What do the drop-down menu options "network is reduced by" mean?

"1: prioritizing close neighbors":
This is the default. Functional links are retrieved and THEN selected by confidence. To prioritize the closest network neighbors of your query, links to them are only filtered once, despite stronger periferal links. Due to this feature, you always have an uninterrupted network path from the query to any other displayed gene.
     "retain at least 5 links per query gene":
With this option, 5 links for each queried protein will be "protected" - shown despite lower confidence.
"2: treating all links equally":
If this option is set, and the number of strongest links is limited, an earlier found link can be later removed if it is weaker than others. You may get an output where some nodes will look isolated from your query! However, you may still consider this useful, as all the nodes are relevant to the query.
"3: ARACNE algorithm":
The authors of this work noticed that in co-expression-based networks the signal, measured by gene-by-gene co-expression, often propagates isotropically, i.e. in all directions rather than down the cascade that we intuitively expect. Network graphs present it as ugly hairballs that have nothing to do with protein complex members, albeit looking so. The ARACNE algorithm takes every basic triangle (proteins A, B, and C are connected with each other), and removes the weakest one of edges {AB, BC, AC}. Even though ARACNE's assumptions (e.g. "the employed co-expression metric is exact", or "the real graph is a tree") won't hold in reality, we find it useful to reduce the number of displayed links, while ALL nodes are kept. Although, other polygons will re-appear - as ARACNE is impotent in respect of larger loops (>3 proteins in a closed chain).

Meanwhile, as you might have observed, FunCoup networks are NOT ONLY co-expression-based. Indeed, they are physical interaction-powered (add here other data sources that often support each other). You would only face unfortunate "hairballs" when you have trusted low-confidence links solely based on indirect evidence, such as co-expression, co-localization etc. To limit the network reduction in ARACNE, you can allow it only:
"4: ...in absence of physical interactions":
In this way, the above mentioned weaker links will be trusted given an observed protein-protein interaction. Its value is set to trust a single observed fact, even in another organism. This solution might make more biological sense.