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I am creating a circular neural network with N nodes. Each node is connected via a send pathway to every other node, and the connection between two nodes has a weight. Any number sent over the connection deteriorates linearly with the weight.

When a value is received by a node, that node takes the value and transmits it along its send paths to the other nodes in the network. Eventually the values deteriorate (at an infinite series). The aim is to find the sum of all the values that pass through a node. Each node takes inputs, which are not added to the sum of the values it has received, but are simply propagated along the network.

(I apologize in advance. I do not know the math syntax. If someone would be so kind as to edit my post, I will review the edits, and learn the syntax.)

These weights can be expressed in a two dimensional matrix. In my example, there are five nodes, and the weights on each of the connections is .5

[[0, .5, .5, .5, .5] [.5, 0, .5, .5, .5].....[.5, .5, .5, .5, 0]]

Rows represent each node, and columns represent each target node. A node sending a value to itself sends with a weight of zero (no transmission occurs).

It is possible to represent the output of a node in a generalized form. Assuming node A, the output weight of the node (effectively the average of the outputs) is $(1/(N-1))({\rm sum\ of\ outputs})=S_a$ where $N$ is the number of nodes, and $S_a$ is Send:A.

Similarly, the inputs of the nodes can be expressed as $(1/(N-1))({\rm sum\ of\ outputs})=R_a$, where $R_a$ is Return:A.

The average of all of the other nodes not included in the connections going to A is $(1/(N^2-3N+2))({\rm sum\ of\ others})=F_o$.

For the following equations, $M$ is the value of the initial input to a single node in the network.

$M(\sum_{i=0}^{\infty} (R_a^i)(S_a^i)+(S_a)(R_a)(F_o^i))$ This equals $M((R_aS_a)/(1-R_aS_a) - (S_aR_aF_o)/(F_o-1))$.

This formula will calculate the value that node A receives as a result of node A's propagation. All is fine and well with this sum, and it works as it should.

The problem is that this isn't the sum of the values node A has received overall, only as a result of node A's input. For instance, if a number were sent to both node B and node A, then the propagation of the number from B would also add amounts to node A's accumulator.

From the above matrix of weights, a new data set can be derived, that holds the send, receive, and other values of each of the nodes.

The goal is this: find a formula or algorithm, which, when given the inputs to each node, and the matrix of weights, computes the infinite series for a node, and returns its total value.

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I have attempted to improve the formatting. I note that $S_a$ is defined as Send:A, and $R_a$ is defined as Return:A, but neither Send:A nor Return:A is defined, so I don't know what they mean. I also note that you have a sum on $i$ which involves $I$ but not $i$. You might want to edit to correct that. – Gerry Myerson Apr 1 '12 at 23:21
Thank you so much! Sa is the send pathways on node A, similarly Ra is the return pathways. – AIGuy Apr 2 '12 at 22:46

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