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Pitman, Marsh-field Google Scholar. Podolny JM Networks as the pipes and prisms of the market. Am J Sociol —60 Google Scholar. Am Sociol Rev — Google Scholar. Acad Manage J — Google Scholar. J Manage Stud — Google Scholar. More exactly, since there may be many length-two paths between u and v , the value obtained by each intermediary, w , is a monotonically decreasing function of the number of such paths. The structural hole model is limited only to intermediaries along length-two paths. An interesting version of the model considers a Harmonic intermediary benefit, in which the value that could be obtained by a direct link between u and v , is instead allocated equally among all intermediaries on length-two paths between them.
In the undirected case, this becomes. Interestingly, for this version of the model, the network as a whole attains the same total value, summed over all intermediaries, from length-two paths, as it would if those end-points were directly connected.
The total value of the network, the sum over all node utilities, is then. A network is formed in which individuals are connected by social links and those interconnections convey on the group as a whole some total productivity or value. Given individual utilities, such as those defined in Eq. For a given value function, that assigns a real number to each network over some fixed number of n nodes, it is interesting to consider its efficient networks, i.
For the conn and the ksh , we have arrived at the value function by summing individual payoffs. It is worth noting, that, for the specialisation of the conn game in which only length-two paths accrue any benefit, i. There are a number of limitations to the conn and ksh models. In particular,. The ksh model only considers length-two paths for indirect benefits. The ksh model allocates the entire indirect benefit to intermediary nodes. This eliminates any personal motivation for a player to form indirect links.
The conn model allocates no benefit to intermediary nodes, ignoring the important role that they play in creating value in the network. In particular, the efficient networks do not contain any triangles, which are known as strong social structures.
We argue that the advantage that a node gains from paths in the network, depends on the quality of the end-points of these paths.
If the end-points are gateways into strong communities, then there is significant advantage, while if the end-points are themselves dead-ends, or have limited reach into the rest of the network, then they yield relatively less value. We illustrate this point in Fig.
Here, we measure the ksh value of a network, as the network is modified to increase its clustering. Specifically, starting with a network with a scale-free degree distribution, we carry out pairwise swaps of edges in the network in such a way that the degree distribution remains fixed, while the clustering coefficient of the network varies.
The interesting features of this plot are where the payoff remains fixed or nearly fixed, while the clustering coefficient decreases. The reduction in clustering coefficient is indicative of intermediaries in structural holes are connecting between ever weaker community structures.
We argue that the payoff of being an intermediary in such a situation should also ideally decrease. We aim to develop a model that accounts for this anomaly and whose efficient networks contain the sort of social structures that we might expect to find in real social networks. Network payoff vs clustering coefficient of network resulted from ksh. The ksh assigns value to bridging, while the conn focuses more on bonding, over direct and indirect links.
Our goal is to propose a new model, that merges the features of the conn and the ksh , to capture both bonding and bridging social capital. We call our model the structural hole connections model shc.
We extend the ksh to longer paths, maintaining the Harmonic allocation of value to intermediaries on these paths. We combine this extended ksh with the conn model, so that value is allocated to both source and intermediary nodes along each path. As will be seen, by maintaining a Harmonic distribution, our extended model retains the same overall value as that of a conn model and hence our model can be understood as a new allocation function for the value in that model.
However, rather than restrict ourselves to the symmetric conn , instead we consider that the benefit is dependent on the end node, v , so that:. Hence, each end-point may value the connection differently. It is a structural measurement of the neighbourhood v. Such a nodal benefit ensures that the anonymity of the network value function is maintained.
That is, that the value remains independent of the node labels. Footnote 2 Several such measures are readily available in the complex networks literature. For example,. Nodes in the network that form many triangles with their neighbours are members of closely knit communities and are hence worth to connect with, either directly or indirectly. Another measure, which considers the density of triangles, rather than a simple count, is the clustering coefficient:.
Another possible measure is:. The smaller the constraint, the better a node can act as a structural hole broker. On the other hand, such a broker would like to connect to constrained nodes, as they are members of strong communities. The ksh assumes that intermediaries connecting nodes u and v , that are not directly connected, receive the value that would otherwise go to the end-points.
The value is assigned entirely to the intermediary, while the conn assumes that nodes obtain value for other nodes to whom they have indirect, as well as direct, connections. Finally, we extend the ksh to longer paths. We retain the Harmonic benefit allocation used in the ksh , so that the full value of an indirect link is retained in the network, but is allocated between intermediary and source nodes.
