Introduction
Related work
Method
EEG coherence
Data representation and EEG coherence network
a | b | c | d | e | f | g | h | i | j | k | l | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
a | 0 |
0.65
| 0.10 | 0.10 |
0.64
|
0.60
|
0.20
| 0.10 |
0.30
|
0.23
| 0.10 | 0.10 |
b |
0.65
| 0 | 0.10 | 0.10 |
0.63
|
0.63
|
0.21
| 0.10 |
0.32
|
0.33
| 0.10 | 0.10 |
c | 0.10 | 0.10 | 0.10 | 0 | 0.10 | 0.10 | 0.10 | 0.10 | 0.10 | 0.10 | 0.10 | 0.10 |
d | 0.10 | 0.10 | 0.10 | 0 | 0.10 | 0.10 |
0.70
|
0.71
| 0.10 | 0.10 | 0.10 |
0.70
|
e |
0.64
|
0.63
| 0.10 | 0.10 | 0 |
0.62
|
0.20
| 0.10 |
0.33
|
0.20
| 0.10 | 0.10 |
f |
0.60
|
0.63
| 0.10 | 0.10 |
0.62
| 0 |
0.70
| 0.10 |
0.30
|
0.31
| 0.10 | 0.10 |
g |
0.20
|
0.21
| 0.10 |
0.70
|
0.20
|
0.70
| 0 |
0.69
|
0.20
|
0.20
| 0.10 |
0.70
|
h | 0.10 | 0.10 | 0.10 |
0.71
| 0.10 | 0.10 |
0.69
| 0 | 0.10 | 0.10 | 0.10 |
0.72
|
i |
0.30
|
0.32
| 0.10 | 0.10 |
0.33
|
0.30
|
0.20
| 0.10 | 0 |
0.32
| 0.10 | 0.10 |
j |
0.23
|
0.33
| 0.10 | 0.10 |
0.20
|
0.31
|
0.20
| 0.10 |
0.32
| 0 | 0.10 | 0.10 |
k | 0.10 | 0.10 | 0.10 | 0.10 | 0.10 | 0.10 | 0.10 | 0.10 | 0.10 | 0.10 | 0 | 0.10 |
l | 0.10 | 0.10 | 0.10 |
0.70
| 0.10 | 0.10 |
0.70
|
0.72
| 0.10 | 0.10 | 0.10 | 0 |
Community clique detection
Community structure
Community clique detection method
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Ci contains a sorted list of the vertices in community i;
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L(v) is the community label of vertex v;
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Hi contains a list of vertices (sorted by vertex number) that are connected to each of the vertices in Ci;
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Ri contains a list of vertices which have at least one Voronoi neighbor in Ci.
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Initialization (lines 1-7).-Initially, every node of the network is a singleton community (line 3).-The set of vertices which are connected to vertices of community Ci is identical to the set of vertices which are connected to the node v=V(i) (line 5)-The set of vertices which are connected Voronoi-neighbours of vertices of community Ci is the set of vertices that are connected Voronoi-neighbours of the node v=V(i) (line 6).-Set flag as true (line 8). flag is used to indicate that the modularity can be improved. If flag is false, there is no improvement of modularity.
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Main Procedure (line 8-line 37). This consists of the following steps.-Take the ith node v=V(i) from V. If after removing v from its original community CL(v) any pair of remaining vertices is not Voronoi-connected, nothing is done, and the procedure continues with a new node (line 13). Otherwise, set the maximal gain of modularity maxΔQ to zero, and take the jth community Cj.-In case Cj is empty or v is not connected to any nodes of Cj or v has no Voronoi neighbors in Cj, nothing is done, and the procedure continues with a new community (line 16). Otherwise, compute the modularity gain ΔQ (line 17).-If the current gain ΔQ is higher than maxΔQ, which means the modularity can be improved, set the label of the current community j to the destination community label d to which community the node v will move (line 21). Otherwise, nothing is done, and the procedure continues with a new community.-After all communities are traversed, select the first community which has the highest ΔQ, and update community CL(v) by removing node v from its original community CL(v) (line 26); replace HL(v) by vertices that are connected to each node of the updated community CL(v) (line 27); replace RL(v) by the vertices that are connected Voronoi neighbours of nodes of the updated community CL(v) (line 28); move node v into the destination community Cd (line 29); replace HL(v) by its intersection with the set of vertices that are connected with v in the coherence graph (line 30); replace RL(v) by its union with the vertices that are connected Voronoi neighbours of v (line 31); v receives label d (line 32).-This procedure is repeated until no improvement is obtained, which means flag=false after all nodes have been processed.
