The complexity of comorbidity distribution and pathogenesis in the elderly based on a dependency network simulation

Data processing

We utilized tracking data from CHARLS for the years 2015 and 2018. In 2015, we had 20,042 samples, which included 9,585 males and 10,457 females. In 2018, the sample size decreased slightly to 19,435, with 9,205 males and 10,230 females. After excluding records with missing personal, family, or community identification numbers, as well as year of birth data, we obtained 15,299 samples that were traceable across both 2015 and 2018. Of these, 7,417 were male, and 7,882 were female.

Dependency network

A relational network is a structure where the state of a node depends on other nodes, represented by a “dependency graph” in graph theory. In such a network, nodes typically represent program results or tasks, and edges indicate data transmission or dependencies. In UML (Unified Modeling Language) modeling, a dependency shows that a change in class A affects class B, meaning B depends on A. For example, traveling from city A to city C via city B can be represented by nodes A, B, and C, with directed edges A→B and B→C. The relationship B→C relies on A→B, highlighting the importance of connection order and edge weight in dependency-based networks.

The transition from being healthy to having a single chronic disease, and eventually multiple diseases, is a time-dependent, multi-state process with state dependency. Direct transitions from a healthy state to multiple diseases are rare, with most following a path from one chronic disease to several. We aim to use dependency networks to model this process, capturing two key aspects: (1) individuals’ health states change with age, and (2) they eventually stabilize in certain disease states.

In the dependency network, disease status changes were represented by nodes, which reflected multidimensional disease states rather than specific chronic diseases. For n chronic diseases, there are theoretically \(\:{2}^{n}\) possible health states, ranging from completely healthy (n-dimensional 0) to all diseases present (n-dimensional 1). A comorbidity network in this space should have \(\:{2}^{n}\) nodes, each representing a different disease state. Transitions between these states were shown as directed edges, and to capture the sequential dependency, we used time-stamped data. For example, if an individual moved from being healthy in 2015 to having hypertension in 2018, a directed edge from the healthy node to the hypertension-only node showed the transition.

The edge weight in this directed network was replaced by the frequency of state transitions in the sample, reflecting the dependence between two states. This model no longer views chronic disease progression as irreversible but build a relationship network based solely on state transitions in the tracking dataset, generating directed edges for reversible processes as well.

Centrality measures

Degree centrality measures a node’s connections, closeness centrality its ability to disseminate information, betweenness centrality its role in information transmission, and eigenvector centrality its overall importance.

Degree centrality \(\:d(\nu)\) represents the number of direct connections between a disease and others. Diseases with high degree centrality are more interconnected. Degree centrality is split into in-degree \(\:{d}_{in}\left(\nu\:\right)\) and out-degree \(\:{d}_{out}\left(\nu\:\right)\). High in-degree indicates a stable disease-absorbing state, while high out-degree suggests an unstable state with a higher likelihood of transitioning to other diseases. We also consider edge weights by calculating the weighted in/out degree centrality.

Closeness centrality \(\:C\left(\nu\:\right)\) is the reciprocal of the sum of the shortest path lengths from a node to all others. A high value indicates shorter distances to other nodes, making it easier to connect with other diseases and highlighting its diagnostic importance. Inbound closeness centrality \(\:{C}_{in}\left(\nu\:\right)\) reflects how easily other diseases transition to this state, showing strong integration. Outbound closeness centrality \(\:{C}_{out}\left(\nu\:\right)\) reflects how easily this disease transitions to others, showing strong influence. The calculation also considers edge weights, where higher weights reduce path cost, facilitating transitions.

Betweenness centrality \(\:B\left(\nu\:\right)\) measures how often a node acts as an intermediary in the network’s shortest paths. A disease with high betweenness centrality plays a key role in connecting other diseases. The calculation also accounts for edge weights, using the inverse of the edge weight as the path cost.

Eigenvector centrality measures a node’s influence, where connections to high-scoring nodes contribute more to its score. A high score indicates a node is linked to many influential nodes. This centrality highlights a node’s environment, with higher values indicating proximity to sources of risk and requiring extra precautions.

1. 1.

   \(\:\:PageRank\left(\nu\:\right)\): Since eigenvector centrality is suited for undirected graphs, many nodes in directed graphs have an eigenvector centrality of 0. PageRank improves on this by adapting eigenvector centrality for directed graphs.

2. 2.

   \(\:\:Hub\left(\nu\:\right)\) and \(\:Authority\left(\nu\:\right)\): These recursive centralities measure two types of links. A node’s hub score is the sum of its successors’ authority scores, and its authority score is the sum of its predecessors’ hub scores. Both hub and authority scores sum to 1, with the hub score indicating an important sender and the authority score indicating an important receiver.

The in_comprehensive centrality index is expressed as:

The out_comprehensive centrality index is expressed as: