledge [3,44]. For example, transferring knowledge regarding software installation will be costly when knowledge seeking and sourcing VKOs do not have basic common knowledge such as knowledge of the operating systems [12,61]. In contrast, if the two have too much common knowledge (that is, the knowledge profiles of the two VKOs are almost identical), there would be little knowledge to be transferred.What is necessary, therefore, is to identify the optimal amount of common knowledge between two VKOs, which becomes a complementary requirement for efficient knowledge transfer. In this section, we measure the commonness of knowledge (knowledge similarity) and its effects on knowledge agents.
Consider a VKOwith a knowledge profile noted by a vector Pi = [ki1, ki2,…, kiT],where kit (≥0) is the amount of knowledge on topic t, while T is the total number of topics. The magnitude of a knowledge profile is denoted by pi = ||Pi||, which assesses the quantity of knowledge in Ci. Assume that pi N 0 as each VKO maintains a certain amount of knowledge in at least one topic. Let Sij and Dij denote knowledge similarity, the normalized degree of knowledge commonness between by Ci and Cj, andknowledge distance, the degree of difference of knowledge (transferable knowledge) between Pi and Pj, respectively. Sij and Dij between the knowledge profiles Pi and Pj can be defined as follows [74];
where Sij ranges from 0 to 1. If any two VKOs have common knowledge, there must be at least one common topic (t N 0) and both kit and kjt are positive. Sij = 0, iff there is no common knowledge, and Sij = 1 when Pi = λPj. There should be transferable knowledge, if Pi ≠ Pj. In other words, if knowledge of two VKOs is of high similarity but no distance, it indicates that there is no knowledge to be transferred. Similarly, if it is of high distance but no similarity, it indicates that the two VKOs
Figure 5 The Curvilinear relationship among the number of participating knowledge agents (Wij), knowledge similarity (Sij) and knowledge magnitudes (pi,pj)
may not need knowledge of each other. Knowledge similarity is a qualitative measure of likelihood of knowledge transfer, while knowledge distance is a quantitative one. Sij in conjunction with Dij, therefore, can be meaningful proxies of the amount of transferable knowledge between Ci and Cj.
Furthermore, it is reasonable to conjecture that the more similar the knowledge profiles of two VKOs and the more transferable knowledge between the two, the more knowledge agents who would engage in knowledge transfer activities between them [29]. Studies showed that a VKO tends to find and connect to another one with which the knowledge transfer cost can be optimized [74], and this cost declines as knowledge similarity between them is escalated. The rationale behind this argument is rather simple: Common knowledge lowers the knowledge transfer costs and lower transfer cost attracts knowledge consumers. A study of knowledge transfer in a multi-organizational network suggests that effective inter-organizational knowledge transfer requires relatedness in knowledge content among business units and the network of lateral relations that facilitates task units to acquire related knowledge [31]. Studies in sociology also showed that people tend to feel more comfortable with engaging in community activities when they find communality [64].
Since the degree of commonness and distance of knowledge profiles are important factors influencing the number of knowledge agents (wij) between Ci and Cj, we obtain the following equation:
where α, β, and c are positive constants. Applying the exponential function to both sides of Eq. 12, we obtain
The number of expected knowledge agents w = wij is a function of knowledge similarity S = Sij and distance Dij 9 which is a function of pi and pj. Based on Eq.13, Fig. 5 depicts the increasing number of knowledge agents between two VKOs in relation to changing Sij and pi.
As the magnitudes of two knowledge profiles Pi and Pj converge, the curve of w tends to decrease after an optimal point of S. Otherwise, w increases monotonically as S increases. That is, when the magnitudes of knowledge profiles are the same, increasing their similarity S will eventually lead to identical knowledge profiles and there would be no knowledge to be transferred, whereas a certain degree of asymmetry between knowledge profiles may cause monotonic increase in the activities of knowledge agents as the similarity S increases. Based on this rationale, the following proposition, therefore, can be obtained.
Proposition 5. The number of knowledge agents between two VKOs is increasing and concave with respect to the knowledge similarity S initially. However, as the knowledge profiles of the two VKOs converge (that is, the difference of magnitudes of the two knowledge profiles decreases as shown in Fig. 5 (1) → (4)), it turns to a decreasing function after a certain optimal point of S.


