To analyze the dynamical interaction network in urban traffic, this study focuses on the road network within Beijing fourth Ring Road. This road network is divided into 20 predefined hexagonal regions (Figure 1a). We use 1-min interval speed records from floating vehicles collected during 17 workdays in October 2015. Raw speed data from over 33,000 road segments are aggregated to construct a global traffic state for each time step (Figure 1b). Each global traffic state is represented as a 20-dimensional binary vector of regional states, defined by a congestion proportion f and a regional jam threshold l (see Data and methods).

We apply a pairwise maximum entropy model [15] (Equations 9, 10) to infer regional interaction networks over 24 consecutive 1-h time windows (7:00–18:30, with half-hour sliding intervals). The 24 time windows mentioned below refer to these consecutive 1-h intervals. For each time window, the model input includes global traffic states (Figure 1b) from all time steps within that window over the 17 workdays (see Methods). The resulting interaction network (Figure 1c) consists of nodes representing the 20 predefined road regions (Figure 1a), and edges defined by the inferred model parameters J _ij = J _ji . These interaction networks are weighted and heterogeneous. The absolute value |J _ij | represents interaction strength, and the sign distinguishes mutual congestion propagation (J _ij >0) and mitigation (J _ij <0) [16]. This maximum entropy model infers a local interaction network capturing the statistical distribution of system-level macroscopic performance (Appendix A). By accounting for the collective behavior of the entire system, it characterizes how local interactions shape global state distributions.

Next, we compare interaction networks across traffic periods with different efficiency levels. Traffic efficiency (see Methods) is measured by the mean relative speed v (Equation 7) and the mean size of the largest functional cluster G (Equation 8) [6]. These metrics vary considerably across the 24 time windows and display a clear pattern of morning peak, noon off-peak, and evening peak periods (Figure 2d). Based on this, we choose three typical time windows for comparison: 8:00–9:00 in morning peak, 12:00–13:00 in noon off-peak, and 17:30–18:30 in evening peak. As shown in Figures 2a–c, the spatial distributions of the top 5% strongest interaction edges are very different in these three time windows. Compared to noon off-peak period (Figure 2b), peak periods (Figures 2a, c) have more strong positive links, suggesting a greater likelihood of congestion propagation between regions. Moreover, the strong positive links during peak periods tend to cluster in space. For instance, in the morning peak (Figure 2a), Region 5 has strong positive connections with three other regions. In the evening peak (Figure 2c), Region 12 exhibits a similar clustering pattern. This spatial clustering suggests the emergence of critical regions during peak hours. In these regions, localized congestion is more likely to spread and trigger large-scale gridlock [17], thereby reducing overall network efficiency. These findings imply that temporal variations in traffic efficiency (Figure 2d) may be influenced by structural dynamics of the interaction networks. This highlights the need to further investigate their dynamical characteristics, extending the previous model [15] that focusing on a single typical time window.

To quantify macroscopic dynamics of interaction networks, we analyze their structural differences across distinct traffic periods. Specifically, we compare interaction networks during the morning peak, noon off-peak, and evening peak periods (Figure 2d), both within and across these periods. Structural dissimilarity is measured using the Jaccard index, which is the ratio of common edges with identical signs to the total number of edges (Equation 11). A low Jaccard index indicates a large structural difference, while a high index suggests great structural similarity. We compute Jaccard indices for all intra-period and inter-period interaction network pairs, generating empirical distributions of these indices. To assess these distributions, we perform a one-tailed Mann-Whitney U test [18] (see Methods) to compare the empirical Jaccard index distribution with that of shuffled edge-sign networks. A one-tailed p-value below the predefined significance level indicates that the empirical Jaccard indices are statistically significantly greater than that of the shuffled networks, suggesting that empirical interaction networks exhibit stronger structural similarity than expected by chance. In addition, we report Cliff’s Δ (Effect size; Equation 12) to quantify the extent to which the structural similarity in empirical interaction networks exceeds that of the shuffled baseline. Larger positive values of Cliff’s Δ indicate a greater proportion of empirical Jaccard indices exceeding those from shuffled networks.

