Decentralized Networks

A solution to the acyclic graph problem has been found! This post adds to the continuing thread on modeling social organizations with Neural Networks (post 1) (post 2) (post 3)

The Dependency Problem

The issue with cyclic neural networks is dependencies. If we say Agent A factors in information from Agent B when deciding what message to transmit, but Agent B factors in messages from Agent A to make its decision, then we end up with an infinite loop of dependencies. One solution is to kickstart the system with a “dummy value” for Agent B (something like “On iteration 1, Agent B always transmits 0”), but this is clunky, difficult to perform in a library like Tensorflow, and still doesn’t mesh well with arbitrary evaluation order (for each iteration, do you evaluate A or B first?).

Instead, we can bypass the problem with a one-directional loop. The trick is as follows:

  1. Agent A0 sends a message (not dependent on B)
  2. Agent B0 receives A0’s message, and decides on a message to send
  3. Agent A1 receives B0’s message, and factors it (along with the messages A0 received) in to deciding on a message to send
  4. Agent B1 receives A1’s message, and factors it (along with the messages B0 received) in to deciding on a message to send

We have now created a dependency tree where A can rely on B’s message, and B can rely on the message generated in response, but all without creating an infinite loop. When judging the success of such a network, we look only at the outputs of A1 and B1, not their intermediate steps (A0 and B0).

If it’s absolutely essential you can create a third layer, where A2 is dependent on the message sent by B1, and so on. As you approach an infinite number of layers you get closer and closer to the original circular dependency solution, but using more than two or three layers usually slows down computation considerably without yielding significantly different results.

Social-Orgs, Revisited

With the above solution in mind, let’s re-evaluate the previous social-group problems with two layers of Agents instead of one. Each layer can send all of the data it’s received, with no communications cost or noise, to its counterpart one layer up. This effectively creates the A0 and A1 dynamic described above. When evaluating the success of the network we will look at the accuracy of environmental estimates from only the outermost layer, but will count communications costs from all layers.

Multilayer simple trial

Tada! A social organization that doesn’t revolve around A0 reading every piece of the environment itself!

Note: In the above graph, most nodes only have a 0 or 1 layer, not both. This is because the other layer of the agent does not listen to anything, and is not shown in the graph. More complex examples will include both layers more frequently.

The result is still unlikely - all information passes through A2 before reaching the agents (even A0 gets information about three environment nodes through A2) - but it’s already more balanced than previous graphs.

Next Steps

A better evaluation algorithm is needed. With the two-layer solution there is no longer a requirement for centralization - but there is no incentive for decentralization, either. A real human organization has not only total costs, but individual costs as well. Making one person do 100 units of work is not equivalent to making 10 people do 10 units of work. Therefore, we need a cost algorithm where communications become exponentially more expensive as they are added to a worker. This should make it “cheaper” to distribute labor across several workers.

Attribution

This post is based off of research I am working on at the Santa Fe Institute led by David Wolpert and Justin Grana. Future posts on this subject are related to the same research, even if they do not explicitly contain this attribution.

Posted 7/9/17