Knowledge is considered one of the most valuable resources for the growth and sustained competitive advantage of the modern organizations [49,55]. Because organizations do not have all knowledge necessary within their organizational boundary, they need to somehow search for external knowledge sources beyond the boundary. One efficientway of acquiring knowledge fromexternal sources is to utilize virtual knowledge networks [7,29].
The importance of VKON cannot be overstated. When well structured, it would play as one of themost efficient mechanisms for locating and transferring knowledge. An indispensable benefit of a VKON is that it creates an opportunity for knowledge workers to go beyond interaction with content and people. Even more dramatic is the availability and amount of knowledge offered. Combining content management with cross-organizational online collaboration at all levels of the business can help unite organizations and give them greater insight and perspective, which, in turn, exponentially increases the value of intellectual capital [60]. It is important to note that amajority of innovations have taken place across disciplines or organizational boundaries [41]. Researchers have agreed that collaboration across organizational boundaries is a true driving force of competitive advantage [12]. To address this urgent issue, National Science Foundation (NSF) has called for proposals for discovering advanced ways to facilitate cross-disciplinary intellectual collaboration using virtual organizational network.10 We believe that successful formation of knowledge networks will significantly increase the benefits and decrease the costs to knowledge organizations.

The result of this study provides a set of important implications regarding cross-organizational knowledge transfer using VKON. Theorem 1 can be used as a general guiding principle for maximizing the capacity of knowledge transfer, which eventually offers the most benefits to VKON. This, however, may not be sufficient to warrant the optimal status of the network. Theorem 3 suggests that the flow of knowledge in a VKON can be even further improved by placing a new knowledge agent between two VKOswhere information transfer capacity is at minimum. Theorem 2 offers an alternative measure assessing the knowledge transfer capacity between two VKOs. It is worth noting that while the measures of knowledge flow capacity from the prior studies merely rely on endogenous variables of the two related VKOs [65], Eq.4 enables us to assess the knowledge transfer capacity between two VKOs using network complexity. VKOs may need to be merged at times in order to enhance the capacity of knowledge transfer in the entire network. An important question to a VKON manager is when two VKOs should bemerged. Theorem 2 provides a set of criteria for merging VKOs. A general guideline is that merging two VKOs with a high degree of knowledge transfer capacity will decrease the complexity of the entire VKON, while merging two with a low degree of knowledge transfer capacity will improve it. Using Proposition 4, we can identify essential knowledge agents who play a critical role in knowledge transfer of thewhole VKON. The findings of this paper will provide a set of useful guidelines for managing a VKON.
The notion of centrality is derived from local information around a given node and does not provide sufficient information regarding global properties of the network. Unlike other centrality measures, information centrality adopted in this paper begins with global invariants of the network and extracts local information. Hence, one may speculate the possibility of retrieving global properties of the network upon the underlying meaning of this centrality. This study reveals this hidden mystery (Theorem 2) and provides how this intriguing property can be concretely implemented (Theorem 3) for the purpose of efficient global stabilization by strategically distributing knowledge agents. Therefore, we claim that the main contributions of this paper relate global organizational strategy rather than micro management of a network.
Our macro perspectives on virtual knowledge network are differentiated by three points from the ones of the prior studies investigating within-organization knowledge transfer. One, the prior studies mainly focused on the interaction at an individual level within a single community boundary, and thus supplied limited information in relation to VKON. This study examined the relationships between/among VKOs by constraining knowledge agents, and illustrated the dynamics of inter-community knowledge transfer.
Two, despite the enormous benefits of VKON, the success and failure of an implementation depend primarily on the social context in which they are used [3]. We have taken into account the role of common knowledge in the virtual knowledge transfer process. Common knowledge accumulated in an organizational boundary has been regarded as the organizational memory or learning capacity [48]. While the prior studies of virtual communities have emphasized merely on the quantity of knowledge transferred through a network structure, it is a pioneering attempt to consider the role of common knowledge of virtual communities as a determinant of knowledge transfer capacity.
Three, one unique phenomenon in the evolution process of a VKO is that they evolve by merging two VKOs to improve the capacity of the knowledge flow. We investigated the impacts of merging VKOs on knowledge transfer of the entire network.
Finally, network complexity and knowledge transfer capacity are crucial indicators of the knowledge transfer efficacy in a VKON. There are conventional complexity and centrality measures devised from the perspective of offline individual networks (c.f. [23]). The applications of these measures, however, are limited to the context of an offline individual network. We reinterpreted those measures by reflecting unique characteristics of virtual knowledge organizations; the merge ability [63] of VKOs. The other aspect of VKOs, the split ability of a VKO, should be further investigated in the future research.
We attempted to develop new VKO-level knowledge transfer capacity assessment methods using network complexity. In addition, the effects of network complexity on the capacity of knowledge transfer in VKON were examined. The validity of developed measures needs to be empirical tested in future research.