To summarise, in the shc model, a node obtains value from the network. In the following, we write the utility of a node w in the graph, by considering these three types of benefit. That is:. While it is beyond the scope of this paper to explore in detail the efficient networks of the shc , in Fig.
It may be observed that, in the second case, efficient networks containing triangles are found, as only nodes connected to triangles have a non-zero nodal benefit. Our primary interest in this paper is to use the shc game as a means of defining a structural hole centrality measure that can identify nodes in a social network with high social capital.
This total network value is obtained as a sum over all the paths in the network of the path-length discounted benefits obtained from end-points of those paths. The question of a fair allocation of such network value has been addressed in works such as [ 24 ]. One approach is to identify desirable properties of the allocation function and determine an allocation that satisfies those properties.
Two desirable properties of a fair allocation are that it be component additive , that is, the value generated by any connected component in a network should be allocated among the nodes in that component; and that it satisfy equal bargaining power , that is, that if two nodes, u and v are connected, then the change in the value allocated to node u when the edge u , v is removed, should equal the change in the value allocated to node v.
Equal bargaining power says that the pair of nodes each benefit or suffer equally from the addition of a link between them. The Myerson allocation will often allocate high value to intermediate nodes, as they are crucial for the creation of value on paths that traverse them. Hence we define the structural hole centrality measure, shce , as the payoff of the shc game. To parameterise the cost, we stick with a fixed cost c for every link, and note that the total value in the network is zero when.
The parameters of the shce are summarised in Table 1. Nevertheless, the shce is not identical to either measure. The difference in the measures is illustrated for the Minnesota road network, shown in Fig. The plot shows the tied rank of the measures, where nodes with largest centrality value have rank n and nodes with smallest have rank 1. The Spearman rank correlation of shce with closeness and betweenness is not particularly strong for these settings.
However, the Katz allocates its value solely to the source nodes on such paths and so cannot be used as a measure of bridging capital. It is instructive to compare the shc allocation of value to that of the Myerson value in a simple network, with a constant benefit function. In Fig. By counting all shortest paths in this network, we can find the total network value as. The Myerson value allocates the value of each path evenly among all the nodes along the path, since each node is equally responsible for bringing that value to the network.
We can arrive at the Myerson allocation as. Myerson and shce values on a simple path. If a fifth node is added in order to produce a second length-three path connecting the end-points, as shown in Fig. Both methods give higher weight to node 3 than nodes 2 and 5, since the value remains in the network if either one of these is removed.
Again, in Fig. In the case of the triangle benefit, value is concentrated on the nodes that form the single triangle in the network nodes, 1, 4 and 13 , for both measures. The Myerson gives higher values to peripheral nodes 7 and 8, since these nodes add to the value of the network by linking to nodes with non-zero benefit. The overall rank correlation of the shce and Myerson is 0.
When all nodes have the same benefit, high Myerson values attach to nodes 3 and 5 that add value to the network by forming the path that connects the nodes in the lower left corner to the rest. But, again Myerson credits the peripheral nodes 7, 8 and 9 because they too add to the overall value in the network. The highest correlation 0. From these examples, it is clear that there is no best value of the shce parameters, in the fairness sense from which the Myerson is derived.
But it is also generally the case that some settings of the shce parameters can achieve centrality scores that correlate strongly with the Myerson. The shce relies on the analyst to determine an insightful allocation of the value in the network by adjusting its parameters, while the Myerson provides a single best allocation in some well-defined sense.
We note however that work such as [ 24 ] argues that the fairness criteria of the Myerson may not be appropriate, depending on the context in which the strategic game is analysed.
The work of [ 11 ] is the most comprehensive recent study of network centrality measures. This work examines the correlations between 17 centrality measures across a large range of different graphs, drawn from different application domains. To obtain a good understanding of where the shce fits in relation to other metrics, it is worthwhile applying this same analysis to the shce. Following the work of [ 11 ], we evaluate the proposed shce centrality measure using a subset of the CommunityFitNet corpus of networks [ 25 ] which, in total, contains real-world networks drawn from the Index of Complex Networks ICON [ 26 ].
The CommunityFitNet corpus includes a variety of network sizes and structures. Our analysis assumes unweighted, simple, undirected networks.
