I see. No, the household is literal in this case. The head of the family would be what you're describing; they live elsewhere, in the leading bloodline's villa outside the city.

[Although Kratos has never met him, blood father or not.]

The subsistence bloodlines live separately, so the households need someone who heads it on the main family's behalf. That would be Nan.

[Who is now visible in the first courtyard, in fact, with a hat on her head while bent over the herb garden the staff keeps in the lead building. The other two are for training. She straightens when she hears their footsteps, dark hair faintly greying around the temples, but eyes still sharp. They soften when they see Kratos, narrow when they see Yuan.]

Zelus wasn't carrying tales, I see.

Good day, Nan.

[Kratos meets her at the edge of the garden to kiss her cheek. She humphs, but it's fond.]

Those had better not be amphoras of wine for the two of you.