Metro Systems: Analysis & Urban Impact

• Metro systems are urban heavy rail transit infrastructures characterized by network topology, demand-responsive operations, and significant urban policy effects.
• Research employs graph-theoretic, statistical, and machine-learning methods to analyze robustness, passenger flow, and network expansion planning.
• Studies focus on optimizing operational control, safety via RFID systems, and predictive maintenance to enhance performance and urban mobility.

Metro denotes the urban heavy rail transit system and, in the research literature, a class of infrastructures that can be formalized as graphs, dynamical systems, sensing platforms, and urban policy instruments. Across the cited work, metro systems are modeled both as physical networks of stations and track segments and as demand-responsive transport systems whose robustness, controllability, passenger assignment, expansion, visualization, and wider urban effects are measurable with graph-theoretic, statistical, and machine-learning methods (Wang et al., 2015, Dees et al., 2021).

1. Topological representations and metric structure

A standard abstraction models a metro as an undirected graph $G(N,L)$, where $N$ is the number of nodes and $L$ is the number of links. In the robustness literature, nodes are transfer stations plus terminals and links are track segments; the adjacency matrix $A\in\{0,1\}^{N\times N}$ has entries $a_{ij}=1$ iff stations $i$ and $j$ are directly connected, and the Laplacian is $Q=\Delta-A$ with $\Delta=\mathrm{diag}(d_i)$ and $d_i=\sum_j a_{ij}$ (Wang et al., 2015). A related graph-theoretic formulation for traffic modeling uses $N$0, adjacency matrix $N$1, degree matrix $N$2, and graph Laplacian $N$3, while allowing node attributes such as zone number, average entries, exits, and geographic co-ordinates, and edge attributes such as line identity, physical distance, and number of parallel lines (Dees et al., 2021).

Several complementary spatial representations are used. In $N$4-space, each station is a vertex and an edge links two stations if they are consecutive on some line; in $N$5-space, the vertices are still stations but any two stations on the same line are connected by an edge, so each line becomes a clique (Li et al., 2014). These representations support different transport questions. For directly connected stations in $N$6-space, the $N$7-Wasserstein distance satisfies $N$8, and the mean $N$9 is computed over all edges. In $L$0-space, the mean pairwise transport cost $L$1 relates closely to the average number of transfers $L$2 (Li et al., 2014).

Metric structure is also studied through fractal dimension and graph energy. For the 28 metros examined in the optimal-transport study, the fractal dimension $L$3 ranged from about $L$4 for Montreal to $L$5 for New York, and the energy cost per unit area obeyed the empirical laws

$L$6

The same study reports that New York and Berlin consistently occupy the top two positions in ranking lists formed from normalized transport-related quantities (Li et al., 2014).

Historical growth introduces another structural regularity. Analyses of old metro maps show that Pearson degree correlation grows increasingly from initially negative values toward positive values over time and in some cases becomes decidedly positive. Among the canonical forms examined, only a few types patterned after a wide area network with a core-periphery structure show similar positive-trending degree correlation as network size increases; this suggests that large metro systems either are designed or evolve into the equivalent of message carriers that seek to balance travel between arbitrary node-destination pairs with avoidance of congestion in the central regions of the network (Whitney, 2012).

2. Robustness, vulnerability, and safety

Robustness in metro networks is explicitly multi-metric. A basic theoretical indicator is the robustness indicator

$L$7

where $L$8 is the cyclomatic number, interpreted as the number of independent cycles or total alternative paths. A second metric is the effective graph conductance $L$9, defined via the effective graph resistance

$A\in\{0,1\}^{N\times N}$0

with $A\in\{0,1\}^{N\times N}$1 the nonzero Laplacian eigenvalues. The first metric captures cycle-based redundancy; the second incorporates both the number of parallel paths and their lengths, favoring short and multiple routes and therefore fast transfers (Wang et al., 2015).

Experimental robustness is quantified through node-removal simulations. Let $A\in\{0,1\}^{N\times N}$2 be the size of the largest connected component after $A\in\{0,1\}^{N\times N}$3 nodes are removed and $A\in\{0,1\}^{N\times N}$4. Then $A\in\{0,1\}^{N\times N}$5 is the smallest $A\in\{0,1\}^{N\times N}$6 such that $A\in\{0,1\}^{N\times N}$7, and $A\in\{0,1\}^{N\times N}$8 is the smallest $A\in\{0,1\}^{N\times N}$9 such that $a_{ij}=1$0. Simulations use both random failures and targeted attacks, the latter removing the remaining node of highest degree while breaking ties randomly (Wang et al., 2015).

