Near repeat burglary chains: describing the physical and network properties of a network of close burglary pairs.

Dr Michael Townsley, UCL Jill Dando Institutem.townsley@ucl.ac.uk

Outline

• 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.

Space (m)

Tim

e (w

ks)

10

20

30

40

50

500 1000 1500 2000

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

• 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)

in

out

0 1 N

01

N

in

out

0 1 N

01

N

sinks

in

out

0 1 N

01

N

sinks

sour

ces

in

out

0 1 N

01

N

sinks

sour

ces

paths

in

out

0 1 N

01

N

sinks

sour

ces

paths

amplifiers

bottlenecks

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)

0 10 20 30 40 50 60

0.0

0.1

0.2

0.3

0.4

0.5order

number of events per chain

Den

sity

0 10 20 30 40

0.00

0.05

0.10

0.15

0.20

lifetime

weeks per chain

Den

sity

0 5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

area

area per chain (sq.km)

Den

sity

0.0 0.2 0.4 0.6 0.8 1.0

0

1

2

3

4

eccentricity

eccentricity per chain

Den

sity

observed in-degree prob distribution

0 5 10 15

0.0

0.1

0.2

0.3expected in-degree prob distribution

0 5 10 15

0.0

0.1

0.2

0.3

observed out-degree prob distribution

0 5 10 15

0.0

0.1

0.2

0.3expected out-degree prob distribution

0 5 10 15

0.0

0.1

0.2

0.3

Node motif summaries

160 180 200 220

0.00

0.01

0.02

0.03

0.04

(a) amplifiers (in < out)

N

Den

sity

160 180 200 220

0.00

0.01

0.02

0.03

0.04

(b) bottlenecks (in > out)

N

Den

sity

80 100 120 140

0.00

0.01

0.02

0.03

0.04

(c) isolates (zero degree)

N

Den

sity

120 140 160 180 200

0.00

0.01

0.02

0.03

0.04

(d) paths (in = out)

N

Den

sity

150 160 170 180 190 200

0.00

0.01

0.02

0.03

0.04

(e) sinks (out = 0)

N

Den

sity

140 150 160 170 180 190 200

0.00

0.01

0.02

0.03

0.04

0.05

(f) sources (in = 0)

ND

ensi

ty

in

out

0 1 N

01

N

sinks

sour

ces

paths

amplifiers

bottlenecks

(a)

0

5

10

15

0

5

10

15

1.0

1.5

2.0

2.5

in degreeout degree

adju

sted

obs

exp

ratio

0.0

0.5

1.0

1.5

2.0

2.5

3.0

(b)

in degree

out d

egre

e

0

5

10

15

0 5 10 15

(a) all events (b) close pairs (600 m, 14 days)

y

density surfaces conditioned by node motifamplifier

x

bottle neck

x

y

isolate

x

y

path sink

y

source

y

1400 1600 1800 2000

(a) number of edgesD

ensi

ty

0.0016 0.0020

(b) network density

density score

Den

sity

1000 2000

(c) number of triangles

Den

sity

0.60 0.65 0.70 0.75 0.80

(d) triangle %age

percentage

Den

sity

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.