- Regular article
- Open Access
The interhospital transfer network for very low birth weight infants in the United States
- Munik Shrestha^{1, 2},
- Samuel V. Scarpino^{1, 2, 3, 4},
- Erika M. Edwards^{5, 6, 7}Email author,
- Lucy T. Greenberg^{5} and
- Jeffrey D. Horbar^{5, 6}
- Received: 2 February 2018
- Accepted: 27 July 2018
- Published: 10 August 2018
Abstract
Very low birth weight (VLBW) infants require specialized care in neonatal intensive care units. In the United States (U.S.), such infants frequently are transferred between hospitals. Although these neonatal transfer networks are important, both economically and for infant morbidity and mortality, the national level pattern of neonatal transfers is largely unknown. Using data from Vermont Oxford Network on 44,753 births, 2122 hospitals, and 9722 interhospital infant transfers from 2015, we performed the largest analysis to date on the interhospital transfer network for VLBW infants in the U.S. We find that transfers are organized around regional communities, but that despite being largely within state boundaries, most communities often contain at least two hospitals in different states. To classify the structural variation in transfer pattern amongst these communities, we applied a spectral measure for regionalization and found an association between a community’s degree of regionalization and their infant transfer rate, which was not utilized in detecting communities. We also demonstrate that the established measures of network centrality and hierarchy, e.g., the community-wide entropy in PageRank or betweenness centrality and number of distinct “layers,” within a community, correlate weakly with our regionalization index and were not significantly associated with metrics on infant transfer rate. Our results suggest that the regionalization index captures novel information about the structural properties of VLBW infant transfer networks, have the practical implication of characterizing neonatal care in the U.S., and may apply more broadly to the role of centralizing forces in organizing complex adaptive systems.
Keywords
- Neonatology
- Network Science
- Hospital Transfer Networks
- Healthcare Policy
1 Introduction
Although very low birth weight (VLBW) infants, i.e. babies weighing less than 1500 grams at birth, accounted for only 1.4% of all births in the United States (U.S.) in 2015, they accounted for 52% of all infant deaths [1]. These extremely fragile infants require specialized care in a neonatal intensive care unit (NICU).
The creation of regional systems for perinatal care, first proposed in 1975 [2], envisioned three levels of care including NICUs. The complexity of a patient’s needs was meant to determine the hospital where the mother or infant received care. The goal was to improve outcomes for high-risk pregnant women and VLBW infants by ensuring access to high-quality, economically-efficient care for all mothers and their newborns [2].
In the ensuing 40 years, however, systems of perinatal care developed based on financial incentives, geography, patient preferences, and hospitals’ interests in establishing NICUs to attract young families [3, 4]. This growth has led to the deregionalization of care, the proliferation of smaller maternity centers and NICUs, and the uneven distribution of perinatal resources unrelated to regional requirements [5–8]. Regionalization models, regulations, and measures of risk-appropriate care for high risk infants vary widely among states [4, 9–13].
One effect of deregionalization—coupled with the unavoidable fact that emergency deliveries happen, sometimes far from home—is that VLBW infants are not always born at a hospital capable of providing the appropriate level of care for their entire stay, and require transport to a different facility. There is substantial evidence suggesting that birth at a hospital with an appropriate NICU and care at a NICU with an adequate volume of VLBW infants are associated with improved survival [9, 14–19].
Recently, the American Academy of Pediatrics and the American College of Obstetrics and Gynecology developed new functional classifications of facilities that provide hospital care for pregnant women and newborn infants and recommend regionalized systems of perinatal care based on these classifications [9, 20, 21]. In the American Academy of Pediatrics guidelines, the highest level (III or IV) NICUs can care for VLBW infants throughout their entire birth hospitalization. Mothers should be transferred to such hospitals prior to birth, or VLBW infants born at a hospital with a lower level NICU should be transferred to a hospital capable of providing the appropriate level of care. Understanding how hospitals work together in regionalized networks to care for infants is an important part of that effort [22, 23].
In this study, we perform a network analysis on a U.S. national-level population of VLBW infants using data from Vermont Oxford Network. Our approach first identifies groups of highly connected hospitals, i.e. communities of hospitals, as measured by VLBW infant transfers. We then quantify the degree to which these communities are different in terms of regionalization by defining a regionalization index as the entropy in singular values of a normalized adjacency matrix. An important feature of our method is that it retains information about the frequency and direction of VLBW infant transfers, which advances our understanding of VLBW infant transfer networks and our capacity to study weighted, directed networks. By characterizing network regionalization, we find that the degree of regionalization within communities varies across the U.S. and that this variation is associated with key information on infant care. Our results further highlight the growing utility of applying network science approaches to hospital care networks.
