From: Non-backtracking cycles: length spectrum theory and graph mining applications
Symbol | Definition |
---|---|
π_{1}(X,p) | The fundamental group of X with basepoint p |
[c] | Homotopy class of loop c |
\(\mathcal {L}, \mathcal{L'}\) | Length spectrum function, relaxed length spectrum |
Conv(X) | If X is a graph, Conv(X) is its 2-core |
G=(V,E) | An undirected graph with node set V and edge set E |
n,m | Number of nodes and number of edges of a graph G |
e, e^{−1} | A directed edge e=u→v and its inverse e^{−1}=v→u |
NBC | Non-backtracking cycle, in which no edge is followed by its inverse |
B | 2m×2m non-backtracking matrix of a graph |
B ^{′} | 2n×2n matrix whose eigenvalues are the same as those of B, save for ±1 |
λ_{k}=a_{k}+ib_{k} | k-th largest eigenvalue, in magnitude, of B |
Re(λ),Im(λ) | Real and imaginary parts of the complex number λ |
P,Q | n×2m directed incidence matrices of a graph |
p _{ k} | The fraction of a graphs’ nodes with degree k |
γ | The degree exponent in the case when p_{k}∼k^{−γ} |
〈k^{i}〉 | i-th moment of the degree distribution p_{k} |
deg(u) | Degree of node u |
nnz(A) | Number of non-zero elements of binary matrix A |
NBD(G,H) | Non-backtracking distance between graphs G and H |
d({λ_{k}},{μ_{k}}) | Distance between eigenvalues λ_{k} and eigenvalues μ_{k} |
r | Number of eigenvalues computed from one graph |
\(\rho = \sqrt {\lambda _{1}}\) | Magnitude threshold for eigenvalue computation |
r _{0} | Number of eigenvalues whose magnitude is greater than ρ |
σ | Spread parameter of the RBF kernel |