Network dynamics can also be analyzed backwards in time. Let us assume that we have observed or measured that network at instant t is in a given state. We can then compute state distribution probability for instant t − 1, that is we can compute the law by which states at instant t − 1 might have caused the state actually observed or measured at instant t. This is provided by defining a *state backward-transition* matrix B, describing probabilities obtained inverting through Bayes rule the relations between events.

Its elements bij are:

$$b_{i j}(t)\stackrel{\circ}{=}p(X_{t-1}=x_{j}\,|\,X_{t}=x_{i})$$
that can be written as 
$$b_{i j}(t)={\frac{p(X_{t-1}=x_{j},\,X_{t}=x_{i})}{p(X_{t}=x_{i})}}$$
and applying again Bayes rule we have
$${\mathrm{and~applying~ag}}$$
$$b_{ij}(t)=\frac{p(X_{t}=x_{i}\,|\,X_{t-1}=x_{j})p(X_{t-1}=x_{j})}{p(X_{t}=x_{i})}=\frac{s_{jl}p(X_{t-1}=x_{j})}{p(X_{t}=x_{i})}=\frac{\mathbf{p}_{t-1}(j)s_{jl}}{\mathbf{p}_{t}(i)}=\frac{\mathbf{p}_{t-1}(j)s_{jl}}{\mathbf{p}_{t-1}\cdot\mathbf{S}^{i}}$$  If at instant $t-1$ state probability distribution is uniform then last formula becomes 
$$b_{i j}(t)={\frac{s_{j i}}{\sum_{k}s_{k i}}}$$

$$(1)$$

Note that if state probability distribution is uniform then state backward-transition matrix B is a kind of transpose of the state transition matrix S. Note also that while S is time constant, B is not so, in general.

Row Bi(t) of the state backward-transition matrix B provides the conditional probability distribution p(Xt−1 | Xt = xi) describing network state at the instant *previous* to the one the network is in state xi.

## 3 Effective Information 3.1 Introduction

Effective information can be informally described as the quantity of information on possible predecessors of current states acquired *additionally* from actually measuring the current network state with respect to what can be acquired from the knowledge of state transition matrix only. We propose calling it experimental information or *Galileian information*, given the emphasis it gives to experimentally acquired knowledge with respect to purely theoretical knowledge. Here quantity of information is intended in the standard sense of the Shannon's Information Theory.

The main question effective informations answers to is: if network observation finds that its current state is xi, which is the additional knowledge provided by this measure with respect to what can be known on the network by its state transition matrix only, i.e. without knowing which is the current state of the network?

Still remaining at the informal level this additional knowledge can be described as the reduction in uncertainty provided by the actual measurement with respect to the uncertainty existing on the basis of the state transition matrix only.