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Citation:Verma, Arvind and S. K. Lodha, "A Typological Representation of the Criminal Event."Western Criminology Review3(2). [Online]. Available: http://wcr.sonoma.edu/v3n2/verma.html.

The criminal event has five dimensions: space, time, law, offender, and target or victim. These five components are necessary and sufficient conditions for the existence of a crime. However, criminological research has typically focused on one or two dimensions instead of analyzing all the dimensions simultaneously. This paper introduces a theoretical framework to deal with multiple dimensions of the complex crime phenomenon. Using the concept of metric spaces, the notion of distance and topology are introduced in the space of criminal events. This mathematical technique allows one to consider clustered events simultaneously in space, in time, law and other dimensions. It describes how crime clusters form ‘hot spots’ in the spatial dimension as well as ‘burning times’ in the temporal space and ‘stinging laws’ among the thousands of statutes. The paper illustrates these ideas with different kinds of metric spaces and provides a unifying framework for analyzing criminal events. The mathematical representation presented in this work also allows different ways of visualizing and analyzing the criminal events and associated patterns that transcend the spatial boundaries of crime. This mathematics enables deeper insight into the complex question of crime and its distribution.

**Keywords:** criminal events,
topology, hot spots, burning times, stinging laws

Crime is a complex event and can be described by the five dimensions of law, offender, target and/or the victim, place and time of the incident. These five components are a necessary and sufficient condition, for without one, the other four, even together, will not constitute a criminal incident (Brantingham and Brantingham 1991). Despite the obvious multi-faceted nature of crime, scholars and practitioners often attempt to study them separately. For instance, lawyers and political scientists focus on the legal dimension; sociologists, psychologists and civil rights groups generally look to the offenders and victims, while geographers concentrate upon the location of the event. Perhaps, a major reason for such parallel inquiries into the problem of crime is the inter-disciplinary approach that is so common in criminology. These separate inquiries are converging into a distinct discipline, but this is a recent development. Even today the subject is taught in departments of sociology, political science and even public or social affairs. Accordingly, the attempt and desire to integrate all the components into one comprehensive treatment is still in its developmental stage.

This paper introduces the mathematics of metric topology. This approach provides a conceptual framework for analyzing different dimensions of crime both separately and simultaneously. We demonstrate that many disparate techniques that are currently employed for analyzing criminal events can be viewed as special cases of a unified framework. As a particular example, we argue that the concept of hot spots is relative to a metric topology and that criminal incidents form similar aggregations in other dimensions too. "Hot spots" or "burning times" are really clusters of crimes in different dimensions, namely, space and time. We further describe how different metrics introduce new views of "hot spots." Our paper presents some topological techniques to integrate the different dimensions of crime and suggests applications for further research in criminology. We contend that metric-based topological approach opens up the possibility of analyzing and visualizing criminal events in novel and useful ways. This approach is likely to provide deeper insights into the distribution of crime and its control mechanism.

**THE MULTIFACETED DIMENSIONS OF
CRIME**

Although criminology maintains a singular focus on crime, this view is multidimensional. Different schools and researchers within this discipline study different features related to the concept, nature, prevention and control of crime. For instance, the location in space and time is an essential feature of any criminal event. The two components are embryonically linked and must be considered simultaneously. Yet, a great deal of scholarly attention, after the groundbreaking work by the Chicago school (Shaw and McKay 1969), has remained focused upon the spatial domain, especially the so-called "hot spots" (Sherman, Gartin & Burger 1989; Koper, 1995; Mazerolle, 1995; Nasar and Fisher, 1993; Weisburd and Green, 1995). The development of Geographical Information Systems (GIS) and the popularity of crime mapping among criminal justice practitioners has further emphasized the attention on the spatial dimensions of crime (Buckley, 1996; Harries 1999). In this somewhat exclusive pursuit of the identification and nature of hot spots, the significance of temporal and other dimensions has received less attention (see Langworthy and LeBeau 1990; LeBeau 1994; Kposowa and Breault 1998).

This should not be so, since the
concept of criminality of place is obviously linked to the theory of
routine activities (Cohen and Felson, 1979). As also pointed out
quite early by Hawley (1950), in any criminal situation the
space-time rhythm is a crucial factor. Wood (1991: 91) too suggests
that the commission of crime is "... very much a matter of knowing
*where* to go, just as ... knowing *when* to do it" (our
italics). Undoubtedly, the criminality of a place can be alleged for
only some specific time period; at other periods this place is as
"normal" or crime free as any other (Rengert and Wasilchick 1985). We
may sum this up by stating that such criminogenic spots are "hot"
only during definite times. In other words, hot spots go together
with "burning" times.

