Fig. 1

a Example of HIN composed of multiple node types, represented by diverse shapes, an multiple link types. Nodes are already grouped by shapes. b Its associated network schema composed by five nodes and twenty links. Each node corresponds to a set of nodes in the corresponding HIN. In the same way, each link is a set of links in the corresponding HIN. See for instance the paths and meta-path of length two in blue \(\blacksquare \rightarrow \blacktriangle \rightarrow \bigstar \); the blue paths are said to satisfy the blue meta-path. c Illustration of the problem statement. For each pair of nodes in (\(\blacksquare \),\(\blacklozenge \)), there is possibly a link connecting them. The link weight is approximated by a linear combination of the path-constrained random walk results, i.e., probability distributions of being at a particular node. Roughly speaking, the probabilities resulting from the random walk constrained by the target meta-path , denoted by PCRW, are expressed as a linear combination F of probabilities resulting from the random walks constrained by three different meta-paths \({\mathcal {P}}_1 = \blacksquare \rightarrow \blacksquare \rightarrow \blacklozenge \), \({\mathcal {P}}_2 = \blacksquare \rightarrow \blacktriangle \rightarrow \blacklozenge \), \({\mathcal {P}}_3 = \blacksquare \rightarrow \blacktriangle \rightarrow \bigstar \rightarrow \blacklozenge \), denoted by PCRW\(({\mathcal {P}}_1)\), PCRW\(({\mathcal {P}}_2)\) and PCRW\(({\mathcal {P}}_3)\), respectively, whose real-valued coefficients are \(\beta _{{\mathcal {P}}_1}, \beta _{{\mathcal {P}}_2}\) and \(\beta _{{\mathcal {P}}_3}\), respectively, plus a possible independent term \(\beta _0\), that is to say \(F((\blacksquare ,\blacklozenge );{\mathcal {E}}_{{\mathcal {P}}}) = \beta _0 + \sum _{{\mathcal {P}} \in {\mathcal {E}}_{{\mathcal {P}}}} \beta _{{\mathcal {P}}}\, \text {PCRW}({\mathcal {P}})\). One can see that other meta-paths exist between nodes in (\(\blacksquare \),\(\blacklozenge \)). The problem is to identify the “best” \({\mathcal {E}}_{{\mathcal {P}}}\) and a linear function F with respect to PCRW