Table 5 Dense formulation of mean-weighted functional motif adjacency matrices
From: Motif-based spectral clustering of weighted directed networks
Motif | C | C ′ | M func |
|---|---|---|---|
\(\mathcal {M}_{\mathrm {s}}\) | G+GT | ||
\(\mathcal {M}_{\mathrm {d}}\) | \(\frac {1}{2} G_{\mathrm {d}}\) | ||
\(\mathcal {M}_{1}\) | JT∘(JG)+JT∘(GJ)+GT∘(JJ) | \(\frac {1}{3} \left (C + C^{\mathsf {T}}\right)\) | |
\(\mathcal {M}_{2}\) | JT∘(JdG)+JT∘(GdJ)+GT∘(JdJ)+JT∘(JGd)+JT∘(GJd)+GT∘(JJd)+Jd∘(JG)+Jd∘(GJ)+Gd∘(JJ) | \(\frac {1}{4} \left (C + C^{\mathsf {T}}\right)\) | |
\(\mathcal {M}_{3}\) | J∘(JdGd)+J∘(GdJd)+G∘(JdJd)+Jd∘(JdG)+Jd∘(GdJ)+Gd∘(JdJ)+Jd∘(JGd)+Jd∘(GJd)+Gd∘(JJd) | \(\frac {1}{5} \left (C + C^{\mathsf {T}}\right)\) | |
\(\mathcal {M}_{4}\) | Jd∘(JdGd)+Jd∘(GdJd)+Gd∘(JdJd) | \( \frac {1}{6} C\) | |
\(\mathcal {M}_{5}\) | J∘(JG)+J∘(GJ)+G∘(JJ)+J∘(JGT)+J∘(GJT)+G∘(JJT)+J∘(JTG)+J∘(GTJ)+G∘(JTJ) | \(\frac {1}{3} \left (C + C^{\mathsf {T}}\right)\) | |
\(\mathcal {M}_{6}\) | J∘(JGd)+J∘(GJd)+G∘(JJd)+Jd∘(JTG) | Gd∘(JTJ) | \(\frac {1}{4} \left (C + C^{\mathsf {T}} + C' \right)\) |
\(\mathcal {M}_{7}\) | J∘(JdG)+J∘(GdJ)+G∘(JdJ) | Jd∘(JGT)+Jd∘(GJT)+Gd∘(JJT) | \( \frac {1}{4} \left (C + C^{\mathsf {T}} + C' \right)\) |
\(\mathcal {M}_{8}\) | J∘(GJn)+G∘(JJn) | Jn∘(JTG)+Jn∘(GTJ) | \(\frac {1}{2} \left (C + C^{\mathsf {T}} + C' \right)\) |
\(\mathcal {M}_{9}\) | J∘(JnGT)+G∘(JnJT)+Jn∘(JG)+Jn∘(GJ)+J∘(GTJn)+G∘(JTJn) | \(\frac {1}{2} \left (C + C^{\mathsf {T}}\right)\) | |
\(\mathcal {M}_{10}\) | J∘(JnG)+G∘(JnJ) | Jn∘(JGT)+Jn∘(GJT) | \(\frac {1}{2} \left (C + C^{\mathsf {T}} + C' \right)\) |
\(\mathcal {M}_{11}\) | Jd∘(GJn)+Gd∘(JJn)+Jn∘(JdG)+Jn∘(GdJ)+J∘(GdJn)+G∘(JdJn) | \(\frac {1}{3} \left (C + C^{\mathsf {T}}\right)\) | |
\(\mathcal {M}_{12}\) | Jd∘(JnG)+Gd∘(JnJ)+Jn∘(JGd)+Jn∘(GJd)+J∘(JnGd)+G∘(JnJd) | \( \frac {1}{3} \left (C + C^{\mathsf {T}}\right)\) | |
\(\mathcal {M}_{13}\) | Jd∘(GdJn)+Gd∘(JdJn)+Jn∘(JdGd) | \(\frac {1}{4} \left (C + C^{\mathsf {T}} \right)\) | |
\(\mathcal {M}_{\text {coll}}\) | Jn∘(JGT) | \(\frac {1}{2} \left (C + C^{\mathsf {T}} \right)\) | |
\(\mathcal {M}_{\text {expa}}\) | Jn∘(JTG) | \(\frac {1}{2} \left (C + C^{\mathsf {T}} \right)\) |