A network approach to stroke systems of care

Abstract

The landscape of stroke systems of care is evolving as patients are increasingly transferred between hospitals for access to higher levels of care. This is driven by time-sensitive disability-reducing interventions such as mechanical thrombectomy. However, coordination and triage of patients for such treatment remain a challenge worldwide, particularly given complex eligibility criteria and varying time windows for treatment. Network analysis is an approach that may be applied to this problem. Hospital networks interlinked by patients moved from facility to facility can be studied using network modeling that respects the inter-dependent nature of the system. This allows understanding of the central hubs, the change of network structure over time, and the diffusion of innovations. This topical review introduces the basic principles of network science, and provides an overview on the applications and potential interventions in stroke systems of care.

Increasingly, the care of stroke patients occurs in a connected system of care.^1,2 There is an ongoing consolidation of hospital systems, a pattern of increasing frequency of patient transfers, and a changing landscape of hospital connections in response to the evidence for mechanical thrombectomy.³ The 2015 publication of benefit for mechanical thrombectomy,^4-7 an intervention that is only available at the most specialized centers, has compelled reorganization of stroke systems of care in order to ensure timely and equitable access to the treatment.⁸ Recent simulation modeling studies have taken a network-oriented approach to considering stroke systems of care, with the aim of optimizing local and national systems for accessibility to endovascular intervention.^9,10 Yet stroke systems are complex. Acute treatment options and patient eligibility are challenging to determine in the prehospital setting as they are dependent on patient characteristics, clinical and time factors, as well as complex imaging criteria. Achieving optimized systems that maximize patient benefit will likely vary based on regional characteristics and hospital resources.

Hospitals do not exist as silos in the care of stroke patients, but rather, as part of interconnected networks of multiple hospitals. Interdependence between hospitals is a product of shared ownership, shared providers, shared patients, and even shared technology, such as telemedicine. Patient sharing occurs as a product of patient referral patterns, patient transfer between hospitals, and patient care seeking patterns. Yet the importance and meaning of these connections between hospitals is not well-understood. There is a need for evidence-based understanding of inter-hospital relationships, of how hospitals are connected in stroke systems of care, and of how the interconnected nature of hospitals influences health care delivery and patient outcomes at both sending and receiving sites. Furthermore, the extent to which these interhospital connections may be leveraged to optimize knowledge translation is not well known. In order to effectively disseminate interventions and innovations, it is important to understand the degree of system integration and whether connections between hospitals are meaningful conduits for transmission of best practices and quality of care.¹¹

Network science, a new interdisciplinary science, is ideal for studying these types of systems. The data required for network construction and the methods for making sense of these systems are increasingly available. Yet, as intuitive as networks can be in their interpretation, there are caveats and challenges as well. For example, individual observations in a network are typically interdependent, with the consequence that standard statistical methods, which assume independence of observations, are often not appropriate. The purpose of this topical review is to introduce some of the basic principles of network science to provide an overview on the potential for the approach as a tool to characterize the structure and to learn about the meaning of inter-hospital relationships, as well as to propose a framework for future applications of network analysis in the study of stroke systems of care.

What Is Network Science?

Network science is a field which has developed in the last twenty years for the study of complex networks as diverse as computer networks, technological networks, biological networks, and social networks. It involves the construction, analysis, and modeling of networks using nodes to represent distinct system actors (elements or individuals) and edges to represent the connections between nodes. The origins of network science are in graph theory in mathematics and date as far back as the 1700s to the Swiss mathematician Leonhard Euler.¹² Unlike typical analyses in epidemiology or clinical research, which are predicated on the assumption of independence of observations, network analysis attempts to capture the structure of interactions among interdependent actors. In many ways, network analysis enables a more realistic description of interdependent systems, such as a group of hospitals, in which the actors are interconnected by virtue of sharing of patients, resources, and physicians. In network analysis, this interdependence is directly described, modelled, and used in interpretations.

