Minor Updates to rsfcNet

This adds the feature of a soft thresholding procedure for converting signed correlation networks into positive-valued networks. Some additional features are also added to eigenvector centrality that allow different normalization options. This is important if you wish to compare scores across networks.

Version 1.0

First major release

rsfcNet Public Release Preview

Version 1.0 is nearly complete. All functions should be working (though please alert me to any issues) but package documentation needs some refining (spell checks, etc).

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Introduction

Graph theory is an important and versatile approach to studying the connectivity of different brain regions and can be applied to any number of neuroimaging methods, including structural MRI, functional MRI, diffusion tensor or kurtosis imaging, EEG, and MEG. A graph is simply a matrix that quantifies the connections between individual members of some group of interest. Each member is called a node (or vertex) and the connection between each node is an edge. Such graphs can be binary, where each edge is either a 1 or a 0, or weighted, where the strength of connection can vary. Furthermore, weighted networks can be signed, as in the case of a correlation matrix. Correlations (or partial correlations) quantify a primary type of connectivity between given pair of nodes, but when considering many variables higher-order relationships can illuminate aspects of the network in question that bi-variate relationships fail to capture. However, correlations are not the only means by which one can define edges in a network.

Most graph theoretic metrics are centrality measures, which aim to identify through various methods the most important members of a network. Which measure is most appropriate depends on the type of network being characterized, the modality of being important (for example, a CEO of a company may have fewer direct connections than others within a company but nevertheless occupy a privileged and important place in the company network), and the means by which edges are represented.

Graph Theory Measures

Betweenness Centrality

Nodes with a high betweenness centrality are considered important because they comprise pathways which control information flow. Betweenness centrality shows which nodes act as "relays" or "hubs" between nodes in a network by measuring number of times a node lies on the shortest path between other nodes. Betweenness centrality is considered a more global measure of centrality because it attempts to estimate a node's importance to network integrity. Although a popular measure in resting state fMRI research, its reliance on shortest paths makes it an inappropriate measure for systems like the brain which uses a parallel distributed means of information flow (where information does not deterministically follow a shortest path).

Closeness Centrality

Closeness Centrality is a measure where each node’s importance is determined by closeness to all other nodes. Closeness is defined as the reciprocal of the sum of shortest path lengths (the distance) between node_i and all other nodes.

$$ \frac{1}{∑_{j\neq i}^N d(n_i,e_{ij})} $$

Closeness centrality estimates how fast the flow of information would be through a given node to other nodes. Nodes with high closeness centrality may have better access to information at other nodes or more direct influence on other nodes. Although it relies on the calculation of shortest paths, it is more applicable to parallel diffusion networks than Betweenness centrality. It is still apt for parallel diffusion networks because closeness centrality is measuring closeness to any other node, rather than treating the node as a junction between two nodes as in Betweenness Centrality (Borgatti, 2005).

However, see current centrality for an analagous spectral measure even more appropriate to parallel diffusion networks.

Communities ; Community Structure

See Modules.

Current Centrality (Circuit-Flow Closeness Centrality)

Circuit flow closeness centrality, or current centrality, is a method of calculating the closeness more appropriate to networks where information does not travel in a necessarily serial fashion. In an analogy to electricity flowing through a series of circuits, closeness is defined as the conductance between two nodes, which is the inverse of the resistance distance. Current centrality is then the average conductance between two nodes. Current centrality is a spectral measure calculated using the Moore-Penrose inversion of the graph Laplacian. Current centrality is also known as information centrality.

Diversity Coefficient

A measure that characterizes the degree to which a node is connected to the entire network (high Diversity coefficient) or only within a module (low Diversity coefficient). Low Diversity nodes with high within-module z scores are considered provincial hubs (important for within-module communication) and high Diversity nodes with high within-module z-scores are considered connector hubs (important for inter-module communication).