On 33 worldwide metro systems, the theoretical metrics correlate with different empirical regimes. $a_{ij}=1$1 strongly positively correlates with both $a_{ij}=1$2 measures, with $a_{ij}=1$3; $a_{ij}=1$4 strongly positively correlates with $a_{ij}=1$5, with $a_{ij}=1$6; and $a_{ij}=1$7 and $a_{ij}=1$8 are negatively correlated, with $a_{ij}=1$9. The paper’s interpretation is that networks with high conductance maintain at least $i$0 connectivity under both random failures and attacks, whereas networks with many cycles require removal of a larger fraction of nodes before total disintegration (Wang et al., 2015). Illustrative targeted-attack thresholds emphasize the distinction: Rome has $i$1 and $i$2; Cairo and Marseille have $i$3 and $i$4; Seoul has $i$5 and $i$6, making it the most “ultimate” robust in that comparison (Wang et al., 2015).

Vulnerability can also be localized at stations. Degree, closeness, and betweenness centralities are used to identify critical nodes, with the interpretation that a high-$i$7 station lies on many shortest paths and its disruption forces many commuters to reroute, while a low-$i$8 leaf-station, if cut off, disconnects whole sub-trees. In the London Underground Zones 1–2 case, the top-3 betweenness stations were Bank, Canary Wharf, and Oxford Circus (Dees et al., 2021). Robustness improvement can then be posed as a $i$9-edge augmentation problem: find the smallest set of new edges $j$0, disjoint from the existing edge set, such that the augmented graph is $j$1-edge-connected, or equivalently raise algebraic connectivity $j$2 subject to geographic constraints $j$3 (Dees et al., 2021).

Safety has also been treated as a local control problem rather than solely a network-topology problem. An RFID-based accident-prevention model equips the train with an RFID reader and a control circuit with a microcontroller, and places active RFID tags at curves, slopes, station approaches, tunnel walls, on the rear car bodies of other trains, and potentially on passengers as moving-obstacle detectors. The microcontroller maps tag IDs to condition profiles and adjusts motor or wheel speed accordingly. In a laboratory prototype with 60 runs per scenario, the RFID-system-activated case recorded 2 incidents and 58 safe runs, while the deactivated case recorded 14 incidents and 46 safe runs, a relative reduction in incidents of approximately $j$4 (Sahba et al., 2018).

A recurrent misconception in this literature is that metro robustness is a single scalar property. The reported negative correlation between $j$5 and $j$6 indicates the opposite: adding many long loops can improve ultimate fragmentation resistance while lowering conductance, whereas highly centralized or star-like structures can improve initial connectedness while reducing cycle-based redundancy (Wang et al., 2015).

3. Train dynamics and operational control

Operational metro theory often begins with discrete-event constraints on train departures. In the max-plus framework for a linear line with $j$7 segments and $j$8 trains, the essential variables are departure time $j$9, arrival time $Q=\Delta-A$0, dwell time $Q=\Delta-A$1, run time $Q=\Delta-A$2, travel time $Q=\Delta-A$3, safe-separation time $Q=\Delta-A$4, and headway $Q=\Delta-A$5. Demand-dependent control is introduced through

$Q=\Delta-A$6

and

$Q=\Delta-A$7

where $Q=\Delta-A$8 is built from arrival and departure rates and boarding and alighting capacities (Schanzenbacher et al., 2018).

Under mild technical conditions, the dynamics collapse to a Max-Plus linear system with closed-form steady-state headway

$Q=\Delta-A$9

and corresponding frequency

$\Delta=\mathrm{diag}(d_i)$0

with $\Delta=\mathrm{diag}(d_i)$1. These expressions partition the $\Delta=\mathrm{diag}(d_i)$2-plane into three regimes: free-flow, capacity-limited, and congested. The free-flow phase is governed by total cycle time, the capacity-limited phase by the slowest station or run combination, and the congested phase by block occupancy as $\Delta=\mathrm{diag}(d_i)$3 (Schanzenbacher et al., 2018).

