Belief propagation for networks with loops https://advances.sciencemag.org/content/7/17/eabf1211.abstract
> Our method operates by dividing a network into neighborhoods (20). A > neighborhood Ni(r) around node i is defined as the node iitself and > all of its edges and neighboring nodes, plus all nodes and edges along paths > of length r or less between the neighbors of i. See Fig. 1 for examples. The > key to our approach is to focus initially on networks in which there are > no paths longer than r between the neighbors of i, meaning that all > paths are inside Ni(r). This means that all correlations between spins > within Ni(r) are accounted for by edges that are also within Ni(r), which > allows us to write exact mes-sage passing equations for these networks. > Equivalently, we can de-fine a primitive cycle of length r starting at node i > to be a cycle (i.e., a self-avoiding loop) such that at least one edge in the > cycle is not on any shorter cycle beginning and ending at i. Our methods are > then exact on any network that contains no primitive cycles of length greater > than r + 2. > ... > Having defined the initial neighborhood Ni, we further define a neighborhood > Nj ∖ i to be node j plus all edges in Nj that are not > contained in Ni and the nodes at their ends. Our method involves writing the > marginal probability distribution on the spin at node i in terms of a set of > messages received from nodes j that are in Ni, in-cluding nodes that are not > immediate neighbors of i. (This contrasts with traditional message passing > in which messages are received only from the immediate neighbors of i.) > These messages are then, in turn, calculated from further messages j receives > from nodes k∈Nj ∖ i and so forth. -- ↙↙↙ uǝlƃ - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . FRIAM Applied Complexity Group listserv Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com FRIAM-COMIC http://friam-comic.blogspot.com/ archives: http://friam.471366.n2.nabble.com/
