Community structure in networks is often a consequence of homophily, or assortative mixing, based on some attribute of the vertices. For example, researchers may be grouped into communities corresponding to their research topic. This is possible if vertex attributes have discrete values, but many networks exhibit assortative mixing by some continuous-valued attribute, such as age or geographical location. In such cases, no discrete communities can be identified. We consider how the notion of community structure can be generalized to networks that are based on continuous-valued attributes: in general, a network may contain discrete communities which are ordered according to their attribute values. We propose a method of generating synthetic ordered networks and investigate the effect of ordered community structure on the spread of infectious diseases. We also show that community detection algorithms fail to recover community structure in ordered networks, and evaluate an alternative method using a layout algorithm to recover the ordering.