Social network analysis is a related but distinct field in the social sciences. In contrast to network science, which relies heavily on the mathematical tools of statistical physics and primarily employs mechanistic models to investigate networks, social network analysis uses primarily statistical models and focuses on properties of actors and their relationships with other actors. Social network analysis dates back to the 1930s, when Jacob Moreno began studying the dynamics of social interactions within groups of children.¹³ The study of social networks has primarily dealt with individuals and social behaviors. For example, in medicine social network analysis has been applied to patients to illustrate how obesity, smoking behavior, drug and alcohol use, and exercise behaviors spread through social ties between connected individuals.^14-19 In stroke, social network analysis has been used to illustrate how stroke patients’ social networks decline after a stroke.²⁰ Network analysis has also been used to study professional networks among physicians, variation in physician patient-sharing networks, and how these networks influence care patterns, cost and intensity of care.^21-24 Administrative datasets are often used to build networks, whether by identifying providers who care for common patients to build patient-sharing networks, or hospitals that care for common patients to build patient transfer networks (Figure 1). Methods of network analysis have also been widely adopted in other fields. Just a few examples include its application in the study of information flow on the Internet,²⁶ of neurons in a neuronal network, and of social-ecological systems and climate change.²⁷ Despite the differences in their intellectual traditions, there is considerable overlap between network science and social network analysis, and both approaches are relevant to the study of stroke systems of care. Henceforth, we will use “network analysis” and “the study of networks” to refer to both network science and social network analysis. We do this to avoid confusion, mainly because the brevity of this review is not suitable for a sufficiently nuanced treatment that would be required to appreciate the differences of these two fields.

Figure 1. Network of U.S. Hospitals based on inter-hospital transfer of Medicare patients²⁵.

Most analyses of network data presume interdependence of actors and consider an actor’s position in the network, existing relationships, and the influence that these factors have on the actor’s behavior. Ties between individuals in a social network may enable emergence of a property beyond the constituents of the individual unit, or may represent a form of social capital that may facilitate the flow of information and influence, which may be harnessed to affect outcomes, including health-related outcomes.²⁸ In fact, in a study of networks at the physician level, knowledge sharing was shown to contribute to quality-related performance sharing; social influence has also been tied to physician performance; and features of networks have been linked to patient outcomes.²⁹

How is a Network Characterized?

The Table provides definitions for key terms in network science. Individual actors in a network, also often called a graph, are referred to as nodes. Nodes may be individual people (e.g., patients or providers) or organizations (e.g., hospitals). Each node may have covariates associated with it, and these nodal covariates, also called nodal attributes, may be continuous, ordinal, or categorical. A network may have all nodes of a single category, termed a unipartite network, or may have nodes of more than one category. For example, a bipartite network refers to a network with two types of nodes such that each edge connects a node of one type with a node of another type. As an example of a bipartite network or graph, consider the example of Figure 2, which illustrates how a bipartite network of physicians and patients may be used to obtain a unipartite network of doctors using a simple projection operation.

Figure 2. A Bipartite Network used to Construct a Unipartite Network.

The construction of a projected unipartite network of physicians from a bipartite network of patients and physicians. First, doctor-patient relationships are identified and the bipartite network is depicted (upper left). Next, the unipartite network is derived by identifying which doctors are connected through shared patients (upper right). Mathematically, this is obtained by multiplying the adjacency matrix for the bipartite graph by its transpose, as illustrated.²¹ Other projections are possible, but this simple method is commonly used.

The degree of a node in a network is defined as the number of connections or ties it has (Figure 3). A node with many connections is said to have a high degree. Nodes with high degrees are often influential in networks and are described as having high centrality. Identifying high centrality nodes is important for understanding how processes evolve with respect to a network, for example, in the spread of disease or diffusion of innovation. Although degree is a commonly used centrality measure, and one that is easily interpreted, there is a large collection of different centrality measures, each capturing a different notion about centrality.^30,31

Figure 3. Key Terms in Network Science.

Panel A illustrates calculation of each nodes degree in an undirected graph. Panel B illustrates clustering, which is the density of connections between the neighbors of a node of interest. In this illustration, the node of interest is the darker central node, and its neighbors are the lighter shaded peripheral nodes. The clustering coefficient indicates the proportion of a nodes’ neighbors with connections between them. Panel C illustrates the detection of communities within a network in which nodes are densely connected to one another but only sparsely connected to other nodes. Panel D illustrates three distinct components in a graph, including the largest connected component which includes the plurality of the nodes in the network.

