While describing natural complex dynamical systems is a notoriously difficult endeavor, the network approach (Boccaletti et al., 2006; 2014; Newman, 2018) has been repeatedly shown to provide novel and important insights into such systems in various research areas ranging from neurosciences (Bullmore and Sporns, 2009; Lehnertz et al., 2014) via genomics (Tyler et al., 2009) and proteomics (Uetz et al., 2000) to ecology (Hegland et al., 2009; Olesen et al., 2011; Delmas et al., 2019; Halekotte and Feudel, 2020), climatology (Donges et al., 2009; Zhou et al., 2015), and sociology (Onnela et al., 2007; Palla et al., 2007). This broad applicability is not least explained by the large manifold of network metrics, describing global aspects to local aspects in network terms, which in principle can be directly related to the properties of the described system. Identifying key network constituents is highly relevant when it comes to improving the understanding and control of networks, as it allows us to gain insights about the importance hierarchy of its constituents with respect to the network structure and dynamics. The characterization of a constituent’s role in the network structure and dynamics can be achieved through different concepts and a growing number of metrics such as centralities. Most of these concepts focus on the description of vertices or groups of such [e.g., hubs (Newman, 2003), hub regions in the brain (Stanley et al., 2013; Chung, 2019), and k-core decompositions (Kong et al., 2019)], while only a few metrics assess the centrality of an edge. We recently proposed a novel strength-based edge-centrality concept (Bröhl and Lehnertz, 2022) and introduced modifications to vertex closeness and vertex eigenvector centrality concepts yielding corresponding edge centrality concepts (Bröhl and Lehnertz, 2019). We demonstrated that these edge centralities—together with edge betweenness centrality (Freeman, 1977; Girvan and Newman, 2002)—provide additional information about the network constituents for various topologies. These four centrality concepts, while different in their definition, can be considered complementary in the description of a constituent’s structural integration in the network.

Many early studies assumed networks to be static; however, the recent paradigm shift toward time-dependent (or evolving) networks (Holme and Saramäki, 2012; Kivelä et al., 2014) allows one to describe many systems more accurately. This particularly holds true for biological networks, such as the brain, for which time dependencies on different scales have been observed. Both exogenous and endogenous biological rhythms are expected to assert influences on the level of the network description (Kuhnert et al., 2010; Lehnertz et al., 2017; Mitsis et al., 2020; Kurth et al., 2021; Lehnertz et al., 2021) and therefore on network metrics such as centralities (Geier et al., 2015; Geier and Lehnertz, 2017; Lehnertz et al., 2017; Lehnertz et al., 2021).

For more than 50 years, it has been known that the circadian rhythm and ultradian rhythms impact electroencephalographic (EEG) signals [see Lehnertz et al. (2021) for a recent overview]. Many former studies, however, were based on EEG recordings that either assessed the dynamics of few brain regions only or/and covered timescales ranging only from few seconds to hours. Here, we extend the recent studies and observations (Spoormaker et al., 2011; Chu et al., 2012; Park et al., 2012; Liu et al., 2015; Farahani et al., 2021) and investigate how biological rhythms, particularly the circadian rhythm (with a period length of approximately 24 h), influence the importance hierarchies of the constituents of evolving functional brain networks. Therefore, we focus on both the networks’ vertices that are associated with the sampled brain regions and networks’ edges that represent time-evolving interactions between brain regions.

We observe contributions of rhythms with period lengths of approximately 24 h (and to a lesser extent from rhythms of approximately 12 h or shorter) in all temporal sequences of importance (centrality values and ranks) of the respective network constituents from each subject. However, and contrary to the expectation, we found that these circadian and ultradian contributions are more pronounced in the sequences of some vertices and edges, i.e., some brain regions, as well as the interactions between their dynamics (see Figure 1).

Interestingly, we also observe that the strength of circadian contributions differs for temporal sequences derived with different centrality metrics. This is to be expected, at least to some extent, since the metrics highlight different structural aspects of a network, such as the path-structure or strength distribution. Nevertheless, these observations suggest that the circadian rhythm affects these different structural aspects. Figure 2 demonstrates that this rather unspecific relation regarding path/strength-based centrality metrics for vertices and edges can be observed in the data from all the investigated subjects. Moreover, it becomes quite apparent from this figure that by combining the results yielded by different centrality concepts, almost all network constituents are impacted by the circadian rhythm in all subjects.

Yet, it can also be observed that there is no trivial relation between the influence of the circadian rhythm (estimated with P ₂₄) and a constituent’s importance (cf. Figure 3). Neither for the most nor for the least important network constituents do we observe a generally specific influence of the circadian rhythm (as well as for ultradian rhythms (data not shown)).

