Figure 1

The values of multi-information, connected information of orders 2 and 3, the pairwise mutual information, and pairwise redundancy for three binary variables, whose probability distribution is given by the logical functions AND, OR, and XOR (with the inputs ${\sigma }_1$ and ${\sigma }_2$ chosen at random), and the case of ferromagnetic interaction, FM.Reuse & Permissions

Figure 2

Correlated information of orders 2 and 3 and the multi-information for three variables whose joint probability distribution is given by noisy logical functions. Each panel presents the $I_C$’s and $I$ values for a noisy version of one Boolean gate (XOR in first row, OR in second, AND in third), as a function of noise amplitude. The three types of noise are output noise (probability of flipping ${\sigma }_3$), input noise (probability of flipping ${\sigma }_1$), and input-dependent output noise (probability of flipping ${\sigma }_3$, given that ${\sigma }_1=1$ and ${\sigma }_2=1$).Reuse & Permissions

Figure 3

Correlated information of orders 2 and 3 and the multi-information, for networks of three binary observable elements, ${\sigma }_1,{\sigma }_2,{\sigma }_3$, with one hidden binary element ${\sigma }_4$. (a) $I_C$’s and $I$ values for a network where the value of ${\sigma }_4$ determines the pairwise interaction between the other elements: if ${\sigma }_4=0$ then ${\sigma }_3=\mathrm{A}\mathrm{N}\mathrm{D}({\sigma }_1,{\sigma }_2)$; if ${\sigma }_4=1$ then the interaction among the observable variables is (pairwise) ferromagnetic with a finite temperature ($\beta =0.1$). Information values are plotted as a function of $\gamma =P({\sigma }_4=0)$. $P({\sigma }_1,{\sigma }_2,{\sigma }_3)$ in (b) same as (a), but for a ${\sigma }_4$-dependent mixture of AND and OR. For this case there is no effective three-body interaction. (c) $I_C$’s and $I$ for the three observable binary variables network, where the full four element network has pure three-body interactions, plotted as a function of the inverse temperature $\beta$.Reuse & Permissions