# Network Analysis

### Heap-Map Results

Head-maps were generated for each individual subject and global graph of both the control and test set across the three threshold values, 0.4, 0.6, and 0.8. Individual subject graphs are included in appendix A3, however the head-maps of Control Subject 4 are included as an example in Figure 4. The global graphs of the control and test set are shown in Figure 5. #### Control Set

• T = 0.4: Greater density of connections towards the frontal region was common to several of the patients. All hubs occurred at the central line or towards the frontal region. The number of hubs varied from 1 to 3 per graph. For graphs that had 3 hubs, the hubs were adjacent to one another, separated by no more than one edge. Although subjects 4 and 5 had high overall connectivity, they each contained 1 hub while subject 9, which had the fewest number of connections had 3 hubs. On the other hand, subjects 8 and 10 both had high levels of overall connectivity and each had 3 hubs. The number of hubs per graph did not appear to be affected by the graphs relative number of connections. Meaning, graphs with relatively fewer edges did not necessarily contain fewer hubs and vice versa.
• T = 0.6: Subjects 1 and 2 had the fewest number of connections and the greatest proportion of islands, but no detectable pattern in the location of the islands. The islands appeared clustered in both subjects, for subject 1, islands occurred along the outer edge at the frontal and right sides while for subject 9, the islands were concentrated at the back. Overall, connections were sparser, but consistently appeared at the frontal region for all patients. Hubs were located in the frontal region for multiple patients. The density of connections varied much more between patients than the variation that was observed for the 0.4 threshold.
• T = 0.8: All graphs except those of subjects 5, 6, and 10 were composed primarily of islands. If edges existed, they appeared mostly in the frontal region. For subjects 5, 6, and 10, islands were concentrated at the back region. The graphs of subjects 1 and 9 consisted entirely of islands. No hubs exist in the back regions, but only go as far back as the midpoint.

#### Test Set

• T = 0.4: The graphs of subjects 1 and 4 were much more sparsely connected than all other graphs, which all have about relatively high numbers of connections. Subject 1 had a concentration of connections at the frontal and right regions while subject 4 appeared to form separate clusters at the front and back left regions. Being the most densely connected, the graphs of subjects 2, 3, 8, 9, and 10, appeared to have a higher concentration of connections in the frontal region. Subjects 4, 6, and 9 were the only subjects that contained islands, but had no more than 2 islands each. All subjects except subject 9, had at least 1 hub. Subjects with hubs tended to have 1 or 2, except for subject 2 which has 3 hubs. The distance between hubs in individual graphs varied from being 1 edge, adjacent, to be separated by up to two edges.
• T = 0.6: The graphs appeared much more sparsely connected compared to the control graphs under the 0.6 threshold. The graph of subject 9 was noticeably much more connected than those of the other subjects and featured a high concentration of connections at the frontal region. The graphs that were the most sparsely connected featured dense clusters located at the frontal center region, while less dense clusters occurred in the back left or right regions. Additionally, as the threshold value increased subgraphs could be identified. Islands tended to be nodes located along the outer ring of electrodes, consistently along the front and/or right edges across all subjects. The number of hubs observed per graph ranges between 1 to 5. Graphs with multiple hubs tended to have fewer connections overall. Hubs were located adjacent to one another and were localized in the central or frontal regions.
• T = 0.8: Most graphs contained a very high percentage of islands with the graphs of subjects 4 and 8 being entirely unconnected. Subjects 1, 3, 7, and 10 consisted entirely of island and hub nodes due to the few number of edges contained in each. Subjects 5 and 9 visibly stood out for having a significantly larger number of connections compared to the rest of the set. However, the connections in subject 5 were concentrated at back right region while those of subject 6 were concentrated in the frontal center region. Subjects 1, 2, 3, 7, and 9 have hubs located at the frontal center region while subjects 5, 6, and 10 had hubs located at the back right region. ### Spacial Connectivity Patterns

