Clustering for heterogeneous information networks with extended star-structure

Abstract

Clustering of objects in a heterogeneous information network, where different types of objects are linked to each other, is an important problem in heterogeneous information network analysis. Several existing clustering approaches deal with star-structured information networks with different central-attribute relations. In real applications, homogeneous links between central objects may also be available and useful for clustering. In this paper, we propose a new approach called CluEstar for clustering of network with an extended star-structure (E-Star), which extends the classic star-structure by further including central–central relation, i.e., links between objects of the central type. In CluEstar, all objects have a ranking with respect to each cluster to reflect their within-cluster representativeness and determine the clusters of objects that they linked to. A novel objective function is proposed for clustering of E-Star network by formulating both central-attribute and central–central links in an efficient way. Results of extensive experimental studies with benchmark data sets show that the proposed approach is more favorable than existing ones for clustering of E-Star networks with high quality and good efficiency.

Appendix

Below we give the detailed derivation of updating equations of cluster assignment and ranking. With Lagrange multipliers $\varvec{\gamma }_t$ and $\{\varvec{\lambda }_h\}_{h\in {\mathcal {H}}}$, the Lagrangian is formed as

$$\begin{aligned} L=&J+\varvec{\gamma }_t^T({\mathbf {U}}_t{\mathbf {1}}-{\mathbf {1}}) +\sum _{h \in {\mathcal {H}}}\varvec{\lambda }_h^T({\mathbf {V}}^T_h{\mathbf {1}}-{\mathbf {1}}) \end{aligned}$$

(24)

According to the constraints in Eq. (2), each central object $x^t_i$ may be assigned a membership of 0 in some clusters, and in each cluster c, a number of objects of each ${\mathcal {X}}_h$ may have a ranking of 0. We assume that ${\mathcal {K}}^{t+}_{i}$ is the set of clusters where $x^t_i$ has positive memberships and ${\mathcal {N}}_c^{h+}$ is the set of objects of type $h \in {\mathcal {H}}$ that have positive rankings in cluster c, i.e.,

$$\begin{aligned} {\mathcal {K}}_i^{t+}&=\left\{ f : u^t_{if}>0\right\} \end{aligned}$$

(25)

$$\begin{aligned} {\mathcal {N}}_c^{h+}&=\left\{ l : v^h_{lc}>0\right\} \end{aligned}$$

(26)

and $|{\mathcal {K}}_i^{t+}|>1$, $|{\mathcal {N}}_c^{h+}|>1$. Based on above definition, we write the summation constraints as

$$\begin{aligned} {\mathbf {e}}^{tT}_i{\mathbf {u}}'^t_i=1 \ \ \forall \ \ i=[n_t] \ \ \text {and} \ \ {\mathbf {e}}'^{hT}_c{\mathbf {v}}^h_c=1 \ \ \forall \ \ h \in {\mathcal {H}}, c=[k] \end{aligned}$$

(27)

where

$$\begin{aligned} {\mathbf {e}}^{tT}_i=[e^t_{i1}, e^t_{i2}, \ldots , e^t_{ik} ], \quad e^t_{if}=\left\{ \begin{aligned}&1 \quad \text {for} \quad f \in {\mathcal {K}}_i^{t+}\\ {}&0 \quad \text {others}\end{aligned} \right. \end{aligned}$$

(28)

and

$$\begin{aligned} {\mathbf {e}}'^{hT}_c=[e'^h_{1c}, e'^h_{2c}, \ldots , e'^h_{n_t c} ], \quad e'^h_{jc}=\left\{ \begin{aligned}1 \quad&\text {for} \quad j \in {\mathcal {N}}_c^{h+}\\ 0 \quad&\text {others}\end{aligned} \right. \end{aligned}$$

(29)

