Attribute subspace-guided multi-scale community detection

Abstract

Community detection is designed to divide a network into multiple subnetworks (communities) with high cohesiveness, which has attracted wide attention in graph analysis. Attributes are typically served as auxiliary side information to improve the quality of community detection. In spite of their effectiveness, they suffer from two limitations: (1) existing methods usually return a single partition of the network by default, which is a critical requirement and not allowing much flexibility; (2) existing approach just globally assigns the same attribute weights to each community. We believe that community detection should be approached from the perspective of attribute subspace with different dimensional correlations. Toward this end, a novel attribute subspace-guided multi-scale community detection method (ASMS) is proposed, which can identify multi-scale communities with personalized subspaces. Specifically, ASMS can output multiple network divisions of different scales, and each subdivision has a distinctive attribute subspace that is used to reveal the inner meaning of that community formation. In particular, we devise three operators to infer the attribute subspaces. Abundant experimental results indicate that ASMS outperforms the existing methods.

Appendix

In Eq. (11), the minimization of H(w) is a constrained nonlinear optimization problem for which the solution is uncertain. For this we introduce the Lagrange multiplication to generate an unconstrained minimization problem as shown in Eq. (19).

$$\begin{aligned} \begin{array}{c} \min {{H}^{\prime }}(w,\delta )=\sum \limits _{i=1}^{C}{\sum \limits _{j=i+1}^{C}{\sum \limits _{t=1}^{F}{{{w}_{t}}}}}{{\left( {{f}_{{{v}_{i}}t}}-{{f}_{{{v}_{j}}t}} \right) }^{2}} \\ +\gamma \sum \limits _{t=1}^{F}{{{w}_{t}}}{{\log }_{2}}{{w}_{t}}-\delta \left( \sum \limits _{t=1}^{F}{{{w}_{t}}}-1 \right) \\ \end{array} \end{aligned}$$

(19)

where $\delta$ is containing the Lagrange multiplier corresponding to the constraint. Set the gradient of ${{H}^{\prime }}(\textbf{w},\delta )$ with regard to ${{w}_{t}}$ and $\delta$ to 0, Eqs. (20) and (21) can be obtained as follows:

$$\begin{aligned} \frac{\partial {{H}^{\prime }}}{\partial \delta }= & {} (\sum \limits _{t=1}^{F}{{{w}_{t}}}-1)=0 \end{aligned}$$

(20)

$$\begin{aligned} \frac{\partial {{H}^{\prime }}}{\partial {{w}_{t}}}= & {} \sum \limits _{i=1}^{C}{\sum \limits _{j=i+1}^{C}{{{\left( {{f}_{{{v}_{i}}t}}-{{f}_{{{v}_{j}}t}} \right) }^{2}}}} +\gamma \sum \limits _{t=1}^{F}{{{w}_{t}}}\left( 1+{{\log }_{2}}{{w}_{t}} \right) -\delta =0 \end{aligned}$$

(21)

Let ${{D}_{t}}=\sum \nolimits _{i=1}^{C}{\sum \nolimits _{j=i+1}^{C}{{{\left( {{f}_{{{v}_{i}}t}}-{{f}_{{{v}_{j}}t}} \right) }^{2}}}}\text { }$, so:

$$\begin{aligned} {{w}_{t}}=\exp \left( \frac{-{{D}_{t}}-\gamma +\delta }{\gamma } \right) =\exp \left( \frac{\delta -\gamma }{\gamma } \right) \times \exp \left( \frac{-{{D}_{t}}}{\gamma } \right) \end{aligned}$$

(22)

According to Eq. (21), $\sum \nolimits _{t=1}^{F}{{{w}_{t}}}=1$, thus we have:

$$\begin{aligned} \begin{array}{c} \sum \limits _{t=1}^{F}{{{w}_{t}}}=\sum \limits _{t=1}^{F}{\exp }\left( \frac{\delta -\gamma }{\gamma } \right) \times \exp \left( \frac{-{{D}_{t}}}{\gamma } \right) \\ =\exp \left( \frac{\delta -\gamma }{\gamma } \right) \times \sum \limits _{t=1}^{F}{\exp }\left( \frac{-{{D}_{t}}}{\gamma } \right) =1 \end{array} \end{aligned}$$

(23)

By Eq. (23), we have $\exp \left( \frac{\delta -\gamma }{\gamma } \right) ={1}/{\sum \nolimits _{t=1}^{F}{\exp }\left( \frac{-{{D}_{t}}}{\gamma } \right) }\;$. Substituting the above results back into Eqs. (22), (24) can be obtained:

$$\begin{aligned} {{w}_{t}}={\exp \left( \frac{-{{D}_{t}}}{\gamma } \right) }/{\sum \limits _{t=1}^{F}{\exp \left( \frac{-{{D}_{t}}}{\gamma } \right) }}\; \end{aligned}$$

(24)