FU detection using the MCB and IWB method
MCB method
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The set compsub contains an increasing or decreasing clique.
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The set currentcand contains the candidates that are a Voronoi neighbor of at least one element in compsub, and only these can be added to compsub at the current step.
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The set complcand is the complement of currentcand in candidates containing vertices that are connected to all vertices in compsub.
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The set not contains vertices that are connected to all vertices in compsub and that were added to compsub previously.
a | b | c | d | e | f | g | h | i | j | k | l | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
a | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 |
b | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 |
c | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
d | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
e | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 |
f | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 |
g | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 |
h | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 |
i | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 0 |
j | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 |
k | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
l | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
IWB method
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Initialization. An edge queue is initialized with edges (corresponding with a significant coherence) between markers, which are defined as nodes having locally maximal average coherence, and their Voronoi neighbors. These edges are sorted in a descending order based on their values. Each marker corresponds to a basin and is assigned a unique label.
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Main Procedure. Remove the first edge, e=(v,v′), from the queue, and determine the label of node v′. In case the node v′ is unlabelled, v′ receives the label of v and the queue is extended with the edges between v′ and its unlabelled connected Voronoi-neighbours, if v′ is connected to every node of the basin where v is in. In case v′ was also labelled, check if the two basins that contain v and v′ can merge into a single basin. If so, then merge them, otherwise nothing is done.
FU visualization
Comparison of methods applied to synthetic EEG coherence networks
Results
Experimental setup
Comparison of methods applied to real EEG coherence networks
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For the young participants, it can be observed that there is no big difference between FU maps obtained from these three methods. In the [1, 3]Hz frequency band, FU maps are very similar for both young participants in terms of, for example, the number of FUs and their location. Similarly, for the [4, 7]Hz frequency band, there are no big differences between the methods either.×
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For the older participants, however, there are large differences between FU maps for the different methods. In the [4, 7]Hz frequency band, the three methods result in a similar number of FUs. In the [1, 3]Hz frequency band for participant 3 both MCB and IWB methods detect two large FUs located anteriorly and posteriorly: FUsMCB1, 5 and FUsIWB1, 3. In the CCB method, however, these FUs are split into small FUs due to a weak inter-community connection. For example, FUMCB5 splits into FUsCCB6 and 7, while FUMCB1 splits into FUsCCB1 and 2. From these splits, we can see that FUsCCB6 and 7 have higher average coherence than FUMCB5, and the inter-FU coherence between FUsCCB6 and 7 is also lower than their average coherence. This is also true for FUMCB1 and FUsCCB1, 2. From a global point of view, FUCCB7 has the highest average coherence, followed by 1 and 2, and there are higher inter-FU coherences among these FUs. For participant 4, the MCB method detects two large FUs located anteriorly and posteriorly, with significant inter-FU coherence between them. The IWB method has a similar result, except for the frontocentral connection. The CCB method, in contrast, finds a total of seven FUs with size above five. Compared to the CCB method, FU 1 obtained by the MCB and IWB methods is split into four FUs 1, 2, 3, 4 in the CCB method due to the weak inter-community connections with each other. FUCCB1 in the CCB method has the highest average coherence among these four FUs. From a global point of view, the two FUsCCB having the strongest connection are 1 and 7, which are located at the frontal and parietal-occipital areas of the brain, respectively. In the [4, 7]Hz frequency band, the MCB and IWB methods produce similar results, except in the frontocentral area of the brain. The main difference between methods is that FUMCB1 splits into FUsCCB1 and 2 in the CCB method due to weak inter-community connections. In addition, the average coherence of FUCCB1 and 2 is higher than FUMCB1. FUMCB6 is an extension of FUCCB7, but it can be seen that their average coherence differs.