The application of graph-theoretic methods to analyze complex social interactions has a long history, with early examples including Read's (1954) study of aboriginal tribes and Sampson's (1969) work on monks. Network science has also been successfully applied to philology and the analysis of historical texts, with examples including the study of Greek and Roman mythology (Choi and Kim, 2007) and ancient Egyptian and Indian texts (Elwert, 2021). In Biblical studies, network analysis has primarily focused on the New Testament and the events surrounding Jesus. This study aims to reveal hidden relationships within the Hebrew Bible's corpus by focusing on the texts' interpretation of the otherworld and afterlife, excluding pseudepigrapha, apocrypha, and texts outside the Jewish tradition. The study constructs a weighted graph where texts are vertices and edges represent shared concepts related to the afterlife. The analysis includes statistical properties, clustering (using the Louvain method), and vertex ranking (using PageRank, HITS, quasi-exponential, and exponential ranking) to identify influential texts and concepts.

The paper reviews existing literature on the application of network science to various fields, highlighting its use in analyzing social structures, human language, and historical texts. It specifically mentions prior work applying network analysis to biblical studies, noting that most research has concentrated on the New Testament and the life of Jesus. The authors point out a relative scarcity of network analyses applied to Old Testament texts and the Hebrew Bible, although they cite a previous study by Tantlevskij et al. (2021) examining religious and philosophical movements in early Judaism using signed networks. The review sets the stage for the current research, which seeks to address this gap by focusing specifically on the Hebrew Bible and the evolution of concepts surrounding the afterlife.

The study identifies 43 texts from the Hebrew Bible that contain concepts related to the otherworld and afterlife. These texts include books of Genesis, Numbers, Deuteronomy, Samuel (1st and 2nd), Judges, 2 Kings, Isaiah (Proto- and Deutero-), Jeremiah, Ezekiel, Hosea, Malachi, numerous Psalms (6, 16, 17, 21, 23, 27, 30, 36, 37, 41, 49, 55, 56, 61, 73, 88, 94, 103, 106, 110, 116, 119, 139, 142, 143), Job, Proverbs, Qoheleth, Daniel, and 2 Chronicles. The researchers then identified 112 text-concept pairs (a full list is available in the supplementary materials). A weighted graph is constructed where texts are vertices and an edge connects two texts if they share at least one concept; the edge weight equals the number of shared concepts. The resulting graph contains 43 vertices and 383 edges, with a total weight of 462. Statistical analysis includes calculating the average degree (approximately 17.8) and graph density (0.424). The Louvain community detection algorithm is used for clustering, aiming to maximize modularity and identify densely connected subgroups of texts. Four vertex ranking methods (PageRank, HITS, quasi-exponential ranking, and exponential ranking) are employed to identify the most influential texts based on their connectivity and the importance of their connections. The analysis considers the concepts associated with the highly ranked texts to reveal patterns and themes in the development of afterlife concepts in the Hebrew Bible.

The analysis of the constructed network reveals several key findings. First, the high graph density (0.424) and average degree (17.8) indicate a significant level of interconnectedness among the Hebrew Bible texts concerning the afterlife. The Louvain clustering method divides the texts into three groups, suggesting distinct thematic groupings in how these texts address afterlife concepts. Group 3 is characterized by concepts like "Ascension to heaven" and "Spirits of the righteous ascending to God," while Group 2 features concepts like "Sheol as a dull existence" and "Lack of contact with God in Sheol." Group 1, however, displays greater diversity in concepts and requires further analysis. The four vertex ranking methods, while employing different algorithms, yield remarkably consistent results for the top five most influential texts. These are (with minor variations in order among methods): Proto-Isaiah, Qoheleth, Job, Proverbs, and 1 Samuel. The number of concepts associated with each text serves as an indirect indicator of influence, but not the sole determinant. Analysis of the key concepts associated with the most influential texts reveals a frequent occurrence of "Spirits of the righteous ascending to God," "Ascension to heaven," "Lord's justice," "Sheol as dull existence," "True immortality," "Human spirit as bearer of selfhood," and "Preservation of spiritual identity in Sheol." The study notes that the importance of a concept is not solely determined by its frequency across texts. The most influential texts are distributed relatively uniformly across the three clusters generated by the Louvain method, suggesting that the clustering does not perfectly correlate with the influence of individual texts. Figure 1 shows the constructed network graph, and Figure 2 displays the distribution of vertex degrees.

The study's findings suggest a significant interconnectedness among Hebrew Bible texts regarding afterlife concepts, indicating a high degree of mutual influence in shaping these theological ideas. The clustering analysis reveals thematically distinct groups of texts, reflecting different perspectives on the nature of the afterlife. The consistent ranking of specific texts across various ranking methods points to the significant influence of certain books in shaping the overall discourse on the otherworld. The identified key concepts reflect an evolution of thought, moving toward an increasing emphasis on the preservation of individual personality (initially spiritual, later also physical through resurrection), alongside a growing ethical component linking afterlife rewards to righteousness and foolishness. Future research could expand the scope to include later texts (apocrypha, pseudepigrapha, Dead Sea Scrolls, New Testament, early Christian and Rabbinic literature), enabling a broader analysis of the evolution of afterlife concepts within the Jewish and early Christian traditions and tracing intertextual influence.

This study demonstrates the utility of network analysis in revealing hidden relationships and patterns within the Hebrew Bible's discourse on the afterlife. The identification of influential texts and key concepts provides valuable insights into the evolution of these theological ideas. Future research should expand the scope to include a wider range of texts and explore the identified mathematical properties of the network in more detail. The use of network analysis shows a novel approach to uncovering hidden interconnectedness and patterns in a complex religious text.

The study's interpretation of concepts is inherently subjective, relying on the researchers' understanding of the texts. The selection of the 43 texts included in the analysis might influence the results, and the inclusion of additional texts could modify the network structure and rankings. The method relies on the accuracy and consistency of the text-concept pairs identified by the researchers. While the different ranking methods show considerable convergence in identifying the most influential texts, the exact order and relative importance within the top-ranking set can vary slightly across methods. Further research is needed to investigate the robustness of these findings to changes in parameters and methodologies.