Trophic coherence, a measure of a graph’s hierarchical organisation, has been shown to be linked to a graph’s structural and dynamical aspects such as cyclicity, stability and normality. Trophic levels of vertices can reveal their functional properties, partition and rank the vertices accordingly. Trophic levels and hence trophic coherence can only be defined on graphs with basal vertices, i.e. vertices with zero in-degree. Consequently, trophic analysis of graphs had been restricted until now. In this paper we introduce a hierarchical framework which can be defined on any simple graph. Within this general framework, we develop several metrics: hierarchical levels, a generalisation of the notion of trophic levels, influence centrality, a measure of a vertex’s ability to influence dynamics, and democracy coefficient, a measure of overall feedback in the system. We discuss how our generalisation relates to previous attempts and what new insights are illuminated on the topological and dynamical aspects of graphs. Finally, we show how the hierarchical structure of a network relates to the incidence rate in a SIS epidemic model and the economic insights we can gain through it.
Trophic coherence, a measure of a graph’s hierarchical organisation, has been shown to be linked to a graph’s structural and dynamical aspects such as cyclicity, stability and normality. Trophic levels of vertices can reveal their functional properties, partition and rank the vertices accordingly. Trophic levels and hence trophic coherence can only be defined on graphs with basal vertices, i.e. vertices with zero in-degree. Consequently, trophic analysis of graphs had been restricted until now. In this paper we introduce a hierarchical framework which can be defined on any simple graph. Within this general framework, we develop several metrics: hierarchical levels, a generalisation of the notion of trophic levels, influence centrality, a measure of a vertex’s ability to influence dynamics, and democracy coefficient, a measure of overall feedback in the system. We discuss how our generalisation relates to previous attempts and what new insights are illuminated on the topological and dynamical aspects of graphs. Finally, we show how the hierarchical structure of a network relates to the incidence rate in a SIS epidemic model.
Trophic analysis exposes the underlying hierarchies present in large complex systems. This allows one to use data to diagnose the sources, propagation paths, and basins of influence of shocks or information among variables or agents, which may be utilised to analyse dynamics in social, economic and historical data sets. Often, the analysis of static networks provides an aggregated picture of a dynamical process and explicit temporal information is typically missing or incomplete. Yet, for many networks, particularly historical ones, temporal information is often implicit, for example in the direction of edges in a network. In this paper, we show that the application of trophic analysis allows one to use the network structure to infer temporal information. We demonstrate this on a sociohistorical network derived from the study of hadith, which are narratives about the Prophet Muhammad’s actions and sayings that cite the people that transmitted the narratives from one generation to the next before they were systematically written down. We corroborate the results of the trophic analysis with a partially specified time labelling of a subset of the transmitters. The results correlate in a manner consistent with an observed history of information transmission flowing through the network. Thus, we show that one may reconstruct a temporal structure for a complex network in which information diffuses from one agent to another via social links and thus allows for the reconstruction of an event based temporal network from an aggregated static snapshot. Our paper demonstrates the utility of trophic analysis in revealing novel information from hierarchical structure, thus showing its potential for probing complex systems, particularly those with an inherent asymmetry.
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