As shown in Figures 3a–c, intra-period interaction networks exhibit significantly high structural similarity, with most Jaccard indices exceeding 0.6 and some surpassing 0.7. Statistical comparisons with shuffled null networks confirm the stability of within-period structures (morning peak: p < 0.001, Cliff’s Δ = 0.905; noon off-peak: p < 0.001, Cliff’s Δ = 0.889; evening peak: p < 0.001, Cliff’s Δ = 0.871). In contrast, inter-period interaction networks show lower Jaccard indices (about 0.5 in Figures 3d–f), indicating greater structural dissimilarity. Notably, peak-off-peak comparisons (Figures 3e, f) show p-values greater than 0.1 and very low effect sizes (Cliff’s Δ < 0.16), providing no statistical evidence that empirical interaction networks exhibit greater structural similarity across peak and off-peak periods than shuffled networks. These results suggest that interaction networks remain stable within each period but show structural shifts between periods, particularly between peak and off-peak periods. High within-period similarity indicates consistent regional interaction patterns, maintaining relatively consistent traffic behavior under stable traffic conditions. In contrast, lower between-period similarity reflects interaction reconfigurations, which drive transitions between distinct traffic states.

To investigate the microscopic dynamics of interaction networks, we examine the fluctuations of strongly interacting edges over consecutive time windows. First, we rank all edges based on their average interaction strength (⟨|J _ij |⟩) across the 24 time windows. As shown in Figure 4a, edges in the top 30% by ⟨|J _ij |⟩ are considered high-strength edges, while the rest are classified as low-strength edges. The threshold for high-strength edges is ⟨|J _ij |⟩₍₃₀₎ = 0.299, meaning that most edges (70%) have relatively low average interaction strength smaller than that. To quantify the fluctuations of these high-strength edges, we measure two aspects: strength volatility and sign volatility. Strength volatility (CV(|J _ij |) in Equation 13) is the coefficient of variation of |J _ij | across the consecutive time windows, with higher CV(|J _ij |) values indicating greater fluctuations in interaction strength. Sign volatility (SF(J _ij ) in Equation 14) is the number of sign flips in J _ij across the consecutive time windows, with more flips indicating higher instability in interaction sign.

As shown in Figure 4b, high-strength edges exhibit a wide range of both strength and sign volatility. Specifically, CV(|J _ij |) ranges from 0.8 to 2.1, while SF(J _ij ) ranges from 2 to 12. We classify edges as stable if both volatility measures are below their mean values, and as unstable if both volatility measures are greater than or equal to their mean values. Among these high-strength edges, stable edges account for 24.6%, while unstable edges make up 19.3%. Notably, stable high-strength edges have a higher proportion of negative interactions occurring in more than 50% of the time windows (purple points in Figure 4b), compared to unstable high-strength edges. This indicates that stable high-strength edges consistently suppress congestion. Spatial analysis further reveals that some stable negative high-strength edges (e.g., edge between regions 3 and 17 in Figure 4c) span longer distances than unstable ones (Figure 4d). These long-range, negative-effect stable edges form a geographically dispersed network. Showing negative interactions across distant regions, they may reflect negative feedbacks to prevent congestion from spreading into widespread gridlock [17], demonstrating the inherent resilience [19] of urban traffic.

After analyzing the dynamics of macroscopic structure and microscopic edge, we further investigate the evolution of structural features in dynamical interaction network. We quantify this evolution using four structural metrics (denoted as P + , P + s , Q p w , and k n w ; see Methods), which capture both network-level and node-level properties of the weighted and heterogeneous interaction networks. For each of the 24 interaction networks (corresponding to the 24 time windows), we compute these metrics, which reveal distinct temporal patterns and exhibit strong interpretability (Figure 5).

The proportion of positive interactions ( P + ; Equation 15) quantifies the overall positive polarity of interactions within the network. A higher P + value indicates more mutual reinforcement between regions, increasing the likelihood of congestion propagation. As shown in Figure 5a, P + is significantly higher during the morning and evening peak hours compared to noon off-peak. Specifically, P + reaches approximately 0.6 in the morning peak, 0.5 in the evening peak, and 0.3 during noon off-peak. This suggests that peak hours have more positive interactions, facilitating congestion spread. A similar trend is observed for the proportion of strong positive interactions ( P + s , calculated from subnetwork with |J _ij | >0.2): peak values are much higher than off-peak (Figure 5b), enhancing congestion spread effects.

The weighted modularity of positive interactions ( Q p w ; Equation 16) reflects whether positive links tend to form community-like structures. A higher Q p w value implies that congestion propagation is more likely to remain within specific regional clusters rather than spread globally. As shown in Figure 5c, Q p w is much higher during noon off-peak compared to peak hours. Most Q p w values are around 0.4 during noon off-peak, while they are about 0.2 during peak hours. This indicates that positive interactions during off-peak periods are more likely to form community structures, helping to confine congestion within local regions. In contrast, during peak hours, positive interactions are less likely to form such structures, resulting in widespread congestion.

The average node strength of negative interactions ( k n w ; Equation 17) represents the strength of inhibitory effects between regions. A higher k n w signifies stronger suppression of congestion. As shown in Figure 5d, k n w is lower during peak hours (around 1.5) and higher during off-peak hours (approaching 6.0). This demonstrates that inhibitory effects of regions are significantly stronger during off-peak periods, preventing congestion from spreading between regions.