Similarly, the nature of the law and its impact upon the criminal incident has also been largely a unidimensional approach. The legal inquiry has focused upon the broad principles of jurisprudence rather than in association with the actors and their actions. Obviously these are important to the occurrence of the criminal event as pointed out by Black (1989). Moreover, there are a very large number of criminal statutes, but only a few are violated repeatedly. Offenses such as theft, burglary and assault are more common than homicide, forgery of bank notes or conspiracy to overthrow the state. For instance, of the 36,796,000 criminal victimizations reported to the National Crime Victimization Survey (NCVS) in 1996, theft constituted 57% and assaults 20% (Bureau of Justice Statistics, 1998). Clearly, there are some laws that are violated more than others. Furthermore, many researchers have pointed out that the offender and his/her victim share many characteristics (Ellenberger, 1955; Fattah, 1991; Karmen, 1990; Schultz, 1968). Thus, there is a need to consider these various dimensions simultaneously and establish an interdependent relationship for any meaningful study of crime.

**Motivation**

The common methodology for analyzing the multifaceted nature of crime has largely been multivariate statistical techniques. It is of course well known that Quêtêlet (1984) used statistics to study crime patterns in Europe almost two centuries ago. At present, multivariate statistical techniques are the backbone of the quantitative paradigm and an essential part of the curriculum in any criminal justice or related graduate school. Techniques such as Multivariate Analysis of Variance (e.g. O’Quin and Vogler 1989); Discriminant Function Analysis (e.g. Williams, Singh and McGrath 1984); Factor Analysis (e.g. Longshore, Turner and Stein 1996); Cluster Analysis (e.g. McGurk, McEwan and Graham 1981); Log-Linear Analysis (e.g. Crowley and Adrian 1992); Multi-Dimensional Scaling (e.g. Smith, Smith and Noma 1986) and the more sophisticated Structural Equations Analysis (Liu and Kaplan 1999) are routinely used by criminal justice scholars in the examination of the crime phenomenon.

Several of these statistical
techniques, such as Cluster Analysis, Factor Analysis and
Multi-Dimensional Scaling, attempt to identify important crime
clusters by grouping similar variables or cases together. The advent
of crime mapping has also increased the popularity and usage of these
statistical techniques. These applications are driven by the power of
statistical inference, which is especially suited to the study of
criminal behavior where it is impossible to know and study the entire
population. However, statistical models of the crime phenomenon are
difficult to visualize and interpret. For instance, the latent
structure derived from Factor Analysis is difficult to interpret.
Similarly, Cluster Analysis helps in identifying the hot spots from a
crime map, but presenting them in a realistic layout in all its
dimensions remains a formidable challenge. This paper expands the
present techniques in an important way. By the use of metric spaces
it opens the possibility of defining proximity in ways that are not
based upon Euclidean distance alone. Indeed, Euclidean distance
between two crimes may not be an appropriate notion of distance if
there are physical obstacles between the two locations (see Figure 1
below). Similarly, the proximity between two times may be less
important than the cyclic nature of time depending simply upon the
daily, weekly, or seasonal cycle, or may depend on something more
complex such as the routine activities surrounding the event. In the
law, offender and target/victim dimensions, the notion of similarity
or closeness of two objects is even more complex. The key point is,
that independent of the crime dimension one is interested in, there
is not *the* notion of distance or similarity but several
different possible measures of similarity that different theories and
approaches advocate. The metric space approach provides a direct
language to the researcher to specify when two criminal events should
be considered similar to one another.

