Interactions of information transfer along separable causal paths

Complex systems arise as a result of interdependences between multiple variables, whose causal interactions can be visualized in a time-series graph. Transfer entropy and information partitioning approaches have been used to characterize such dependences. However, these approaches capture net information transfer occurring through a multitude of pathways involved in the interaction and as a result mask our ability to discern the causal interaction within a subgraph of interest through specific pathways. We build on recent developments of momentary information transfer along causal paths proposed by Runge [Phys. Rev. E 92, 062829 (2015)] to develop a framework for quantifying information partitioning along separable causal paths. Momentary information transfer along causal paths captures the amount of information transfer between any two variables lagged at two specific points in time. Our approach expands this concept to characterize the causal interaction in terms of synergistic, unique, and redundant information transfer through separable causal paths. Through a graphical model, we analyze the impact of the separable and nonseparable causal paths and the causality structure embedded in the graph as well as the noise effect on information partitioning by using synthetic data generated from two coupled logistic equation models. Our approach can provide a valuable reference for an autonomous information partitioning along separable causal paths which form a causal subgraph influencing a target.

Figure 1

Illustration of a three-variable dynamical system: (a) the process network graph illustrating the lagged interaction of three variables $X, Y$, and $Z$, where the numbers on the directed links represent the lagged time step; (b) the time-series graph corresponding to the process graph, with the causal path $C_{X_{t-2}\to Z_t}$ whose directed links are highlighted by brown arrows; (c) the nonseparable causal paths $C_{X_{t-1}\to Z_t}$ and $C_{X_{t-2}\to Z_t}$; (d) the separable causal paths with independent pathways $C_{X_{t-1}\to Z_t}$ and $C_{Y_{t-3}\to Z_t}$; and (e) the separable causal paths with overlapping pathways $C_{X_{t-2}\to Z_t}$ and $C_{Y_{t-3}\to Z_t}$ (brown and blue nodes, source nodes; black nodes, target nodes; brown and blue links, the corresponding causal paths from the sources to the target; and nodes with black circles, nodes affecting the two separable causal paths and the target node).

Figure 2

Illustration of the Markovian conditions for the causal subgraphs comprising one causal path and two separable causal paths shown in Figs. 1 and 1, respectively. (a) Conditions for the causal path $C_{X_{t-2}\to Z_t}$ (brown solid circles denote $P_{Z_t}\setminus C_{X_{t-2}\to Z_t}$, where $\setminus$ represents the subtraction operation for sets; brown dashed circles denote $P_{C_{X_{t-2}\to Z_t}}$). (b) Conditions for the separable causal paths $C_{X_{t-1}\to Z_t}$ and $C_{Y_{t-3}\to Z_t}$ (the black dashed circle denotes ${\vec{W}}_1$, the brown solid circle ${\vec{W}}_2$, and the blue solid circles ${\vec{W}}_3$).

Figure 3

Common driver model. (a) Process network graph representation, where the numbers on the directed links represent the delayed time step. (b) Time series graph representation, where brown and blue circles are the source nodes (i.e., $X_{t-1}$ and $Y_{t-1}$), the black circle is the target node (i.e., $Z_t$), the brown circle represents the nodes in $P_{C_{X_{t-1}\to Z_t}}\setminus C_{Y_{t-1}\to Z_t}$, and the blue circle refers to the nodes in $P_{C_{Y_{t-1}\to Z_t}}\setminus C_{X_{t-1}\to Z_t}$. Also, $P_{Z_t}\setminus (C_{X_{t-1}\to Z_t}\cup C_{Y_{t-1}\to Z_t})$ is empty in this case. Therefore, the condition set $\vec{W}$ is only $\left\{V_{t-2}\right\}$.

Figure 4

Estimated interaction information as well as redundant, synergistic, and unique information from both MPID (left column) and PID (right column) for the sources $\{X_{t-1},Y_{t-1}\}$ and the target $Z_t$, based on Eq. (24) with a simulation time length of 10000 for each combination of $c_{\mathrm{VX}}$ and $c_{\mathrm{VY}}$.

Figure 5

Illustration of the time-series graphs and the averaged information partitioning for the two cases in Eq. (33) (i.e., cases SI and NI) when the coupling strength $\epsilon =0.5$: time-series graphs of cases (a) SI and (c) NI [blue and brown nodes, the source nodes; black node, the target node; blue and brown directed links, the links in the causal paths of the two sources; gray solid nodes, the conditioning nodes in $\vec{W}$ for MPID in Eq. (10)] and pie charts of the averaged synergistic, redundant, and unique information percentages from PID and MPID of cases (b) SI and (d) NI, with the legend showing the averaged values along with the corresponding standard deviations in parentheses ($U_{{\lambda }_1}$ and $U_{{\lambda }_1,c}$, the unique information of $X_{1,t-1}$, and $U_{{\lambda }_2}$ and $U_{{\lambda }_2,c}$, the unique information of $X_{2,t-1}$).

Figure 6

Illustration of the time-series graphs and the averaged information partitioning for the two cases in Eq. (34) (i.e., cases SC and NC) when the coupling strength $\epsilon =0.5$: time-series graphs of cases (a) SC and (c) NC [blue and brown nodes, the source nodes; black node, the target node; blue and brown directed links, the links in the causal paths of the two sources; gray solid nodes, the conditioning nodes in $\vec{W}$ for MPID in Eq. (10)] and pie charts of the averaged synergistic, redundant, and unique information percentages from PID and MPID of cases (b) SC and (d) NC, with the legend showing the averaged values along with the corresponding standard deviations in parentheses ($U_{{\lambda }_1}$ and $U_{{\lambda }_1,c}$, the unique information of $X_{1,t-1}$, and $U_{{\lambda }_2}$ and $U_{{\lambda }_2,c}$, the unique information of $X_{2,t-1}$).

Figure 7

Averaged percentages of total information, PID, and MPID for the four cases based on the two coupled logistic equations (33) and (34) in terms of different SNRs. The first and second columns plot the percentages of total information in Eqs. (36) and (37) (red line) and the similarity functions in Eq. (35) between the first source and the target, ${\lambda }_1\&tar$ (black line marked with triangles); the second source and the target, ${\lambda }_2\&tar$ (black line marked with solid circles); and the two sources, ${\lambda }_1\&{\lambda }_2$ (black dashed line), for PID and MPID, respectively. The first and second columns plot the synergistic (orange line), the redundant (black line), and the unique information (blue and sienna dashed lines) for PID and MPID, respectively.