Supplemental Material 1
Topological properties of the network
Measures of functional connectivity
Connectivity degree is one of the most basic and important measures of network
analysis. The degree Ki of a node i is defined as the number of connections to that node.
Nodes with a high degree are interacting with many other nodes in the network. The degree K
of a graph G is the average of the degrees of all nodes in the graph:
𝐾=
1
∑ 𝐾𝑖
𝑁
𝑖𝜖𝐺
Density can be defined as the fraction of present connections to possible connections and
represents the total ‘wiring cost’ of the network.
Next, the weighted connectivity strength Si of each node in de network was computed
as the sum of weights (connection density) of all the connections of a node i, providing
information on the total level of weighted connectivity of a node. Formally:
𝑆𝑖 = ∑ 𝑤𝑖𝑗
𝑗𝜖𝑁
In turn, the total connection strength S of the graph was computed as the sum of Si for all
nodes.
𝑆=
1
∑ 𝑆𝑖
𝑁
𝑖𝜖𝑁
Measures of functional segregation
Measures of segregation quantify the presence of densely interconnected groups of
brain regions, known as clusters within the network. The clustering coefficient Ci is defined as
the number of existing connections among the node's neighbors divided by all their possible
connections:
𝐶𝑖 =
𝐸𝑖
𝐾𝑖 (𝐾𝑖 − 1)⁄2
where Ei is the number of existing connections among the node's neighbors. The clustering
coefficient C of a network is the average of the clustering coefficient of all nodes:
𝐶=
1
∑ 𝐶𝑖
𝑁
𝑖𝜖𝐺
Local efficiency Ei_loc of a node i is related to the clustering coefficient and can be calculated
as:
𝐸𝑖_𝑙𝑜𝑐 =
1
1
∑
𝑘𝑖 (𝑘𝑖 − 1)
𝐿𝑗,𝑘
𝑗,𝑘𝜖𝐺𝑖
where the subgraph Gi is the set of nodes that are the direct neighbors of the node i and in
which Lj,k is the smallest number of edges that must be traversed to make a connection
between node j and node k. Ei_loc reveals how much the network is fault tolerant, showing how
efficient the communication is among the first neighbors of the node i when it is removed.
The mean local efficiency of a graph is defined as:
𝐸𝑙𝑜𝑐 =
1
∑ 𝐸𝑖_𝑙𝑜𝑐
𝑁
𝑖𝜖𝐺
which is the mean of the local efficiency of all the nodes in the graph.
Measures of functional integration
Measures of functional integration characterize the ability to rapidly combine
specialized information from distributed brain regions and are commonly based on the
concept of a path, with shorter paths implying stronger potential for integration.
The mean shortest path length Li of a node i is:
𝐿𝑖 =
1
∑ 𝐿𝑖,𝑗
𝑁−1
𝑖≠𝑗𝜖𝐺
The characteristic path length L of a network is the average of the shortest path length
between the nodes:
𝐿=
1
∑ 𝐿𝑖
𝑁
𝑖𝜖𝐺
The average inverse shortest path length is a related measure known as global efficiency of a
network (Latora & Marchiori, 2001).
Small-world brain connectivity
Small-world networks are defined as networks that are significantly more clustered
than random networks, yet have approximately the same characteristic path length as random
networks (Watts & Strogatz, 1998), that is
𝛾 = 𝐶 𝑟𝑒𝑎𝑙 ⁄𝐶 𝑟𝑎𝑛𝑑 > 1
𝜆 = 𝐿𝑟𝑒𝑎𝑙 ⁄𝐶 𝑟𝑎𝑛𝑑 ≈ 1
where the 𝐿𝑟𝑒𝑎𝑙 and 𝐶 𝑟𝑒𝑎𝑙 are the characteristic path length and clustering coefficient of the real
network, the 𝐿𝑟𝑎𝑛𝑑 and 𝐶 𝑟𝑎𝑛𝑑 are the mean characteristic path length and clustering coefficient
of an equivalent random network, i.e. a random network with the same number of nodes,
edges, and degree distribution as the real network (Maslow & Sneppen, 2002; Sporns et al.,
2004). In order to obtain the values for the random network parameters in the above
equations, we applied a method estimating these random values whilst preserving the
principal characteristics of the real network. If the mean vertex degree k of a graph G is
defined as k = 2*E/N, where E is the number of edges and N is the number of vertices of G, a
random graph that is similar to G can be defined with the same mean vertex degree k and
number of vertices N as G. Watts and Strogatz (1998) defined a reliable estimate of the mean
cluster coefficient of a random graph with mean vertex degree k and number of vertices N as
Crand = k / N. This estimate can be used to normalize the actual mean cluster coefficient of G
for the calculation of ‘small world character’ of a network. Similarly, Watts and Strogatz
(1998) found an estimate of the mean path length of a random graph Lrand = ln(N)/ln(k).
Fronczak et al. (2004); however showed that this is actually an estimate of the diameter (i.e.,
the maximal shortest path length). They also derived an analytic solution for the mean path
length of a random graph with mean vertex degree k and number of vertices N as Lrand =
((ln(N) - γ) / ln(k)) + 0.5 (where γ is Euler's constant). This estimate can then be used to
normalize the actual mean path length of G with regard to calculation of the ‘small world
character’ of a network.
Measures of centrality
Measures of centrality are based on the idea that central nodes participate in many short paths
within a network and consequently act as important controls of information flow (Rubinov &
Sporns, 2010). A sensitive measure is betweenness centrality bi of a node i is defined as:
𝑏𝑖 =
1
(𝑁 − 1)(𝑁 − 2)
∑
ℎ,𝑗𝜖𝐺
ℎ≠𝑗,ℎ≠𝑖,𝑗≠𝑖
𝜌ℎ𝑗(𝑖)
𝜌ℎ𝑗
in which 𝜌ℎ𝑗 is the number of shortest paths between nodes h and j and 𝜌ℎ𝑗(𝑖) is the number of
shortest paths between nodes h and j that pass through node i.
The nodes with the largest betweenness centrality were considered to be pivotal nodes (i.e.,
hubs) in the network. Specifically, nodes were identified as hubs in the network if bi was
greater than 1.5 times the average betweenness centrality of the network (He, Chen, & Evans,
2008; Shu et al., 2009).
Supplemental Material 2
Each of the graph theory metrics have been computed across 5 connectivity threshold
levels, i.e., p = 0.0001, 0.0005, 0.001, 0.005 and 0.01. This revealed that the statistical results
of topological changes between TBI patients and controls were stable and essentially
unaffected by manipulating the network density, reinforcing the robustness of the obtained
findings.
p-values 0.0001 0.0005 0.001 0.005
Small world
Normalized clustering coefficient
gamma
Normalized path length lambda
Connectivity degree
Connectivity strength
Density
Efficiency
0.01
p>0.10
p>0.10
p>0.10
p>0.10
p>0.10 p>0.10 p>0.10
p>0.10 p>0.10 p>0.10
p>0.10
p<0.01
p<0.01
p<0.01
p<0.05
p>0.10
p<0.01
p<0.01
p<0.01
p<0.05
p>0.10
p<0.01
p<0.01
p<0.01
p<0.05
p>0.10
p<0.01
p<0.01
p<0.01
p<0.05
p>0.10
p<0.01
p<0.01
p<0.01
p<0.05
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