Table 1 Notation

G = (V, E)

Deterministic graph or backbone graph

V, E

Sets of nodes and edges

\(N = |V|,\,L = |E|\)

Numbers of nodes and edges

g = (U, F)

Motif, or connected subgraph

\({{\bar{d}}}=L/N\)

Average degree of each node

\(\mathbf{R}, {{\mathcal {R}}}, {\bar{\mathbf{R}}}\)

R-dimensional vectors of N nodes, i.e., \(N \times R\) matrix

R

Number of role patterns

\(\mathbf{C}, {{\mathcal {C}}}, {\bar{\mathbf{C}}}\)

\(N \times N\) similarity matrix

\(\mathbf{H}, {{\mathcal {H}}}, {\bar{\mathbf{H}}}\)

\(N \times K\) affiliation matrix

K

Number of role clusters

\(\varGamma (v)\)

Set of adjacent nodes of node v

\({{\mathcal {G}}} = (G, \mathbf{p})\)

Uncertain graph

p(e), p

Edge-existence probability

\(G_s = (V, E_s)\)

Sampled graph of an uncertain graph

S

Number of samples

Pr[G]

Occurrence probability of graph G

\(\delta ()\)

Kronecker delta function

\({{\mathcal {D}}}_{s,s'}\)

Set of edges that appear in \(G_s\) but not in \(G_{s'}\), and vice versa.