Using cognitive services 

This is similar in architecture to the connectivity aspect, with the small difference being that the application complements its abilities by using the cognitive services of the cloud. 

In order to elucidate, let us consider you are creating a survey application and you want to perform sentiment analysis of feedback submitted to the application via a simple web frontend. 

Now, to write a full fledged sentiment analysis application is no easy feat, so you decide to outsource this to the public cloud, say Google in this instance. This will be an example of using the public cloud for only the cognitive features. 

The following architecture might help reinforce our understanding: 

As we can see in the previous diagram, the application server is simply making API calls to the endpoints of a few public cloud services and augmenting its capabilities multi-fold. 

In these cases, even a private network is not required as these could be accessed securely (using HTTPs) over the internet and also through the enterprise proxies, with little to no modification. What also should be noted is the fact that it is pay per use, so we don't end up paying anything if certain features in the software are not getting used.