We present and develop the theory of 3-way networks, a type of hypergraph in which each edge models relationships between triplets of objects as opposed to pairs of objects as done by standard network models .

We explore approaches of how to prune these 3-way networks, illustrate their utility in comparative genomics and demonstrate how they find relationships which would be missed by standard 2—way network models using a phylogenomic dataset of 211 bacterial genomes.

In order to address this, we have developed a new three-way similarity metric and constructed three-way networks modelling the relationships between 211 bacterial genomes.

For both the Sorensen Index and the Czekanowski Index, the union of the 3-way best-edge network and the 2-way MST was calculated, resulting in a combined network model .

These networks, when used to model the phylogenomic relationships between 211 bacterial species revealed relationships between the species which were not found when using standard 2-way network models .

Network models are a useful reductionist approach for modelling complex systems.

Thus networks model a system in a pairwise manner, breaking a system down into individual parts (nodes), modelling relationships between pairs of these individual parts (edges) and then reconstructing the system as a network [1].

We then apply a 3-way network model to a set of 211 bacterial genomes, modelling the similarities between the bacteria on a whole genome scale, (based on gene family content), and compare the resulting 3-way networks to those obtained using standard 2-way network models .

With the aim of modelling higher order relationships than simply pairwise relationships, we define 3-way networks as network models of ternary relationships, i.e.

We find the evolution of narrow waists in a wide range of evolutionary parameters, in both linear and nonlinear multilayered network models .

Finally, we asked whether the present mechanism would apply in a nonlinear network model .

Simulations of multilayered network models evolving towards input-output goals

The second (hidden) layer of the network models the association cortex, and contains regular units (circles in Fig.

Earlier neural networks models used “backpropagation-through-time”, but its mechanisms are biologically implausible [77].

Several models addressed how neural network models can store F1 and compare it to F2 [46—48].