Superlinear urban scaling by functional organization: A metabolic interpretation of sectoral water consumption

Prevailing view asserts that the disproportionately greater productivities of larger cities, or superlinear urban scaling, are the result of human interactions channeled by urban networks. But this view was established by considering the spatial organization of urban infrastructure and social networks—the urban “arteries” effects—but neglecting the functional organization of urban production and consumption entities—the urban “organs” effects. Here, adopting a metabolic view and using water consumption as a proxy for metabolism, we empirically quantify the scaling of entity number, size, and metabolic rate for the functionally specific urban residential, commercial, public or institutional, and industrial sectors. Sectoral urban metabolic scaling is highlighted by a disproportionate coordination between residential and enterprise metabolic rates, attributable to the functional mechanisms of mutualism, specialization, and entity size effect. The resultant whole-city metabolic scaling exhibits a constant superlinear exponent for water-abundant regions in numerical agreement with superlinear urban productivity, with varying exponent deviations for water-deficient regions explainable as adaptations to climate-driven resource constraints. These results present a functional organizational, non-social-network explanation of superlinear urban scaling.

Figure 1

Sectoral urban metabolic scaling. Sixteen-year (2005–2020) time series of best-fit slopes $b$ and intercepts $a$ for sectoral vs. whole-city (fractional) scaling of (a) entity number, (b) size, (c) size per entity, (d) water consumption (as metabolic rate), and (e) water consumption per entity for cities in WI, PA (both humid), UT (extensively dry), and CA (partially dry). Superlinear urban scaling of consumption with population results from the inverse of the sublinear residential fractional scaling [(d), panel 5, indicated by the arrow], given that residential consumption is proportional to residential population under water-abundant (humid) conditions.

Figure 2

Data (colored circles, each representing a city) and fits (solid black lines) for the scaling of (a)–(d) sectoral cumulative water consumption and (e)–(g) sectoral per-entity water consumption with whole-city water consumption for cities in WI, PA (both humid), UT (extensively dry), and CA (partially dry). Histograms show log-transformed distributions of whole-city consumptions with log-space normal distribution fits (red curves). Highlighted data points indicate the functionally specialized cities of Madison (MAD; public), Kohler (KOH; industrial), Macungie (MAC; industrial), and Gettysburg (GET; commercial). For clarity, the plots for the residential, commercial, public, and industrial sectors are shifted on the vertical axis by 2.5, 5, 7.5, and 10, respectively. The largest city in each state is labeled: Milwaukee (MIL), Philadelphia (PHI), Salt Lake City (SLC), and Los Angeles (LA).

Figure 3

Rank-size distributions of cities in (a)–(d) sectoral cumulative water consumption and (e)–(g) sectoral per-entity water consumption shown as the ranked probability ($y$ axis) of finding consumption sizes equal to or greater than a given consumption size ($x$ axis) for cities in WI, PA, UT, and CA. Solid lines are lognormal distribution fits (average $R^2=0.99$; two-sample Komogorov-Smirov test for lognormalcy, average $p=0.92$). Rank locations of the specialized cities and the largest city in each state (see Fig. 2 caption) are indicated. For clarity, the plots are relatively shifted on the horizontal axis.

Figure 4

Data (colored circles, each representing a city) and fits (solid black lines) for the scaling of whole-city water consumption with city population for cities in WI, PA, IL, MO, NY (all humid), UT (extensively dry), and CA (partially dry). Histograms show log-transformed distributions of city populations with log-space normal distribution fits (red curves). Dashed lines are references of linear scaling ($b=1$). The largest city in each state is labeled: Milwaukee (MIL), Philadelphia (PHI), Chicago (CHI), Kansas City (KAN), New York City (NYC), Salt Lake City (SLC), and Los Angeles (LA). Insets show scaling of residential sectoral consumption with population (Fig. S21 [40]).