5. A Novel Comprehensive Index of Network Position and Node Characteristics in Knowledge Networks: Ego Network Quality

Note that connectivity is used here in a broader sense than in graph theory. In graph theory connectivity refers to the number of vertices the removal of which disconnects the graph. In our case, this term refers to a similar concept but with a less strict definition. By connectivity we simply mean the extent of ties connecting a given group of vertices.

Before moving forward, though, we have to make a terminological clarification. Generally the term ‘neighbourhood’ of a node refers to the group of nodes connected directly to a specific node. In our study by neighbourhood we mean not only the directly but also the indirectly connected nodes. As this definition in itself would mean that the term neighbourhood refers to the totality of nodes in the graph, we refine the definition and use the more specific term ‘neighbourhood at distance d’ which refers to the nodes exactly at distance d from a specific node.

In this chapter we use a non-weighted algorithm for the calculation of geodesic distances, i.e. the distance of two nodes is regarded as the number of ties connecting them, irrespective of the weights associated with these ties.

The weighting factor is defined to be unity for d = 1 and descending towards zero as d increases. There is no unique best choice with regards the decay function. We present some illustrative simulations related to the choice of the decay function later in this chapter.

Division by two is required because matrix A is symmetric, and thus we can avoid duplications in the counting. This division is not required in the first term because the definition there counts only links from distance d − 1 to distance d and not vice versa.

It is worth devoting a word to the inclusion of distance-crossing ties (the first term in the expression). Our intuition behind the concept of Local Connectivity is that collaboration among partners enhances knowledge sharing and this leads to a better environment for knowledge creation. In the case of the direct neighborhood, the links connecting the node in question and its neighbors are clearly relevant in the general case of weighted ties: the amount of knowledge learnt from the immediate partners depends on the intensity of interactions with those partners. On the other hand we argue that in our concept the question is how dense the tissue of the network around the node is. We are going to attach less weight to this connectivity the farther away it is from the node, but the main point is that better connectivity among nodes is of higher value, and this connectivity is not necessarily restricted to connectivity among nodes at a specific distance.

It is easy to see that using (identical) knowledge levels different from unity would change the results by a multiplicative constant compared to the situation with the normalized levels.

Eigenvector centrality is defined by the following recursive concept. Let x _i denote the centrality of node i and let this centrality be determined by the cetralities of adjacent nodes: \( {x}_i=1/\lambda {\displaystyle \sum_j}{a}_{ij}{x}_j \). Written for all nodes we end up with the matrix equation x = 1/λ Ax, which is an eigenvector problem. The eigenvector corresponding to the largest eigenvalue (which rules out x _i s of opposite signs) gives the required centrality measures. It is easy to see that this recursive definition discounts the centrality value of distant nodes exponentially (given that λ > 1). In addition, if we consider the partners’ centrality indices identical, the centrality index of node i is proportional to its degree, whereas relaxing the assumption of identical centrality measures in the direct neighborhood but retaining it in the consecutive ones, the index for node i turns out to be the sum of degrees of direct partners, and so on. This is not to prove that the expression in Eq. 5.9 and eigenvector centrality are the same, but the underlying concepts have common characteristics.

The sparse network is simulated at 5 % density and the dense network at 30 % density. These two values were picked as follows. Density 5 % is the threshold approximately at which random networks of size 100 become connected, so that the whole network is likely to be connected at 5 % density. The 30 % density value corresponds to the density of interregional co-patenting networks as presented in Sebestyén and Varga (2013).

Note that these illustrations are created for the case when ENQ is calculated with linear distance weights and homogenous knowledge levels across the nodes. Linear distance weights are chosen because in this case the weight of GE in the ENQ index is the highest (see later), thus the differences in the GE element are captured the best in this case.

