Fig. 2: CORNETO unifies network inference methods from prior knowledge and omics, enabling multi-sample network inference through the lens of network flows and mixed-integer optimization.

The framework operates on a context-free PKN \({\mathcal{H}}=({\mathcal{V}},{\mathcal{E}})\) that contains known biological interactions and additional knowledge such as stoichiometric information, gene–protein reaction rules, type of interaction and so on, depending on the type of prior knowledge used. The framework supports multiple types of prior knowledge, including undirected, directed, signed directed graphs and hypergraphs. These graphs can be obtained from databases or be defined by the user. Every network inference method in CORNETO is implemented in terms of data mapping, graph transformations and conversion into a constrained optimization problem using mathematical programming. a, Omics data are transformed and mapped onto the context-free PKN, resulting in n annotated graphs, one per sample or condition. b, Graphs are transformed by the method to optimize the structure based on the input data and to adapt the graph to be suitable for converting it into an optimization problem using network flows. c, The network inference method is formulated as a constrained optimization problem, using flow and indicator variables created by CORNETO on top of the transformed graphs. The regularization term (orange) is added to the objective function to promote structured sparsity across all samples, enabling joint inference. d, The optimization problem is solved by one of the many mathematical solvers supported by CORNETO. This results in n optimal inferred networks, with proofs of optimality provided by the solvers. e, An example of a signalling network with two samples, where we use only information about receptors (top vertices) and TFs (bottom nodes), to infer the networks propagating the signal. Red colour indicates activation and blue inhibition. Single-sample optimization leads to two networks not sharing any edge, where the size (number of edges) of the union is six, whereas the joint (multi-sample) optimization finds two networks that share one edge, leading to a solution with only five edges from the PKN that correctly explains the observed activations and inhibitions.