The individual subject graphs featured connectivity patterns of three types: small-world, clustered, and random. While clustered graphs featured nodes that shared the most connections with their immediate neighbors, small-world graphs also featured connections between distant nodes. We defined a graph to be random if there were a high number of edges that included both small-world and clustered connections. Many of the subgraphs identified across both data sets resembled acyclic, bipartite graphs, in which every node of one set is connected to every other node in the other set. A common subgraph was the “star,” a bipartite graph where one node has a connection to every node in another set. While the graphs across the 0.4 threshold appeared mostly random, the graphs of the 0.6 and 0.8 thresholds appeared much more clustered with small-world connections appearing on an individual subject basis. As threshold value increased and graphs grew sparser, they began to exhibit small-world connections in both the control and test set. Compared to the control set at the 0.4 threshold, fewer of the test set graphs were randomly connected. In the test set at the 0.4 threshold, sparser graphs appeared clustered with either small-world connections or distinct bi-partite subgraphs. At the 0.4 threshold, the variability in the number of connections on a per test subject was much higher than for the control set. The variation in the density of connections increased as threshold value increased, making the density of connections appear less homogenous in the control set especially.

### Network Statistics

After generating graphs for every control and test patient across each threshold value, we evaluated a set of summary statistics used in network analysis across the test and control sets. The network metrics included in the results tables are defined below:

• Hub: In graph theory, a hub is defined as a node that has a number of outgoing edges that are significantly greater than the average number of outgoing edges across all nodes. To identify hubs, the average number of outgoing edges was calculated for each graph, and hubs were defined by nodes that had a number of outgoing connections that was at least 1 standard deviation above the average.
• Island: An island is a node that has no outgoing edges and is therefore unconnected from the rest of the nodes.
• Average number of edges: The number of connections per node was kept track of for each trial, stored in a 16 by 10 (number of nodes by number of trials) matrix. The sum of the matrix across the number of trials gave a vector of the total number of connections per node over the trials. To get the per-node average, we divided this vector by the number of trials. To get the number of connections for that patient, we calculated the sum of the per-node average
• Characteristic Path Length: The characteristic path length of a graph describes the median distance from a certain vertex to any other vertex in the graph. The idea of the statistic is to gauge a rough estimate of how long it takes to get from any given location in the graph to another. The equation for the characteristic path length is defined below: where G is the graph, V(G) is the set of all nodes in the graph, n is the number of nodes in G, and dG(u,v) is the distance between nodes v and w. A predefined function in the CNM Matlab toolbox was used to calculate this statistic.

• Average Clustering Coefficient: The average clustering coefficient of a graph measures the degree to which nodes in a graph cluster together. Specifically, it is the average of the local clustering coefficients of all nodes in the graph. A local clustering coefficient of a node u can be thought of as the percentage of the neighbors of u that are connected to each other as well. A predefined function in the CNM Matlab toolbox was used to calculate this statistic. For average clustering coefficient, the input to the function is the adjacency matrix of a graph, and the output is the average clustering coefficient of the graph. If nodes have a degree of 0, they are ignored, which is why ‘N/A’ (not a number) appears in some of the results.

#### Average number of edges

The average number of edges per patient graph decreased as the value of the edge threshold value increased across both the control and test data sets. The relatively low standard error indicates that the overall network connectivity is consistent within in each set of subjects. Similarly, the standard error was acceptably low for the test data set. The standard error decreased as the value of the edge threshold increased. For all thresholds, the test set had a significantly lower average number of edges compared to the control set; all values for the test set were smaller than the lowest value of the control set for a threshold value of 0.8. This indicated that the relationships among channel pairs were sparser or less active across test patients.

