# Urban topology and dynamics can assess green areas importance

Jacopo Moi  DSMN and ECLT, Ca’Foscari University of Venice
Fondazione Futuro delle città, Florence, Italy  Leonardo Chiesi  University of
Florence, Florence Italy
Fondazione Futuro delle città, Florence, Italy  Gherardo Chirici
<EMAIL_ADDRESS>University of Florence, Florence Italy
Fondazione Futuro delle città, Florence, Italy  Saverio Francini  University
of Florence, Florence Italy
Fondazione Futuro delle città, Florence, Italy  Costanza Borghi  University of
Florence, Florence Italy
Fondazione Futuro delle città, Florence, Italy  Paolo Costa  University of
Florence, Florence Italy
Fondazione Futuro delle città, Florence, Italy  Bianca Galmarini  University
of Florence, Florence Italy
Fondazione Futuro delle città, Florence, Italy  Guido Caldarelli
<EMAIL_ADDRESS>DSMN and ECLT, Ca’Foscari University of Venice,
Italy
Institute of Complex Systems CNR (ISC CNR), Depart. Phys. ”Sapienza”
University Rome, Italy
London Institute for Mathematical Sciences, Royal Institution, London UK

###### Abstract

Green areas are a crucial element in evolution of a city, contributing to
improve citizens’ life, to reduce effects of climate change, and to make
possible the survival of other species in urban areas. Unfortunately, the
above effects are difficult to assess quantitatively for regulators,
stakeholders and experts, making troublesome the planning of city development.
Here we present a method to estimate the impact of these areas in the city
life based on the network topology of the city itself and on a simple model of
dynamics on this structure. Movements between various areas of the city are
simulated by means of an agent-based biased-diffusion process where citizens
try to reach the nearest Public Green Area (PGA) from their position and the
model is fed with real data about the density of populations in the cases of
study. Firstly, we define a centrality measure of PGA’s based on average
farness measured on the city network; this approach outperforms information
based on the simple topology. We then improve this quantity by taking into
account the occupation of PGA’s, thereby providing a quantitative measure of
PGA usage for regulators.

Suggested keywords

## Introduction

Urban green spaces, which encompass parks, gardens, urban forests, and other
forms of vegetation, provide cities with a diverse array of environmental,
climate, economic, social, and health benefits. From an ecological point of
view, the extension of urban green infrastructures, improves biodiversity and
ensures the proper functioning of ecological systems, fostering the prosperity
of various animal species [1]. Green areas also provide a series of ecosystem
services (such as air purification, carbon sequestration, urban cooling,
runoff control, noise reduction, and habitat maintenance [2]) essential for
city life. Finally. green spaces improve city resilience, particularly against
the adverse effects of climate change, and enhance the overall quality of the
urban environment. The economic significance of green spaces is evident as
well, as greening is frequently linked to higher home prices and increased
property values, and serves as an effective strategy for branding and
attracting international capital investments [3]. However, beyond the
ecological and the economic realm, the value of green spaces and particularly
the Public Green Areas (PGA) becomes even more important when viewed from the
citizens’ perspective. A robust and ecologically healthy natural component is
crucial for health and well-being. Enhancing environmental conditions
positively impacts life, particularly in cities where pollution, noise, and
heat pose significant challenges. Furthermore, it is now clear how direct
contact with nature is a vital condition to achieve an healthy urban life
reducing stress, anxiety, and depression, as well as reduced risks of obesity,
cardiovascular diseases, and respiratory illnesses [4, 5, 6, 7, 8]. Among less
immediate benefits, exposure to nature facilitates personal growth and
learning, favours prosocial behaviours and cooperation [9], enhances social
cohesion [10] and environmental awareness. The structure, size, connectivity,
and biodiversity of green spaces are determinants for the provision of
benefits [5]. However, health and social benefits also hinge on the dose of
nature experience [11], thus its frequency, duration, and recurrence. The
variation in these factors raises equity concerns, as differences in access
among different segments of the population can compromise the fair
distribution of health benefits. Consequently, distributive and accessibility
considerations regarding green spaces in cities emerge as indispensable
components for informing a just green planning. We present here a metric based
on statistical physics framework to assess the relevance of PGAs in city life.
The metric takes into account position and shape of areas with respect to the
rest of the city and the facility to access these areas by the population. A
way to assess quantitatively the accessibility of these areas is to measure
their centrality in the city structure. The approach of Statistical Physics is
particularly suitable in this case, given the large sizes and the strong
heterogeneity of the urban systems [12]. More particularly complexity science
[13, 14] and concepts developed in network theory[15], as the centrality
mentioned above, are the most effective ways to achieve a green area metric.
Indeed, by reducing the complexity of cities[16] to a web of city blocks in
the form of a network we can identify even social patterns of urban city life
that, when quantified, appear to follow clear mathematical laws[17, 18].
Recent research on human interactions in cities embraced all these ideas,
trying to produce models that, starting from different data, create
comprehensive multilevel networks[19] that can help understand the phenomenon
better than a digital twin[16]. Indeed, with such a macroscopic level of
description, it is possible to detect a series of “universal” patterns
emerging in cities worldwide[20], indicating that the size of a city –
measured by its population – is a primary determinant of its characteristics.
Through this quantity, it is possible to assess the importance of the various
zones of the city and the other relative properties as the connectivity, the
clustering, the assortativity effects[21], etc. In detail, we consider the
network structures obtained by Voronoi tessellation for three Italian cities
of different sizes and different populations (high, medium, medium/small):
Milan, Florence and Mestre, respectively. On these structures, we evaluate the
accessibility of PGAs at the city block level by utilizing both a traditional
network centrality measure and modified versions that account for spatial
interaction and population density. On these areas, we first perform a
centrality analysis, and then, with the help of a simple dynamical model, we
obtain a computational measure of the usability of such areas.

