Time‐varying nodal measures with temporal community structure: A cautionary note to avoid misinterpretation

Abstract In network neuroscience, temporal network models have gained popularity. In these models, network properties have been related to cognition and behavior. Here, we demonstrate that calculating nodal properties that are dependent on temporal community structure (such as the participation coefficient [PC]) in time‐varying contexts can potentially lead to misleading results. Specifically, with regards to the participation coefficient, increases in integration can be inferred when the opposite is occurring. Further, we present a temporal extension to the PC measure (temporal PC) that circumnavigates this problem by jointly considering all community partitions assigned to a node through time. The proposed method allows us to track a node's integration through time while adjusting for the possible changes in the community structure of the overall network.

node has the same edges, but the communities are different, entailing that the PC changes.
Considering the above, it is understandable that many previous studies have applied the PC through time while using temporal communities (e.g., Betzel, He, Rumschlag, & Sporns, 2015;Fukushima, Betzel, He, & van den Heuvel, 2018;Fukushima & Sporns, 2018;Pedersen, Omidvarnia, Jackson, Zalesky, & Walz, 2017;Rizkallah et al., 2019;Shine et al., 2016;Shine, van den Brink, Hernaus, Nieuwenhuis, & Poldrack, 2018;Tanimizu et al., 2017;Xie et al., 2018). However, a problem with the interpretation of the PC emerges when comparing two (or more) snapshots of a network with different community partitions. When community boundaries are allowed to fluctuate, the PC estimates through time are calculated relative to different community contexts. Here we argue that calculating PC per time point with a temporal community structure does not necessarily quantify its intended property of integration. As a result, the crucial link between the quantified measure and its mapping to the empirical phenomenon breaks down. Consequently, definitive conclusions cannot be drawn about what is happening in the brain.
We demonstrate this problem on toy network examples and then using resting-state fMRI data. Finally, we propose a new method-the temporal PC (TPC)-that takes into account various community partitions calculated over time.

| Data used and data accessibility
We used data from the Midnight Scan Club resting-state fMRI (Gordon et al. 2017) that is publicly available onopenneuro.org (ds000224). The data is available after both preprocessing and denoising steps have been performed (see Gordon et al. 2017 for details of these steps). The data consists of ten subjects that underwent ten resting-state fMRI runs. One subject was excluded due to substantial artefacts. We extracted time series from 200 functionally-defined parcels (Schaefer et al.2018).
F I G U R E 1 Different ways to calculate the participation coefficient (PC) through time. The PC for the shaded node is below each network/ temporal snapshot. The border of each node and the colored backgrounds show the assigned community of each node. In this example, there are two different possible edge weights. (a) Two examples of the PC, illustrating how the measure is calculated relative to the community partition. (b) An example of PC calculated when applying a static community template across multiple temporal snapshots (PC S ). (c) An example of PC calculated when applying a temporal community partition to multiple temporal snapshots (PC T ). PC values between time points cannot be directly compared due to differences in community structure obtained for each time point. (d) An example showing that changes in community partitions over snapshots may result from changes in edges that do not directly connect to the node of interest. The difference in community partition is the result of changes in the nodes in the blue community. These changes affect the PC of the shaded node despite the fact that there was no change in its connectivity

| Time-varying connectivity estimation
We estimated time-varying connectivity using weighted Pearson correlations for each time point. Weights were determined by using the Euclidean distance between all other time points. Briefly, the method creates a T × T matrix containing the distance between all nodes at each time point. At t, the corresponding row of the distance matrix becomes a weight vector for the connectivity estimates at t. The weight vector (W) is then are first converted into a similarity matrix (1-W) and then scaled between 0 and 1. This vector is then used as the weights in the covariance matrix to create a weighted Pearson estimate of each edge at each time point (see Thompson, Brantefors, & Fransson, 2017 for more details). Comparatively, the sliding window approach uses "temporally nearby" time points to support its connectivity estimates when estimating the covariance, this method uses "spatially nearby" time points to support its connectivity estimate (see  for illustrations). We selected this method of functional connectivity estimation because it performs well (second place) at tracking a fluctuating covariance through time in simulations (Thompson, Richter, Plavén-sigray, & Fransson, 2018). The method that was ranked first-the jackknife correlation-was not chosen here because it calculates a time series of "differences in connectivity" which has a different interpretation than most other methods and deemed inappropriate for the method-independent conclusions that we intend to make.
Prior to calculating the Louvain community detection and the PC, all edges below 0 were set to 0. While there are ways to calculate the PC and communities with negative edges, this is a common step that many of the cited studies we are addressing do.

| Quantifying community structure
We calculated the temporal communities using the Louvain algorithm (Blondel, Guillaume, Lambiotte, & Lefebvre, 2008) with a resolution parameter of 1. Temporal consensus clustering was performed by assigning communities at time t-1 with the same label as the community at time t that had the smallest Jaccard distance (Lancichinetti & Fortunato, 2012). We also calculated static functional connectivity using Pearson correlations and a static community partition with the same parameters as the temporal communities.

