How the Width of Links in a Sankey Diagram Represents Flow Magnitude and Preserves Conservation

A standard flowchart maps the sequence of a process, showing where a system moves from one step to the next. A Sankey diagram maps the sequence but adds a strict physical constraint: the width of every line must be exactly proportional to the quantity flowing through it. If a pipeline splits into three smaller branches, the combined width of those three branches must equal the width of the original pipeline. This visual enforcement of the conservation of mass and energy transforms a simple map into a rigorous accounting system.[5][7]

The mechanism relies on nodes and links. Nodes represent states, locations, or stages in a process, while links represent the flow between them. The fundamental rule governing these charts is that the sum of the widths of all incoming links to a node must equal the sum of the widths of all outgoing links, minus any explicitly graphed losses. This ensures that no data is magically created or destroyed within the visualization, a principle first codified in 1898 by Irish engineer Captain Matthew Henry Phineas Riall Sankey.[5][6]

Captain Sankey originally developed the diagram to illustrate the thermal efficiency of a real steam engine, visually demonstrating how much heat was converted into useful work and how much was lost to the environment. Today, energy and sustainability analysts rely on this exact framework to map national power grids and exergy flows. A 2014 framework established standardized methods for using Sankey diagrams to track both energy, which is conserved at 100 percent, and exergy, the useful portion of energy that is consumed, allowing engineers to pinpoint exactly where thermodynamic losses occur in industrial plants.[4][5]

The application has expanded significantly into architecture and building performance simulation. Researchers at Carleton University have investigated how Sankey diagrams can enhance the design process by visualizing heat transfers, HVAC energy consumption, and thermal losses through building envelopes. By mapping these flows proportionally, architects can instantly see whether a building's primary energy drain is poor insulation or inefficient lighting, driving more targeted sustainability interventions during the early phases of a project.[3]

Beyond physical energy, the conservation principle is now being applied to human populations. Clinical researchers use Sankey flow diagrams to track symptom trajectories in older adults with advanced cancer. In this context, the "flow" is the number of patients moving between different states of symptom severity over time. The proportional width of the links allows oncologists to visualize the dominant pathways of disease progression and symptom burden, providing a stark, quantitative picture of patient outcomes that traditional survival curves cannot capture.[1]

As datasets grow more complex, the diagrams are evolving to handle higher dimensionality. Hybrid Sankey diagrams developed at the University of Cambridge integrate multidimensional data to better understand resource use. These advanced models allow analysts to overlay additional variables—such as carbon intensity or financial cost—onto the standard flow magnitude, using color gradients or interactive filtering while maintaining the foundational rule of proportional link width.[2]

The evidence supporting the utility of Sankey diagrams is robust across disciplines, but the format has distinct limitations. Because the human eye struggles to accurately compare the areas of curved, diverging paths, Sankey diagrams are highly effective for showing massive disparities but less precise for comparing closely matched values. Furthermore, if the underlying data contains measurement errors or fails to account for all inputs and outputs, the strict conservation rule forces the diagram builder to either introduce a "dummy" node for unaccounted losses or break the visual integrity of the chart.[6][7]

While the cited academic literature provides rigorous mathematical frameworks, the authors present their findings through technical documentation rather than direct public commentary, leaving the charts to speak for themselves. The enduring power of the Sankey diagram lies in its refusal to let data hide. By anchoring data visualization to the physical laws of conservation, it forces a complete accounting of the system being measured. Whether tracking the thermal efficiency of a 19th-century steam engine or the symptom progression of modern cancer patients, the proportional width of the links ensures that every input is tracked to its ultimate destination.[1][5][7]