The bigraph theory is a relatively young, yet formally rigorous, mathematical framework encompassing Robin Milner's previous work on process calculi, on the one hand, and provides a generic meta-model for complex systems such as multi-agent systems, on the other. A bigraph $F = \langle F^P, F^L\rangle$ is a superposition of two independent graph structures comprising a place graph $F^P$ (i.e., a forest) and a link graph $F^L$ (i.e., a hypergraph), sharing the same node set, to express locality and communication of processes independently from each other. In this paper, we take some preparatory steps towards an algorithm for generating random bigraphs with preferential attachment feature w.r.t. $F^P$ and assortative (disassortative) linkage pattern w.r.t. $F^L$. We employ parameters allowing one to fine-tune the characteristics of the generated bigraph structures. To study the pattern formation properties of our algorithmic model, we analyze several metrics from graph theory based on artificially created bigraphs under different configurations. Bigraphs provide a quite useful and expressive semantic for process calculi for mobile and global ubiquitous computing. So far, this subject has not received attention in the bigraph-related scientific literature. However, artificial models may be particularly useful for simulation and evaluation of real-world applications in ubiquitous systems necessitating random structures.