2 Related work
Although the application of network science methodologies to large, empirical patient transfer networks has only recently been possible, similar approaches have been previously applied to NICUs in California [24] and adult intensive care units nationally [25].
A rich literature on detecting the presence of hierarchical and/or centralization structure in networks exists, which could be extended to study regionalization. For example, methods exist to perform hierarchical decomposition and link prediction in complex networks [26], for identifying interactions between groups in multilayers networks [27], determining the flow-hierarchy of a network (defined as the fraction of edges in the community that are not in the cycle graph [28]), and detecting overlapping communities in hierarchical networks [29]. Additionally, fluctuations or variation in measures such as PageRank and betweenness centrality (both terms we define in the methods section) are often applied to measure hierarchy and centralization in networks [30–32]. Lastly, the role of hierarchical structure has also been studied in the context of gene interaction [33], brain [34] and hydrological [35] networks. Our measure of regionalization closely maps on to established, network-science metrics for describing both the centralization and degree of hierarchical structure in networks.
3 Materials and methods
3.1 Data source: Vermont Oxford Network
Vermont Oxford Network is a voluntary collaboration of health care professionals around the world dedicated to improving the quality, safety, and value of care for newborn infants and their families [36]. More than 700 NICUs in the U.S. participate in the Vermont Oxford Network database. Vermont Oxford Network members submit standardized data for infants with a birth weight of 401 to 1500 grams or a gestational age of 22 weeks 0 days to 29 weeks 6 days who are born in the member hospital or admitted within 28 days of birth without first having been discharged home.
For this study, we included 9722 infants who were born from January 1, 2015, to December 31, 2015, submitted by 702 hospitals in the United States. The additional 1420 hospitals included in our analysis were not members of VON, but transfered an infant to and/or received an infant from VON member hospitals. All hospitals that contributed finalized data for the study period were included. Local staff collected data using uniform definitions that did not change during the study period [37]. All data underwent automated checks for quality and completeness at the time of submission. The University of Vermont and Northeastern University Committees on Human Research considered use of the Vermont Oxford Network Research Repository not human subjects research for this study. Consent for the Research Repository was waived by the University of Vermont Committee on Human Research and the Northeastern University Institutional Review Board. All data were de-identified.
Members submitted the exact name, city, and state of sending or receiving hospitals for infants who were transferred into or out of their hospitals. The number of transfers in this report includes all transfers to a VON member hospital within 28 days of birth, or from a VON member hospital on or before first birthday and prior to being discharged home. Infants transferred between two VON members within 28 days are counted at the receiving hospital. Multiple eligible transfers of a single infant are counted separately. Transfers are not included if the sending or receiving hospital is not recorded. For a complete list of participating facilities, please see Additional file 1.
3.2 Network methods
Here we provide additional details on the network methods used in our analysis.
Hierarchical layers. We applied Blondel and Guillaume et al.’s approach to modularity maximization to find communities [39]. One immediate application of their approach is that it naturally outputs the number of “layers” present in each community. In other words, for each community, to determine the number of “layers,” we simply count the number of times the Guillaume et al. algorithm identified a local maxima and thus collapsed all nodes into a single hierarchical layer. Critically, this approach does not require pre-specifying the number of communities, nor does it require optimizing a set of tuning parameters.
PageRank centrality. PageRank centrality was originally developed by Page et al. [40] as scalable algorithm to rank web pages based on the network structure of links to and from the web pages. Its centrality score for node i is proportional to how often a random walker is at i, given it could either randomly, with probability γ, go to nodes that it points to, or with probability (\(1-\gamma\)) it could hop to any node in the graph with equal probability. In our calculation, \(\gamma=0.85\). The entropy in PageRank then relates to the uncertainty of a random walker undertaking such a walk in the network. In our study, we calculate the PageRank entropy within a community, so random walkers would be restricted to moves, which keep them in the same community.