## Methods

###  .1 Voronoi Tassellation and Networks

We base our analysis on weighted spatial networks created through the
tessellation [22] of three Italian cities: Mestre (Venice), Florence, and
Milan. The weighted spatial network, denoted as $G(V,E)$, comprises nodes
($V$) representing centroids, and weighted edges ($E$) capturing their
interactions. Nodes are defined as $V={v_{1},v_{2},...,v_{N}}$, and edges are
represented as $E={(v_{i},v_{j},w_{ij})}$, with $w_{ij}$ representing the
Euclidean distance between adjacent nodes $v_{i}$ and $v_{j}$. In our network
representation, we have two types of nodes: city blocks (denoted as $b$) with
associated population data from the census[23], and Public Green Areas (PGA,
denoted as $p$). We take this data from “OpenStreetMap”[24]. Each city
displays distinct features in population density and urban layout. City of
Mestre has a concentrated, recent development, while Florence shows an uneven
distribution of population and city blocks some of which with hundreds of
years of story. Milan, a densely populated metropolis, exhibits a more regular
urban structure with ancient and new buildings and with varying population
densities. To evaluate the importance of block nodes relative to the set of
PGAs nodes, we employed various measure of network centrality. In particular,
we shall focus on the idea that the probability of performing a walk between a
block and a PGA follows a well-defined probability distribution that depends
on both network topology and edge length as cost. In the following, we shall
look for the ensemble of “low cost” paths, from any city block to the PGA’s.
Furthermore, PGA nodes that are the target of these walks will be modeled in
the first instance as ’sink’ of infinite capacity and in the second instance
with a finite capacity given by their walkable area.

###  .2 Taking into account the social network

Cities, in general, are characterized by different population densities per
block, which inevitably play a determining factor in characterizing the use of
and accessibility to certain services. This is well known even before the
advent of complex networks, from the principle of human behaviours[25], to
models explicitly pointing out the role of distance in social
interactions[26], to the advantageous concentration of diverse businesses and
individuals in urban areas promoting economic vitality through closeness [27]
over time. All the possible social contacts and relationships with
transportation and supply networks, concur in creating a series of different
co-evolving structures describing the evolution of life in cities. The
usability of PGA’s is inherently tied to their physical proximity to the
population, a factor that, in turn, significantly influences their potential
use. In other terms, the quantity of population physically served by each
green space is a determinant for rating its role in service providing and its
likelihood to use. According to the least effort hypothesis [25], proximity
has a pivotal role in shaping behaviours and choices, and consequently, social
interactions. In the present study, we advance a dynamic network-based model
that posits people density and proximity as primary determinants of agency. At
the same time, we acknowledge the intricate interplay of numerous other
variables, here unexplored and more difficult to model, influencing the use of
public green areas, including design, governance mode, maintenance level,
biodiversity, quality, dimension, but also the non-normativity and functional
indeterminacy[28].