12.Concluding remarks

During the last two decades, many research and development organizations involving innovative tasks began an obvious and swift transformation toward using VKOs [39]. Recently, the speed of organizational transformation has been considerably improved due to environmental and cultural transformation using virtual knowledge networks. To our knowledge, this is the first study employing the notion of the network theory in the context of a virtual knowledge organization network.
This new perspective on knowledge creation and transfer across virtual communities is important not only because it offers a macroscopic view of knowledge economics, but also because it provides a new opportunity for modern society to speed up the process of interdisciplinary knowledge creation. We attempted to further extend the theory of virtual knowledge communities and intercommunity knowledge transfer from a macroscopic perspective. While the founding of this study presents the dynamics of global level of VKON, further research should be conducted with variables reflecting VKO's local characteristics. For example, a method of sequence data classification [18,21] would be a useful tool to conduct an extended examination with VKO's local properties.
The implications of studies in the area of VKOs, in general, are bounded as most studies merely focus on the knowledge sharing process at nodal (focusing on the behavior of a single community) or dyadic (focusing on the relational behavior of a pair) levels. It is important to note that knowledge creation and transfer activities are no longer bounded within a VKO, but have evolved into further sophisticated forms of transfer through a cross-VKO network. As cross-community knowledge flows become more complex with increasing number of VKOs, it is imperative to understand the dynamics of VKO networks from a macro perspective. While the analysis in this study is essentially mathematical, both the future research based on quantitative and qualitative methods will recognize the potential for models.


This work was financially supported by the National R&D project of "Development of Technology for support to deep seawater industry" supported by the Korean Ministry of Oceans and Fisheries, and conducted by Seung Kyoon Shin. This work was also partially supported by a grant from the Simons Foundation #245994 to Woong Kook while he was at the University of Rhode Island

Appendix A

Proof of Theorem 2. Consider a case that i = 1 and j = 2, since the complexity of network does not depend on how the nodes are ordered, the proofs for other cases are the same after relabeling the nodes as necessary. As preliminary facts, we note that if J is an n × n matrix with all entries equal to 1, then it follows from Temperley's formula [69] that the determinant of B = J + L(N) equals det (B) = n2 · c(N). The Laplacian matrix L(N / e12) is obtained from L(N) by replacing Row 2 by Row 1 + Row 2 and Column 2 by Column 1 + Column 2 and then by deleting Row 1 and Column 1. As C = B−1 = (cij) is the inverse matrix of B and B(i|j) is obtained from B by deleting row j and column i, we have
for each i and j. Then,
Now, from the multi-linearity of the determinant function and the construction of L(N / e12) from L(N), we get
where K is an (n − 1) × (n − 1) matrix with the first row equal to (4, 2, 2, …, 2) and all of the remaining rows equal to (2, 1, 1, …, 1). Hence, it follows by a similar argument for det(B) that
Finally, we have
Proof of Theorem 3. By replacing N with N / eij in Eq. (3.2), we obtain
where the sum is over all spanning trees T′ in N / eij. Note thatwij is not included in any c(T′). There is a bijective matching between β(N, eij), the set of all spanning trees T containing the arc eij, and β(N / eij); the matching is given by T → T / eij. Hence, we have,
Furthermore, it is clear that ∂wijc T π ήΌ0 if T ∉ β(N, eij) because such T will not contain wij. Therefore,
From Theorem 2 and the fact c(N / eij) = ∂wijc N π ή, we conclude that Iij is minimal (among all dyadic knowledge transfer capacity), if and only if ∂wijc N π ήis maximum among all partial derivatives of c(N). This implies that a marginal increase of c(N) is maximized when an additional knowledge agent is placed between Ci and Cj with minimum Iij.


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