Betweenness centrality. Whereas PageRank centrality is based on a random walker following random paths through the network, betweenness centrality is computed from deterministic walks between all pairs of nodes. More specifically, the betweenness centrality \(b_{i}\) of node i is the fraction of shortest paths between all pair of nodes in the network that pass through node i. In other words, the betweenness of node i, i.e. \(b_{i}\), is the probability that the shortest path between a randomly selected origin node j and destination node k will “pass through” node i. Again, the entropy in betweenness then relates to uncertainty in a random walker undertaking such a walk in the network. Similar to the PageRank entropy, we calculate the betweenness entropy within a community.
Flow hierarchy. Flow hierarchy ζ of a network, is computed exactly as defined in [28], but very briefly is the fraction of edges, which are not in a cycle.
Reciprocity. Reciprocity is the fraction of edges that are of cycle of length two. Note that reciprocity is related to the flow hierarchy because the fraction, θ, of edges that are in cycles of length greater than two can be computed as \(\theta=1-\zeta-\alpha\).
4 Results
Briefly, their approach is a bottom-up, multilevel clustering algorithm where each node is moved to other communities iteratively in order to maximize the local contribution to the modularity function. When the algorithm reaches a local maximum, all nodes within the community are mapped onto a single node while tracking the weights of associated edges, and the process of searching for better modularity scores continues to the next “level.” As a result, the algorithm naturally identifies both the communities and the number of hierarchical layers in each community. However, as we outline below, the number of “layers” detected via this method provided an overly coarse description of the VLBW infant transfer networks, which was largely redundant with established centrality measures, e.g., PageRank and betweenness centralities.
Hospital and community characteristics with the interquartile range (IQR), i.e. the interval between the 25th and 75th percentiles
Overall | By community | By hospital | |
---|---|---|---|
Median (IQR) | Median (IQR) | ||
Infants (total) | 44,753 | 787 (400–1325) | 51 (25–95) |
Infants transfered | 9722 | 178 (80–272) | 11 (6–19) |
Hospitals | 2122 | 40 (22–63) | – |
Level III or IV hospitals | 27% | 25% (17–33%) | – |
States covered (inc D.C.) | 51 | 2 (1–4) | – |
4.1 Structural variation within VLBW communities
To measure structural variation across these 50 communities, we employ a continuous index for the degree of network regionalization. More formally, for a graph G, defined by a normalized adjacency matrix A whose elements \(A_{ij}\) are the fraction of transfers from hospital j to hospital i, we define regionalization \(\chi(G) \rightarrow \mathbb {R^{+}}\) as the entropy in the normalized singular values of A. Note then that the entropy is minimum for star-graph, i.e., \(\chi(G)=0\), and maximum or \(\chi(G)=\log(N)\) for cycle graphs separating star-like graphs from cycle graphs. We provide more details on the computation of this metric—along with its performance on stylized networks—in the methods section.
From a network science perspective, the regionalization index provides both a measure of how fast a random walker moving through the graph mixes to its stationary distribution and how fast groups of nodes (as indicated from the component of singular vectors) mix to the leading singular value. This approach of including the whole spectrum of the graph resonates with analogous spectral methods used to detect synchronization time scales in hierarchical networks [41]. However, VLBW infant transfer networks are weighted, directed, asymmetric, and have adjacency matrices with complex eigenvalues. To simplify the analysis, while still preserving aspects of the relevant structure, we applied a straightforward transformation, \(A^{T}A\), to construct a symmetric matrix where all of the eigenvalues are real. These eigenvalues are also the square of the singular values of A and thus preserve the relevant structure of the original graph.
Network Metrics by Community. (Network measures are defined in the methods)
Regionalization χ(G) | Hierarchical layers | PageRank | Betweenness | Reciprocity (α) | Flow Hierarchy (η) | |
---|---|---|---|---|---|---|
Median | 2.22 | 2 | 2.77 | 1.34 | 0.19 | 0.77 |
Mean | 1.99 | 2.28 | 2.62 | 1.29 | 0.20 | 0.75 |
IQR | (1.48–2.63) | (1–3) | (2.05–3.27) | (0.86–1.74) | (0.11–0.25) | (0.60–0.88) |
Min./Max. | (0–3.50) | (1–6) | (0.65–3.77) | (0.00–2.66) | (0.00–0.63) | (0.31–1.00) |
Association between proportion not transfered and community-level network metrics. In the regression models, we controlled for variation in the number of hospitals and the fraction of VON member hospitals within each community
Metric | Adjusted \(R^{2}\) | p value |
---|---|---|
Regionalization Index | 0.346 | <0.001 |
PageRank Entropy | 0.158 | 0.085 |
Betweenness Entropy | 0.164 | 0.071 |
Flow hierarchy | 0.129 | 0.228 |
Reciprocity | 0.152 | 0.103 |
5 Discussion
Using a U.S. national-level data set containing 9722 VLBW infant transfers among 2122 hospitals, we found a variety of different types of communities, varying from highly centralized to highly regionalized. That we found strongly regionalized networks is perhaps not surprising given the financial incentives of the health care market, the differences in state regulations governing NICU expansion, and geography [3, 4, 6, 10–13]. Our communities were largely organized around state boundaries, but 25% of communities overlapped at least four states and 8% of all transfers occurred between hospitals in different states. The interstate nature of transports highlights the importance of states working together to coordinate policies in these regional NICU networks to improve efficiency and quality of care.