Furthermore, we find that different centrality metrics identify different constituents as the most important (on average over the whole observation time, in line with previous studies) (see, e.g., Kuhnert et al. (2012); Bröhl and Lehnertz (2019); Bröhl and Lehnertz (2022); and the references therein). One needs to take into account that the constituents deemed the most important on average do not always coincide with the constituents that are deemed the most important for the largest fraction of the recording time (cf. Figure 4). Likewise, constituents, for which the temporal profiles of importance are impacted strongly by the circadian rhythm, neither coincide with those constituents that are deemed the most important on average nor with those constituents that are deemed the most important for most of the recording time. This discrepancy cannot be traced back to the ceiling or floor effects, resulting from the definitions of their respective centrality metrics. Overall, we observe a rather unspecific influence of primarily the circadian rhythm on many structural aspects of network constituents: each brain region (vertex) and even interactions between such regions (edges) appear to be influenced, at least to some extent, in their structural integration.

In order to improve on the findings achieved so far, we investigate whether there exists a day/night pattern in the temporal evolution of the importance of vertices and edges (cf. Figures 1A, C). Interestingly, we observe that the largest differences in the importance of network constituents between night- and daytimes are related to very distinct brain areas along with their interactions (see Figure 5). These vertices and edges not only exhibit the largest change in centrality values when functional brain networks transit from night- to daytimes but are also further identified as the most important constituent on average and for the largest fraction of the total recording time. While betweenness centrality highlights bilateral frontotemporal vertices and edges, closeness centrality highlights the predominantly left temporoparietal vertices and edges. Both strength-based centralities (eigenvector centrality and nearest-neighbor centrality) predominantly highlight the left temporoparietal and left parietooccipital vertices and edges. Independent of the employed centrality metric, temporoparietal network constituents (vertices T7, T8, and P7 as well as their associated edges) are identified as the most important during daytime (here, 12:00 to 16:00 h). In contrast, during nighttime (here, 24:00 to 4:00 h), the importance shifts to parietal network constituents (vertices P3, P4, and PZ as well as their associated edges). Apart from this night–daytime-related spatial shift of the importance of a few network components, our findings also point toward a key arrangement of connected vertices and edges, relating to a subnetwork that comprises vertices T7, P7, P3 PZ, and P4 together with their associated edges. This subnetwork, which is slightly more dominantly located on the left brain hemisphere, is possibly involved in ongoing activities during wakefulness and sleep.

We investigated how the circadian rhythm impacts the time-dependent importance hierarchy of the vertices and edges of an evolving functional brain network. We employed different path- and strength-based centrality metrics for vertices and edges to comprehensively characterize the importance hierarchy of these network constituents. At the single-constituent level, we observed that their time-dependent changes in importance are clearly modulated by the circadian rhythm although to a varying degree. Irrespective of their interindividual variabilities, we observed pronounced differences in the constituents’ importance hierarchy when contrasting data obtained during daytimes and nighttimes. This may point to a local, circadian rhythm-driven modulation of the dynamics of various brain regions alongside their interactions. These brain regions form vital and fundamental subnetworks within the evolving functional brain networks.

The subnetwork highlighted with betweenness centrality comprises temporofrontal brain regions from both hemispheres. It is rather unexpected that the subnetworks, as highlighted with closeness, eigenvector, and strength/nearest-neighbor centrality metrics (but not with betweenness centrality), are largely overlapping, despite the fact that the different centrality metrics assess different structural aspects of network constituents. This subnetwork is predominantly restricted to the temporoparietal brain regions, with a left-hemispheric dominance during the nighttime. However, whether the subnetworks observed here are related to the resting-state network needs further investigation (Raichle, 2015).

Studies revealed that the hippocampus, deep inside the temporal lobe, and the visual cortex are simultaneously involved in the reactivation of coherent memory traces during sleep, which points toward a contribution to the memory consolidation process (Prabhakaran et al., 2000; Albouy et al., 2013). The interaction between those brain regions might possibly relate to the T7–P7 (–T8) structure in the observed vital subnetworks as these vertices and edges, associated with these electrodes and interactions between the sampled brain regions, are deemed more important in general and for a larger fraction of time during the nighttime compared to the daytime.

During the daytime, the vertices and edges associated with the parietal lobes (PZ, P3, and P4) are deemed more important and for a larger fraction of time compared to the nighttime. These areas consolidate spatial and visual information and integrate perceptions with other sensory inputs, resulting in the recognition of the trajectories of moving objects. These areas also mediate proprioception (perception of the position of the body in space) and are involved in skills such as arithmetic, writing, left–right orientation, and finger perception (see Rizzolatti et al. (1997) for an overview). Since these functions may also be involved during dream phases (which account for approximately 25% of the sleep period), these vertices and edges are nonetheless important during nighttimes although for shorter periods of time.

Overall, we observe that circadian (and ultradian) biological rhythms strongly influence the importance hierarchy, as assessed with different centrality concepts, of the constituents in time-dependent functional brain networks. These observations highlight, for each employed centrality concept, distinct subnetworks in evolving functional brain networks. The structural composition of these networks, however, largely coincides, which points toward the existence of a vital and fundamental subnetwork that is rather generally involved in ongoing brain activities.