A related line model without passenger-demand control is stable and converges to stationary regimes with a unique asymptotic average growth rate, but introducing a naïve demand-dependent dwell law makes the traffic unstable. The subsequent control model reverses the sign of the passenger term so that dwell-time feedback contracts large gaps rather than amplifying them; the authors report that the controlled dynamics are stable, converge to stationary regimes with a unique asymptotic average growth rate, preserve the max-plus capacity up to a demand threshold, and then degrade gracefully as demand rises further (Farhi et al., 2016).

Graph-diffusion models provide a different operational description. For morning rush hour in the London Underground, commuter movement is modeled through a discrete-graph version of Fick’s law. If $\Delta=\mathrm{diag}(d_i)$4 is the observed net flow vector and $\Delta=\mathrm{diag}(d_i)$5 is the resident population around each station, then

$\Delta=\mathrm{diag}(d_i)$6

This recovers a spatial pattern of residential versus business areas from station-level entry and exit data (Dees et al., 2021).

4. Passenger assignment, crowding, and demand forecasting

Automated Fare Collection data are central to modern metro inference, but they are incomplete by design. For each trip, AFC systems directly provide tap-in and tap-out timestamps and stations; the train and route chosen by a passenger are unknown. Zhao et al. formulate this as a probabilistic route-choice problem for OD pairs with multiple effective routes, dividing the day into time-slots and assigning route-choice probabilities $\Delta=\mathrm{diag}(d_i)$7. Segment travel and waiting components are governed by densities $\Delta=\mathrm{diag}(d_i)$8, and the likelihood

$\Delta=\mathrm{diag}(d_i)$9

is optimized by an EM-type procedure (Zhao et al., 2016). The Shenzhen validation uses 60 days, 4 million cards, and 300 million tap events, and reports, for example, that on the WuHe$d_i=\sum_j a_{ij}$0YanNan OD pair about $d_i=\sum_j a_{ij}$1 still choose the longer single-transfer route, confirming a transfer penalty (Zhao et al., 2016).

Crowding-aware assignment leads to a more coupled formulation. In the Hong Kong MTR study, path shares $d_i=\sum_j a_{ij}$2 follow a C-logit model, but the delay rate $d_i=\sum_j a_{ij}$3 must remain consistent with network loading under finite train capacities, timetable constraints, and First-In-First-Board rules. Because the resulting optimization has a non-analytical constraint and a non-linear equation constraint, it is decomposed into three sub-problems: assignment and crowding update, rough path-share estimation by quadratic programming, and C-logit parameter update by weighted maximum likelihood (Mo et al., 2020). In the 18:00–19:00 peak case study, model-implied arrival and boarding counts at Admiralty northbound matched AFC counts within less than $d_i=\sum_j a_{ij}$4 error, and the network-level matching error was $d_i=\sum_j a_{ij}$5 lower than survey-derived path shares and $d_i=\sum_j a_{ij}$6 lower than uniform path shares (Mo et al., 2020).

Forecasting in-situ density extends this pipeline from route inference to segment occupancy. The PIPE framework first predicts the time-dependent OD matrix $d_i=\sum_j a_{ij}$7 and then estimates route travel times with truncated normal mixture distributions fit by EM. Expected onboard passenger count on edge $d_i=\sum_j a_{ij}$8 at time $d_i=\sum_j a_{ij}$9 is computed as

$N$00

On Singapore MRT data comprising 102 stations, 7 lines, 114 edges, and 251 million trip records, random forest produced the lowest one-step-ahead OD-cell MSE at $N$01, compared with $N$02 for the vanilla model, $N$03 for ARIMA, and $N$04 for LSTM; for alighting-rate prediction, the full PIPE model achieved MSE $N$05, compared with $N$06 for random forest and $N$07 for the vanilla baseline (Tian et al., 2020).

Spatial-temporal deep learning introduces richer graph structure. PVCGN models a metro system through a physical graph, a similarity graph based on DTW similarity of historical inflow and outflow series, and a correlation graph derived from OD ratios. These graphs feed a Graph Convolution Gated Recurrent Unit, while a Fully-Connected GRU captures global temporal tendencies; both are embedded in a Seq2Seq architecture (Liu et al., 2020). On SHMetro, PVCGN achieved $N$08-minute-ahead RMSE $N$09, MAE $N$10, and MAPE $N$11, improving over DCRNN at $N$12, $N$13, $N$14 and GCRNN at $N$15, $N$16, $N$17. The same framework also reduced $N$18-minute online OD MAPE from approximately $N$19 to $N$20 (Liu et al., 2020).