The second important point of
departure that the current work opens up is the possibility of
mapping crime in innovative ways. The most popular technique to map
crime has been to tie the occurrence of the crime to its location.
This approach gives supreme importance to the spatial dimension of
the crime since it is easy to place it on a two-dimensional map.
However, criminal events occurring close together still need not be
similar for these may differ in time, law or other dimensions.
Several visualization techniques such as star plots, Chernoff faces,
profile plots, and scatter plots have been proposed to depict
multi-dimensional data on a two-dimensional layout or screen (Cluff
1991). Recently developed information visualization techniques have
also been adopted to visualize a wide variety of objects including
documents, family history, users’ interaction patterns with the
internet, spreadsheets, relational databases and hierarchical graphs
(Roth and Keim 2000). An important by-product of this approach is
that the mapping has transcended the spatial boundaries. It is now
becoming possible to depict any abstract object such as the contents
of a document or characteristics of a person on a two-dimensional
screen. More important for the purposes of analysis, the researcher
should have the flexibility to develop any notion of similarity and
then depict two similar objects together in the graphical mode. The
metric topology provides us with a tool to (i) define similarity
between criminal events in various ways that incorporate and unify
several existing approaches and open the possibility for many
innovative ones, and (ii) map similar criminal events closer together
for the analysis and understanding of the crime and its distribution.
In this paper, we introduce and apply the metric topology to define
and explore the notion of similarity between criminal events. The
metric topology and the crime mapping approach together provide a way
for *data to speak for themselves* [our italics] as
suggested by Maltz, Gordon and Friedman (1991).

**Topology**

Topology is a well-established mathematical field and has found wide applications in subjects ranging from anthropology to astronomy. A literature review reveals that Lewin (1936) pointed towards the applicability of topological spaces in psychology while Brantingham and Brantingham (1975) have used the technique to construct "connected" spaces among residential blocks. Their technique identifies "edges" in an urban layout that are formed by combining the residential blocks on the basis of their common characteristics. The topological spaces so generated were used to identify the high, medium or low regions of crimes. Quite significantly, crime was found to be greater near boundary areas than in "interior" regions of these topological spaces. They have also suggested a process of "regionalization" using topological techniques (1978). The postmodernist literature has also suggested many interesting applications of topology in criminological inquiry (Baker 1993; Milovanovic 1996, 1997; Pepinsky 1991; Young 1991).

Abstract topology is qualitative mathematics (Mansfield, 1963). It is concerned with the intrinsic qualitative properties of spatial configuration that are independent of size, location or shape. In this work, however, we focus on metric topology that is quantitative and introduces the notion of distance between objects. In our case, the objects are the criminal events. A metric space topology can assist in examining the relationships between the various dimensions of crime where the concept of 'space' may be modified according to the need. The definition need not be limited to the geographical space but could be extended to social relations, conceptual interactions and temporal connections. There is no restriction upon the researcher for the property common to all objects of topological study, that is, they are sets. The elements of the set may be anything -- crime sites, criminal events, police functions, a time period, a group of police officers, a set of laws, a gang of offenders or a group of victims. Thereafter, the rules for union and intersection of these sets could be devised to examine the criminal event from different perspectives.

**Metric Spaces and
Topology**

A metric space consists of two
objects: a set of elements that is not an empty set and a distance
function ‘d‘ that is defined between any two objects of
this set. Consider any two elements of the set. The distance
function, d, describes how similar or different these elements are.
The distance is large if the elements are dissimilar, and the
distance is small if the elements are similar. This distance function
must satisfy three conditions in order to meet our intuitive notion
of distance. If these three conditions are satisfied, this distance
function is said to be a *metric*.

The first condition, referred to as positivity, states that the function always has positive values, except when the two points coincide if the distance is zero. The second condition, referred to as symmetry, states that the distance from x to y is the same as the distance from y to x. The third condition, referred to as the triangular inequality, states that the distance between any two points x and y is always less than the sum of the distance between the first point x and any third point z and the distance between the third point z and the second point y.

These three conditions can be defined more formally as follows. A function ‘d’ from an ordered set of pairs of elements of X to real numbers, that is, d: X x X --> R is defined to be a metric if it satisfies three conditions:

(i) Positivity: d (x, y) = 0, and d (x, y) = 0 if and only if x = y;(ii) Symmetry: d (x, y) = d (y, x); and

(iii) Triangular inequality: d (x, y) < d (x, z) + d (z, y).

The metric is often referred to as the distance function. However, there can be many different notions of metrics or distance functions associated with the same set. The notion of metric space or distance function introduces a topology or shape on the set of all the elements. As an example, if we take a square piece of paper and identify the left and right edges of the paper, the shape changes from a square to a cylinder. In the old shape, events that are just right to the left edge and those that are just left to the right edge are far apart. However, in the new shape of cylinder, these events are viewed as close together. This happens because the distance functions in the two shapes or topologies are different. The new shapes allow us to cluster the events in new ways and to think about them differently.

