Quantifying spatial heterogeneity through random walks

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