I spent a couple hours digging into this question. I'll leave it as an exercise of the reader to explore that rabbit hole; but I'll explain highlights here.

First, the relevant blog post: https://news.blizzard.com/en-us/article/24226326/reap-the-flesh-harvest#Cycle

Swiss Tournaments are frequently used in Chess Tournaments, but also other online games. A presumption of the competitors is there is a high number (more than 4) and the skills are wide. The goal of the tournament style is players complete the tournament in a final place similar to their seed spot. Ideally higher seeded players aren't knocked out too early compared to a single elimination tournament like March Madness. Also if player is mid-seeded they finish in a middle rank, even if their first battle was against a higher seeded player.

Best implementation of Swiss Tournament sees 8, 16, 32, 64… teams. But variations are available for other sized groups.

How is a Swiss Tournament set up:

1) Rank all competitors by their expected finishing place. Can be done many ways, such as finishes in previous tournaments, or others players placement of a player. Getting the exact place isn't as important as an approximation.

2) Set brackets as one team from top half, against one team from bottom half. I have seen this does in two different styles: Highest vs Lowest or Closest Match. In Highest vs Lowest, the highest ranked team is most likely going to win their first pairing. In “Closest Match”, brackets are selected in corresponding spots to ensure “closest match”. This would ensure a fairer fight in round 1.

For consistency and ease of discussion, let's assume 8 ranked players and paired “closest match”:

3) Results are given points according to tournament rules. In round 2, winners bracketed together, and then non-winners. Each bracket group is set either High v Low Rank, or Closest Match. Assume higher seeded teams end up winning.

4) After round 2, Seed 1 beat Seed 3, but Seed 4 beats Seed 2. Their ranking would be Seed 1 (2-0), Seed 4 (2-0), Seed 2 (1-1), and Seed 3 (1-1) (tie breaker on win records determined by seed position). Similar results in lower bracket: Seed 5 loses to Seed 7. Seed 6 beats Seed 8. Their final positions are Seed 6 (1-1), Seed 7 (1-1), Seed 5 (0-2) and Seed 8 (0-2).

5) To determine ultimate Tournament winner the competitors are regrouped: (2-0), (1-1), and (0-2). Special Note: Seed 2 already played Seed 6 in round 1, duplicates competitions are avoided when possible.

Tournament 1st and 2nd is determined by 2-0 group. 3rd and 4rd are winners of 1-1 groups (tiebreaker by seed), 5th & 6th are losers (tiebreaker by seed). 7th&8th is determined by the 0-2 group.

In larger tournaments, additional round brackets are set up for teams 3-0, 2-1, 1-2, 0-3 as needed; or higher as needed . The number of rounds for number of competitors is as follows:

When even pairing isn't possible, the highest seed from the next lowest group is moved up to complete brackets. In the event no one is left for pairing, the competitor is given a BYE for the round. A BYE is awarded only once to a single competitor.

Out of all transparency, at the time of this posting, I am working on assumptions, and what is posted in the Blog from Sept 23.

Assumption 1: The blog post suggests, there will be two Tournament Brackets, Top 10 Shadow Clans and Everyone else.
Clans that are ranked in the top 10 are placed in a protected matchmaking pool
Assumption 2: Brackets are set “Closest Match” to keep pairings most interesting.
Assumption 3: especially for the first week of the new Immortal Reign, Initial seeding is based on Shadow Ranking. But this could still work if seed ranking is totally random.
Assumption 4: Per Swiss Style Tournaments, when a score group has an odd number of clans, the Clan with the best Shadow rank in the next lower scoring group (1W-1L vs 2W-0L) will be moved up to complete brackets.

Facts: Clans will earn 3 points for a win, and 1 point for a loss.
Facts: Each Clan will need 30 players to register.
Facts: Registration will be automatic, and Clan Leader must intentionally turn Registration to Manual.
Facts: where possible, teams will not battle the same team multiple times.

Based on the blog post we know there will continue to be Tuesday/Thursday cadence. It’s likely that the week’s Battles will be determined by Win-Loss points, and Shadow Rank. In the first week of the Immortal Reign, Battle Brackets will have everyone. In either example of Best versus Worst, or Most Fair, you will see Clans currently rank 11-20 get put against clans with far less experience, size, and power. After the first week, we should start seeing Clans of Similar power competing in fairer and fairer matches.

Based on Assumption1, Top10 Clans will compete solely against each other for top spots. The Thursday Battle will be composed of a top5 competing against a rank 6-10, in 5 brackets. The winners of each will pair off again on Saturday, with the highest ranked clan from the loser-group filling in the brackets.

For the Top10 Brackets with 10 clans, according to the Swiss Style tournament, ultimate Ranking will be completed after the 4th round (2nd week). In other tournaments these competitors would be eliminated from that tournament, it’s possible they will be removed from the Top10 and put in to non-top10 brackets… this would allow Clans ranked 11-15 to have an opportunity to move up. This is personal opinion however, we will see how reality pans out.

Compared to the Current model of Shadow War, this is a marked improvement, as after the first few Wars clans will start competing against same-sized clans. In the current Shadow War competition many clans are playing Top10 clans weekly and getting decimated. Similarly, Top10 clans are only competing against non-Top10 until the penultimate Battle after the Challenge of the Immortal.