The mathematical description of how a metric space introduces a shape or topology is well known, but complex. For the sake of completeness, we describe this process briefly. We begin by presenting the mathematical definition of the topology. A topology consists of two objects: a non-empty set X and a set T of subsets of X that satisfies two conditions:

(i) The union of any members of T is also a member of T; and(ii) The intersection of any finite number of members of T is also a member of T.

These two properties are also referred to by stating that the set T is ‘closed’ under arbitrary unions and finite intersections. The members of the set T are referred to as ‘open’ sets of the topological space (X, T). Any subset Y of X, that is a complement of an open set, is referred to as a closed set. A metric space automatically defines a topology on its underlying set X. Furthermore, as in the case of metric spaces, the same underlying set X can admit many different topologies too.

Given a metric space (X, d), one can introduce a topology (X, T) as follows. First, define any open sphere S (x, r) of radius r centered on point x. This is the set of all points y that are less than r units away from x: {y: d (y, x) < r} and belong to T. Second, define any set Y as open (belonging to T), if given any point y in Y, one can always find an open sphere S (y, r) of some radius r around y, that is contained in Y. With these definitions, it follows that the arbitrary union of open sets is always open, and the finite intersection of open sets is always open (Simmons, 1963).

Although the abstract process
defined above may yield very complex shapes in general, in the
examples detailed below, we are interested in several concrete
illustrations of this process. We will present several examples that
are already being used by the criminologists to demonstrate that the
proposed framework unifies several existing techniques of analyzing
many different dimensions of crime. We also discuss some new distance
functions that allow us to cluster criminal events in novel ways. We
will discuss the utility of this approach again later in this
work.

**Example 1: Euclidean Topology**

Consider a set X that consists of
criminal incidents in the everyday concept of geographical space.
Without loss of any information, we can consider the location of
these incidents to be points in this space. Although the coordinates
of these points can be latitude and longitude, for small distances
(small compared to the scale of the earth), it suffices to consider
the Cartesian coordinates of the points. Any standard mapping system
invokes such a coordinate system and assigns a unique coordinate to
every place of incident (Harries 1999). Consider any two criminal
incidents that are marked on the crime map with points x and y having
coordinates (x_{1}, y_{1}) and (x_{2},
y_{2}) respectively.

On this coordinate system we can define the Euclidean metric as

d (x, y) = square root of [(x_{1}-x_{2})^{2}+ (y_{1}-y_{2})^{ 2}].

This definition satisfies the three conditions -- positivity, symmetry and triangular inequality -- and therefore qualifies as a metric. The metric introduces a topology on the space, known as Euclidean topology. This metric topology is the familiar Cartesian coordinate system that is extensively used in crime pattern analyses (for example, Block and Block, 1995; Brantingham and Brantingham, 1984; Read and Oldfield, 1995). The distance between any two points in this topology is measured by the shortest distance on the map irrespective of the terrain.

**Example 2: Manhattan
Topology**

One disadvantage of the Euclidean topology is that in practical terms the shortest Euclidean distance is not always available. For example, when the path is obstructed by the presence of buildings, the shortest path may be around it rather than through it. This difficulty may be overcome by introducing another metric and topology on the same space, known as Manhattan topology. Here the Manhattan metric is introduced as follows:

d (x,y) = | x_{1}-x_{2}| + | y_{1}-y_{2}|.

Again, this definition satisfies the three conditions and therefore qualifies as a metric. Under this topology, one can travel only along horizontal and vertical directions. Here, the open sets are unions of open rectangles (that is, the boundary lines are not included in these rectangles).

This notion of distance in either Euclidean or Manhattan topology allows one to characterize the notion of hot spots in the spatial dimensions. Hot spot is a "cluster" of spatial points where many crimes occur. The notion of cluster depends upon the notion of closeness or distance. In other words, one can speak about hot spots in the context of a metric space. If the "topological distance" between two criminal incidents is "small" then these are considered part of the same cluster. For instance, hot spots referred to by Mazerolle and Terrill (1998) and McEwen and Taxman (1995) are Euclidean hot spots since these are the points on a map that are in close proximity to one another.

**Example 3: Street
Topology**

Consider a connected street network on a two-dimensional map as in the figure below:

The distance between any two points is defined as the shortest path along this street network. Thus, between any two incident points A and B, we define the shortest path as

where d_{i} s are the
distances measured along different possible streets connecting A to
B.