###  .3 Probability of paths to PGAs nodes

The accessibility to Public Green Areas (PGAs) is defined as the probability
of a block resident’s access to PGAs, primarily based on the distance between
the block and PGAs. To derive a probability distribution for each block, we
employed the bag-of-path approach[29], which employs the principles of random
walk theory within network analysis. A random walk on a network is a
stochastic process that starts from a given vertex $v$, and then selects one
of its neighbors according to a probability distribution. A naturally defined
distribution arises by imposing a uniformly likely passage on the edges. This
means to consider the degree of the node $P_{v\rightarrow
u}=\frac{1}{\text{Deg}(v)}$, if and only if $u,v$ are connected. The latter
relationship can be calculated for each pair of nodes $u,v$ and stored in a
transition matrix $\mathbf{P}_{\text{ref}}$. A typical process in networks is
when specific nodes are made adsorbing, causing random walks from node $i$ to
halt indefinitely at node $j_{\text{ads}}$. This properties can be modelled by
placing $p_{j_{ads},j_{ads}}=1,p_{j_{ads},i}=0$ in the transition matrix. In
this case, a quantity of interest is the expected number of times that a walk
from any node $i$ reaches one of the absorbing nodes ${j_{\text{ads}}}$. This
is computed for each starting node by summing all possible paths of length
$t$, accounting for the adsorbing property of selected nodes[30]:

$\mathbf{Z}=\sum_{t=0}^{\infty}\mathbf{P}_{\text{ref}}^{\>t}=(\mathbf{I}-\mathbf{P}_{\text{ref}})^{-1}$
(1)

The $\mathbf{Z}$ matrix is called the fundamental matrix and each entry
$Z_{ij}$ contains the expected number of times that a random walk starting in
$i$ visits one node $j$ before being adsorbed. Since edge weights represent
Euclidean distance in our spatial networks, we aim to calculate the
probability of observing a path from a block node $b$ to one of the PGA nodes,
treating them as absorbing, with path cost influenced by edge weights. The
objective is to obtain a probability distribution where longer paths are
penalized in terms of cost. Assuming path costs equal path length, a
statistical ensemble minimizing the expected total cost
$\mathbb{E}[c(\mathcal{P})]$ can be derived [29, 31]. This leads to the
identification of a new transition matrix, incorporating the natural matrix
$\mathbf{P_{\text{ref}}}$ and exponentially attenuated edge costs (a sort of
Boltzmann’s weight). This represents a system of nodes considered to be
partially absorptive i.e. the probability is not conserved on paths:

$\mathbf{W}=\mathbf{P_{\text{ref}}}\circ e^{-\gamma\mathbf{C}}$ (2)

Here

  * •

the symbol $\circ$ corresponds to the Hadamard’s matrix product[32] where
entries of the matrices are multiplied cell by cell;

  * •

$\gamma$ (corresponding to an inverse temperature) is a parameter controlling
the ensemble bias toward shortest paths;

  * •

$\mathbf{C}$ is the cost matrix equivalent to the network’s weight matrix; the
fundamental matrix $\mathbf{Z}$ is derived from $\mathbf{W}$ as in Eq.(1) and
it naturally defines the partition function of this random walk system.

Leveraging this property allows defining the probability of observing a direct
path from a block $b$ to a PGA node $p$ (but not vice versa) by first
diagonalizing

$\mathbf{Z^{h}}=\frac{\mathbf{Z}}{\text{Diag}(\mathbf{Z})}$ (3)

and then row-normalizing:

$\mathbf{Z^{\textbf{PGA}}}_{i,j}=(\mathcal{P}(b=i,p=j))=\frac{Z^{h}_{i,j}}{\sum_{k\in{\text{PGA}}}Z^{h}_{i,k}}$
(4)

$P(\mathcal{P}(b,p))$ thus corresponds to the probability of reaching the PGA
$p$ (made implicitly adsorbing) from block $b$ without being killed during the
walk. The matrix $\mathbf{Z^{\textbf{PGA}}}$ collects the probabilities
calculated above for every blocks.