It is important to highlight that the proposed model is not capable of capturing all characteristics of the empirical knowledge networks one encounters in practice. For example, the networks generated are characterized by one core group and multiple cores are not accounted for. Also, hierarchical structures often found in real networks are not present in the simulated structures. The goal, however, is not to provide a network model which generates topologies that precisely reflects empirical ones, but to establish a relatively simple method to span a reasonably wide range of network structures and to test the behaviour of the ENQ index under these structures. On the other hand, the choice of the underlying network model seems reasonable as it comes up with topologies reflecting those characteristics often found in reality. First, it accounts for preferential attachment in its intermediate range (which is found to be a robust driving force behind real world networks) and second, it also accounts for centralized structures with connected cores and marked periphery which is a typical pattern in knowledge networks. Additionally, although less relevant from an empirical point of view, but as an extreme case the random topology is accounted for.

The network size is 100 and the density is 28 % in this specific illustration (corresponding to the empirical network analyzed by Sebestyén and Varga (2013)) but further simulations show that the tendency visible in the figure is robust across different network sizes and densities.

Average path length is the average of the shortest paths measured between every pair of nodes in the network. The clustering coefficient measures the density of the direct neighborhood of a node and the average clustering coefficient is simply the mean of these local coefficients (see Wasserman and Faust (1994) for details).

Take the star network as an extreme example. In this topology average path length is somewhat smaller than two as the majority of the nodes are at distance two from all other nodes except from the central one and the central node is at distance one from all other nodes.

The empirical network has an average path length of 1.78 whereas the corresponding random network (with the same size and density) has a path length of 1.72 (not significantly different from the empirical number). As a consequence, with regards to the path lengths, this empirical network can be positioned on the left hand side of Fig. 5.2. The clustering coefficient of the empirical network is 0.66, 2.35 times higher than the coefficient of 0.28 characterizing the corresponding random network, thus from the clustering point of view, the empirical network is situated around 0.6 on the horizontal axis of Fig. 5.2. This shows that the network model can reflect empirically relevant topologies throughout its interval from random to centralized structures.

The figure illustrates the results of a simulation with networks of size 100 and density 30 %. For all structures 100 independent runs were executed and then averaged. The results shown are robust for networks with different sizes and densities.

Given a specific structural setting along the horizontal axis of Fig. 5.3 between random and centralized topologies, moving one step in either direction resulting in a different structural setting leads to the same absolute change in the ENQ index irrespective of the choice of the decay function.

See Fig. 5.5 and the explanation in the Appendix.

It is known from graph theory that the number of connected components in a graph is given by the multiplicity of the zero eigenvalues of the Laplacian matrix of the graph. The Laplacian matrix is simply the difference of the diagonal degree matrix (with node degrees on the diagonal) and the adjacency matrix of a graph. (see e.g. Godsil and Royle 2001). Taking then the node-generated subgraphs spanned by the nodes at specific distances from the node in question and using the Laplacian method, we can easily calculate the number of connected components, although closed formula cannot be given.

Although many of the results in this field show that a position in structural holes contribute to better performance in a diversity of fields (e.g. Hopp et al. (2010), Kretschmer (2004), Donckels and Lambrecht (1997), Zaheer and Bell (2005), Powell et al. (1999), Tsai (2001), Burt et al. (2000), Burton et al. (2010)), there is still evidence on the opposite (Salmenkaita 2004; Cross and Cummings 2004). Rumsey-Wairepo (2006) argues that the two structural settings are complementary to each other rather than substitutes in explaining performance. In general, it seems that different structural dimensions can be important for different networks. When information flows and power is important, structural holes indeed provide better position, however, as in our case, if knowledge production is in the focus, exclusion resulting from structural holes may be harmful and cohesiveness meaning better interaction may have positive contribution.

Further simulations showed that the results are robust for altering the size of the network (the tendencies are better illustrated by larger networks – this is why we used size 500, but are qualitatively the same for smaller networks). Sparse networks mean 5 % density while dense networks 30 % density as before, and for each structure 100 independent simulations were executed and the results averaged.

See Varga et al. (2013) and Sebestyén and Varga (2013) for further details on data and methodology.