No, you cannot convert a Cypher to Big O notation.
Cypher does not describe how to fetch information, only what kind of information you want to return. It is up to the Cypher planner in the Neo4j database to convert a Cypher into an executable query (using heuristics about what info it has to find, what indexes are available to it, and internal statistics about the dataset being queried. So simply changing the state of the database can change the complexity of a Cypher.)
A very simple example of this is the Cypher Cypher 3.1 MATCH (a{id:1})-[*0..25]->(b) RETURN DISTINCT b
. Using a fairly average connected graph with cycles, running against Neo4j 3.1.1 will time out for being too complex (Because the planner tries to find all paths, even though it doesn't need that redundant information), while Neo4j 3.2.3 will return very quickly (Because the Planner recognizes it only needs to do a graph scan like depth first search to find all connected nodes).
Side note, you can argue for BIG O notation on the return results. For example MATCH (a), (b)
must have a minimum complexity of n^2 because the result is a Cartesian product, and execution can't be less complex then the answer. This understanding of how complexity affects row counts can help you write Cyphers that reduce the amount of work the Planner ends up planning.
For example, using WITH COLLECT(n) as data MATCH (c:M)
to reduce the number of rows the Planner ends up doing work against before the next part of a Cypher from nm (first match count times second match count) to m (1 times second match count).
However, since Cypher makes no promises about how data is found, there is no way to guarantee the complexity of the execution. We can only try to write Cyphers that are more likely to get an optimal execution plan, and use EXPLAIN/PROFILE to evaluate if the planner is able to find a relatively optimal solution.