We only consider networks with a single connected component and also reject any other networks for which any of the analysed centrality measures fails to compute.
Footnote 3 There remains networks, on which our analysis is performed, which come from 6 different domains see Table 2 , with a range of nodes from 8 to average and a mean sparsity of 4. This statistic is chosen in [ 11 ] on the basis that relationships between measures can be nonlinear, though they are generally always monotonic. The centrality measures that we compare against are listed and defined in Table 3. From their definitions, the connections to the shce are apparent.
In particular, shce relies on values measured along shortest paths, similarly to the cc , hc , bc and kc. In Figs. Figures 8 and 9 contain the analogous boxplots for the case of the triangle benefit function. At the same time, we see a strengthening of the correlation to the cc , bc and hc that value short connections from source nodes to other nodes in the network. On the other hand, when a cost for link formation is introduced Figs. We can see that the shce becomes less well-correlated with standard centrality measures as a mixture of benefits Figs.
We also see less strong correlations with the standard centrality measures when the triangle benefit function is used. It should be noted that, particularly, for some of the smaller networks in the dataset, these can be a high fraction of nodes that are not incident on any triangles, reducing the benefit of connecting to them to zero. Similar to correlation analysis between centrality measures in [ 11 ], in Fig. In addition to correlation between shce and other centrality measures, we also examined the association between network properties and the CMC for different networks.
We used following six out of the eight global network properties used for the similar analysis in [ 11 ]: assortivity, connection density, clustering, global efficiency, majorization gap, and spectral gap.
In particular, objective of this analysis to examine how the shce relates to the network topology as well as how it is compared relative to other centrality measures. Before results of this analysis are discussed, we briefly remind ourselves the definitions of network topological properties that were used in the analysis.
Clustering is the number of closed triangles in the network. The efficiency measure defined by [ 27 ] is the inverse of path connecting two nodes in the network and at global scale global efficiency is the average of efficiency for all the nodes in the network [ 28 ]. The majorization gap is the difference between empirical network and idealised threshold network [ 29 ]. It is calculated as difference in network degree sequence and its corrected conjugate sequence.
Networks with high majorization gap will be distant from a threshold network and have lower CMCs [ 11 ]. Finally, the spectral gap is the difference between moduli of two largest eigenvalues of the adjacency matrix. It quantifies the extent to which a network being sparse and well connected at the same time [ 11 ]. The lower triangle in each subplot indicates the Spearman correlation between CMC and the network property. The upper triangle indicates if this correlation was significant grey or not while.
Overall, it may be concluded that the shce behaves in an expected manner and aligns with other centrality measures to a greater or lesser extent, depending on the setting of its parameters. We illustrate an application of the shce in the analysis of the social network of Norwegian boards of directors introduced in [ 30 ].
This set of networks were originally used to analyse the social capital of women directors in Norway. We take the May one-mode dataset in which actors correspond to board members and a link between a pair of actors exists in the network if they are members of a common board. We extract the largest connected component of this network, which consists of nodes and edges.
Thus, we allow long paths up to 10 connections to impact on the shce and discount according to path length relatively slowly. We focus on how the profile can allow broad categories of actor to be identified.
Examples of the three different profile types, are illustrated in Fig. Proportion of male and female actors ordered by peak position in shce Profile. The purpose of this example is to illustrate the potential of the shce to shed light on issues of social capital in social networks.
We do not offer definitive conclusions and refer readers to [ 30 ] for a deep sociological analysis of these networks. However, we do contend that the shce can yield deeper insights, in comparison to the betweenness centrality measure that was exploited in the original study. This paper has extended the state-of-the-art on strategic network formation by proposing a new utility with associated formation game, that generalises and combines the previously proposed conn and ksh network formation games.
While we have shown some examples of efficient networks that emerge from this game, the main focus of this paper has been on a new centrality measure, that is defined as a fixed point of the linear system that spreads the benefit associated with each node in the network, among those nodes that connect to it along geodesic paths.
The new centrality measure has the advantage of the Katz measure in that it depends on the connecting paths, rather than simply on path-lengths. But, more particularly, it is parameterised in a way that allows the analyst to control the way nodes are valued according to their bonding and bridging capabilities.
We have benchmarked the new measure against a number of other common centrality measured and showed its application on some example networks. In future work, we will provide a more detailed analysis of the bonding and bridging game and identify the structures that emerge as stable networks from this game.
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