Exogenous covariates have more limited but measurable effects. In hourly passenger-flow forecasting with weather variables, weekday performance improved only marginally, while weekend performance improved more clearly. For the Bagging Regressor, weekday MAE changed from approximately $N$21 passengers per hour to approximately $N$22, whereas weekend MAE fell from approximately $N$23 to approximately $N$24; barometric pressure had the strongest individual effect, especially at low-volume or tourist stations during weekends (Hu et al., 2022).

5. Digital infrastructure, maintenance, and metro-linked analytics

Predictive maintenance research treats the metro as an instrumented cyber-physical asset. The MetroPT dataset was collected in 2022 with an urban metro public transportation service in Porto and includes analog sensor signals, digital signals, and GPS information sampled uniformly at $N$25 Hz. The public subset covers January–June 2022 and contains $N$26 points in a single time-stamped CSV with $N$27 columns: $N$28 analog, $N$29 digital, and $N$30 GPS (Veloso et al., 2022). The benchmark framing emphasizes online anomaly detection and horizon failure prediction, with the company requirement that warnings be raised at least $N$31 hours before the observed non-operational point of the train (Veloso et al., 2022).

The same dataset summary reports several baseline methods. A rule-based system, One-Class SVM, Random Forest, and LSTM autoencoder are listed, with the LSTM autoencoder achieving precision $N$32, recall $N$33, F1 $N$34, and AUC $N$35, while detecting failures on average $N$36 hours before the non-operational point (Veloso et al., 2022). The paper’s broader implication is methodological rather than system-specific: metro maintenance data are multimodal, noisy, and highly imbalanced, with only three labeled failures in six months (Veloso et al., 2022).

Digital metro analytics also extend beyond operations into spatial decision support. RailEstate is a web-based system for metro-linked property analysis in the Washington metropolitan area. It integrates Zillow monthly home-value index data at ZIP level from 2000 to 2025, WMATA GTFS feeds, and ZIP-code boundary GeoJSON; normalizes them into six tables; and stores them in PostgreSQL/PostGIS hosted on Supabase (Chang et al., 29 Oct 2025). Spatial GiST and attribute B-tree indexes yield sub-second query latencies for point-in-polygon and distance filters over 300K+ rows, while a natural-language interface routes React ChatWidget input through FastAPI to a LangChain SQLDatabaseChain using GPT-4o-mini, sanitizes the SQL, executes it through Supabase’s RESTful API, and formats results into prose (Chang et al., 29 Oct 2025).

Forecasting in RailEstate relies on Zillow’s proprietary Home Value Forecast through the recursive update

$N$37

and the system also allows future substitution of ARIMA, linear regression, or LSTM models (Chang et al., 29 Oct 2025). In this literature, metro accessibility is not merely a transport variable; it is also an explanatory variable for neighborhood-level housing dynamics.

6. Expansion planning, service design, and cartography

Metro expansion is a transport network design problem that has increasingly been cast as sequential decision-making. MetroGNN formalizes metro network expansion as a Markov decision process on an urban heterogeneous multi-graph whose nodes are regions and whose edges encode spatial adjacency and OD intensity. At each step the action adds one new region node and one edge, and the reward is the incremental gain in satisfied OD demand (Su et al., 2024). The attentive policy network scores candidate nodes using node embeddings from a graph neural network, then masks infeasible actions under geometric constraints. On Beijing with budget $N$38, MetroGNN achieved satisfied OD flow $N$39, compared with $N$40 for DRL-CNN, a $N$41 improvement; averaged over budgets and two cities, it yielded $N$42 higher satisfied OD than the best non-RL baseline (Su et al., 2024).

