Transcript chs2007 6810

Near repeat burglary chains: describing the
physical and network properties of a network
of close burglary pairs.
Dr Michael Townsley,
UCL Jill Dando Institute
[email protected]
Outline
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Near repeat victimisation literature
Poly–order near repeats (chains)
Physical properties of near repeat chains
Network properties of near repeat chains
Modelling of near repeat chains
Near repeat victimisation literature
• Research shows prior victimisation gives an
elevated risk of future victimisation, but this
declines over time.
• New research indicates the same finding for
targets near prior victims, with identical time
signature.
• We call this near repeat victimisation
Near repeat pairs – pairs of events that occur
close in space and time
For N events generate the complete set of pairs (So
N*(N-1)/2 pairs)
For each pair:
– Calculate the spatial distance
– Calculate the temporal distance
Tabulate the number of pairs occuring at different
space-time thresholds.
50
Time (wks)
40
30
20
10
500
1000
1500
2000
Space (m)
p
0.005
p
0.01
p
0.05
p
0.05
Poly–order near repeats (chains)
• So far, most treatments are a-spatial and a-temporal
• Want to look at the spatial distribution of these near
repeats in order to ascertain further patterns.
• For example, like to know whether near repeats tend to be
‘linked’ to form chains with each other more often than
‘ordinary’ pairs, or even if near repeat chains continue to
propagate over long time periods or are short lived and
ephemeral.
A near repeat chain is defined to be any group of
events (crimes) where each member is close in
space and time to at least one other member of the
chain
(a)
close in space
(b)
close in time
(c)
close in space and time
Using graph theory to specify near repeats
• The events (crimes) are called nodes
• When two nodes are near repeats they are
connected by an edge.
• By considering the temporal order of the events
the graph can be specified as being directed.
• Near repeat chains are therefore directed
walks/paths/chains (sequences of alternating
nodes and edges)
Descriptive statistics
Physical properties
– Chain lifetime – duration of chain
– Chain area – size of min. spanning ellipse and
eccentricity
Network properties of near repeat chains
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Node degree – in-degree and out-degree
Degree distributions
Node motif – classification of nodes
Chain order – nodes/chain
Chains/network
Triangles
Node motif distribution/network
source
(in-degree = 0, out-degree > 0)
amplifier
(in-degree < out-degree)
path
(in-degree = out-degree)
isolate
(in-degree = out-degree = 0)
sink
(in-degree > 0, out-degree = 0)
bottle neck
(in-degree > out-degree)
N
out
1
0
0 1
in
N
N
out
1
0
sinks
0 1
in
N
out
sources
N
1
0
sinks
0 1
in
N
1
0
pa
th
s
out
sources
N
sinks
0 1
in
N
N
1
0
pa
th
s
out
sources
amplifiers
bottle
necks
sinks
0 1
in
N
Data
• Two years burglary data (N=951 events, ~450K
pairs)
• Space threshold 600 metres
• Time threshold 14 days
• Generated 2007 close pairs
Methods
• Generated expected distribution via resampling
(999 iterations + obs = 1000 sample size)
• Constructed adjacency matrix (951-by-951) where
entry ij = 1 if close in space and time, but 0 if not
• Descriptive statistics are either summary
measures or many values
Results for the observed data - general
• 951 events formed 264 distinct chains.
• Predominantly small in size (about 100 chains
were comprised of single events – i.e. order 1).
• Relatively short-lived; the vast bulk expired within
three weeks.
Results (red = observed, black = expected)
order
lifetime
0.5
0.20
0.4
Density
Density
0.15
0.3
0.2
0.10
0.1
0.05
0.0
0.00
0
10
20
30
40
50
60
0
10
20
30
number of events per chain
weeks per chain
area
eccentricity
40
1.0
4
Density
Density
0.8
0.6
0.4
3
2
0.2
1
0.0
0
0
5
10
area per chain (sq.km)
15
20
0.0
0.2
0.4
0.6
eccentricity per chain
0.8
1.0
observed in-degree prob distribution
expected in-degree prob distribution
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
0
5
10
15
0
observed out-degree prob distribution
0.3
0.2
0.2
0.1
0.1
0.0
0.0
5
10
15
10
15
expected out-degree prob distribution
0.3
0
5
0
5
10
15
Node motif summaries
0.04
0.04
0.03
0.03
0.03
0.02
Density
0.04
Density
Density
(c) isolates (zero degree)
(b) bottlenecks (in > out)
(a) amplifiers (in < out)
0.02
0.01
0.01
0.01
0.00
0.00
0.00
200
180
160
200
180
160
220
80
220
140
120
100
N
N
N
(d) paths (in = out)
(e) sinks (out = 0)
(f) sources (in = 0)
0.05
0.04
0.04
0.04
0.02
0.03
Density
Density
0.03
Density
0.02
0.02
0.03
0.02
0.01
0.01
0.01
0.00
0.00
0.00
120
160
140
N
180
200
150
160
170
N
180
190
200
140
150
160
170
N
180
190
200
N
1
0
pa
th
s
out
sources
amplifiers
bottle
necks
sinks
0 1
in
N
(b)
(a)
3.0
15
2.5
2.0
2.0
1.5
1.5
10
out degree
adjusted obs exp ratio
2.5
1.0
1.0
15
5
15
10
out degree
10
5
5
0
0.5
in degree
0
0
0.0
0
5
10
in degree
15
(b) close pairs (600 m, 14 days)
y
(a) all events
density surfaces conditioned by node motif
isolate
y
bottle neck
y
amplifier
source
x
x
y
sink
x
y
path
(b) network density
Density
Density
(a) number of edges
1400
1600
1800
2000
0.0016
0.0020
density score
(d) triangle % age
Density
Density
(c) number of triangles
1000
2000
0.60 0.65 0.70 0.75 0.80
percentage
Summary
• Some consistency of result with null hypothesis
• Differences observed for node motifs
• Limitations in scope (one site, one pair of selected
thresholds)
Future directions – statistical modelling of
pair and chain dynamics
• Some work on near repeat pair consistency by
method of entry, point of entry and time of day
• p* models allow node attributes to be used as
covariates for predicting the likelihood of
connections between nodes
• Hierarchical p* models allow parameter estimates
to be computed at the chain level.