Connections between nodes are referred to as ties or edges, and mathematically they are described as a pair of nodes. Connections between nodes may be structural (e.g., roads between hospitals), may indicate interactions (e.g., patient transfer between hospitals), or may indicate dependence (e.g., the delivery of care at Hospital X depends on factors at Hospital Y). A tie may be described dichotomously (exists or not), or a tie may have an associated value or weight. For example, a network may contain ties of varying strength, indicating the intensity of the connection between the two nodes. As an example, Hospital X may share patients with both Hospital Y and with Hospital Z. The connections between Hospital X with Hospitals Y and Z may be represented dichotomously or may be represented with weights based on the number of patients shared between hospitals. If Hospital X shares 50 patients with Hospital Y and shares 100 patients with Hospital Z, the use of weighted ties would convey the information that the tie strength between Hospitals X and Z is greater than the tie strength between Hospitals X and Y. We may carry this example further by considering the direction of patient movement, assigning directions to the ties. If all of 50 of the shared patients between Hospitals X and Y are transferred from Hospital X to Hospital Y, these hospitals have a single tie between them. In contrast, if Hospital X transfers 90 patients to Hospital Z, and Hospital Z transfers 10 patients to Hospital X, then Hospitals X and Z have two ties between them to indicate a reciprocal relationship. Both networks are termed directed networks. In a directed network, any two nodes may have zero, one, or two ties between them, where two ties indicate a reciprocal relationship. In contrast, in an undirected network two nodes may be connected or unconnected, but there is not a direction ascribed to the connection. A network may also have different types of ties, which is termed a multiplex network, or different layers of connections, termed a multilayer network.³² An example of a multilayer network is a patient transfer network in which patients are transferred by air or by ground.

Network topology refers to the mathematical structure of a network and network topology, or network structure, is distinct from how a network is visualized. Networks are mathematically high-dimensional objects, yet typical visualization strategies have to reduce the dimensionality of a network to two. As a result, any given network structure will have many different possible visualizations or arrangements of nodes and edges on paper or screen. In network visualizations, network nodes are usually represented as points or circles, with lines between nodes to indicate ties between nodes. In most network visualizations the placement of nodes does not carry any specific information, but rather, the information is in the connections that exist, or its topology.

The concept of a clustering coefficient, or density, is used to describe the degree to which small local groups of nodes are clustered together into groups or cliques (Figure 3). Clustering in a network may be described on a local or a global level. The so-called local clustering coefficient is a property of an individual node and it describes how closely connected its neighbors are.³³ Because clustering coefficient is defined as a proportion, its value ranges from 0 to 1, where 0 indicates that the neighbors of a given node are not connected with one another whereas a value of 1 indicates that all neighbors of a given node are connected to all other neighbors. To characterize the clustering of an entire graph, one can either compute the average of node-level clustering coefficients or use formulations of the clustering coefficient that apply to the entire network as a whole.

If a network has discrete components such that it can be “broken apart” into pieces without breaking a tie, these discrete components are referred to as connected components, often called just components (Figure 3). Components have one or more nodes and are composed of disjointed sets of nodes (i.e., no shared nodes across components). The largest connected component designates the largest component of a network and contains a plurality of the network nodes. Network communities are entities between the microscopic scale (e.g., degree) and macroscopic scale (e.g., component). Although there are many different mathematical ways to define network communities, essentially all of them consider a network community to be a set nodes that are densely interconnected but are only sparsely connected to nodes in other communities; network communities are often thought to be associated with functional units in the network.³⁴ Identification of network communities is known as community detection, which is a vibrant subfield within network science.^35,36 Community detection is somewhat analogous to the various clustering techniques that are used in multivariate analysis, however the critical difference is that the absence of a natural metric (in the mathematical sense) makes it a much more difficult problem. Some of the many applications of community detection methods include the study of the relationship between geographical constraints and social group structure,³⁷ and identification of naturally occurring physician networks (Figure 4).²²

Figure 4. An Example of Physician Networks in 2 Communities using Community Detection Methods.