The diversity coefficient is given by the following formula:

$$ h_i = -\frac{1}{log(m)} \sum_{u=1}^{N_M} \Bigg( \left ( \frac{s_{iu}}{s_i} \right ) \cdot log \left ( \frac{s_{iu}}{s_i} \right ) \Bigg) $$

Rubinov & Sporns (2011) give a generalization of this to signed networks, where the diversity is calculated separately for positive and negative edge weights. These are then combined by the following formula, where $s_i^{-}$ and $s_i^{+}$ are respectively the strength of the negative and positive connections in the network. This is offered in the rsfcNet package.

$$ h_{i}^{*} = h_{i}^{+} - \Bigg( \frac{s_i^{-}}{s_i^{+}+s_i^{-}}\Bigg) h_{i}^{-} $$

Degree Centrality

Degree centrality, also known as just degree, is simply a count of the number of non-zero connections a node has. If the nodes in a network have weighted edges, the weighted degree, also called strength, is simply the sum of a node's edge weights. Degree and Strength are considered local measures of centrality because only a given node's immediate connections are considered.

Delta Centrality (Energy)

Delta centrality measures the change in a global property of the graph that occurs due to the deletion of a node or edge (Fornito et al, 2016). Implemented in this package is delta energy, which tracks the change in graph energy due to each of the ith nodes being deleted. See Graph Energy for more information. Also see Laplacian Centrality and Vitality, another delta centrality measure.

Neighbor centrality

Neighbor centrality is a measure of the average degree or strength of the edges of a node's neighbors. Neighbor centrality shows which nodes are connected to well connected nodes. This offers an improvement over degree since a low-degree node with connections to high degree nodes may have a central role in the network. Like leverage centrality and Laplacian centrality it considers not only the immediate environment of a node but an intermediate space between the local neighborhood and global embeddedness. However, this is conceptually distinct from leverage centrality, which defines importance as being connected to nodes with fewer connections of their own.

Eigenvector Centrality

The eigenvector centrality is the ith entry (for the ith node) in the principal eigenvector, that is, the eigenvector belonging to the largest eigenvalue of a network. Eigenvector centrality differs conceptually from degree or strength. A node with many connections does not necessarily have a high eigenvector centrality. For example, a node may have many very weak connections that yield a large value for strength/degree. Likewise, a node with high eigenvector centrality may have a low degree but be well connected to a small number of important nodes. Eigenvector centrality is a spectral measure appropriate for networks where parallel diffusion occurs.

Fiedler Value

The Fiedler value is the second smallest eigenvalue of the Laplacian representation of a graph. The closer the Fiedler value is to zero the more easily the graph can be split into separate components unconnected to each other. The Fiedler value is also known as the algebraic connectivity of a graph (Mohar, 1991). Hence the Fiedler value can be used as a measure of a network's robustness to becoming disconnected.

Graph Energy

Graph energy was originally applied in organic chemistry to quantify the stability of molecular orbitals associated with pi-electrons (Li, Shi, & Gutman, 2012). The graph energy informs about the connectivity of the graph as a whole. A graph energy of zero means the nodes are not connected at all. The Graph Energy is calculated simply by summing the absolute values of the eigenvalues of a matrix:

$$ E(G) = \sum{|\lambda_i|} $$

Laplacian Centrality

Laplacian centrality is a spectral graph theory measure and a member of the delta centrality family (where centrality is defined as the change in some graph-level measure due to the deletion of a node). Here the graph level measure of interest is the Laplacian energy of a graph, which is defined as the sum of squared eigenvalues of the graph Laplacian. This measure not only takes into account the local environment immediately around it but also the larger environment around its neighbors. It is an intermediate between metrics that assess a node's position in the whole network (such as eigenvector centrality) and the local neighborhood (such as strength).

The Laplacian energy can be calculated as the sum of the sums of squared degrees (weighted degree or binary) for each node and twice the sum of squared edge weights for each edge in a graph.

$$ E_{L}(G)=∑_{i=1}^nd_i^2+2∑_{i<j}w_{i\text{,}j}^2 $$

The Laplacian centrality for a $node_i$ is then the difference in Laplacian graph energy between the full graph and the graph where $node_i$ is deleted.

$$ \Delta E_{L} (G) = E_L (G) - E_L(G_{-node_i}) $$

Leverage Centrality

Leverage centrality defines importance as being connected to other nodes who in turn have only fewer connections.

$$ l_i = \frac{1}{k_i} \sum_{j \in N_i} \frac{k_i - k_j}{k_i + k_j} $$

It was proposed by Joyce et al (2010) and inspired by the fact that neural connections integrate information from their connections. If a single cell (or region) synapse...