To obtain these results on VLBW infant transfer network structure, we developed and applied a network science measure able to distinguish networks based on their degree of regionalization, which also accounts for the weighted, directed, and asymmetric nature of VLBW transfers. We verified this metric using data from all infants in the detected communities, finding the percent of infants not transferred was associated with greater regionalization scores.
Our study differs from Kunz et al. [24], who characterized NICU referral networks in California, and Iwashyna et al. [25], who described adult critical care transfer networks nationally. Kunz et al. collapsed all transfers between hospitals into a single edge and made the network symmetric by removing the directionality of transfers, then used the Fortunato community detection algorithm [42] to identify communities. Iwashyna et al. used PageRank [40] to calculate centrality. These methods, while valid, do not make use of all of the information provided by a hierarchical network in which transfers are weighted, directed, and asymmetric. Additionally, given our findings that many VLBW infant transfer communities span multiple states, obtaining an accurate picture of network structure necessitates national-level data. Therefore, our results likely provide the first picture of national-level VLBW infant hospital transfer networks.
There is a rich and growing body of literature on detecting hierarchical structure in networks [43–45]. Our work advances from earlier studies on identifying the presence of hierarchy in networks by defining a metric of regionalization structure, which provides a fuller description of the network than many of the standard centrality measures. Briefly, Corominas-Murtra [30] maps the definition of hierarchy to three dimensional points (Orderability O, Feedforwardness F and Treeness T), Luo & Magee [28] defines hierarchy as the fraction of edges that are not a part of the cycle graph, and Czégel & Palla [46] measure hierarchy as the normalized variance in the stationary distribution of a random walker on a network [28, 30, 46]. Clearly, Czégel and Palla’s approach is closely related to ours and will be nearly identical to the PageRank entropies we report for each community [46]. Most similar to our work is a methodology proposed for using eigenvalues to reveal the time-scale of synchronizing dynamics in hierarchical networks [41]. Our approach is able to detect regionalization structure for asymmetric, weighted, and directed graphs. Additionally, the regionalization index we derived provides novel structural information about a network, as compared to established network science metrics. Critically, our regionalization metric captured a relationship between network structure and the fraction of infants who are born into a hospital community but never transfered, which was not captured by the established metrics used in our analysis, which suggests that the regionalization metric contains novel information about network structure.
In 2015, VON members submitted data on nearly 90% of the VLBW infants born in the US. Most VON members are Level III or IV hospitals; however, in this analysis, members tracked the exact names of hospitals where infants originated or were sent, which included non-VON members. Although non-VON members do not report data to VON, we did know their levels of care. The only NICUs completely absent from our analysis were non-VON members that never transferred infants or exclusively transferred infants to other non-VON members. Since the VON data contains information on nearly all VLBW infants, an interesting avenue of future research would be to compare the nearly complete VON data to alternative, but less complete, data sets. Medicaid would provide information on the entire U.S., and as we have shown, understanding how communities cross state lines is vital. Additionally, many areas have systems of care where infants are born at a hospital capable of providing the appropriate level of care, or where transfers are not an option. We need to understand the systems of care in areas that do not have transfer communities and compare infant outcomes between different types of systems. Insurance markets inevitably play a large role, another area for future research.
One aspect of the VLBW transfer network that could interest the broad network science community—and separates these networks from physical structures such as the power grid or hydrological networks—is that transfers largely exist because infants were transferred up to a higher level of care, transfered laterally to a hospital with a similar level of care but different services, or were transferred down to a lower level of care before discharge home. As a result, from a complex systems perspective, the VLBW infant transfer network is best considered an emergent property of bottom-up (e.g., hospital capacity) and top-down (e.g., state-level policies on transport reimbursement) processes, but one that may still be strongly constrained by geography. Therefore, an implication of our results is that both top-down and bottom-up effects might have measurable impacts on emergent network structure, a debate currently ongoing in both the ecological [47] and complex adaptive systems [48] literature.