Note that if there are physical barriers between two points A and B that are close to each other in the usual Euclidean sense, they can still be far apart under this notion of street distance (see Figure 2). This notion of distance indeed satisfies the three conditions -- positivity, symmetry and triangular inequality -- to qualify as a metric.

We refer to the topology introduced by this metric as the street topology because the open sets here follow the street network. The Euclidean topology and Manhattan topology are really idealizations of street topology, where it is assumed that there are no physical barriers or that the street network is aligned only horizontally and vertically. Street topology is more true to the criminal events than either the Euclidean or the Manhattan topology for an offender is likely to escape by the shortest available path. As seen in Figure 2, crime incidents A and B would appear to belong to the same cluster, but due to the presence of an object [sports stadium] one needs to go around it. This becomes considerably longer than the distance between A and C. Street connectivity defines a very different set of movement patterns than the ones assumed under the direct line-of-sight movements. This concept is significant since the criminogeneity of a place is dependent upon its accessibility.

Hot spots under the Euclidean topology may not cluster under the street topology. For instance, if crimes concentrate on two sides of an impenetrable barrier as displayed above then these must be considered as belonging to different clusters. This has important implications for policing. A group of criminal incidents on a crime map may give the appearance of hot spots if these are spatially clustered. However, a careful scrutiny of the area may reveal that all these places are not related due to the absence of convenient travel paths. Before finalizing plans for some proactive initiatives, such as directed patrolling, a proper configuration of the topological spaces needs to be developed since necessarily these incidents are being driven by different sets of situations.

**Example 4: Real-Time
Topology**

In the same set X of criminal
incidents, the focus could also be upon the time of occurrence. Now,
every incident has a time stamp associated with it, and this temporal
dimension can be represented in a unit consisting of 6 coordinates of
the year, month, date, and time (in hours, minutes and seconds).
However, all these 6 coordinates can be reduced to a simple
coordinate time as measured in seconds from some arbitrarily chosen
datum, for example, January 1, 1999, 00:00:00. Time is a negative
coordinate if it falls before that datum. The real-time metric
distance between two incidents occurring at different time periods,
say time elements t_{1} and t_{2} can then be defined
as:

This metric introduces a topology
on the time, whereby the time element can be viewed as an infinite
line. Currently, this is the most popular representation of the time
element in the crime mapping literature. In this metric space,
"burning times" will be those periods when a large number of crime
incidents occur around a small time period. Thus, the multiple
murders at Coleman High School, the attacks on the US embassies in
Africa or Benjamin Smith killing two people and injuring nine others
within two days (*Hoosier Times,* 1999) are examples of
clustered incidents sharing similar temporal
characteristics.

**Example 5: Daily-Cycle
Topology**

The set X may be analyzed still differently in terms of its time units. On this set we introduce the daily-cycle metric as follows:

Since one day consists of 86,400 seconds, in this metric, two time elements are close to one other if and only if they are close in the sense of happening at the same time of any day (24-hour cycle) irrespective of the year, month or date. This metric introduces a topology on the time, whereby the time element will be viewed as a circle or a ring of twenty-four hour units. This daily-cycle topology is useful in identifying the daily periodicity in the occurrence of criminal activities. The "burning times" for this topology are the criminal activities occurring on similar hours of any day. For instance, late night drunken bouts, burglaries during mid-afternoon hours and pick-pocketing on rush hour transit systems are some such examples of daily periodicity of crimes that will show up as "burning times" in this topology.

**Example 6: Weekly Cycle
Topology**

One can also introduce the
weekly-cycle metric on the temporal set of criminal incidents as: d
(t_{1}- t_{2}) = | (t_{1}- t_{2}) mod
14515200 |.

Since one week consists of 14,515,200 seconds, in this metric, two time elements are close to each other if and only if they are close in the sense of happening at the same time of the same day of the week, irrespective of the month or year. This metric introduces a topology whereby the time element is still viewed as a circle or a ring, albeit the circumference is much bigger. This topology is useful in identifying the weekly periodicity of criminal activities. The "burning times" are specific days of the week when crime peaks due to some cyclic routine activities. For instance, Friday and Saturday nights are marred by drunken fights, weekends are filled with noisy football parties and Mondays witness illegal absenteeism on a large scale. If criminal incidents are aggregated on a weekly basis then these periods are the burning times. Other forms of periodicity like monthly or seasonal cycles can be represented similarly.