###  .4 Network centrality

Network centrality[33] measures the “importance” of a given vertex with
respect of the others in the network. Various centrality measures are
possible, according to what we consider important. The Farness Centrality of a
node in a graph is a measure of the total distance of a given node to a subset
of nodes in the graph. It is defined as:

$C_{f}(b)=\sum_{p\in\\{\text{PGA}\\}}d_{SP}(b,p)$ (5)

Where $d_{SP}$ is the weighted distance (i.e. the shortest path) between nodes
$b,p$. Visually, this calculated metric is akin to computing the sum of
distances within a circle that stretches to encompass the farthest nodes,
treating each path equally as contributing to the node’s centrality. However,
in an urban context, it is not necessarily taken for granted that the actual
contribution of the shortest-path length is proportional to its distance.
Consider, for example, the case where a block has equidistant accesses to a
set of PGAs (i.e. PGAs arranged on a circumference); within this metric, this
is the most favoured block. In contrast, a block close to the circumference
will experience a greater Farness but a larger set of short paths toward near
PGAs, which turn out to be more likely to be crossed. This observation
suggests that, given a distance-dependent probability distribution
$P(d_{SP}(b,v))$ of observing a path from block $b$ to the set of PGAs, an
Average Farness defined as follows is more informative:

$\langle C_{f}(b)\rangle=\sum_{v\in\\{\text{PGA}\\}}d_{SP}(b,v)P(d_{SP}(b,v))$
(6)

Where $P(d_{SP}(b,v))$ is a suitable probability distribution that describes
the ’importance’ of the paths $b,v$. In so doing, we will obtain values of
centrality that range from $0$ to the largest possible distance. For further
applications, we might need a normalized version $C^{N}_{f}$ of this class of
centralities. In general, we shall transform those quantities by using the
following formula (Max-Min)

$C^{N}_{f}=\frac{\text{max}(C_{f})-C_{f}}{{\text{max}(C_{f})-\text{min}(C_{f})}}$
(7)

####  .4.1 Average farness involving low-cost paths

Here is defined the Average Farness of a block $b$ toward the whole set of
PGAs, by resorting to the directed hitting-paths probability matrix
$\mathbf{Z^{\textbf{PGA}}}$:

$\begin{split}\langle
C_{f}(b)\rangle=\sum_{v\in\\{\text{PGA}\\}}d_{SP}(b,v)P(d_{SP}(b,v))&=\\\
\sum_{v\in\\{\text{PGA}\\}}d_{SP}(b,v)Z^{\text{PGA}}_{b,v}\end{split}$ (8)

This measure describes how much a block is locally far from the rest of PGAs
in accordance with Eq.(6). This is fair because a walker will experience only
the situation from his starting block. A conceptual limitation of this metric
is based on defining PGA nodes as infinitely adsorbing and thus thought of as
inherently having infinite capacity. As introduced, PGAs possess a finite
walkable area, which gives rise to access competition among residents. In this
case, the quality of a block is best described by an occupancy probability
distribution $P_{\text{PGA}}^{\>b}$ regarding the probability that residents
from the block $b$ will access the set of PGA nodes, given the different path
probability toward these.

####  .4.2 Average farness from PGAs occupancy distribution

This centrality is based on the PGA occupancy probability distribution
$P_{PGA}^{b}$ per block $b$. This distribution simply represents the frequency
of residents of block $b$ who have accessed the i-$th$ PGA. This distribution
is inserted into the Average Farness expression:

$\begin{split}\langle
C_{f}(b)\rangle_{PGA}=\sum_{v\in\\{\text{PGA}\\}}d_{SP}(b,v)P_{\text{PGA}}^{\;b}(v)\end{split}$
(9)

To model the distribution, we presumed a latent competition among city
residents for access to PGAs. The distribution is then sampled from the
collective outcomes of independent agents seeking low-cost paths to PGAs.
Below the agent model is introduced and details onto the calculation of
$P_{\text{PGA}}$ are then provided.