To advance our understanding of top-down vs. bottom-up processes in hospital transfer networks more specifically, and complex systems more generally, an important extension of this work is to analyze how the VLBW infant transfer network structure has changed over time; in particular, whether changes in state-level policies on transport or changes in reimbursement and financial incentives are associated with the types of changes in network structure predicted by our results. However, one important policy implication of the current research shows that VLBW transfers cross state borders while states policies/regulations governing maternal and neonatal transport do not [11]. The associations between existing state laws and community structure remain an important part of future research.
6 Conclusions
This study represents the first attempt to analyze the weighted, directed, and asymmetrical nature of VLBW transfers in the U.S. We developed and applied a spectral hierarchy measure in networks, which we termed a regionalization index, and found that regionalization correlated with empirically known information about infants in the detected communities. While there is still more to learn about perinatal care networks, our results contribute to what is known about the organization of neonatal care in the U.S. and may more broadly apply to the role of hierarchical forces in organizing complex adaptive systems.
Declarations
Acknowledgements
We thank Vermont Oxford Network members that contributed data to this study. We acknowledge our colleagues in the Network Science in Neonatology Working Group for their valuable discussions that deepened our understanding of the applications of network science in neonatology: Jochen Profit, Henry Lee, Sarah Kunz, Marinka Zitnik, Jure Leskovec, Ciaran Phibbs, Scott Lorch, Douglas Staiger, Jeannette Rogowski, and Jeffrey Gould.
Availability of data and materials
The infant-level data, including data on transfers, is protected by U.S. privacy laws and legal membership agreement. Individuals interested in using VON data are asked to follow our Policy and Guidelines for Collaborative Research using the Vermont Oxford Network Databases, available at https://public.vtoxford.org/wp-content/uploads/2017/02/Vermont-Oxford-Network-Policy-and-Guidelines-for-Collaborative-Research-2017.pdf. To further strengthen both the utility and reproducibility of our study, we have posted code associated with our analyses to the following github repository: https://github.com/Emergent-Epidemics/VON_NICU_2018.
Funding
SVS and MS acknowledge funding support from the University of Vermont and Northeastern University. The funders played no role in the study design. JDH and LTG are employees of Vermont Oxford Network. EME is an employee of the University of Vermont funded by Vermont Oxford Network.
Authors’ contributions
All authors collaboratively designed and performed the research, contributed new analytic tools, analyzed data, and wrote the paper. All authors read and approved the final manuscript.
Competing interests
The authors declare no conflicts of interest exist.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- United States Department of Health and Human Services, Centers for Disease Control and Prevention, National Center for Health Statistics, Division of Vital Statistics. Linked birth/infant death records, 2007 to 2015, 2017. http://wonder.cdc.gov/lbd-current.html, accessed January 10, 2018
- Ryan GM Jr et al. (1975) Toward improving the outcome of pregnancy: recommendations for the regional development of perinatal health services. Obstet Gynecol 46(4):375–384 Google Scholar
- Goodman DC, Fisher ES, Little GA, Stukel TA, Chang C-h, Schoendorf KS (2002) The relation between the availability of neonatal intensive care and neonatal mortality. N Engl J Med 346(20):1538–1544 View ArticleGoogle Scholar
- Lorch SA, Myers S, Carr B (2010) The regionalization of pediatric health care. Pediatrics 126(6):1182–1190 View ArticleGoogle Scholar
