A page has been removed from Google due to DMCA, are its inbound links still valuable?

Okay. This will be one of those cases where I will work through the process so that it becomes clearer. It will be somewhat long, but hopefully not painfully long.

Let's start at the beginning shall we?

Starting with what we know about how Google works based originally with the research paper by Brin and Page back in 1997, we know a few things things that are very likely still in play today.

Google has a URL in it's index and fetch queue and fetches the page. The code for the page is stored within its database for various forms of processing. One of the processes would be to find new links. Any link that has been found by Google will be first located in the link index if it exists. It it does not, then the link will be added to the link table and added to the fetch queue.

Any link within the link table has at least these elements, the link URL, the source URL, and the link text. It is likely that there are other data elements, however, these do not advance the discussion. Any link that is added to the link table has a verified the source URL, but not necessarily the target URL. Using relational databases as an example, the source and target URLs could be a URL ID within the URL table and a join table would join the link table source URL and target URL elements using an ID back to the URL table. Confused? Don't be.

For any case where the target page has not been fetched, the link within the link table is said to be a dangling link. Once the page is fetched, then the link within the link table is complete. If the target page does not exist, then the link within the link table is a broken link. Simple?

Only complete links can pass value. The PageRank algorithm requires a complete link in order to calculate value. All dangling and broken links stops any calculations using the link. Previously, PR was a recursive process that would calculate link values using the link table over and over until the value that can be adjusted to any link falls within a numerical value that is so small that it is effectively not going to make a difference. I am sure this still occurs as a house keeping process. However, PR today is calculated using another method similar to hops in a network that measures distance from one page to another with relative importance. It is based upon the trust network model which is how the original PageRank model was designed to emulate. A link is a trust vote from one entity to another. While it gets more complicated than this, you get the picture. It effectively does the same thing as the recursive process using a more real-time calculation though likely less precise but precise enough to be reliable. This requires complete links since trust values (using the trust network model) cannot be passed if trust is not established. Remember that a link is a trust vote or link in the trust network model. PageRank is represented as a trust value in a trust network.

Now that you understand links and how important they are, lets move on.

For a search engine, it does not make sense to remove any URL. If a URL does not exist within the URL table, then you cannot know anything about the URL and would be at a loss. URLs are likely not deleted generally unless it makes sense to, for example, if the URL no longer exists. However, when a page is set to NOINDEX, then the search engine has been explicitly instructed NOT TO index the page. Since a web page within the index consists of two things, a URL and the HTML source code, NOINDEX effectively removes the page at this point. Links to a NOINDEX page are at least dangling.

Now that you know what an indexed page looks like, lets move further.

There are many ways that a search engine will penalize a web page or site. One is delisting. This is the most severe of all penalties and takes a long time to recover. This category of penalty you can evidence since the page will not and cannot be found. As well, Googles Search Console will, in a round about way, let you know that pages are being delisted. Of the remaining penalties, the penalties are applied in SERP filters.

When a search query is performed, there is actually several queries against the index at once that are then blended into a result set based upon a portion of the algorithm. The remaining algorithm, that we often refer to as a single entity, is a series of relatively simple SERP algorithms. The primary algorithms of which will reorder the result set based upon more real-time metrics such as trends. Of the algorithms, the ones that remove entries from the result set or seriously downgrade the placement of an entry within the result set are called filters. One which is applied is the filter that handles DMCA as evidenced with ...we have removed 1 result(s) from this page...

So now that you know how penalties are applied, are links, PR, and DMCA filters connected?

With this, we know a filter has been applied, however, this has nothing to do with the link index which is how PageRank is calculated. It is as far removed from the link/PR process as it can get. Links and PR are at the beginning of the of the indexing process whereas removing the DMCA penalized page is at the end of the query process. In fact, these are two completely separate engines. So while a page may be removed because of a DMCA complaint, it is not actually removed from the index and therefore links to and from the page are still calculated.

Clear as mud? I hope that I explained this well. Please let me know if I can clarify something for you.

[Update]

An exception that does not apply to the OP's scenario.

@StephenOstermiller brings up a good point that does not undermine the above, however, I would like to add it for completeness.

As you well know, scoring a site or page within search requires many factors. While this is not as technical or mystical as you may imagine, it is still a lot or factors to weigh. I forgot about the effect of trust scores mostly because it did not apply in the OP's case. So I am adding it here.

Clearly there are sites that are up to no good such as, spam sites. Within this classification of sites are sites that are habitual abusers of copyright content. This was a huge problem many years ago where content scrapers would build sites off of your hard work. For a long time, nothing was done. The sites with original content would lose out to the scraper sites fairly consistently. I should know. I had two PR 8 sites that lost nearly all of it's traffic due to scraper sites with absolutely no recourse.

However things have changed. And it has only largely been about four years since the significant changes have begun.

For these special classification of sites, the sites trust score can be significantly reduced. This is well known. It takes years to rebuild trust scores and for some sites, this may never happen. Why, for example, do you think domain monetizers are so willing to thoroughly trash a site with hundreds of thousands waiting in the wings for the same abuse? It is because the reality is that a domain can ruin it's value beyond redemption.

There are many factors that go into establishing trust. I will not get into that here. However, do know that trust is a major component of building rank for any site.

That said, for any site that is a serious violator of the DMCA with a fairly extensive track record, would see a serious knock in it's trust score. This is not the scenario the OP is describing. However, it is the scenario I am assuming here.

Links and establishing PageRank have more than one component. One is PageRank (authority) of the page itself. For highly authoritative pages, there is an authority cap. A PR 8 page will not share a value of 8 amongst the links on that page. This is part of the original PageRank algorithm intended to put a more natural curve into PR. Otherwise, it would be nearly impossible for a new page to compete against a page with high authority even after a long period of time. The value of the link itself is scored using several factors including the semantic value of the link text, link URL, location of the link (prominence), the semantic value of the content block that contains the link if it applies, etc. All links are scored from 0 to .9. The calculation of the authority and link score is the value passed by any link.

Well and good. So how does this effect a site that is a significant violator of the DMCA?

The value of any inbound link would not necessarily be effected by the trust score of the target site since the links value comes from the source site. However, any outbound link could be. The authority of any site that is a significant DMCA abuser would be effected by the trust score. Afterall, authority comes from trust. So in this way, the value of an inbound link would not be passed through outbound links without being degraded depending upon the trust score.

This changes the answer somewhat.

While it does not apply to the OP's scenario, there is a scenario where an inbound link value is not completely passed through the site with a DMCA violation. However, this is a hard case and therefore the threshold before this happens is significant.

HAHA I think I actually, fully, understand this question now :/

It is YOU who was served the notice, the link to YOUR site is hidden in GOOGLE search, the FILE is still on YOUR site and there a lot of OTHER LINKS pointing to YOUR site, and do THOSE links carry weight to YOUR site when showing up on GOOGLE linking to YOUR site AND the LINKS have the OMITTED notice info at the bottom of Goggle search. :P Am I right?

In light of that --

My original answers would have been great help for the people filing the DMCA takedown notices OH! -drumroll

So here is my answer: No and hell NO!

Those specific links, are basically untrusted and you should be concerned that they still link to your site. You might as well do the disavow move and protect yourself. For real.

Furthermore, it should be the other site owners that should be more concerned and why auditing your backlinks can be very important. A history of this type of association where the author is posting these type of links will have bad effects on them.

You may consider letting the other people know they have bad content links pointing to YOU and possibly hurting THEM.

Honestly, that type of content is not worth having for you or them and you should never rely on those types of links to benefit your SEO.

Short term decisions can have long term effects.

I was wondering why you kept leaking out info a little at a time.