These four panels illustrate Hospital Networks and Community Networks in Tallahassee, FL (Panels A and B), and in Norfolk, VA (panels C and D). Points are physicians and colors indicate hospital/community. Lines are connections with ≥ 10 shared patients. The left-hand column (panels A and C) illustrate the networks in Tallahassee and Norfolk based on hospital affiliation. The right-hand columns (panels B and D) illustrate the networks in Tallahassee and Norfolk based on community detection methods.²²

Analysis of networks requires its own methods. Standard statistical methods are inadequate for studying network data because the interdependence of outcomes in networks invalidates standard inference frameworks. Most statistical models assume observations to be independent, but assuming independence of observations would entirely discount the interdependence between nodes that creates the network structure. Instead, network and graph theory are applied to the investigation of systems with a large number of interacting components.³⁸ Network science (vs. social network analysis) uses mechanistic models that are able to capture the structure of networked systems often very precisely using only a small number of (often) domain specific mechanisms, but inference and model selection frameworks for these types of models are in their infancy. All approaches to modeling networks are grounded in empirical data, make use of computational models, and are highly visual.³⁹ In social network analysis, modeling tends to focus on describing and predicting linkage patterns between actors formed via repetitive exchanges, such as information transmission and resource sharing.^40,41 For example, models in the exponential family of random graphs (ERGM) use modeling to predict links between nodes, while accounting for the interdependence in the system.⁴² Applications of network analysis enable the examination of structural relationships, the examination of influence within networks, the examination of information diffusion in networks, and the examination of diffusion of innovation.

Homophily, or assortative mixing, describes the phenomenon that connected nodes tend to be similar to one another. Homophily appears to be present in almost all social networks, although it can manifest itself in other types of networks as well. While there are many possible origins of homophily, there are three principal mechanisms that often can give rise to homophily. One mechanism is selection, meaning that characteristics of individual nodes drive the formation of links, i.e., nodes select their neighbors based on their properties. A second mechanism that leads to homophily is social influence, which refers to the phenomenon that the existence of a tie modifies the properties of nodes, making them more similar to one another. To contrast selection and influence, under selection nodal attributes drive the evolution of edges; ties between nodes develop as a result of characteristics of the node. In contrast, under influence, edges drive the evolution of nodal attributes; existing connections between nodes lead to connected nodes becoming more similar to one another. A third mechanism leading to homophily is confounding, meaning that an external factor influences the evolution of nodal attributes, edges, or both, so that nodes end up being connected to other similar nodes as a result of upstream, external factors.

Applications of Network Science

The tools of social network analysis and network science are useful for characterizing and understanding structural relationships in organizational networks,³⁹ and for studying system-level questions such as the efficiency of patient transfer patterns.^43,44 Work in critical care and cardiology has applied network methods to inform system improvements in critical care networks,^43,45 and to determine how an optimized network with maximally efficient transfer patterns could lead to reductions in mortality for cardiac patients.⁴⁴ The organization of hospitals in the care of stroke patients has not been fully characterized, but the examples from critical care and cardiology suggest that understanding hospital connections may be a valuable tool for strengthening stroke systems of care. As we consider how hospitals may be connected to one other, for example through patient transfer, we might consider how these inter-hospital connections influence the delivery of care to patients with stroke. Network analysis methods will provide a useful toolbox for studying and understanding these processes.

The network approach has been applied to inform care system development in stroke as well. Simulation modeling has been used to determine how to maximize accessibility to comprehensive stroke centers at the population level,⁹ and to study the effects of changes in the inter-hospital transfer network on stroke patients’ eligibility for endovascular intervention and clinical outcomes.¹⁰ Both in applications at the individual and at the organizational level, there is significant potential for the application of network analysis toward the aim of achieving safer, higher quality healthcare.⁴⁶ At the individual level, network methods have been used to characterize the relationship between physician network properties and the value of health care delivery.^23,47 Similar approaches may be considered to characterize the relationship between hospital network characteristics and stroke patient outcomes.

Of course, in addition to these hospital- and system-level examples, network analysis also provides the opportunity to better understand dynamics at the individual level. For example, electronic health record and administrative data have been used to identify collaborative patient-sharing networks among providers (Figure 5).^48,49 This approach has then been applied to identify characteristics of high-performing teams, to describe the relationship between physician networks and cost and quality of care, and to characterize provider relationships and patient outcomes.^47,50

Figure 5. Patient-Sharing Network of Stroke Patients in California.