- Gould JB, Marks AR, Chavez G (2002) Expansion of community-based perinatal care in California. J Perinatol 22(8):630 View ArticleGoogle Scholar
- Rayburn WF, Richards ME, Elwell EC (2012) Drive times to hospitals with perinatal care in the United States. Obstet Gynecol 119(3):611–616 View ArticleGoogle Scholar
- Kastenberg ZJ, Lee HC, Profit J, Gould JB, Sylvester KG (2015) Effect of deregionalized care on mortality in very low-birth-weight infants with necrotizing enterocolitis. JAMA Pediatr 169(1):26–32 View ArticleGoogle Scholar
- Harrison WN, Wasserman JR, Goodman DC (2017) Regional variation in neonatal intensive care admissions and the relationship to bed supply. J Pediatr Google Scholar
- Stark AR (2004) Levels of neonatal care. Pediatrics 114(5):1341–1347 View ArticleGoogle Scholar
- Nowakowski L, Barfield WD, Kroelinger CD, Lauver CB, Lawler MH, White VA, Ramos LR (2012) Assessment of state measures of risk-appropriate care for very low birth weight infants and recommendations for enhancing regionalized state systems. Matern Child Health J 16(1):217–227 View ArticleGoogle Scholar
- Okoroh EM, Kroelinger CD, Lasswell S, Goodman DA, Williams A, Barfield W (2016) United States and territory policies supporting maternal and neonatal transfer: review of transport and reimbursement. J Perinatol 36(1):30–34 View ArticleGoogle Scholar
- Brantley MD, Davis NL, Goodman DA, Callaghan WM, Barfield WD (2017) Perinatal regionalization: a geospatial view of perinatal critical care, United States, 2010–2013. Am J Obstet Gynecol 216(2):185 View ArticleGoogle Scholar
- Kroelinger C, Okoroh E, Goodman D, Lasswell S, Barfield W (2017) Comparison of state risk-appropriate neonatal care policies with the 2012 aap policy statement. J Perinatol. p. epub ahead of print Google Scholar
- McCormick MC, Shapiro S, Starfield BH (1985) The regionalization of perinatal services: summary of the evaluation of a national demonstration program. JAMA 253(6):799–804 View ArticleGoogle Scholar
- Paneth N, Kiely JL, Wallenstein S, Marcus M, Pakter J, Susser M (1982) Newborn intensive care and neonatal mortality in low-birth-weight infants: a population study. N Engl J Med 307(3):149–155 View ArticleGoogle Scholar
- Rogowski JA, Horbar JD, Staiger DO, Kenny M, Carpenter J, Geppert J (2004) Indirect vs direct hospital quality indicators for very low-birth-weight infants. JAMA 291(2):202–209 View ArticleGoogle Scholar
- Phibbs CS, Baker LC, Caughey AB, Danielsen B, Schmitt SK, Phibbs RH (2007) Level and volume of neonatal intensive care and mortality in very-low-birth-weight infants. N Engl J Med 356(21):2165–2175 View ArticleGoogle Scholar
- Lasswell SM, Barfield WD, Rochat RW, Blackmon L (2010) Perinatal regionalization for very low-birth-weight and very preterm infants: a meta-analysis. JAMA 304(9):992–1000 View ArticleGoogle Scholar
- Lorch SA, Baiocchi M, Ahlberg CE, Small DS (2012) “The differential impact of delivery hospital on the outcomes of premature infants,” Pediatrics pp peds–2011 Google Scholar
- Barfield WD, Papile L-A, Baley JE, Benitz W, Cummings J, Carlo WA, Kumar P, Polin RA, Tan RC, Wang KS et al. (2012) Levels of neonatal care. Pediatrics 130(3):587–597 View ArticleGoogle Scholar
- Menard MK, Kilpatrick S, Saade G, Hollier LM, Joseph GF, Barfield W, Callaghan W, Jennings J, Conry J, American College of Obstetricians Gynecologists et al. (2015) Levels of maternal care. Am J Obstet Gynecol 212(3):259–271 Google Scholar
- Rashidian A, Omidvari A, Vali Y, Mortaz S, Yousefi-Nooraie R, Jafari M, Bhutta Z (2014) The effectiveness of regionalization of perinatal care services—a systematic review. Public Health 128(10):872–885 View ArticleGoogle Scholar
- Lorch SA (2015) Ensuring access to the appropriate health care professionals: regionalization and centralization of care in a new era of health care financing and delivery. JAMA Pediatr 169(1):11–12 View ArticleGoogle Scholar
- Kunz S, Zupancic J, Rigdon J, Phibbs C, Lee H, Gould J, Leskovec J, Profit J (2017) Network analysis: a novel method for mapping neonatal acute transport patterns in California. J Perinatol 37(6):702–708 View ArticleGoogle Scholar
