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In a modern computer such as a smartphone, what is the relative energy used by the CPU vs the screen (and controller) and network? Clearly the answer to this question is (in so many ways) "it depends" but let's say that this is for a high end phone circa 2018 and somebody is tapping around the internet reading articles.

What's the biggest consumer of energy?

Or put another way, if we were to remove the network and the CPU from the equation, i.e., the phone is just displaying a static image, how much would this affect the battery life?

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    \$\begingroup\$ Android tells you the "Use since last full charge" for several items including display on the battery status screen. Did you check what information is available on your phone? \$\endgroup\$
    – Transistor
    Feb 11, 2018 at 23:10

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The question is actually the most important one when designing phone platform and selecting components. Fortunately there are thousands of studies on the subject, because of its importance on battery life. See, for example, this research paper, which is one of 800,000 in Google list for "smartphone power consumption analysis". The power partitioning obviously depends on dozen of parameters like screen backlight brightness, proximity of GSM receiver to cell tower, background tasks and subscription services, etc. etc. In particular case of web browsing, they got the following power partition:

enter image description here

As Transistor commented above, you can use power statistics reported by your phone under your specific conditions.

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