###  .5 Agent-based model

We simulated the behaviour of the population (agents) with a process akin to
diffusion. The diffusion is not an ordinary random diffusion (where people
would wander around with no direction), rather in this agent-based model, each
agent acts as a biased random walker, aiming to stop at a PGA capable of
accommodating the agent’s searched area, denoted as $f_{mq}$. From this
parameter we then obtain $MaxOcc_{i}=\frac{Area(PGA)}{f_{mq}}$, which
represents the maximum number of agents that can be hosted by the $i$-th PGA
in the city. Agents possess a limited amount of energy (i.e. total length
walked) expressed in meters. During the walk, each agent incurs an energy loss
$\Delta E$ equal to the length of the crossed edge. There are three possible
final states for an agent: (i) staying in a PGA node, (ii) reaching a PGA
without staying there, or (iii) running out of energy without reaching a PGA.
As introduced, agents compete for occupation of the PGAs. When an agent
reaches a PGA, the decision-making process involves testing
$\mathcal{X}>Occ_{i}$, where $Occ$ is the actual occupancy within a PGA
expressed as the ratio of the current number of agents who stopped in the PGA
to the maximum occupancy. We decided to evaluate the case where $\mathcal{X}$
follows a uniform distribution $\mathcal{X}\sim U(0,1)$. If the outcome is
positive, the agent stops its walk in the PGA (i). Otherwise, it continues the
walk from the PGA node (ii). In the case of state (iii), the agent restarts
the walk from its original node. In this way, the agents follow low-cost paths
leading to PGAs, determined by the optimization of a ’reward’ function
$\phi_{i}$ for each block $i$. The $\phi$ is derived directly by the matrix
$\mathbf{Z}$ in the following way:

$\phi_{i}=\mathbf{e_{i}}^{\text{T}}\mathbf{Z}\mathbf{e}_{\\{v\\}}$ (10)

Where $\mathbf{e_{i}}$ is the vector indexing the $i$-th block. The reward
function $\phi$ is the sum of the expected number of times that the $i$-th
block can potentially visit the set of PGA nodes. In that sense, nodes that
possess a high reward have a better quality in terms of low-cost paths than
ones with a lower one. The $\phi$ function increase along paths leading to
PGAs, with PGA nodes possessing the highest $\phi$ across the network. Under
this approach, each node in the network is informed about its neighborhood
quality of PGAs in terms of distances. The transition probability employs the
Monte Carlo sampling strategy:

$P_{i\rightarrow j}=\frac{e^{-\beta\Delta\phi_{ij}}}{\sum_{k\in
N(i)}e^{-\beta\Delta\phi_{ki}}}$ (11)

where $N(i)$ is the neighbors set of the node $i$. An agent walk thus
interpolates between a random walk and a walk that follows low-cost paths but
without a PGA chosen in advance. In any case, as we shall see, green areas
close to a block in terms of distance are more likely to be reached by
walkers, and conversely, occupation of distant ones is a rare event triggered
by overcrowding of nearby ones. To account for population density disparities
within the cities, agents are further distinguished by unique starting nodes
allocated in proportion to the population distribution across the blocks. Each
simulation was repeated by varying the $f_{mq}$ parameter across a range of
values. This approach enables an investigation of agent dynamics across a
spectrum, transitioning from scenarios abundant in walkable green areas to
those with a scarcity. At the end of each simulation, agents’ paths were
systematically collected to extract the occupancy probability distribution
$P_{PGA}(b,f_{mq})$ per block $b$, which is dependent from the parameter
$f_{mq}$. To recover a comprehensive distribution, the marginal distribution
was then recovered:

$P_{\text{PGA}}^{b}=\sum_{f_{mq}}P_{\text{PGA}}(b,f_{mq})$ (12)

##### Cumulative density

PGAs act as sinks that absorb a quantity of material (i.e., agents) from
blocks at different distances. To understand the relation between population
density and PGAs, a simple metric was defined:

$\rho_{>}(m)=\frac{1}{N_{a}}\sum_{p\in\\{\text{PGA}\\}}\sum_{j\in\\{V_{b}^{p}\\}}\mathbf{1}\left\\{d_{\text{SP}}(p,j)\leq
m\right\\}$ (13)

Where $N_{a}$ stands for the total number of agents, $V_{b}^{i}$ the set of
blocks from which the PGA $j$ has sourced a number of occupants, and $d_{SP}$
the weighted shortest distance between the PGA and the $j$-th block. This
metric is normalized between 0 and 1, so it allows us to explore the
Cumulative density of inhabitants that reached PGAs at different distances
with respect to the $i$-th bins $m$.

Figure 1: Main elements of the method devised in this work. From left: farness
centrality calculated from shortest paths; average farness obtained by path
weighting (bag-of-path approach); reward function and agent-based simulations.