#### Percentage and frequency of hubs

The percentage of hubs across was relatively consistent across all thresholds for both the control and test set. The standard error was relatively low, and did not appear to correlate with differing threshold values. The percentage of hubs fell within the range of approximately 15-19% for both the control and test set. Although the percentage of hubs was largest for the 0.6 threshold for each data set, the range of the percentage of hubs was only 1.2% and 4.4% for the control and test sets, respectively. The percentage of hubs appeared to be independent of the threshold value chosen and whether or not the subject was healthy or in a coma state. Similarly, the percentage of hubs for the Global Control graph appeared to be consistently low across all thresholds. Both the Global Control and Global Test graph had almost a third of the mean percentage of hubs of each set; both of which had 6.25% hubs for threshold values of 0.4 and 0.6 and no hubs for a threshold value of 0.8. The significant decrease in the number of hubs can be attributed to the high variability of the hub nodes for each individual control subject. In other words, when taking the average across all control subjects, the connections around hubs become sparser unless a hub is common to a significant portion of the patient set. The Global Control graph had hubs at nodes 5 and 8 for threshold values 0.4 and 0.6, respectively, and no hubs for a threshold value of 0.8. For the control set with a threshold value of 0.4, node 12 appeared as a hub for 6 of the subjects, node 9 in 5 subjects, nodes 5, 6, 8, and 15, in 3 subjects and nodes 3 and 11 in 2 subjects. For the control set with a threshold value of 0.6, nodes 9 and 11 appeared as a hub for 4 of the subjects, nodes 1, 5, 12, and 13 in 3 subjects, nodes 6 and 8 in 2 subjects and nodes 7 and 10, in 1 subject each. For the control set with a threshold value of 0.8, node 11 appeared as a hub for 5 of the subjects, node 1 in 4 subjects, nodes 10, 12, and 14, in 3 subjects, nodes 5, 8, and 15 in 2 subjects and node 6 in 1 subject. The Global Test graph had hubs at nodes 11 and 8 for threshold values 0.4 and 0.6, respectively, and no hubs for a threshold value of 0.8. For the test set with a threshold value of 0.4, node 8 appeared as a hub for 8 of the subjects, node 9 in 6 subjects, node 6 in 4 subjects, node 11 in 3 subjects, nodes 4 and 12 in 2 subjects, and nodes 3, 5, 7, and 14 in 1 subject each. For the test set with a threshold value of 0.6, node 8 appeared as a hub for 6 of the subjects, nodes 5 and 12 in 4 subjects, nodes 11 and 9 in 3 subjects, nodes 4 and 6 in 2 subjects, and nodes 1, 3, 7, 10, 14 and 16 in 1 subject each. For the test set with a threshold value of 0.8, nodes 5, 8, and 13 appeared as a hub for 3 of the subjects, nodes 11 and 16 in 2 subjects, and nodes 6, 7, 9, 10, and 12 in 1 subject each.

#### Percentage and frequency of islands

The percentage of islands increased as the threshold value increased for both the control and test sets. The relatively low standard error indicates that this behavior is consistent for each set of subjects. With respect to threshold value, the number of islands increased as threshold increased; this trend was opposite to that of the average number of edges. This is due to the expected decrease in edges when using a higher threshold value, so that a greater number of channel pairs appear to be uncorrelated. The range in the percentage of hubs was 75% and 76.3% for the control and test data sets, respectively. This is a much larger range than that of the percentage of hubs. Thus, the unconnected nodes in a graph can be subject to higher variability than those that are hubs. For the Global Control graph, islands for the 0.6 threshold value, consisted of nodes 1, 3, 4, 7, 13, and 16. The lack of islands for threshold 0.4 is due to the higher connectivity expected of lower threshold graphs. For the 0.8 threshold, no islands appeared which can be attributed to the lack of common islands across the control set. In calculating the frequency of island nodes across each of the data sets, the island nodes were only listed if the graph was considered to be sufficiently connected. The criteria required that greater than half of the nodes be connected, such that graphs with an island percentage of 50% or greater were not included in our analysis. The criteria was chosen under the assumption that an island node was considered significant if it occurred in a “reasonably connected” graph. Under this requirement, all graphs under the 0.8 threshold do not have individual islands listed. For the control set under the 0.6 threshold, nodes 4, 7, and 16 occurred as an island in 3 subjects and nodes 2, 3, 5, 6, and 13 in 1 subject each. For the control set under the 0.8 threshold, all graphs except those of CP5 and CP10 had over 50% or greater islands. For the test set under the 0.4 threshold, all graphs except those of TP4 and TP9 had 50% or greater islands. For the test set under the 0.6 threshold, node 16 occurred as an island in 4 subjects, nodes 1 and 10 in 3 subjects, node 14 in 2 subjects, and nodes 2, 3, 4, 5, 11, and 15 in 1 subject each. Additionally, the Global Test graph had 50% or greater islands for both the 0.6 and 0.8 threshold and no islands under the 0.4 threshold. #### Average clustering coefficient and characteristic path length

