Dynamic networks

Dynamic networks are considered in a so-called module representation. In this setting modules are linear time-invariant (single input, single output) systems that appear on the links in a graph, connecting nodes that represent node signals as time series.

A dynamic network is characterized by the equation

$$w(t) = G(q)w(t) + R(q)r(t) + H(q)e(t)$$

with:

• $w$: a column vector of internal node variables, $w_1 ... w_L$, each being a time series;
• $t$: the discrete-time variable;
• $r$: a vector of external excitation variables;
• $e$: a vector of white noise (disturbance) variables;
• $G$: the dynamic network matrix; a rational matrix indicating which node gets input from which other nodes in the network;
• $R$: a dynamic transfer matrix, representing the mapping from external excitation variables to internal node variables;
• $H$: a dynamic transfer matrix, modeling the disturbances that are present on the node variables.
• $q$: the time-shift variable, $q w(t) = w(t+1)$;

In the beta-version of the Toolbox mainly structural properties of the network are considered, implying that $G, H, R$ are represented by binary adjacency matrices, indicating which link/module is present (=1) or not (=0), i.e. if there is a link from $w_k$ to $w_n$ then $G_{nk} = 1$.

External signals $r$, if present, are typically available to the user, together with a selection of node variables in $w$.

Example of an identification problem in dynamic networks:

• For a given topology, evaluate under which conditions on the presence/measurability of $r$ and $w$ signals, we can estimate the dynamics of a module or of the full network.

Reference:

• P.M.J. Van den Hof, A. Dankers, P. Heuberger and X. Bombois (2013). Identification of dynamic models in complex networks with prediction error methods - basic methods for consistent module estimates. Automatica, Vol. 49, no. 10, pp. 2994-3006.