Then, the Minimum Information Partition *(MIP*) is defined as the partition providing the minimum value for the integrated information after the normalization process, that is

$$P_{M P}(t,V,v_{h})\triangleq\arg\operatorname*{min}_{P}\left\{{\frac{\phi(t,V,P,v_{h})}{N(t,V,P,v_{h})}}\right\}$$

The above formula has been defined by Tononi for generic partitions, but in all of its papers and here it is only discussed the case of bi-partitions, i.e. partitions in two subsets.

Integrated information φ for subset V , in state vh at instant t, is now formally defined as the value of the partition-dependent integrated information computed on MIP, that is

$$\phi(t,V,v_{h})\triangleq\phi(t,V,P_{M I P}(t,V,v_{h}),v_{h})$$

And it is now possible to formally define the value of integrated information for the whole system X. A
subset V ⊆ X having φ > 0 is called *complex*. If it is not a proper subset of another subset with a larger φ it is called *main complex*. The value of integrated information of X, in state xi at instant t, is defined as the value of integrated information of its main complex of maximum value.

$$\phi(t,x_{i})\triangleq\operatorname*{max}_{V\subset X}\;\phi(t,V,P_{M I P}(t,V,v_{h}),v_{h})$$

The value of integrated information averaged over all states of the system is provided through the state distribution probability pt(·), that is

$$\phi(t)\triangleq\sum_{x_{i}\in X}\phi(t,x_{i})\,p_{t}(i)$$

## 5 Integrated Information In Disconnected Systems

Intuitively, any system having a partition in two independent subsets, i.e. that can be partitioned in two subsets such that no node in a subset affects the state value of nodes in the other subset, should have zero as value of its integrated information.

We now give a formal proof of this property, to the best of our knowledge never appeared in the literature. We consider the value of integrated information assuming at instant t − 1 the system has a uniform state probability distribution, consistently with discussion in [1]. Remember that for a subset V
of the system X in state xh we use vh as a shorthand for Vxh, the restriction of xh to nodes in V .

Theorem 1 (Integrated information in a disconnected network) Let A′ and A′′ be two disjoint subsets of a network X, A′ ∪ A′′ = V ⊆ X. Let us denote with vh the current state of V *, and with* a
′

h e a
′′

h the current states of subsets A′ and A′′*, respectively.*
For each state vh and time instant t *it is*