  Many real-world applications give rise to large heterogeneous networks where
nodes and edges can be of any arbitrary type (e.g., user, web page, location).
Special cases of such heterogeneous graphs include homogeneous graphs,
bipartite, k-partite, signed, labeled graphs, among many others. In this work,
we generalize the notion of network motifs to heterogeneous networks. In
particular, small induced typed subgraphs called typed graphlets (heterogeneous
network motifs) are introduced and shown to be the fundamental building blocks
of complex heterogeneous networks. Typed graphlets are a powerful
generalization of the notion of graphlet (network motif) to heterogeneous
networks as they capture both the induced subgraph of interest and the types
associated with the nodes in the induced subgraph. To address this problem, we
propose a fast, parallel, and space-efficient framework for counting typed
graphlets in large networks. We discover the existence of non-trivial
combinatorial relationships between lower-order ($k-1$)-node typed graphlets
and leverage them for deriving many of the $k$-node typed graphlets in $o(1)$
constant time. Thus, we avoid explicit enumeration of those typed graphlets.
Notably, the time complexity matches the best untyped graphlet counting
algorithm. The experiments demonstrate the effectiveness of the proposed
framework in terms of runtime, space-efficiency, parallel speedup, and
scalability as it is able to handle large-scale networks.

Social and Information Networks;Discrete Mathematics;Data Structures and Algorithms;Machine Learning