| Static PC
The PC is defined by Guimerà & Nunes Amaral, (2005) as: where i is a node index and N M is the number of communities. k is is the within-community degree, and k i is the overall degree of node i.
Here, we see temporal subscripts for the participation and the degree of the nodes. Note that, there is only one community partition used for all time points.
2.6 | The PC through time with temporal communities (PC T ) The PC with temporal communities is: Above, we are summing over the communities N mt which is the number of communities found at time point t. This method uses a community partition calculated separately per time point.

| Temporal PC
The TPC that we introduce is: Here, we have added that all temporal community partitions are considered for each time point. See the results section for the motivation behind the TPC.
We abbreviate the different PC methods as follows: "the static participation coefficient" (static PC), "the participation coefficient per time point with static communities" (PC S ), "the participation coefficient per time point with temporal communities" (PC T ), and "the temporal participation coefficient" (TPC).

| Additional network measures
We also related the different participation measures to different network estimates throughout the article to help interpret what the changes in participation means for the network. The additional measures we used were: (a) flexibility (the percentage of times where a node changes its community, Bassett et al., 2011); and (b) within-module degree z-score (z; z-score of node's degree within its community, Guimerà & Nunes Amaral, 2005). We calculated z in two ways: using static communities (z S ) and using temporal communities (z T ). We combined the different z estimates with their methodological participation counterpart (i.e., z S is used with P s ).

| Illustrating the methodological choices do not impact the conclusion
To ensure that our concerns about PC T generalize across different methods, we replicated part of the analysis (Figure 5a) by changing some methodological choices: (a) the time-varying connectivity estimation method, and (b) the community detection algorithm. As an alternative time-varying connectivity estimation method, we used the multiplication of the temporal derivative (MTD) method (Shine et al., 2015). This method multiplies the temporal derivatives of each pair of time series and applies a smoothing parameter that averages over ±4 time points.
When changing the community detection algorithm, we applied the "temporal community by trajectory clustering" method (TCTC) (Thompson et al., 2019, with parameters: ϵ: 0.5, σ: 5, τ: 5, κ: 1). Since this community detection is multilabel, we forced a single label partition by applying the Louvain algorithm (resolution parameter: 1) to drive each node into a single community. A possible objection to this criticism of PC T is that the temporal communities are calculated on the edges themselves, entailing an interconnection between the community context and edge context of a node. This objection does not adequately take into account how communities are generally calculated. Communities take into account the "global edge context" (i.e., all edges in a network and how they relate to each other) whereas the PC only considers the "local edge context" (i.e., all edges connected to one node). There is no necessary relationship between these two (exemplified in Figure 1d). A node's strength can increase with no effect on the community partition. Alternatively, a node can change its community assignment with no change to its own edges.  (Figure 2b). Note how PC T assigns increased participation, and thus interpreted as having higher integration, to the node marked in blue in Figure 2b. The difference between the network at Time 1 and Time 2 is the blue community which has split into two smaller communities. Following the interpretation that integration represents the information processing across different communities (e.g., quote by Power et al. 2013), it is hard to consider a split in community structure due to a decrease in the magnitude of edges as increased integration.

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Likewise, the temporal community of the red node has extended in the second time point (Figure 2b, Time 2). This extension of the community can be interpreted as the red node sharing similar information with more nodes due to the increase in edge weights (i.e., integration).
However, PC T will ascribe a low score and interpret less integration at A possible critique of the foregoing analysis is that the differences we highlight reflect a well-known negative relationship between PC and z and we are merely illustrating small deviations between the two methods while preserving the static relationship. However, our argument is that different time points receive substantially different interpretations between the two PC methods, and not to draw any conclusion about the overall negative relationship between the differences of PC and z.
To clarify this, we found that when difference in PC is large, then either PC T or PC S will have high participation (Figure 2d-g). Thus, we see that In sum, we have shown that there is a divergence between the methods. This divergence entails that PC S and PC T give different interpretations about a node's role in the network through time. Further, we have shown that differences in PC T across time do not map to increases in integration. Thus, given the PC's long history with being used to quantify the amount of integration in the brain, using PC T will lead to misinterpreting results. Importantly, PC S does not suffer from this problem as it uses the same reference community. However, the temporal community information is discarded, leading us to our possible modification of PC to allow for temporal communities.

