This provides a set of Python functions that implement concepts as proposed in the corresponding study about spatial effects in network analyses (see DOI: 10.1209/0295-5075/100/28002). The fundamental idea is to quantify the influence of the spatial embedding on network measures. This is done by constructing an ensemble of spatially embedded random networks (SERN) that share the same spatial embedding and link probability depending on link length as the original network of interest. A network measure on such a set of surrogate networks quantifies how that given network measure is affected by the spatial embedding.
The provided set of Python functions are explained in the following example of the global flight network. Considered are all flights from openflights.org as an undirected network and the focus is on the network measure eigenvector centrality as implemented by graph tool and how it is influenced by the location of airports (spatial embedding).
For this we import all functions from sern.py as well as graph tool and a few other standard Python modules.
import sern as sn
import numpy as np
from matplotlib import pyplot as pl
from scipy.stats import percentileofscore
import graph_tool.all as gt
Now we can load all airports from the file nodes and all flight routes from the file links.
ids, lon, lat = np.loadtxt('nodes', unpack = True)
links = np.loadtxt('links', dtype = 'int')
Then we construct the adjacency matrix and edge list by:
A, b = sn.AdjacencyMatrix(ids, links)
lon, lat = lon[b], lat[b]
n = A.shape[0]
A[np.tril_indices(n)] = 0
edges = np.transpose(A.nonzero())
A = A[np.triu_indices(n, 1)]
Using graph tool we construct a graph object and measure the eigenvector centrality of the original network:
g = sn.Graph(edges, n)
_, v = gt.eigenvector(g)
vo = np.array(v.a)
Note that graph tools is just an example, see sern_example_minimal.py for an example without graph tools and the network measure degree centrality.
In order to derive the link probability depending on the spatial length of links we need to compute all spatial distances between airports and round them to appropriately scaled integers. This allows us to use distances as indices for arrays.
D, x = sn.IntegerDistances(lat, lon, scale = 50)
Note that integer distances are not only rounded to integers but also logarithmically scaled. I.e., distances are logarithmically binned. The also returned x array gives for each distance bin index the original distance in kilometers. With the integer distances D and the adjacency matrix A (both in upper-triangle format) we can derive the link probability for each distance bin:
p = sn.LinkProbability(A, D)
For your personal example you should check the number of bins and the noise of the estimated link probability. You can use the x array returned by IntegerDistances() as the correct x-axis array for the link probability p (see also link_prob_show.py):
pl.semilogx(x, p, 'bo')
pl.ylabel('Link probability given distance')
pl.xlabel('Distance [km]')
Finally, we construct an ensemble of 1000 surrogate networks and measure the eigenvector centrality on each of them.
M = 1000
var = np.zeros((M, n))
for i in range(M):
e = sn.SernEdges(D, p, n)
g = sn.Graph(e, n)
_, v = gt.eigenvector(g)
v = np.array(v.a)
var[i,:] = v
In vo we have the original measure for each airport and in var we have a distribution of the measure for each airport. In sern_example.py the eigenvector centrality of the original network (A) is shown together with the expected eigenvector centrality due to the spatial embedding (B), as well as the difference between the two (C), and the percentiles of the measure on the original network in the distribution of measures in var (D). I.e., A shows vo, B shows var.mean(axis = 0), C shows vo - var.mean(axis = 0), and D shows [percentileofscore(var[:,i], vo[i]) for i in range(n)].
One interpretation of these results could be, that due to the high density of airports in Europe and the US (east coast) the expected eigenvector centrality is high given the observed link probability depending on distance (see B). However, not all airports actually have such a high centrality (see blue airports in C), but a few hubs have even higher centralities (see red airports in C). In terms of percentiles the ensemble of SERN can also be seen as a NULL model. In that sense many airports in coastal regions have a higher eigenvector centrality than expected (see red airports in D).
The measure eigenvector centrality is just a place holder, everything done above could also be done with another measure such as shortest-path edge betweenness, closeness centrality, random walk betweenness. For a minimal example using degree centrality see sern_example_minimal.py. A simple MPI version of the degree centrality example is given by sern_example_mpi4py.py.