Some definitions for brain connectivity analysis
Component: A component is a set of nodes, for which every pair of nodes is joined by at least one path.
Connectedness: A connected graph has only one component. A disconnected graph has at least two components.
Cycle: A cycle is a path, which links a node to itself.
Degree: The degree of a node is the sum of its incoming (afferent) and outgoing (efferent) connections. The number of afferent and efferent connections is also called the in-degree and out-degree, respectively.
Density (edge density): Proportion of edges (or arcs) existing in the network to the number of all possible edges (arcs).
Distance: The distance between a source node j and a target node i is equal to the length of the shortest path.
Distance matrix: The entries dij of the distance matrix correspond to the distance between node j and i. If no path exists, dij = ∞.
Effective connectivity: Describes the set of causal effects of one neural system over another (Sporns et al. Trends in Cognitive Sciences, 2004). Thus, unlike functional connectivity, effective connectivity is not "model-free", but requires the specification of a causal model including structural parameters. Experimentally, effective connectivity may be inferred through perturbations, or through the observation of the temporal ordering of neural events.
Exponential graph: Erdoes-Renyi random graph (Erdoes, 1960) with binomial degree distribution that can be fitted by an exponential function (Poisson distribution).
Functional connectivity: Captures patterns of deviations from statistical independence between distributed and often spatially remote neuronal units, measuring their correlation/covariance, spectral coherence or phase-locking. Functional connectivity is time-dependent (hundreds of milliseconds) and "model-free", i.e. measuring statistical interdependence (mutual information), without explicit reference to causal effects.
Graph: Graphs are a set of n nodes (vertices, points, units) and k edges (connections, arcs). Graphs may be undirected (all connections are symmetrical) or directed. Because of the polarized nature of most neural connections, we focus on directed graphs, also called digraphs. In addition, graphs are simple, that means, multiple (undirected) edges between nodes or loops (connections of one node to itself) do not exist.
Hodology: The study of pathways. The word is used in several contexts. (1) In brain physiology, it is the study of the interconnections of brain cells. (2) In philosophy, it is the study of interconnected ideas. (3) In geography, it is the study of paths.
Linear graph: Graph with many linear chains of nodes which can be detected by the clustering coefficient being lower than the density (Kaiser & Hilgetag, Physical Review E, 2004).
Path: A path is an ordered sequence of distinct connections and nodes, linking a source node j to a target node i. No connection or node is visited twice in a given path. The length of a path is equal to the number of distinct connections.
Random graph: Also called Erdoes-Renyi graph. A graph with uniform connection probabilities and a binomial degree distribution. All nodes have roughly the same degree ('single-scale').
Scale-free graph: Graph with a power-law degree distribution (Barabasi & Albert, Science, 1999). 'Scale-free' means that degrees are not grouped around one characteristic average degree (scale), but can spread over a very wide range of values, often spanning several orders of magnitude.
Small-world graph: A graph where the clustering coefficient is much higher than in a comparable random network but the characteristic path length remains about the same (Watts & Strogatz, Nature, 1998). The term small-world was coined by the notion that any two persons can be linked over few intermediate acquaintances (Milgram, Psychology Today, 1967).
Spatial graph: Graphs or networks that extent in space, that means that every node has a spatial position. Spatial graphs are usually two- or three-dimensional but more dimensions are possible (Watts, 1999).
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Replies to This Discussion
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Permalink Reply by Ingo Bojak on -
Curious that you've left out
Structural connectivity: Physical or synaptic contacts between neural units.
It seems to me that this is rather fundamental as far as actual neuroscience is concerned... -
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Permalink Reply by Saeideh Bakhshi on -
Some more network theory concepts that might be useful in brain connectivity analysis (most of them collected from BCT, sporns et al):
Shorcuts: are central edges which significantly reduce the characteristic path length in the network
The characteristic path length is the average shortest path length in the network.
The global efficiency is the average inverse shortest path length in the network.
The eccentricity of a graph vertex v in a connected graph G is the maximum graph distance between v and any other vertex u of G. For a disconnected graph, all vertices are defined to have infinite eccentricity.
Graph diameter: The maximum eccentricity is the graph diameter.
Graph radius: The minimum graph eccentricity is called the graph radius.
The assortativity coefficient is a correlation coefficient between the degrees of all nodes on two opposite ends of a link. A positive assortativity coefficient indicates that nodes tend to link to other nodes with the same or similar degree.
The global efficiency is the average of inverse shortest path length, and is inversely related to the characteristic path length.The local efficiency is the global efficiency computed on the neighborhood of the node, and is related to the clustering coefficient.
Transitivity is the ratio of 'triangles to triplets' in the network (A classical version of the clustering coefficient).
Matching index: For any two nodes u and v, the matching index computes the amount of overlap in the connection patterns of u and v. Self-connections and u-v connections are ignored. The matching index is a symmetric quantity, similar to a correlation or a dot product.
The optimal community structure is a subdivision of the network into nonoverlapping groups of nodes in a way that maximizes the number of within-group edges, and minimizes the number of between-group edges.The modularity is a statistic that quantifies the degree to which the network may be subdivided into such clearly delineated groups.
Participation coefficient is a measure of diversity of intermodular connections of individual nodes.The within-module degree z-score is a within-module version of degree centrality.
Closeness centrality: In the network theory, closeness is a sophisticated measure of centrality. It is defined as the mean geodesic distance (i.e., the shortest path) between a vertex v and all other vertices reachable from it:CC=\frac{\Sigma_{t \in V\\v}d_G(v,t)}{n-1}
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