##  I Results and Discussion

###  I.1 Average Farness centrality

Initially, the hitting path probability matrix $\mathbf{Z^{\mathbf{PGA}}}$ was
derived for the three specified cities. The $\gamma$ parameter was thus set to
$\gamma=$8.3\text{\times}{10}^{-4}$$, which corresponds to the ensemble of
paths given the reference 15-minutes distance of $1200\text{\,}\mathrm{m}$.

(a) Mestre

(b) Firenze

(c) Milano

Figure 2: Average Farness centrality $\langle C_{f}\rangle$ mapped onto the
respective spatial networks. PGA nodes are are shown (red).

We show in Fig. 2 the depicted Average Farness following the formulation in
Eq. (8) weight the accessibility by tempering the measure with the proximity
of the block to nearby PGAs. Note that the different spatial scales of the
three city clearly positively affect this measure and reveal the spatial
organization of the PGAs across the city network. In detail, for Mestre, most
of the blocks across the city exhibit an highly equitable accessibility to
PGAs, when considering distance as the main characteristic of quality. We can
explain the spatial origin of this phenomenon from the organization of PGAs
into small clusters close together, which allows the blocks to feel their
proximity. For Florence, on the other hand, this consideration fails when
applied to its city center. Here, most of the high-centrality blocks lay in
two large clusters on opposite banks of the river, corresponding to highly
serviced spatial zones in terms of green areas. We put in relation this
feature with the presence of environmental constraints on the planning of
these areas, which appear naturally confined to only one part of the city. For
Milan, high quality blocks are concentrated near smaller PGA clusters. In this
case, the spatial scale of the city, which is considerably larger than
Florence and Mestre, affects the centrality. We measure that few blocks are
highly served ($\langle C_{f}\rangle\sim 0.95$) while most retain a value
around $\langle C_{f}\rangle\sim 0.75$. In essence, these observations reveal
a kind of neighborhood planning at block-level PGAs in the city of Mestre and
a rather different arrangement in the city of Florence, governed by
morphological conditions. In the city of Milan, they reveal a quasi-uniform
arrangement of PGAs although arge areas are missing green zones. This modified
version of the Farness Centrality metric, better defined in Methods, improves
the classical Farness Centrality used in this field. Indeed instead of
computing centrality from a disk of infinite amplitude we now consider the
probability of observing low-cost paths heading to the respective PGAs. In
fact, the latter is constrained by both the finite nature of the spatial
network and the unphysical idea of assigning uniform weights to all paths to
PGAs. However, this latter measure does not incorporate information about the
area of the PGAs and considers these as equally contributing to the quality
assessment, therefore avoiding the intrinsic scarcity that characterizes PGAs.
Moreover, as anticipated, urban blocks are characterized by different
population densities, and neither of these measures account for it.

###  I.2 Average farness from occupancy distribution

In this section, we report the results of the average farness from occupancy
distribution obtained by the agent-based model. Before starting computer
simulations, we extract from the $\mathbf{Z^{\mathbf{PGA}}}$ (derived in
Subsection I.1) the corresponding $\phi$ values for each block node within
every city. The agent-based simulations have been performed with an amount of
$45000$ agents for each of the three cities, corresponding to $20\%$ of the
total population in the case of Mestre, $4\%$ for Milan, and $12\%$ for
Florence. Each city was simulated by setting $f_{mq}$ parameter to different
values, including the ’ground’ case in which the total capacity of PGA nodes
is much larger than the number of agents. We set every agent to the same
velocity of $1.33\frac{m}{s}$ and simulations were run for $5000$ steps, which
is a sufficient amount of time to allow the set of agents to occupy their
place. For agents, we set the maximum $E$ to $5000\text{\,}\mathrm{m}$ and the
$\beta$ parameter was set to 1, in order to not further bias the sampling.

(a) Mestre

(b) Firenze

(c) Milano

Figure 3: Average Farness centrality $\langle C_{f}\rangle_{PGA}$ from agent-
based model mapped onto the respective spatial networks. PGA nodes are shown
(red).

