The heterogeneity of a spatially-embedded complex system quite often

carries important information about the function of the system as a

whole. This is the reason why the quantitative characterisation of

complex spatial patterns has received much attention in different

fields, from urbanism to neuroscience, from geography to economics,

from transportation to engineering. A particularly interesting problem

in this area is the quantification of spatial segregation, i.e., the

tendency of people to cluster around uniform patches of residential

settings. In this work we represent the adjacency relations between the

neighbourhoods (census areas) of an urban system as a

spatially-embedded graph $G(V,E)$. We assume that the population of

the city is divided into $C$ classes (which could correspond to ethnic

groups, income or education levels, interests, etc.), so that each

neighbourhood $i\in V$ is assigned to a vector $x_i\in \mathbb{R}^{C}$

whose components are the number of people of each class living in $i$.

We propose to characterise the heterogeneity of the distribution of

classes across the city by looking at the statistical properties of

the trajectories of random walks on that graph. In particular, we

consider a uniform random walk over the graph $G$, and we look at the

time series $Y_i(t)$ of the total number of different classes seen by

the walker up to time $t$ when it starts from node $i$. For each node

$i$, we define the Class Coverage Time $CTT(i)$ as the expected number

of steps after which $Y_i(t)=C$ for the first time. The main

hypothesis is that if a neighbourhood $i$ belongs to a segregated

cluster, the corresponding class coverage time $CTT(i)$ will be

larger. In this sense $CCT(i)$ quantifies the local level of

segregation of an area. By looking at ethnic segregation in Greater London,

we show that the proposed approach provides interesting quantitative insight

on the level of segregation of different areas of a spatial system

# Quantifying spatial heterogeneity through random walks

Room:

1

Date:

Tuesday, September 25, 2018 - 12:15 to 12:30