Functional connections between and within brain subnetworks under resting-state

The focus of this paper is on the functional role of brain regions focusing on their modular architecture and individual variability. Our main assumption is that the more variable anti-correlation patterns reflect random connections, while the more conserved ones play a functional role. Within this framework, we expanded on previous results using a different database and a different methodological approach. Aiming to identify the role of specific functional connections within a global network organization which includes subnetworks, we found that the fronto-parietal module acts as the main source of anti-correlations. In addition, the pre-frontal regions (namely: frontal middle, frontal middle orbital, frontal inferior triangular) and the parietal inferior region are highly conserved and, at the same time, act as highly connected nodes, thus confirming their importance in functional modulation.

1 Metrics and jargon used in this paper.
Basic Notation: N = set of all nodes in the network; n = total number of nodes in a network; k = node degree; l = total number of links in a network.
• Rich-club coefficient: Networks having a relatively high rich-club coefficient are characterized by many connections between nodes of high degree [1].
where E >k = number of links within nodes n >k , having node degree higher than k.
• Small-world phenomenon: It is generally defined in terms of segregation-integration balance. In the present paper segregation and integration were evaluated through local efficiency and global efficiency, respectively.
• Efficiency: It has been proposed to evaluate the information exchange within a network [2]. The concept can be applied at both local and global scales. Local efficiency = measure of the information exchange between each node and its neighbors [3]: where a ij = binary value equal to 1 if a link between i and j node does exist; d ij = shortest path length between nodes i and j. Global efficiency = measure of the information exchange across the whole network • (Dis)assortativity: Pearson correlation coefficient used to evaluate up to what extent nodes associate with other nodes sharing similar or different characters. In a disassortative network high degree nodes are connected, on average, to nodes with low(er) degree and, on average, low degree nodes are connected to high(er) degree nodes.
• Modularity: A module is a set of nodes densely connected internally. If the nodes can be grouped into potentially overlapping and highly interconnected sets, the concept allows for a coarse-grained, and thus simplified, description of the network and of its community structure. The presence of community structures in networks, including non-overlapping modules, is based upon the evaluation of the Q index [4].
• Q index: Q is defined as the normalized fraction of links in a module minus the expected number in an equivalent random network [4]: where a ij = binary value equal to 1 if a link between i and j node does exist; k i and k j = node degree of nodes i and j; m i and m j = module of i and j; δ m i m j = Kronecker delta with value 1 if m i = m j , 0 otherwise.

• Functional cartography:
Method able to extract information from the topology of a complex network by means of the z and P indexes [5].
Within module node degree (z i ): measure of the intra-module connectivity, for node i, z i spans from -∞ to +∞ and nodes with z > 2.5 are defined hubs: where: m i = module containing node i; k i (m i )= intra-module node degree of node i (the number of links between i and all other nodes in m i ); < k > (m i ) = average of the intra-module node degree of nodes within-module m i ; σ k(m i ) = standard deviation of the intra-module node degree of nodes within-module m i . Participation coefficient (P i ): measure of the inter-module connectivity, for node i, P i spans from 0 to 1, if the connections are within their own module, or distributed in all modules, respectively: where: M = set of modules; k i (m) = number of links between i and nodes in module m.
• Null statistical models by network randomization: The statistical significance of the results was checked by the straightforward method in [6] whose main advantage is the preservation, in the randomized network, of the degree distribution present in the original network.
2 Anatomical location of brain regions and modules. Color labeling is the same as in Figure 5 in the text, namely: Red = Limbic; blue = Frontoparietal; orange = Basal-ganglia; yellow = Temporo-parietal; purple = Occipital; green = DMN; light blue = thalamus. The numeric labels refer to the 90 ROIs obtained from [7].

3
Overview of the research strategy used in the paper.

Reckoning the parameters of an Adjacency Matrix.
In Figure 3 a simple network is reported (left panel), together with the corresponding Adjacency Matrix (right panel). The following sequence of operations, coded in the form of a MATLAB script, can be used to estimate the network parameters, namely the Richclub, the Local-club and the Feeder-club indexes.
• Count the total amount of links in the network: 40 links, and saved as E t .
• Find nodes having node degree > 2 (green and blue): 20 nodes, and count the number of links between them: 28 links; • Find nodes having node degree > 3 (blue): 4, and count the number of links between them: 4; • Find nodes having node degree > 4: 0; • The number of nodes and of links are saved as N rich and E rich , respectively.
• Find nodes having node degree ≤ 2 (red): 8, and count the number of links between them: 4; • Find nodes having node degree ≤ 3 (red and green): 24, and count the number of links between them: 28; • Find nodes having node degree ≤ 4 (red, green and blue): 28, and counts the number of links: 40; • The number of nodes and of links are saved as N local and E local , respectively.
The values generated by the previous sequence of operations on the basis of the network and of the Adjacency Matrix reported In Figure 3 are contained in Table 1. The MATLAB function able to generate the values is the following: [Rc Lc Fc Erich Elocal Efeeder Nrich Nlocal] = club(R, 2,4) where R is the 28 x 28 matrix of the network in the example and 2 and 4 are the minimum and the maximum node degree values in the network. The corresponding MATLAB script can be obtained from one of the authors (F.P.) upon request. node degree N rich N local E rich E local E f eeder Rc Lc  3.2 Schematics of the analytical steps used in this paper. Figure 4: Flow Chart of the computing strategy followed in this paper. The scripts indicated by asterisks or ($) in the flow-chart have been taken from https://sites.google.com/ site/bctnet/. The original script named 'home made' is reported below.