The mean of the average clustering coefficient for the test subjects decreased as threshold value increased, however the mean for each threshold only took into account the graphs of subjects that did not entirely consist of islands, representing only 20% of the data set for the 0.8 threshold. The standard error was relatively low for both the average clustering coefficient and characteristic path length, but increased slightly as threshold value increased for both metrics. The mean of the average clustering coefficient across test subjects was 0.073 higher than that of the Global Control for the 0.4 threshold, 0.483 higher for 0.6, and not applicable to the unconnected graph of the 0.8 threshold. On average, the average clustering coefficient was higher among the test subjects than was represented by the Global Control. The mean of characteristic path length across test subjects was 0.27 lower than that of the Global Control for the 0.4 threshold, 0.12 lower for the 0.6 threshold, and not applicable for the unconnected 0.8 threshold graph. The difference in characteristic path length between the mean of the test set and that of Global Control was 3 times the standard error for the 0.4 threshold, but slightly less than the standard error for the 0.6 threshold.

The mean of average clustering coefficient among the test subjects was higher than average clustering coefficient of the Global Control. The average clustering coefficient of the Global Control exhibited higher sensitivity to threshold value than the mean average clustering coefficient of the test subjects. While the mean of the average clustering coefficient increased with threshold value, there was no trend in the mean of the characteristic path length with respect to threshold value. Overall, varying threshold value did not highlight strong distinctions between the characteristic path lengths of both data sets. Moreover, the characteristic path length was not as sensitive to the threshold value as was the average clustering coefficient.  The difference in characteristic path length between the Global Control and the mean of the test subjects for the 0.4 threshold was greater than that of the 0.6 threshold, revealing that connectivity differences are more discernable at lower thresholds, which generate more highly connected graphs on average.

#### Relationship between network metrics and spatial network observations

Differences between the data sets were more strongly highlighted by the differences in the number of islands rather than the number of hubs. However, the location of hubs was consistently in the frontal region across both subject sets, while the spatial distribution of islands was much more variable. The head-maps of test patients featured mostly clustered connections while the control subjects formed dense, but random connections. Therefore, it makes sense that the average clustering coefficient was on average higher for the test group than that of the Global Control at the 0.4 and 0.6 thresholds. In general, the average clustering coefficient was highest at the lowest threshold value which was verified by the high number of random connections observed for all graphs at the 0.4 threshold. Although the total number of connections drastically reduced as threshold increased, the average clustering coefficient did not decrease as dramatically. Even though islands comprised the majority of graphs at the threshold value of 0.8, the remaining connected nodes formed bipartite subgraphs thereby conserving the connectivity of neighbors to each other. The average characteristic path length for the control and test subjects was highest at the 0.6 threshold, which verifies the small-world characteristics observed among all subjects at the 0.6 threshold. The standard error of the average number of connections for the test subjects was much smaller than that of the control set, due to the reduced number of connections per graph and the observation that the test graphs featured less variability in the spatial density of connections.