Detecting causality in policy diffusion processes

Abstract

A universal question in network science entails learning about the topology of interaction from collective dynamics. Here, we address this question by examining diffusion of laws across US states. We propose two complementary techniques to unravel determinants of this diffusion process: information-theoretic union transfer entropy and event synchronization. In order to systematically investigate their performance on law activity data, we establish a new stochastic model to generate synthetic law activity data based on plausible networks of interactions. Through extensive parametric studies, we demonstrate the ability of these methods to reconstruct networks, varying in size, link density, and degree heterogeneity. Our results suggest that union transfer entropy should be preferred for slowly varying processes, which may be associated with policies attending to specific local problems that occur only rarely or with policies facing high levels of opposition. In contrast, event synchronization is effective for faster enactment rates, which may be related to policies involving Federal mandates or incentives. This study puts forward a data-driven toolbox to explain the determinants of legal activity applicable to political science, across dynamical systems, information theory, and complex networks.

FIG. 1.

Inter-event times involved in calculating the dynamical delay for ES between law activities of states i and j. Two events at $t_r^i$ and $t_s^j$ can be uniquely interpreted as synchronized if their time lapse is smaller than the dynamical delay, that is, the minimum of time lapses to preceding and succeeding events divided by two, as defined in Eq. (9).

FIG. 2.

Network of influences among states determines their law activity dynamics. (a) An exemplary network of N = 10 nodes and in-degree k = 2 for all nodes. State 9 (green) receives incoming links from state 5 (red) and state 2 (blue). (b) The adjacency matrix A defined in Eq. (15) encodes the network structure displayed in (a), with light gray color indicating the presence of a link. (c) Law activity of state 2 (blue bar) and state 5 (red bar) follows the last law event of state 9 (green bar). (d) Law events in the neighboring states 2 and 5 (blue and red bar in (c)) increase the law event probability of state 9. Law activity was generated for $\Theta =0.0005$ and γ = 1.

FIG. 3.

ROC curves for (a) UTE and (b) ES applied to the exemplary network in Fig. 2(a). (a) Exemplary ROC curves for UTE associated with different realizations of the law activity dynamics follow from Eqs. (21a) and (21b). Each tenth data point on the red curve is marked in black; from right to left, they correspond to the values $z=1,43/45,71/90,2/3,4/9,23/90,2/15,1/15,1/45,$ and 1/90. (b) Exemplary ROC curves for ES associated with different realizations of the law activity dynamics follow from Eqs. (22a) and (22b). Each tenth data point on the red curve is marked in black; from right to left, they correspond to the values $z=1,17/30,3/10,1/5,1/9,1/15,1/45,1/45,1/90,$ and 1/90. (c) Exemplary matrix $I_{X^j\to X^i}$ for directed causal influences based on UTE as defined in Eq. (7) that leads to the curve highlighted in red in (a) via thresholding. Matrix values range from 0 (black) to 0.01 (white). (d) Exemplary matrix $Q_{X^j\to X^i}$ for directed synchronicity based on ES as defined in Eq. (13) that leads to the curve highlighted in red in (b) via thresholding. Matrix values range from 0 (black) to 0.5 (white). Law activity was generated over a time span of 50 000 days and with parameters $\Theta =0.00005$ and γ = 20.

FIG. 4.

Motif networks characterize four principal cases: (a) two states both influence a common neighbor; (b) two states are both influenced by a common neighbor state; (c) three states influence each other in a cycle; and (d) three states influence each other in a chain.

FIG. 5.

Performance of UTE (left column) and ES (right column) for the four motifs in Fig. 4 (top row: motif (a); second row: motif (b); third row: motif (c); and bottom row: motif (d)). All AUC-values are averaged over 10 realizations. Contours highlighted in black identify the area where our methods perform better than chance.

FIG. 6.

Three networks with N = 10 nodes: (a) a regular network; (b) a random network with constant in-degree $k^{\text{in}}=1$; and (c) a random network with constant out-degree $k^{\text{out}}=1$.

FIG. 7.

AUC scores of UTE and ES for the networks depicted in Fig. 6 as functions of the parameters $\Theta$ and γ: a ring network (upper row) and randomized networks with constant in-degree (middle row) and out-degree (bottom row). All AUC-values are averaged over 10 realizations. Contours highlighted in black identify the area where our methods perform better than chance.

FIG. 8.

Three networks with N = 10 nodes: (a) a regular network; (b) a random network with constant in-degree $k^{\text{in}}=3$; and a random network with constant out-degree $k^{\text{out}}=3$.

FIG. 9.

AUC scores of UTE and ES for the networks depicted in Fig. 8 as functions of the parameters $\Theta$ and γ: a ring network (upper row) and randomized networks with constant in-degree (middle row) and out-degree (bottom row). All AUC-values are averaged over 10 realizations. Contours highlighted in black identify the area where our methods perform better than chance.

FIG. 10.

Proposed networks for the 50 US states based on (a) geography and (b) ideology. The nodes represent the US states. AK and HI are omitted for clarity; in the geography-based network, AK and HI are isolated nodes, while in the ideology-based network, AK has seven incoming edges from CO, DE, GA, IN, LA, MO, and NV; and one outgoing edge to WY; and HI has one outgoing edge only to MA.

FIG. 11.

AUC-scores of UTE (left column) and ES (right column) for the networks depicted in Fig. 10 as functions of the parameters $\Theta$ and γ. Networks for the 50 US states based on ideology (upper row) and based on geography (bottom row). All AUC-values are averaged over 10 realizations. Contours highlighted in black identify the area where our methods perform better than chance.

FIG. 12.

Proposed networks for the 50 US states based on UTE (a) and ES (b). Links that are common between the two networks are highlighted in black. Alaska and Hawaii are not explicitly indicated on the map.