Now we derive $\mathbf {u'}^t_{i}$, the membership vector of central object $x^t_i$ in all the k clusters. According to the first order necessary condition

$$\begin{aligned} \nabla _{\mathbf {u'}^t_{i}}L={\mathbf {g}}^t_i-\rho _h\mathbf {u'}^t_i+\gamma ^t_i{\mathbf {1}}={\mathbf {0}} \end{aligned}$$

(30)

which gives

$$\begin{aligned} \mathbf {u'}^t_{i}=\frac{1}{\rho _t}({\mathbf {g}}^t_i+\gamma ^t_i{\mathbf {1}}) \end{aligned}$$

(31)

with

$$\begin{aligned} {\mathbf {g}}_i^{t}=[g^t_{i1},g^t_{i2},\ldots , g^t_{ik}]^T=\sum _{h\in {\mathcal {H}}}\beta _{h}{\mathbf {V}}^T_h\mathbf {r'}^{h}_i \end{aligned}$$

(32)

According to Eq. (27), $\gamma _i^t$ can be calculated and substitute back into Eq. (31) to get

$$\begin{aligned} \mathbf {u'}^t_i=\frac{1}{\rho _t}{\mathbf {g}}_i^t +\frac{1}{|{\mathcal {K}}_i^{t+}|} \left( 1-\frac{1}{\rho _t}{\mathbf {e}}^T_i{\mathbf {g}}_i^t\right) {\mathbf {1}} \end{aligned}$$

(33)

For attribute object $x^p_i$, it is assigned to a cluster where the total ranking of its linked central objects is the highest, i.e.,

$$\begin{aligned} l^p_{i}=\arg \max _{c=[k]} {\mathbf {v}}^t_c\mathbf {r'}^p_i \end{aligned}$$

(34)

In a similarly way, the ranking values of all $x^h_j \in {\mathcal {X}}_h$ in cluster c can be obtained with the following rule

$$\begin{aligned} {\mathbf {v}}^h_c=\frac{1}{\eta _h}{\mathbf {z}}_c^h +\frac{1}{|{\mathcal {N}}_c^{h+}|} \left( 1-\frac{1}{\eta _h}{\mathbf {e}}'^T_c{\mathbf {z}}_c^h\right) {\mathbf {1}} \end{aligned}$$

(35)

with

$$\begin{aligned} {\mathbf {z}}_c^h=[z^h_{1c},z^h_{2c},\ldots ,z^h_{n_hc}]^T=\left\{ \begin{array}{l} \sum \limits _{\nu \in {\mathcal {H}}}\beta _{\nu }{\mathbf {R}}_{\nu }{\mathbf {u}}_c^\nu \ \ \text {for} \ \ h=t\\ \beta _{h}{\mathbf {R}}_{h}^T{\mathbf {u}}_c^t \ \ \text {for} \ \ h\in {\mathcal {A}} \end{array} \right. \end{aligned}$$

(36)

The first term of Eq. (33) decides the membership distribution of each central object in the k clusters while the second term is a normalization to ensure the summation constraint to be satisfied. Similarly, the first term in Eq. (35) decides the distribution of ranking values among objects in ${\mathcal {X}}_h$ in cluster c, and the second term ensures that the sum of rankings of objects of the same type in a cluster is 1.

The last problem left is to decide ${\mathcal {K}}^{t+}_{i}$ and ${\mathcal {N}}^{h+}_{c}$. According to the discussions in Miyamoto and Umayahara (1998) and Mei and Chen (2010), it can be proved that if $c \in K^{t+}_i$, then $\{\forall f \in K^{t+}_i| f: g^t_{if}>g^t_{ic}\}$ and if $j \in N^{h+}_c$, then $\{\forall l \in N^{h+}_c| l: z^h_{lc}>z^h_{jc}\}$. Based on this, ${\mathcal {K}}^{t+}_{i}$ and ${\mathcal {N}}^{h+}_{c}$ can be obtained in an incremental way which is similar as Procedure-K and Procedure-N given in Mei and Chen (2010).