Skip to main content

Dynamic networks

Example dynamic network

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


  • ww: a column vector of internal node variables, w1...wLw_1 ... w_L, each being a time series;
  • tt: the discrete-time variable;
  • rr: a vector of external excitation variables;
  • ee: a vector of white noise (disturbance) variables;
  • GG: the dynamic network matrix; a rational matrix indicating which node gets input from which other nodes in the network;
  • RR: a dynamic transfer matrix, representing the mapping from external excitation variables to internal node variables;
  • HH: a dynamic transfer matrix, modeling the disturbances that are present on the node variables.
  • qq: the time-shift variable, qw(t)=w(t+1)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,RG, 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 wkw_k to wnw_n then Gnk=1G_{nk} = 1.

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

Example of an identification problem in dynamic networks:

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