⚡ electrical engineering

A Multiplex Graph Framework for Vertical Urban Renewal: Rooftop Connectivity and Mobility Optimization in Historic Dense Cities

This study proposes a constraint-driven multiplex graph framework that integrates a modular rooftop sky-network with ground-level streets to significantly reduce travel costs and improve environmental conditions in the historic, high-density urban fabric of Puran Dhaka.

Original authors: Farhan Labib, Maliha Tabassum Maurin

Published 2026-07-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Farhan Labib, Maliha Tabassum Maurin

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine Old Dhaka, a historic part of the city in Bangladesh, as a giant, incredibly crowded puzzle. The streets are so narrow and packed with people and traffic that moving from one place to another feels like trying to run through a maze made of wet spaghetti. The buildings are so close together that you can almost touch the next one, but the ground below is stuck in a traffic jam that never ends.

This paper proposes a clever, three-dimensional solution to this problem: Stop trying to fix the traffic on the ground; start using the sky.

Here is the simple breakdown of their idea, using everyday analogies:

1. The Problem: A "Flat" City is Too Full

The authors looked at a specific area called Chawkbazar. It's so dense that 37.5% of the streets are so clogged with people that walking is a nightmare. You can't just build wider roads because the buildings are historic and protected; tearing them down would destroy the city's soul. It's like trying to widen a hallway in a house that's already full of furniture—you can't move the furniture, so you have to find a new way to walk.

2. The Solution: A "Layer Cake" City

Instead of just one layer (the ground), the researchers suggest turning the city into a three-layer cake:

The Bottom Layer (Ground): The existing streets. We leave these mostly alone.
The Middle Layer (Canopy): A lightweight roof over the streets to give people shade and protection from rain (like a giant, transparent umbrella over the sidewalk).
The Top Layer (The Sky-Net): This is the big idea. They propose building a network of lightweight bridges connecting the rooftops of the buildings.

Think of the rooftops as a new "highway" floating above the chaos. Since the buildings are so close together, you can walk from one roof to another without ever touching the crowded street below.

3. The "Math Magic" (Graph Theory)

To prove this works, the researchers used a mathematical tool called a Multiplex Graph.

The Analogy: Imagine a video game map. Usually, you can only walk on the ground. This model adds a second map on top of the first one (the roofs) and connects them with elevators (vertical stairs).
The Rules: You can't just build a bridge anywhere. The math checks if the buildings are close enough (within 15 meters) and if the height difference isn't too big (less than 3 meters, or one story). It also checks if the roof is strong enough to hold people. If a roof is made of flimsy tin, the math says, "No bridge here, it's too risky."

4. The Results: A Massive Shortcut

When they ran the numbers using real data from Old Dhaka:

The "Ground Only" Trip: Walking just on the streets took a long time and cost a lot of "energy" because of the crowds.
The "Sky Net" Trip: By using the rooftop bridges, the travel time dropped by 58.4%.
The Analogy: It's like being stuck in a traffic jam on a highway, but then realizing you can hop onto a private, empty train track running right above the cars. You get to your destination almost twice as fast.

5. Bonus Benefits: More Than Just Walking

The paper also suggests that this "Sky Net" does more than just move people:

Cooling the City: If they put plants (green roofs) on these bridges, it acts like a giant air conditioner, lowering the temperature of the city by about 9°C (16°F).
Managing Rain: The roofs can catch rainwater, acting like a sponge to prevent flooding, rather than letting it all rush down the streets.
Light: It lets more sunlight into the dark, narrow alleys below.

6. Is it Realistic?

The researchers were careful. They didn't just dream up a fantasy city. They looked at 211 actual buildings in Old Dhaka.

They found that while you can't connect every building to every other building (the "connectivity" is actually quite low, only about 3.3% of all possible connections are possible), the ones you can connect are enough to make a huge difference.
They estimated the cost to be roughly $22–$ 35 million for the whole area, which they suggest could be built in phases, starting with the most crowded spots first.

The Bottom Line

This paper argues that in historic, crowded cities where you can't tear down old buildings to make new roads, the best solution is to look up. By turning rooftops into a connected network of walkways, you can bypass the ground-level chaos, cool down the city, and save time, all while keeping the historic streets exactly as they are. It's a way to give an old city a new "second floor" for movement without touching the "first floor" of history.

Technical Summary: A Multiplex Graph Framework for Vertical Urban Renewal

Problem Statement
Historic urban centers, particularly in South Asia such as Old Dhaka (Puran Dhaka), face critical challenges including extreme population density (locally exceeding 143,000 persons/km²), inadequate pedestrian connectivity, severe shortages of public open space, and recurrent environmental stressors like urban heat islands. Traditional ground-level redevelopment is often impractical due to the risk of irreversible damage to the historical and socio-cultural fabric, as well as the physical constraints of narrow, irregular street networks (often <3–5 meters wide). Existing vertical urban models or elevated pedestrian systems (e.g., in Hong Kong or Minneapolis) lack a cohesive, optimization-driven theoretical framework that integrates ground circulation, rooftop mobility, and environmental functions into a unified system.