Clearly, burning times are a "cluster" of points in time when many crimes occur conjointly. Again, the notion of cluster depends upon the notion of closeness or distance. In other words, one can speak about burning times within a metric space introduced in the time dimension. Burning times in real-time topology are not the same as the burning time in daily or weekly, monthly or even seasonal-cycle topology. Crimes that cluster together around midnight will appear scattered in real time topology but will show up as burning times in the daily-cycle topology.

**Example 7: Law
Topology**

We can now consider a plausible topological representation of the legal dimension of crime. The set of all the laws may be considered as a "legal space" for our representation. Consider the set L of all the rules that constitute the body of the criminal laws applicable to a geographical region under consideration. This set incorporates all the federal, state and local laws including any judgments, decrees and judicial pronouncements that constitute, define and state what crime is in that particular society. For instance, when a person steals a wallet from another person, the crime is identified by the offense of theft and so this statute is an element of the set L. Mathematically, L = {l | l is a set of rules that forms the body of the criminal law}.

The next component of law is the
*quantum* of punishment stipulated for its violation. This is a
natural association since every criminal behavior invites a
stipulated minimum or maximum amount of punishment in the form of
fines or imprisonment. Thus, there is a dollar amount and a period of
incarceration that quantifies the invocation of any violation of the
criminal law. Therefore, amount of punishment can be used in
measuring distance (or seriousness of the offense) between any two
legal violations.

However, there are many other components relevant to the law dimension. Law is a quantitative variable that "increases and decreases, and one setting has more than another." Moreover, it also "varies in time and space" and "varies with who complains about whom, who the legal official is, and who the other parties are" (Black 1976: 3-4). Moreover, law varies with other forms of social control. Invocation of the law by the police is more of an explanation of the behavior of the law than that of the individual police officer. As suggested by Black (1976: 3), law is a quantitative variable that is "known by the number and scope of prohibitions, obligations, and other standards to which people are subject, and by the rate of legislation, litigation, and adjudication." Thus there are several ways in which law can be measured as a quantitative variable. Examples include formal rule violation vs. informal norm violation and characteristics of situation the law is designed to protect (e.g. property rights, personal liberty, physical sanctity, privacy, etc.).

The metric to be constructed in the law dimension depends, as in the case of other dimensions, upon which variables the researcher considers important in analyzing a particular event or a set of events. Here, for illustration purposes, we consider a metric that is a mix of punishment and the damages incurred. To this purpose, we first categorize every law into a finite number of components that are related to the quantum of different forms of punishment. For the purpose of the current discussion, let these components be dollar amount (x), incarceration period (y) and days of community service (z). Based on these three variables alone, the law dimension may be visualized as follows:

Let us now add the additional
dimensions such as the physical damages incurred (p), and the
perceived severity of the crime on a predefined uni-dimensional scale
(q). In order to induce the notion of proximity between any two
points L_{1}: (x_{1}, y_{1}) and
L_{2}: (x_{2}, y_{2}) in this legal space, we
first define the distance in each component in the usual
manner:

d_{1} (x_{1},
x_{2}) = | x_{1} - x_{2} |

d_{2} (y_{1},
y_{2}) = | y_{1} - y_{2} |
.

The "distance" between
L_{1} and L_{2} is now defined as:

where w_{1} and
w_{2} are positive weights associated to each component.
These weights may be assigned on the basis of emphasis that needs to
be laid upon each of the components. For instance, we may decrease
the weights of the fines, imprisonment and incarceration component if
one believes that those are the inappropriate or less important ways
of looking at the law violation. In effect, the weights are
essentially used for emphasizing different variables that constitute
the law dimension. This function introduces a metric and a topology
on the space of laws. Such a topology will distinguish between the
criminal events based upon the perceived severity of the crime,
damages involved and the quantum of punishment given by the criminal
justice system. By using different weights and this notion of
closeness, one can identify a cluster of laws as those that prescribe
similar punishments. In a given police district it is likely that
assaults, disturbances, burglary, motor vehicle thefts and other
thefts are proportionately more than other forms of crimes and will
constitute large clusters in this legal space. Similar to the concept
of hot spots, we may visualize and describe this cluster as
‘stinging laws.’

