Part III built explainable static risk reasoning on a static KG. UAM is **high-frequency dynamic**: positions, battery, weather, and airspace status evolve rapidly. Static graphs capture **state slices**, not continuous evolution. For millisecond conflict detection and cooperative control we cross into Part IV—deep fusion of System 1 and System 2. This chapter lays the foundation: **temporal knowledge graphs (TKGs)** and core mechanisms of **dynamic relational reasoning**.

## 11.1 From static KG to temporal KG

Static KGs use triples $(h, r, t)$—head, relation, tail—often treated as permanently true—adequate for stable structure (e.g., “vertiport X lies in grid Y”).

Traffic facts are mostly **transient**: “UAV A occupies grid 3A” may hold only from $10{:}05{:}00$ to $10{:}05{:}30$. TKGs add time explicitly as **quadruples**:

$$(h, r, t, \tau)$$

where $\tau$ is a timestamp or interval—recording history, current state, and substrate for forecasting graph evolution.

## 11.2 Time, events, and state transitions

In TKGs, time is not a stray tag; it couples to **state change** and **events**.

1. **Temporal modes:**
   * **Time points:** Snapshots or instantaneous events—e.g., radar fixes at $t_0$.
   * **Time intervals:** $[t_s, t_e]$ for sustained states—e.g., temporary restrictions.
2. **Events as first-class nodes:** Landings, handoffs, etc., with time, participants, and location.
3. **State transitions:** Events mutate the graph—edges appear (e.g., “connected to 5G cell”) or disappear as links break.

## 11.3 Dynamic relations and time-varying edge weights

Dense swarms induce **dynamic, nonlinear** interactions; topology shapes collision risk. Model **time-varying connectivity** and **edge weights**:

* **Time-varying connectivity:** Communication or threat edges exist only when distance drops below a threshold; they vanish when aircraft separate.
* **Continuous edge-weight evolution:** Even when present, strength changes—e.g., with distance $d_{ij}(t)$ and relative speed:

$$w_{ij}(t) = \exp(-\lambda d_{ij}(t))$$

Such weights supply quantitative tensors for GNN message passing over continuous time.

## 11.4 Temporal logic and continuous-time modeling

To **reason** about the future, not only **record** the past, add temporal formalisms and neural time coding.

1. **Temporal logic:** At the symbolic layer, linear temporal logic (LTL) or metric temporal logic (MTL) adds operators such as $\square$ (always), $\lozenge$ (eventually), and $\mathcal{U}$ (until). Example safety constraint: **before** battery exhaustion the UAV **must** reach a safe landing zone—compilable to graph validators as hard constraints.
2. **Continuous time and encodings:** Irregular sampling requires mapping time gaps $\Delta t$ to continuous features. A common encoding:

$$\Phi(\Delta t) = \sqrt{\frac{1}{d_T}} [\cos(\omega_1 \Delta t), \sin(\omega_1 \Delta t), \dots, \cos(\omega_k \Delta t), \sin(\omega_k \Delta t)]$$

Combined with neural ODEs, one can interpolate and roll forward continuous motion between sampled graph states.

## 11.5 Event-driven reasoning and dynamic updates

City-scale TKGs face **computational blow-up** if thousands of UAVs mutate the graph every second under full recomputation. Use **event-driven** processing and **incremental updates**:

* **Local graph evolution:** On new telemetry, recompute **affected subgraphs**, not the whole city graph.
* **Streamed messaging:** Trigger GNN or rule layers only when new temporal edges (e.g., “UAV A enters grid B”) cross airspace thresholds.

## 11.6 Fit of dynamic graphs for traffic and flight systems

TKGs suit low-altitude and flight governance because they mirror physics:

1. **Spatiotemporal coupling:** One topology for instantaneous positions, predicted trajectories, and static obstacles.
2. **Causality and traceability:** Time-sliced histories support counterfactual analysis and accountability after incidents.
3. **Bridging symbols and physics:** Continuous trajectories discretize into timed, structured networks—setting the stage for GNN deployment on TKGs and multi-agent coordination in the next chapter.

## Chapter summary

We explained why static KGs miss high-frequency dynamics; TKGs add time via quadruples; events and transitions unify temporal semantics; dynamic relations and time-varying weights quantify interaction strength; temporal logic plus continuous encodings link symbolic time constraints to neural representations; event-driven incremental updates avoid full-graph recomputation; TKGs connect continuous physics to structured reasoning in one state space.

## Key concepts

- Temporal knowledge graph: Dynamic KG with explicit time dimension.
- Event nodes: Reifying processes with time, place, and participants.
- Time-varying edge weights: Relation strength that changes over time.
- Temporal logic: Formal “always / eventually / until” constraints.
- Incremental update: Local recomputation on affected subgraphs only.

## Study questions

1. Why is hanging time merely as a node attribute often insufficient for dynamic reasoning?
2. What does time-varying weighting contribute to multi-agent modeling and risk propagation?
3. Without event-driven incremental updates, what breaks at city-scale TKG operation?

## Case study

UAV transitioning from a normal corridor to an emergency detour corridor—time slices for aircraft, corridor state, and applicable rules, showing state transitions, event triggers, and traceability.

## Figure suggestions

- Figure 11-1: Structural contrast—static KG vs. TKG.

![](../Chart/Figure11-1.png)

- Figure 11-2: Time points, intervals, event nodes, and transitions.

![](../Chart/Figure11-2.png)

- Figure 11-3: Event-driven incremental update over an affected subgraph.

![](../Chart/Figure11-3.png)

## Formula index

- Temporal quadruple: $(h, r, t, \tau)$
- Time-varying weight: $w_{ij}(t) = \exp(-\lambda\,d_{ij}(t))$
- Time encoding: $\Phi(\Delta t)$ (see §11.4)
- Theme: elevating **time** from annotation to **reasoning structure**.

## References

1. Trivedi, R., Dai, H., Wang, Y., & Song, L. (2017). Know-Evolve: Deep Temporal Reasoning for Dynamic Knowledge Graphs. *Proceedings of the 34th International Conference on Machine Learning* (ICML).
2. Kazemi, S. M., et al. (2020). Representation Learning for Dynamic Graphs: A Survey. *Journal of Machine Learning Research*, 21(70), 1–73.
3. Pnueli, A. (1977). The Temporal Logic of Programs. *Proceedings of the 18th Annual Symposium on Foundations of Computer Science* (FOCS), 46–57.
4. Chen, R. T. Q., Rubanova, Y., Bettencourt, J., & Duvenaud, D. (2018). Neural Ordinary Differential Equations. *Advances in Neural Information Processing Systems* (NeurIPS).
5. Rossi, E., Chamberlain, B., Frasca, F., Eynard, D., Monti, F., & Bronstein, M. (2020). Temporal Graph Networks for Deep Learning on Dynamic Graphs. *arXiv preprint arXiv:2006.10637*.