Network topology | |
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Modularity |
Measures non-trivial grouping structure within a network, based on the observed number of edges within a subset of nodes, to the number of edges expected from random assignment \(\mathop \sum \limits_{k = 1}^{k} [f_{kk} \left( G \right) - f_{kk}^{*} ]^{2}\) where \(f_{kk}^{*}\) is the expected value of \(f_{kk}\) under some model of random edge assignment |
Transitivity |
Measures the extent to which nodes in a network cluster together, based on the ratio of the number of triangles and the number of connected triples \(\frac{{3\tau_{\Delta } \left( G \right)}}{{\tau_{3} \left( G \right)}}\) where \(3\tau_{\Delta } \left( G \right)\) is the number of triangles in the graph, and \(\tau_{3} \left( G \right)\) is the number of connected triples |
Density |
Measures the ratio of the number of edges in a graph to the maximum number of possible edges \(\frac{{\left| {E_{H} } \right|}}{{\left| { V_{H} } \right| \left( { \left| {V_{H} } \right| - 1 } \right)/ 2}}\) where |E| is the number of edges and |V| is the number of nodes in the graph |
Average Path Length |
Measures the mean for the shortest paths between all nodes in a network \(_{{ \frac{1}{{n \cdot \left( {n - 1} \right)}} \cdot \mathop \sum \limits_{i \ne j} d\left( { v_{i} ,v_{j} } \right)}}\) where d(v_{i}, v_{j}) is the shortest path between nodes v_{i} and v_{2}, and n is the number of nodes in the graph |
Diameter | Measures the largest distance between any pair of nodes in a network |
\(max_{u, v} d\left( {u, v} \right)\) where d(u, v) is the distance between nodes u and v |
Topic centrality | |
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Betweenness |
The fraction of shortest paths that pass through a node \(\mathop \sum \limits_{s \ne t \ne v \in V} \frac{{\sigma {(}s, t {|} v)}}{{\sigma \left( {s, t} \right)}}\) where \(\sigma {(}s, t {|} v)\) is the number of shortest paths between s and t that pass through v, and \(\sigma \left( {s, t} \right) = \mathop \sum \limits_{v} \sigma {(}s, t {|}v)\) |
Degree |
The number of edges connected to a node \(g\left( v \right) = {\text{deg}}\left( v \right)\) |
PageRank |
A measure of node importance based on the likelihood of reaching a given node when randomly following links within a network \(\alpha \mathop \sum \limits_{j} \alpha_{ij} \frac{{x_{j} }}{L\left( j \right)} + \beta\) where \(L\left( j \right) = \mathop \sum \limits_{i} a_{ij}\) is the number of neighbors of node j, and \(\alpha\) is a damping factor |