Nodes are non-federal hospitals in California. Edges represent one or more stroke patients shared between hospitals. 16 isolated nodes were removed. The network graph naturally shows the central and peripheral (hub-and-spoke) organization of hospitals. Without geocoordinates, it also reveals the geography of California with northern and southern regions.

Network analysis is also valuable tool for the study of diffusion of innovation. Among individuals, local peer influences are important mechanisms to help explain how individuals adopt and utilize new technology.^51-53 An individual’s position in a network is key, with early adopters tending to be more centrally located in their local groups.^52-54 The dissemination of knowledge through a network has been studied on the organizational level as well. For example, a study of the network of organizations involved in patient safety initiatives found that as more partnerships formed in the network, there was increased potential for information flow between organizations.⁵⁵

In considering the potential for diffusion of ideas and innovation between organizations in a network, it is important to place organizations in context. For example, an organizations’ connections will influence its knowledge of peers’ experience with an innovation, and the more integrated that network is, the more opportunity there is for this to happen. By the same token, networks may be vulnerable to the impact of highly influential or central actors, and disengagement of a key node may have ramifications throughout the network. Anderson and Jay have suggested four critical steps for the utilization of networks in the diffusion of medical technology.⁵³ The first step is to determine the network structure and identify central influential actors. The second step is to introduce change to those influential nodes, and the third is to increase ties among potential adopters. Finally, a more individualized approach is necessary for those nodes most on the periphery.⁵³ Among physicians, it has been found that the tendency to be an early adopter of a new technology or innovation is higher among centrally located individuals in a network than it is among those who perceive themselves to be influential.⁵⁶ This finding may be relevant in considering the use of network maps for the dissemination of innovation and best practices in stroke systems of care.

As with all methodological approaches, there are limitations to the application of network science to stroke systems of care as well. First, network methods may be well-suited for statistical descriptions of a system of interdependent actors, and may also be suitable for determining the ideal approach for dissemination of an innovation through a hospital network. However, at this point these advantages are mainly theoretical for stroke networks. We are not aware of any studies that have attempted to compare network structures for predicting which would be more beneficial for patients or more efficient for care delivery. In addition, such models have not been tested against real-world changes. Yet it is possible that stroke systems research will follow the trajectory observed in the study of infectious diseases: network methods were initially used descriptively, but have since been used more fully in the design and analysis of interventions.^57-60

A second limitation is in the extent to which inter-hospital connections may be interpreted. For example, if knowledge translation between hospitals occurs verbally over the phone or via telemedicine, these events are not captured in an analyzable way, and the use of administrative data will be particularly limited in this regard. Despite such data limitations, network analysis has been used in the past to study physician patient-sharing networks, and have yielded useful insights.^21,23,47

Finally, while we have outlined some of the theoretical advantages of the network approach over more standard approaches, it is not yet clear whether using network methods would produce different results, how such results could be used to solve problems such as coordination of care or optimal routing, or finally, how results would inform tangible system changes. While examples may be found, such as the use of network science methods to identify groups of physicians who might readily be able to function as accountable care organizations,²² the applications of network methods to directly informing system and policy change remain less well-defined. Like many statistical approaches, network methods may be able to describe and characterize the nature of stroke systems of care and to inform system improvements, however the methodology, in and of itself, cannot overcome challenges related to regionalization and inequitable distribution of resources without corresponding action and policy change.

Implementation of Network Methods

There are many excellent resources now available for learning more about network science.^31,61-63 The most commonly used programming languages for network science are Python and R. Within Python, the authors recommend the NetworkX package for creation and study of complex networks.⁶⁴ One of the authors has created a publicly-available online Python course, which includes videos introducing network analysis, specifically addressing the use of the NetworkX library, graph visualization, random graphs, and other statistical network approaches.⁶⁵ For users of R, we recommend the igraph network library.⁶⁶ Both NetworkX and igraph are open source and publicly available without charge. Additionally, there are specific packages available for particular network models, for example statnet is a package that has statistical modeling capabilities including models from the exponential random graph model family, latent space, and latent cluster models.⁶⁷