- Iwashyna TJ, Christie JD, Moody J, Kahn JM, Asch DA (2009) The structure of critical care transfer networks. Med Care 47(7):787 View ArticleGoogle Scholar
- Clauset A, Moore C, Newman ME (2008) Hierarchical structure and the prediction of missing links in networks. Nature 453(7191):98–101 View ArticleGoogle Scholar
- Valles-Catala T, Massucci FA, Guimera R, Sales-Pardo M (2016) Multilayer stochastic block models reveal the multilayer structure of complex networks. Phys Rev X 6(1):011036 Google Scholar
- Luo J, Magee CL (2011) Detecting evolving patterns of self-organizing networks by flow hierarchy measurement. Complexity 16(6):53–61 View ArticleGoogle Scholar
- Shen H, Cheng X, Cai K, Hu M-B (2009) Detect overlapping and hierarchical community structure in networks. Phys A, Stat Mech Appl 388(8):1706–1712 View ArticleGoogle Scholar
- Corominas-Murtra B, Goñi J, Solé RV, Rodríguez-Caso C (2013) On the origins of hierarchy in complex networks. Proc Natl Acad Sci 110(33):13316–13321 MathSciNetView ArticleMATHGoogle Scholar
- Johnson S, Jones NS (2017) Looplessness in networks is linked to trophic coherence. Proc Natl Acad Sci 114(22):5618–5623 MathSciNetView ArticleGoogle Scholar
- Corominas-Murtra B, Rodríguez-Caso C, Goñi J, Solé R (2010) Topological reversibility and causality in feed-forward networks. New J Phys 12(11):113051 View ArticleGoogle Scholar
- Yu H, Gerstein M (2006) Genomic analysis of the hierarchical structure of regulatory networks. Proc Natl Acad Sci 103(40):14724–14731 View ArticleGoogle Scholar
- Meunier D, Lambiotte R, Bullmore ET (2010) Modular and hierarchically modular organization of brain networks. In: Frontiers in neuroscience, vol 4 Google Scholar
- Lebecherel L, Andréassian V, Perrin C (2016) On evaluating the robustness of spatial-proximity-based regionalization methods. J Hydrol 539:196–203 View ArticleGoogle Scholar
- Horbar JD, Soll RF, Edwards WH (2010) The Vermont Oxford network: a community of practice. Clin Perinatol 37(1):29–47. Quality Improvement in Neonatal and Perinatal Medicine View ArticleGoogle Scholar
- Horbar JD, Soll RF (2016) 2015 manual of operations, part 2, release 19.0. Vermont Oxford Network Google Scholar
- Girvan M, Newman MEJ (2002) Community structure in social and biological networks. Proc Natl Acad Sci 99(12):7821–7826 MathSciNetView ArticleMATHGoogle Scholar
- Blondel VD, Guillaume J-L, Lambiotte R, Lefebvre E (2008) Fast unfolding of communities in large networks. J Stat Mech Theory Exp 10:10008 View ArticleGoogle Scholar
- Page L, Brin S, Motwani R, Winograd T (1999) The pagerank citation ranking: bringing order to the web. Technical Report 1999-66, Stanford InfoLab. Previous number = SIDL-WP-1999-0120 Google Scholar
- Arenas A, Díaz-Guilera A, Pérez-Vicente CJ (2006) Synchronization reveals topological scales in complex networks. Phys Rev Lett 96:114102 View ArticleGoogle Scholar
- Fortunato S (2010) Community detection in graphs. Phys Rep 486:75–174 MathSciNetView ArticleGoogle Scholar
- Krackhardt D (2014) Graph theoretical dimensions of informal organizations. In: Computational organization theory. Psychology Press, United Kingdom, pp 107–130 Google Scholar
- Trusina A, Maslov S, Minnhagen P, Sneppen K (2004) Hierarchy measures in complex networks. Phys Rev Lett 92(17):178702 View ArticleGoogle Scholar
- Mones E, Vicsek L, Vicsek T (2012) Hierarchy measure for complex networks. PLoS ONE 7(3):e33799 View ArticleGoogle Scholar
- Czégel D, Palla G (2015) Random walk hierarchy measure: what is more hierarchical, a chain, a tree or a star? Sci Rep 5:17994 View ArticleGoogle Scholar
- Lynam CP, Llope M, Möllmann C, Helaouët P, Bayliss-Brown GA, Stenseth NC (2017) Interaction between top-down and bottom-up control in marine food webs. Proc Natl Acad Sci 114(8):1952–1957 View ArticleGoogle Scholar
- Flack JC (2017) Coarse-graining as a downward causation mechanism. Philos Trans R Soc A, Math Phys Eng Sci 375(2109):20160338 View ArticleGoogle Scholar