A later line of work argues that some metro expansion problems are small enough not to require Deep RL. Reformulating the Metro Network Expansion Problem as a Non-Markovian Reward Decision Process, the tabular method defines states as grid cells, actions as eight directions, and uses an 8-bit action mask to prevent revisits, boundary violations, cycles, and U-turns. The framework incorporates Max Efficiency, Equal-Sharing via the Generalized Gini Index, and Rawlsian max-min justice as reward criteria (Michailidis et al., 2 Jun 2026). In Xi’an and Amsterdam, TabularMNEP remained competitive with DeepRL while requiring 25,000 episodes rather than 448,000, reducing total episodes by a factor of $N$43 and total carbon emissions by a factor of $N$44 on average (Michailidis et al., 2 Jun 2026).

Service design also operates at the level of passenger-perceived quality. The DT-IPA model combines Importance-Performance Analysis, CART decision trees, and Analytic Hierarchy Process feasibility scoring. In the Changsha Metro study, 18 service-quality attributes were measured on a 5-point Likert scale from 107 valid responses, Cronbach’s $N$45 was $N$46, and the optimal tree had 11 nodes, 6 leaves, depth $N$47, and cross-validation misclassification error approximately $N$48 (Weiya et al., 2021). Initial Priority-1 attributes were Ticket-and-top-up service, In-station crowding, and Carriage crowding; feasibility and decision-tree rules then demoted Carriage crowding and quantified required improvement amounts for Ticket and Safety under a rule with confidence $N$49 (Weiya et al., 2021).

Metro cartography constitutes another design layer. The mixed-metro-map problem seeks a layout in which a subset of edges approximates a user-supplied guide shape while the remaining edges follow schematic conventions such as octilinearity. The framework defines seven design criteria, from constrained layouts and topographic accuracy to shape representability and hybrid orientation, and solves the problem in three phases: route detection and selection, shape-and-layout deformation, and grid alignment (Batik et al., 2022). End-to-end runtime ranges from approximately $N$50 seconds for Lisbon with 60 stations to $N$51 minutes for Paris with 304 stations. In a user study with 63 participants, over $N$52 correctly traced simple shapes such as the flower, heart, and circle; the “eye” was recognized by $N$53, the more detailed bear by $N$54, and simple shapes improved route-planning accuracy by approximately $N$55–$N$56 while reducing completion time by $N$57–$N$58 (Batik et al., 2022).

7. Urban and socioeconomic effects

Metro systems are studied not only as transport infrastructures but also as instruments that reshape modal split. In a European sample of nearly 400 cities compiled through CitiesMoving.com under the ABC framework, cities were categorized as having a metro, being tram-only, or having neither rail mode. The final sample contains 47 metro cities, 46 tram-only cities, and 285 no-rail cities (Prieto-Curiel, 11 Nov 2025). Population-weighted car shares differ sharply across these groups: $N$59 Among cities with population above 750,000, the car share is $N$60 in metro cities versus $N$61 in non-metro cities, widening the gap to 27 percentage points (Prieto-Curiel, 11 Nov 2025).

The compositional analysis uses Dirichlet regression with a log link, incorporating HasMetro, HasTram, HasRail, $N$62, and interaction terms. The reported coefficients, all with $N$63, show that HasMetro boosts $N$64 and reduces $N$65, especially in larger cities, whereas HasTram has the opposite sign once interactions with size are accounted for (Prieto-Curiel, 11 Nov 2025). The paper interprets this difference through vehicle and network characteristics: metros are high-capacity, high-speed, long-distance carriers that remain attractive as cities grow, while tram systems tend to serve shorter corridors with lower speeds and capacities (Prieto-Curiel, 11 Nov 2025).

Metro accessibility also enters urban housing analysis. RailEstate explicitly treats proximity to metro stations as a driver of residential property demand in the Washington metropolitan area, enabling ZIP-code-level exploration of price trajectories and forecasts around specific stations (Chang et al., 29 Oct 2025). This does not establish a universal causal law, but it does indicate how metro-linked accessibility is being operationalized in spatial analytics systems.

Taken together, the literature presents metro as a layered object. At one layer it is a graph with cycles, conductance, algebraic connectivity, and degree correlation; at another it is a controlled queueing and diffusion system with dwell-time feedback, headway regimes, and passenger assignment; at another it is a platform for forecasting, maintenance, safety, cartography, and reinforcement-learning-based expansion; and at still another it is an urban intervention associated with lower car use and with neighborhood-scale accessibility effects (Wang et al., 2015, Prieto-Curiel, 11 Nov 2025).