Conclusions

In summary, the study of networks has potential to significantly improve research in stroke systems of care. The delivery of patient care does not occur between a patient and a single physician in isolation, but increasingly, care is delivered by teams of providers at multiple sites. Complicated medical conditions require flexibility in team composition and approach.^68,69 Therefore, networks are a valuable tool for investigating connections between these different entities. The study of networks enables an understanding of individuals and hospitals in the context of their peers and their relationships; it provides tools for studying networks in the context of these interdependent relationships; and it gives a framework for studying the diffusion of innovation between hospitals. Understanding peer influences is critical in order to most effectively introduce innovation, with the ultimate end of advancing evidence-based practice. Systems of care may be most effectively improved when we understand a network’s characteristics and functioning, identify key influential actors, and facilitate its improvement. Future research must focus on network study for the purpose of improving stroke systems of care. This may come in the form of harnessing hospital networks to identify influential hospitals and seed quality improvement interventions to be disseminated through the network, or this may come in the form of real-time network diagnosis and adaption, improving ambulance routing algorithms based on hospital characteristics, contemporaneous traffic conditions, hospital crowding conditions, and physician availability. In these and many other ways, the study of networks has potential to add tremendous value to the study and optimization of stroke systems of care.

Table.

Key Terms in Network Science

Key Terms	Definition	Example
Node	An individual actor in a network. Typically represented as a point or a circle.	Individual people (patients, providers) Organizations (hospitals)
Nodal attributes	Covariates associates with a given node. May be continuous, ordinal, or categorical.	Patient age, patient sex, hospital academic status, hospital stroke center status
Tie or Edge	A connection between two nodes that may be structural, relational, or an indicator of dependence; may be described dichotomously or weighted.	A road between hospitals (structural) Hospitals connected by patient transfer (interaction-based, relational). Two hospitals in which the delivery of care at one hospital depends on factors at the other hospital (dependence)
Dichotomous ties	Dichotomous ties are connections that either exist or do not exist.	In a hospital transfer network, Hospitals X and Y share a minimum threshold of patients and are defined as connected.
Weighted ties	Weighted ties are ties with an associated value indicating the relative strength of the connection between nodes.	In a hospital transfer network, Hospital X shares 50 patients with Hospital Y, and the tie is weighted by that number of shared patients.
Degree	The degree of a node is given by the number of connections or ties that it has.	In a hospital transfer network, a hospital that receives patients from three other hospitals has a degree (more specifically, in-degree) of 3.
Centrality	A measure of a node’s importance in a network. There are many ways of characterizing centrality.	A node with a high degree is considered to have high centrality.
Undirected network	A network consisting of ties between nodes without a direction ascribed. Nodes are either connected or not connected.	A hospital network in which hospitals are said to be connected or not connected.
Directed network	A network in which ties are directional; any two nodes may have zero, one, or two ties between them. Two ties between two given nodes indicates a reciprocal relationship.	A hospital network in which hospitals are connected by patient transfer and the direction of a tie indicates the direction of patient movement between hospitals.
Unipartite network	A network in which all nodes are of a single category.	A network of physicians connected by social ties.
Bipartite network	A network in which two different types of nodes are connected with one another.	A network of physicians and their associated patients.
Multiplex network	A network with different types of ties.	A hospital network in which hospitals may be connected by patient transfer (tie type 1), by telemedicine (tie type 2), or by shared ownership (tie type 3).
Multilayer network	A network with different layers of connections.	A hospital transfer network in which patients are transferred by air or by ground.
Network density	The extent to which network nodes are connected, usually quantified as the proportion of potential edges actually present.	See Figure 3
Local clustering coefficient	A property of an individual node describing how closely connected its neighbors are. Given as a proportion ranging from 0-1 where 0 indicates no connections between a node’s neighbors, and 1 indicates that all of a node’s neighbors are also connected to one another.	See Figure 3
Components	Discrete components of a network consisting of 1 or more nodes that are not connected to other nodes in the network such that the network can be “broken apart” without breaking a tie.	See Figure 3
Largest connected component	The largest component of a network, containing a plurality of the network nodes.	See Figure 3
Network community	A set of nodes that are densely interconnected but only sparsely connected to other nodes in the graph.	See Figure 3
Community detection	The process of identifying network communities in a given graph.	Identification of naturally occurring physician networks. See Figure 3