In essence, this metric resolves the lack of valuable information on both
Farness Centrality and Average Farness centrality. In Fig.3 the Average
Farness centrality calculated as in Eq.(9) is reported from the agent-based
simulations described in Methods. Compared to Fig.2 it is possible to note a
partial overlap and expansion of low-quality clusters for the three cities.
For Mestre, as for Florence, it is possible to note the appearance of enlarged
low-quality zones. While for Florence it was also evident from the Average
Farness $\langle C_{f}\rangle$ the presence of local fragility given by the
low presence of PGAs, for Mestre such fragility can be reported mainly for the
modeled competition, where it appears clear that green areas are not
sufficient to ensure equitable access for downtown residents. For Milan, the
analysis reveals a more pronounced accessibility inequality. From Fig.3 it is
possible to note a low-quality zone $\langle C_{f}\rangle_{\text{PGA}}\sim
0.25$ (cyan) spanning the peripheral blocks of the city. The peripheral belt
of blocks within these centrality values matches low-quality clusters found in
spatial studies [34] which assume population density and PGAs total area as
the driving factor of urban green quality. The difference between the quality
of blocks reported for the Average Farness and for the agent-based Average
Farness can be thus understood from the non-cooperative competition of agents
toward the PGAs induced both by the accessibility at different distances. To
explore to which extent agents compete for PGA access, the agent-based Average
Farness centrality $\langle C_{f}\rangle_{\text{PGA}}$ is plotted versus the
Average Farness $\langle C_{f}\rangle$ in Fig. 4.

(a) Mestre

(b) Firenze

(c) Milano

Figure 4: Scatter plot of the agent-based Average Farness centrality vs the
Average farness per block at different $f_{mq}$ (ground, 2.5, 5.0). Red line
correspond to linear fit of the points.

(a) Mestre

(b) Firenze

(c) Milano

Figure 5: Histogram of unnormalized $\langle C_{f}\rangle_{\text{PGA}}$ and
$\langle C_{f}\rangle$ per block expressed in meters at different distance
bins $m$ (i.e. probability density of blocks that fall in the $i$-th bins).

In particular, the figure shows a distinguishable pattern: at
$f_{mq}=\text{ground}$, which corresponds to high PGA capacity, the Average
Farness shows a linear correlation with agent-based one which is broken at
increasing $f_{mq}$. First, this emphasizes that the assumption underlying the
simulations method is verified because under high-capacity conditions agents
sample paths proportional to the probability of accessing the nearest ones,
thus recovering the Average Farness as a proportion of path probability. In
the second instance, it reinforces the observation that, according to this
model, the population gradient together with the increasing scarcity of green
areas modifies the frequency with which agents are accepted by PGAs. Indeed,
histograms reported in Fig. 5 show how events considered ’rare,’ (i.e.
traveling long distances to access a PGA), become increasingly frequent as the
area sought is constrained by the actual provision of green areas. This
results in a distribution more shifted toward higher distance values; this
effect is prominent in the case of Milan. This points out that PGAs become
increasingly correlated with blocks at long distances.

###  I.3 Cumulative density as diffusion analysis

We show in Fig.7 the Cumulative density calculated onto the previous simulated
systems. t is possible to observe that, at increasing $f_{mq}$, the PGAs nodes
source most of the agent’s population at increasing distances. To understand
how this property scales with respects to the 15-minutes distance, we show in
Fig. 6 the fit of the data reported by the cumulative density at increasing
$f_{mq}$. These plots suggest a linear relationship between the two quantities
which decrease with a specific coefficient. The linearity follows the
diffusion/adsorption mechanism, since the simulated copies of the system vary
only by the parameter $f_{mq}$. The flux of agents on edges entering PGAs is
proportional to the probability of observing a given path to them, and the
fraction of agents accepted (i.e., removed from the flux) depends on the
actual occupancy of the designated PGA. A negative change in $f_{mq}$ thus
produces a linear decrease in the fraction of accepted agents. The negative
slope then indicates globally the degree of crowdedness present on PGAs
walkable areas.

Figure 6: Fit of cumulative density at the 15-minutes distance vs different
$f_{mq}$. Fitted points correspond to the ones that intersect the 15-minutes
distance (green line).

(a) Mestre

(b) Firenze

(c) Milano

Figure 7:  Cumulative density for the three cities at different $f_{mq}$.

Finally, we point out that as $f_{mq}$ increases, it is possible that a
fraction of agents may not have reached an available PGA, thus slightly
biasing the results. In this case, though, the simulations were controlled to
minimize this problem. From our simulation we obtain a fraction less than
$2\%$ of the total of agents still wandering at the end of the allowed
exploration time.