Our findings reveal that the proportion and modularity of positive interactions, along with the strength of negative interactions, are important structural features distinguishing peak from off-peak hours. During peak hours, the network is dominated by a high proportion and low modularity of positive interactions, along with weak inhibitory interactions. These structural characteristics reduce the ability of the interaction network to localize congestion, leading to large-scale traffic congestion. In contrast, off-peak periods are characterized by a low proportion and high modularity of positive interactions, and strong inhibitory effects, which help localize congestion and maintain overall traffic efficiency.

To study the interactions between local components of the road network, we focus on regional interactions rather than segment-level interactions to avoid combinatorial explosion in state space. Following our previous work [15], we divide the entire road network into 20 hexagonal regions (Figure 1a), and aggregate road segment states into global traffic states for each time step. The aggregation procedure is as follows.

To quantify the traffic efficiency of each 1-h time window, we compute two metrics: the mean relative speed v , reflecting overall traffic conditions, and the mean size of the largest functional cluster G , which captures the interplay between traffic dynamics and road network topology [6]. Both metrics are averaged over all time steps within a given time window across 17 workdays.

For a given time step t, we define.

(1) Average relative speed v(t) as the mean of the relative velocities of all road segments:

Based on Equations 5, 6, the aggregate metrics for each time window are then computed as: v = 1 T ∑ t ∈ T v t , (7) G = 1 T ∑ t ∈ T G t , (8) where T is the set of all time steps within the time window across all 17 workdays, and T = T is the total number of such time steps.

Here, we infer the interaction network between regions for each time window using a pairwise maximum entropy model [15]. This model has been applied to neuron interaction network [16], brain functional connectivity [26], and genetic interaction networks [27]. The model maximizes (Gibbs) entropy under constraints of first- and second-order statistics, yielding the probability distribution p s k : max H = − ∑ 1 ≤ k ≤ n p s k log ⁡ p s k s . t . ∑ 1 ≤ k ≤ n p s k = 1 ∑ 1 ≤ k ≤ n p s k s k i = ∑ s d ∈ D f s d s d i , 1 ≤ i ≤ m ∑ 1 ≤ k ≤ n p s k s k i s k j = ∑ s d ∈ D f s d s d i s d j , 1 ≤ i < j ≤ m , (9) where the state space consists of n = 2 ^m possible states, with m = 20 representing the number of road network regions, and k denoting a particular state within this space. Specifically, s k represents a global traffic state vector (Equation 4) composed of the states of 20 regions, where s k i = 1 indicates a jam region and s k i = − 1 indicates a free region (Equation 3). s d and f s d denote observed global traffic states and their frequencies in dataset D. Solving the Lagrange function yields the Boltzmann distribution: p s k = 1 / Z e ∑ 1 ≤ i ≤ m h i s k i + ∑ 1 ≤ i < j ≤ m J i j s k i s k j , (10) where h _i captures the regional congestion tendency of region i, J _ij quantifies the pairwise interaction between regions i and j, and Z is the partition function ∑ 1 ≤ k ≤ n e ∑ 1 ≤ i ≤ m h i s k i + ∑ 1 ≤ i < j ≤ m J i j s k i s k j .

For each 1-h time window (e.g., 8:00–9:00), we construct a dataset D comprising global traffic states obtained from 60 time steps within the window across 17 workdays. We then compute the empirical moments ∑ s d ∈ D f s d s d i and ∑ s d ∈ D f s d s d i s d j ​. These moments are used to ensure that the resulting distribution p s k in Equation 10 satisfies the moment constraints specified in Equation 9. The inferred model parameters J _ij are used to define the regional interaction network. Due to the intractability of the state space (2²⁰), we approximate parameter estimation using the minimum probability flow (MPF) algorithm [28].

Model validation evaluates the model ability to reproduce distribution of macroscopic traffic properties [15], such as region-level traffic performance. This performance is defined as the proportion of regions in the largest connected cluster formed by free regions, relative to total regions. As shown in Appendix A, the model accurately fits the 24 time windows, and captures unimodal or bimodal patterns in region-level traffic performance distributions.

To quantify structural dissimilarity of interaction networks within the same period (intra-period) or across different periods (inter-period), we calculate the Jaccard indices for all interaction network pairs and compare their distributions with the null distribution generated by shuffled networks.

To characterize the weighted and heterogeneous properties of interaction networks, we define four structural metrics that reflect topological aspects such as interaction polarity, modularity, and node connectivity. These include both network-level and node-level metrics.