| PC in relation to all temporal communities
Given the substantial evidence for temporal changes in community structure, there is an understandable desire to calculate the PC with fluctuating communities. We present a possible solution to the problem outlined above: the TPC. The crux of the problem is that the PC of a node is relative to the community partition. If instead, each PC estimate considers all possible community partitions that the node has been assigned, then the multiple PC estimate will be comparable across time points as each estimate is now relative to the same community context ( Figure 3). Specifically, TPC considers how a node is participating relative to all possible community structure it can have. In Figure 3, each TPC estimate is calculated relative to both community contexts and then averaged, entailing that the shaded node at the second time point has more participation compared to the first time point (contrast to PC T in Figure 1). As both time points are considered relative to both community contexts, these values can now be compared.
The motivation behind TPC is to have all time points compared to the same community context (like PC S ) but to retain the temporal information inherent in community fluctuations (used in PC T ). Perhaps this logic seems counter-intuitive as it entails using community partitions that do not fully reflect the snapshot. However, this is also the same logic as PC S , as it applies a static community partition to each time point that does not fully reflect the community structure at each snapshot. The only difference here is that TPC utilizes multiple community partitions instead of a single partition.
TPC quantifies a node's activity in relation to all communities that it could potentially be in. Consequently, if a single large edge at t 1 merges two communities and, at t 2 , only that edge decreases such that the community splits, TPC will always assign higher participation to t 1 . Since all time points use the same community information, it is mathematically impossible for any time point that decreases all of its edge weights to get higher participation-this guarantee is not possible for PC T . Thus, TPC can utilize all community contexts but avoids the problematic applications shown for PC T in the previous section.
There is one crucial assumption when applying this method, namely that the community structure can recur again through time.
This assumption means that the same or similar community partitions will be found at later time points. In a network like the brain, this is reasonable, and it is a basic assumption that is made in all psychological experiments where the task is repeated (see discussion for when F I G U R E 3 The temporal participation coefficient (TPC). To calculate the TPC, PC is calculated per time point, taking into account all possible community contexts the node can be in. PC value calculated for each possible temporal community context is shown under each network. These are then averaged together. This fix makes PC of the shaded node comparable across time points as they have been compared against the same community contexts this assumption breaks down). Without this assumption, it becomes unfair to use earlier or later community partitions to analyze a node's PC in a snapshot as if it was in another community.

| Nodes with high static PC change communities the most
We have presented a theoretical problem, and illustrated how it could lead to differences in interpretation. We have also proposed a fix that does not suffer from interpretation problems. However, it could be the case that there is no difference when applying TPC versus PC T .
We begin by asking the question: how are nodes with high static PC affected by the temporal community partitions? If nodes with high participation have little change in their community context, the problem we raise may be redundant. To clarify this, we compared the flexibility with the static PC (Figure 4a,b). Here, we see that nodes with high participation also increase their flexibility. If nodes with high participation always remained in the same communities, calculating PC with temporal communities would be less problematic. Nodes that switch temporal communities have high static PC. We have not proven which participation method should be preferred here. However, this relationship that the changes in communities are affecting nodes with high participation which will help us understand any differences between participation methods. The correlation between PC time series does not mean that the same nodes or time points will be candidate hubs. For each method, we then identified the highest 5, 10, and 20% of values for both for the top time points for each node ( Figure 5d) and when concatenating across all nodes (Figure 5e). If we try and find when each node has its highest participation, we find that PC T has more unique nodes. When pooling all nodes and time points together, the overlap of all three methods reached over 60% with larger thresholds, but was under 40% for lower thresholds. Finally, we also observed that the TPC and PC S overlapped the most (reaching 80% of nodes in some instances and always over 60% when combining the paired and triple intersections). This overlap is reassuring for TPC as we know PC S is a valid method. Moreover, the divergence that happens between the TPC and PC S with static communities is due to the TPC utilizing the temporal community information.  Hence, interpretations of PC T being a measure of integration when comparing multiple time points are, in our opinion, ill-advised. However, averaging over time points is possible for PC T , which multiple studies have done (e.g., in Shine et al. (2016)). This strategy is possible because it is no longer comparing time points with different communities which is unproblematic (but looses the temporal resolution of the PC). In sum, we feel that any quantification of fluctuations of participation through time should use PC S or TPC.

| DISCUSSION
We are not challenging PC T in all use cases; it is mathematically sound when applied to each time point. The problem arises when contrasting values from different time points that are derived on different community vectors. If two snapshots are contrasted with PC T and presented with their respective community differences, then each estimate can be useful for understanding each snapshot (e.g., Figure 1d) which can facilitate understanding about the network.
However, multiple PC T estimates cannot be directly contrasted unless you alter the meaning of integration.
The solution we present, TPC, does not fit all possible use cases.
One limitation is that it can only be applied when the network can return to previous states (the recurrence assumption). Some temporal communities may only be possible after certain events have transpired-for example, during a contagious outbreak, patients (nodes) could form temporal communities in the hospital. Using our proposed TPC on such a dataset would entail that postinfection communities influence preinfection participation estimates, which would be unrealistic. Furthermore, care would also be needed for any of the participation methods if a network bifurcates its community structure between entirely different states that have little or no topographic overlap. Thus, the proposed solution only covers networks which can theoretically return to similar states. This assumption appears reasonable for networks such as the brain. However, quantifying variations in how nodes relate to their community assignments (e.g., Bassett et al., 2011) or using time-varying measures with static communities (e.g., PC S ) may be more prudent analysis alternatives. The ultimate lesson here is that network measures need to be chosen based on the knowledge about the system under investigation and the new information that the measures hope to attain.
Here, the focus has been on temporal communities and its recent application within network neuroscience. However, this can also be a more general warning for such nodal measures that are relative to the community structure when applied in multilayer cases. For example, if calculating PC per task when each task has its own community partition will suffer the same interpretability problems. We hope that this article highlights the problematic nature of quantifying temporal nodal measures relative to a fluctuating temporal community partition. We have offered one possible solution for this problem that utilizes temporal community information that does not suffer from similar issues regarding interpretation.