##  II Conclusion

We presented a study on the usability of Public Green Areas (PGA) on spatial
networks representation of cities. We obtain a quantitative measure of this
feature by using centrality measures taken from network theory. First, we used
the Farness Centrality from the classical point of view. A substantial
modification was then introduced with the concept of Average Farness, by
introducing optimal path distributions. To further characterize centrality
through population difference, we also introduced a simple agent model, where
agents compete in a non-cooperative manner for access to PGAs following low-
cost paths toward these. Then we compared the centralities and we derived
information regarding the effective distance of PGA to city zones. As
explained in the results, classical farness centrality is partially able to
elucidate the quality of blocks with respect to PGA nodes, because these paths
are equally probable and therefore unrealistic since distance is obviously
positively correlated with an augmented energy spending in terms of walks.
Consequently, we defined a more comprehensive measure of the likelihood of
observing a given path to PGA’s. The probability of recovering a particular
path given edge weights allows us to elucidate the spatial relation between
blocks and PGA by escaping a strictly local description. In that sense we can
then give a comparison between the Average Farness and the agent-based method.
We can compare statistically this metric with Average Farness $\langle
C_{f}\rangle$ . In the latter, PGAs occupancy is dependent only on the first
passage probability of random walks starting from a block, thus resulting in a
uniform occupancy distribution and therefore inherently assumed to have
infinite capacity. In the latter, with the introduction of a capacity
constraint on PGA nodes and the optimization of paths by agents, the sampled
occupancy distribution depends on the maximum occupancy of PGAs and on those
paths corresponding to the second pass of agents in case they are rejected by
a PGA. In other words, the Average Farness itself represents an intrinsic
advantage/disadvantage given distances as spatial attributes, while the agent-
based ones weights independently this intrinsic characteristic and the
competition as a function of distance and green areas provision. The latter
metric represents an attempt to dynamically improve urban sustainability
planning, where traditionally quality metrics are often prescribed based on
static spatial attributes, which can fail to grasp the factors that affect PGA
accessibility, actual use and perception. Nonetheless, the agent model can be
further developed, in several directions, such as, for example: by
considering, on one side, the socio-demographic attributes and needs that
affect actual agents’capability and propension to access green spaces; but
also, on the other, by including other PGA dynamic attributes (beyond the
static spatial ones) that affect actual PGA agents use, such as PGA capacity
to provide, for example, sociality, safety, quietness, shadow, perceived
aesthetics or sense of freedom [28]; by incorporating in the model that, as
sociality is among the strongest drives to go to PGA form for most agents, PGA
crowdedness can also be perceived as an incentive rather than an obstacle to
attendance; and, last but not least, by addressing the fact that green spaces
access does not only occur by walking, but also by other means of transport.
The success of this analytical and modelling effort can produce useful data to
inform PGA design, planning, policy making, and management, allowing a greater
and balanced exposure of citizens to the beneficial effects of green spaces
and promoting social justice and equity in the urban environment.

## Acknowledgments

We acknowledge enlightning discussion with Stefano Mancuso.

##  Appendix A Farness centrality

(a) Mestre

(b) Firenze

(c) Milano

Figure 8: Farness centrality $C_{f}$ mapped onto the respective spatial
networks with PGA nodes (blue).

In Fig.8 the Farness centrality is reported. As can be seen from the Figure,
Farness centrality uniquely identifies spatial zones with the minimum
distances among the PGA node set, which is easy to. For Mestre and Milan,
which have a sparse PGAs disposition, the area is identified near the city
center while for Florence it identify zone near biggest PGA clusters. Farness
centrality tends to favor nodes that concentrate most network connections in
different shells, often aligning with spatially central nodes.

##  Appendix B Case studies in detail

The study areas were obtained using the QGis tool [35]. Each area,
representing a city and part of its conurbations, was obtained by intersecting
the administrative boundaries of the cities with the census sections in the
ISTAT dataset. A summary of the main numerical features of interest is given
in Table 1.

Table 1: City Statistics | Extension($Km^{2}$) | Population($N$) |  $N^{\circ}$ PGAs
---|---|---|---
Mestre | 116 | 261.905 | 159
Firenze | 102 | 382.258 | 396
Milano | 181 | 1.352.000 | 753

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