Methodology
The study proposes a three-tier vertical renewal framework modeled as a weighted multiplex graph, $G_{city} = (V_g \cup V_r, E_g \cup E_r \cup E_v, W)$ . The system is tested on a 58,379.75 m² cluster in Chawkbazar, Old Dhaka, comprising 211 building nodes.

Graph Construction:
- Ground Layer ( $E_g$ ): Represents existing streets. Edge weights incorporate Euclidean distance and a congestion coefficient ( $c_{ij}$ ) derived from street width and pedestrian density. The model classifies 37.5% of local streets into high-congestion tiers.
- Rooftop Layer ( $E_r$ ): Represents a modular sky-network for pedestrians and bicycles. Connectivity is governed by strict structural feasibility constraints: a maximum span ( $R = 15.0$ m) and a maximum height difference ( $\Delta h = 3.0$ m). Edge weights include distance and a structural friction factor ( $s_{ij}$ ) penalizing connections to older, less stable iron-sheet roofs.
- Vertical Layer ( $E_v$ ): Represents transitions between ground and rooftop within the same building footprint, weighted by a vertical penalty factor ( $\delta = 3.0$ ) reflecting higher energy expenditure for climbing.
Optimization and Simulation:
- The multiplex network is transformed into a single-layer supra-graph to enable computationally efficient shortest-path analysis using Dijkstra's algorithm.
- Network efficiency ( $\eta$ ) is calculated as the reduction in average shortest-path travel cost compared to a ground-only baseline.
- Environmental simulations assess Urban Heat Island (UHI) mitigation, daylight availability (Vertical Daylight Factor), and rainwater management using the Rational Method.
Validation:
- The model is validated against OpenStreetMap data, field surveys of 50 buildings (assessing roof types and spacing), and manual route tracing.
- Statistical robustness is confirmed via bootstrapping (10,000 iterations) and comparison against stochastic null models, ensuring the efficiency gains are not artifacts of random connectivity.

Key Contributions

Formal Multiplex Urban Model: The paper introduces a constraint-driven graph-theoretic framework that formally integrates ground streets, rooftop connections, and vertical transitions, addressing the fragmentation between mobility, heritage conservation, and environmental planning.
Constraint-Based Feasibility: Unlike theoretical "vertical city" concepts, this model incorporates realistic engineering constraints (span limits, height tolerances, roof structural integrity) to generate a sparse but functionally viable rooftop network (Roof Connectivity Index, RCI = 0.033).
Integrated Environmental-Mobility Assessment: The framework simultaneously evaluates mobility optimization and environmental co-benefits (thermal comfort, daylight, stormwater retention) within a single computational model.

Results

Mobility Optimization: The integration of the rooftop layer reduces the weighted travel cost by 58.4% compared to the ground-only network. The multiplex network achieves an average path length of 77.0 units versus 185.0 units for the ground network.
Network Characteristics: Despite a low Roof Connectivity Index (3.3%), the rooftop layer acts as a critical "sparse" supplement, providing alternative routing that bypasses ground-level congestion. Optimal paths dynamically transition between layers (Ground $\to$ Rooftop $\to$ Ground).
Environmental Impact:
- Thermal: Simulations with 60% green roof coverage and 40% reflective surfaces predict a surface temperature reduction of 9.1°C and an improvement in Physiological Equivalent Temperature (PET) from 32.1°C to 29.8°C.
- Hydrological: The system reduces the runoff coefficient from 0.95 to 0.56, resulting in a 41.1% reduction in peak flow and a retention volume of 198 m³.
- Daylight: Rooftop spaces achieve a Daylight Autonomy (DA) of 60.0%.
Economic Feasibility: A preliminary cost estimation suggests a total capital cost between $22.37M and $35.29M for the full system, with a phased implementation strategy proposed to manage upfront investment.

Significance and Claims
The paper claims that urban performance improvement in dense historic cities is not solely dependent on increasing network density, but on strategic spatial connectivity patterns. The study demonstrates that a reversible, modular, and constraint-driven vertical approach can significantly enhance accessibility and livability without disrupting the existing heritage fabric.

The authors emphasize that their framework offers a computationally optimized, scalable, and reversible strategy for sustainable urban renewal. By proving that a sparse rooftop network can yield substantial efficiency gains (58.4%) and environmental benefits, the study challenges the necessity of ground-level demolition in historic cores. The work positions the multiplex graph model as a tool for "heritage-sensitive densification," enabling functional expansion through vertical layering rather than horizontal sprawl.

The authors maintain a modest stance, acknowledging that the study prioritizes structural representation over large-scale behavioral validation. They note that actual user adoption depends on factors like safety and familiarity, which require future empirical study, and that thermal/daylight models rely on steady-state approximations. However, the core claim remains that the proposed framework provides a theoretically sound and mathematically validated pathway for optimizing mobility and environment in constrained historic urban contexts.