Abstract
While developing the comprehensive index of Ego Network Quality (ENQ) Sebestyén and Varga (Ann Reg Sci, doi:10.1007/s00168-012-0545-x, 2013) integrates techniques mainly applied in a-spatial studies with solutions implemented in spatial analyses. Following the theory of innovation they applied a systematic scheme for weighting R&D in partner regions with network features frequently appearing in several (mostly non-spatial) studies. The resulting ENQ index thus reflects both network position and node characteristics in knowledge networks. Applying the ENQ index in an empirical knowledge production function analysis Sebestyén and Varga (Ann Reg Sci, doi: 10.1007/s00168-012-0545-x, 2013) demonstrate the validity of ENQ in measuring interregional knowledge flow impacts on regional knowledge generation. The aim of this chapter is twofold. First we show that ENQ is an integrated measure of network position and node characteristics very much resembling to the solution applied in the well-established index of eigenvector centrality. Second, we test the robustness of the weighting schemes in ENQ via simulation and empirical regression analyses.
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Notes
- 1.
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.
- 2.
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.
- 3.
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.
- 4.
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.
- 5.
Distances are always measured from node i.
- 6.
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.
- 7.
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.
- 8.
Note, that the weight for d = 1 is unity by definition.
- 9.
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.
- 10.
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.
- 11.
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).
- 12.
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.
- 13.
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.
- 14.
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.
- 15.
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).
- 16.
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.
- 17.
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.
- 18.
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.
- 19.
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.
- 20.
- 21.
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.
- 22.
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.
- 23.
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.
- 24.
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Acknowledgements
The research leading to this paper has received funding from the Hungarian Academy of Sciences (n° 14121: MTA-PTE Innovation and Economic Growth Research Group) and OTKA (OTKA-K101160). The authors also wish to express their thanks to the useful comments by Nicolas Carayol, Claude Raynaut, Frank van Oort and Mario Maggioni.
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Appendix
Appendix
1.1 The Modified Preferential Attachment Model
The model is developed in order to provide a transition from random graphs of the Erdős-Rényi type through scale-free structures to highly centralized networks. The model starts from a network of M nodes connected randomly with average degree D. Then we increase the size of the network step by step from M to N, adding one new node to the network at a time. In each step the new node establishes exactly D links with the existing ones, on the basis of a probabilistic parameter, r. With probability r the new link is attached to the node with the highest degree in the network and with probability 1 − r the new link is attached randomly to any existing node. It is easy to see that using this method we have two parameters, namely M and r, which contribute to the scalefree characteristics of the underlying network. If r increases with a given M, the network moves towards a more centralized structure and vice versa. However, if r is zero, we still do not have a random network for an arbitrary M as the growth of the network in the algorithm still contributes to an underlying asymmetric degree distribution (older nodes tend to have more links than younger ones).
On the other hand, modifying M and r jointly, we can set up a one-dimensional interval from 0 to 1 which moves from random graphs to centralized graphs through scalefree networks. At one end of this scale we have M = N and r = 0, which is a random graph by definition. Then we gradually increase r and at the same time decrease M. As a result, the network structure resulting from the previously described algorithm departs from being random and becomes more centralized. At the other end of the scale we reach the most centralized structure with r = 1 and M = 1. Note however, two things. First, we can express this process with one parameter, say z, ranging from 0 to 1. Then we have r = z and M = z + (1 − z)N as inputs to our model and the value of z expresses the position between random and centralized graphs. Second, the extreme case of z = 1 is not necessarily the star network as if the degree is higher than one, there is a connected core in the network, but it is true that the size of this core is D and all other nodes are linked only to this core.
The model thus has analogous logic to the Watts-Strogatz model (Watts and Strogatz 1998), with random and star-like topologies on the extremes and scalefree structures in between.
1.2 Trade-Off Between Density and Connected Components
We executed a simple simulation on a random network and on a scalefree one (with the Erdős-Rényi and the Barabási algorithms respectively). The networks in these simulations are sparse networks with global density of 3 %. The sparsity is required from a presentational point of view as the higher the overall density, the more neighborhoods are connected and there is less possibility to find unconnected groups in the neighborhoods.
For both networks we calculated the density and the number of connected components in the direct neighbourhoods of every node (taking these neighbourhoods as subgraphs and calculating density and the connected components on these subgraphs). Figure 5.5 plots the calculated density values and connected components for each node. As connected components must be an integer, the data points are arranged in horizontal lines. The figure clearly shows that there is indeed a trade-off between the two values and this trade-off is stronger in the random network. In the scalefree case we rather observe a missing upper triangle in the diagram, which shows that there are no nodes with dense neighbourhoods and many unconnected groups in their neighbourhood, whereas the other three combinations are present. In addition to dense neighbourhoods with few groups and sparse neighbourhoods with many groups, there are nodes the neighbourhoods of which are sparse and characterized by a small number of unconnected groups, which are not present in the random network. The difference between the two network structures stems from the different degree distributions
Trade-off between structural holes and density in random and scalefree networks
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Sebestyén, T., Varga, A. (2013). A Novel Comprehensive Index of Network Position and Node Characteristics in Knowledge Networks: Ego Network Quality. In: Scherngell, T. (eds) The Geography of Networks and R&D Collaborations. Advances in Spatial Science. Springer, Cham. https://doi.org/10.1007/978-3-319-02699-2_5
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