We present Arianna+, a framework to design networks of ontologies for representing knowledge enabling smart homes to perform human activity recognition online. In the network, nodes are ontologies allowing for various data contextualisation, while edges are general-purpose computational procedures elaborating data. Arianna+ provides a flexible interface between the inputs and outputs of procedures and statements, which are atomic representations of ontological knowledge. Arianna+ schedules procedures on the basis of events by employing logic-based reasoning, i.e., by checking the classification of certain statements in the ontologies. Each procedure involves input and output statements that are differently contextualised in the ontologies based on specific prior knowledge. Arianna+ allows to design networks that encode data within multiple contexts and, as a reference scenario, we present a modular network based on a spatial context shared among all activities and a temporal context specialised for each activity to be recognised. In the paper, we argue that a network of small ontologies is more intelligible and has a reduced computational load than a single ontology encoding the same knowledge. Arianna+ integrates in the same architecture heterogeneous data processing techniques, which may be better suited to different contexts. Thus, we do not propose a new algorithmic approach to activity recognition, instead, we focus on the architectural aspects for accommodating logic-based and data-driven activity models in a context-oriented way. Also, we discuss how to leverage data contextualisation and reasoning for activity recognition, and to support an iterative development process driven by domain experts.