## Abstract

The application of social network analysis to the organization of healthcare delivery is a relatively new area of research that may not be familiar to health services statisticians and other methodologists. We present a methodological introduction to social network analysis with a case study of physicians’ adherence to clinical guidelines regarding use of implantable cardioverter defibrillators (ICDs) for the prevention of sudden cardiac death. We focus on two hospital referral regions in Indiana, Gary and South Bend, characterized by different rates of evidence-based ICD use (86 and 66 %, respectively). Using Medicare Part B claims, we construct a network of physicians who care for cardiovascular disease patients based on patient-sharing relationships. Approaches for weighting physician dyads and aggregating physician dyads by hospital are discussed. Then, we obtain a set of weighted network statistics for the positions of hospitals in their referral region, global statistics for the physician network within each hospital, and of the network positions of individual physicians within hospitals, providing the mathematical specification and sociological intuition underlying each measure. We find that adjusting for network measures can reduce the observed differences between referral regions for evidence-based ICD therapy. This study supports previous reports on how variation in physician network structure relates to utilization of care, and motivates future work using physician network measures to examine variation in evidence-based medicine.

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## Funding

This study was funded by U01AG046830 (to J.S.S.).

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## Ethics declarations

### Conflict of interest

All the authors declares that they have no conflicts of interest.

### Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

## Appendix

### Appendix

The centrality measures discussed herein can be calculated for unweighted and weighted networks. We specify the descriptions for weighted networks to reflect our use of a weighted network. Unweighted counterparts can be derived by simplifying the weighted measures in the case when all the weights equal 1. The unweighted measures are presented or described explicitly below when they have a particularly intuitively interpretable form.

The weighted closeness centrality (*C*
^{wc}) is the inverse of the total weighted geodesic distance from a node to all other nodes in the network. Thus, if \(g_{ij}^{w}\) denotes the weighted geodesic distance from node *i* to node *j*, it follows that

Closeness centrality measures both the direct and indirect connections of node *i* to quantify its closeness to all other nodes in the network. Therefore, if node *i* has smaller geodesic distances than all other nodes, it is considered most central in the network and subsequently will have a larger centrality measure. Multiplying \(C_{i}^{wc}\) by *N* − 1 yields the inverse of the average geodesic distance of node i to the other nodes and is the measure used herein. If \(g_{ij}^{w} = \infty\) for any nodes *j* ≠ *i* then \(C_{i}^{wc} = 0\). For studies interested in computing closeness centrality for a network containing multiple components, or subnetworks of nodes with no connecting dyads between them, a common practice is to use the largest connected component of the network (the set of nodes for which a finite length path exists between every pair of nodes) to compute closeness centrality and other network measures that depend on distance measures between nodes.

The weighted betweenness centrality (*C*
^{wb}) measures the relative frequency with which node *i* appears on the weighted geodesic path between all two pairs of nodes (*j*, *k*) such that *j* ≠ *i* and *k* ≠ *i*. A standardized measure of betweenness centrality is obtained by dividing the relative frequency by the total number of geodesic paths that could have included the focal node, yielding the measure:

where \(0 \le \sigma_{jk\left( i \right)} /\sigma_{jk} \le 1\) is the sum of the weights along the weighted geodesic paths between nodes *j* and *k* that pass through node *i*, denoted *σ*
_{
jk(i)}, divided by the sum of the weights along all of the weighted geodesic paths between nodes *j* and *k*, denoted *σ*
_{
jk
}. If there is a unique weighted geodesic path between nodes *j* and *k* then *σ*
_{
jk
} = *g*
^{w}_{
jk
}
and the numerator of *C*
^{wb}_{
i
}
reduces to a binary indicator variable.

Eigenvector centrality (*C*
^{e}) represents the importance of a node based on the importance of the nodes it shares edges with. Let *C* denote a vector of centrality values defined such that the centrality of node *i*, denoted, \(C_{i}^{e}\) is proportional to a linear combination of the centrality of the nodes with whom node *i* is directly connected, implying the mathematical relation

where *W* is the weighted adjacency matrix (equal to *A* for binary networks). Therefore, the eigenvector centrality measure satisfies the matrix equation *WC* = *λC*, which is immediately recognized as being equivalent to the characteristic equation whose solution yields the eigenvalues and eigenvectors of *W*. Intuitively, the solution that best discriminates between the nodes’ positions in the network is the eigenvector associated with the principal (largest) eigenvalue of *W*, representing the axis along which most of the variability in *W* occurs. Furthermore, because *A* is real-valued and square, the Perron-Frobenius theorem implies that the eigenvector associated with the largest unique eigenvalue of *W* contains only positive elements thereby yielding a quantity suitable for use as a centrality measure (Ruhnau 2000). Weighted eigenvector centrality \(WC_{i}^{e}\) for node *i* is therefore defined as the *i*th element of the vector *WC*
^{e} that solves the equation:

The generalization of eigenvector centrality to a weighted network is still the leading eigenvector of the adjacency matrix, but ties that are valued at twice the weight will contribute twice as much to the vertex’s eigenvector centrality (Newman 2004). Compared with closeness and betweenness centrality measures, eigenvector centrality is more informative for binary-valued networks when centrality is driven by differences in degree, and it is more informative in situations where a high degree node is tied to many low degree nodes or vice versa (Bonarich 2007).

Another important network measure is network clustering. The clustering coefficient is a measure of how complete the neighborhood of the node is (Latapy et al. 2008) and does not directly involve the focal node nor its edges. The igraph package uses the weighted clustering coefficient as defined by Barrat et al. (2004):

where \(d_{i}^{w}\) is the strength of node *i*, *d*
_{
i
} is the degree of node *i*, *w*
_{
ij
} is the weight of the edge between nodes *i* and *j*, and *a*
_{
ij
} are elements of the adjacency matrix. The unweighted counterpart to this can be calculated intuitively with the following definition:

In the case of a binary-valued network, the clustering coefficient reduces to the ratio of the number of triangles (or closed triads) involving node *i*, denoted *t*
_{
i
}, divided by the number of two-paths with node *i* at the apex (a “two-star”), denoted *d*
_{
i
}(*d*
_{
i
} − 1). In other words, \({\text{ClustCoef}}_{i}\) is the ratio of the sum of the weighted products over the closed triads to the sum of the weighted products over all open or closed triads with respect to node *i* (the weighted products reduce to binary indicators in the case of binary networks). A network with a high level of clustering implies if nodes *i* and *j* have highly weighted edges to node *k*, it is more likely *i* and *j* will also have highly weighted edges (or have ties in a binary network) than in a network with little clustering.

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Moen, E.L., Austin, A.M., Bynum, J.P. *et al.* An analysis of patient-sharing physician networks and implantable cardioverter defibrillator therapy.
*Health Serv Outcomes Res Method* **16, **132–153 (2016). https://doi.org/10.1007/s10742-016-0152-x

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### Keywords

- Social network analysis
- Centrality
- Degree distribution
- Structural equivalence
- Exponential random graph model
- Evidence-based medicine
- Implantable cardioverter defibrillators