**Example 8: Socio-Economic and
Demographic Topology**

To illustrate the power of the topological concepts, we now apply topology to socio-economic and demographic variables. Brantingham and Brantingham (1978) have used one such technique to cluster homogeneous neighborhoods where homogeneity is considered in terms of socio-economic variables. The concept may be extended to analyze criminal events in terms of the components of offenders, victims and/or targets. We will illustrate this with a slightly hypothetical, yet an important example, using the value of the house as an underlying variable. Apparent house-value is said to be an important factor in choosing targets (Rengert and Wasilchick, 1985). Consider a neighborhood consisting of residences of different valuations. This situation is depicted in Figure 3A where the areas of the houses reflect their proportionate values. For simplicity, we consider residences in three values (areas) only. The targeted (burgled) houses are shown marked in this figure.

Let us introduce the following
function called a ‘value distance' between the houses: d
(v_{1}, v_{2}) = | v_{1} - v_{2}
|.

Here, v_{1} and
v_{2} represent the actual value in dollar amounts. This
‘value’ metric introduces a topology that allows us to view
the neighborhood as a re-ordered set where houses with similar values
are close together.

This conceptual view is a distortion of the physical view of the neighborhood. Nevertheless, this conceptual view is justified by clustering the houses of similar values together. This topology suggests that hot spots of burgled houses may be based not on some kind of spatial-proximity but rather on the value-proximity. This understanding may enable the police to identify the modus operandi of burglars who target specific houses on the basis of their values rather than accessibility.

In a similar fashion, we may consider socio-economic characteristics, family size, mean educational levels, etc., for family units to develop representations that are more meaningful than the common spatial ones. Topological representation in terms of these variables can help in understanding how the households are realistically situated in a neighborhood and may suggest a cognitive picture that is different from the simple spatial one. This is useful since selection of targets by offenders may be guided by factors other than spatial proximities (Brantingham and Brantingham 1993; Miethe and Meier 1994). Therefore, topological representations will be a useful way of identifying the opportunities that are comprehended cognitively.

**Example 9: Topology for
Offenders, Victims and Targets**

Let O be the set of all people involved in one or more criminal incidents and living in the city or police district under consideration. This is a subset of the set of general population living in that area, which also comprises children, seniors and handicapped people, many of whom may not be associated with criminal behavior. A member of O, designated as o, is then some person who has been involved in a specific criminal incident as an offender. In mathematical notation O = {o | o is a person who has violated a particular criminal law}. This set may be considered as a group of offenders who have violated some provisions of the criminal law.

Note, every person (above the statutory minimum innocence age limit) conceivably does violate some section of the law at one point or other. This violation may be as insignificant as speeding, throwing litter or getting involved in a minor physical assault with another person. The assumption that every person does infringe one of the thousands of criminal statutes during the time frame under consideration is not unreasonable (Felson 1994). In fact, a career criminal (Gottfredson and Hirschi, 1986) may violate more than one law, and so each such offender may be further characterized by the specific legal codes infringed by him or her. Thus,

Moreover, every such offender and for that matter, any person can also be characterized by the socio-economic, demographic and geographical factors specific to that individual. For example, variables like job status, annual income, ethnicity and home address can assist in identifying the individuals. As described in examples 7 and 8 above, each of these variables could be used to describe a metric topology upon this set of people. The weighted metric as described for Law topology in example 7 can then be used to identify the offenders distinctly.

The possibility is that those offenders, who live close to one another, or share common interests and move in common circles, end up violating similar laws and perhaps acting in cohesion. For instance, Miethe and McCorkle (1998) report that the offender profile of persons arrested for robbery display the common characteristics of being male, African-American, young, having prior arrest record, felony record, being city based, and an offense generalist with some specialization and opportunism. In fact, a gang of offenders is a common example of this similarity. In a multi-dimensional space there are groups of offenders who either act together or behave in similar fashion. It is this kind of proximity that identifies them as a gang member.

The same arguments can be applied
to the case of victims or targets. Again, let V = {v (l_{1},
l_{2}, ... l_{n}) | v is a victim under some
provision of a law}, a person who has suffered because of a
particular illegal activity (Karmen 1990). Each element of V is a
single person or group of people who have been victimized in
different ways. For instance, V_{1}, a subset of V, may
represent a set of all the female victims, while another subset
V_{2} is a singleton, consisting of a particular senior lady
whose car has been stolen. Again, every such victim can be similarly
characterized by the socio-economic, demographic and geographical
factors specific to that individual. Thus, variables like education,
family income, age and home address will characterize every victim.
In an analogous manner, each of these variables could be used to
describe a metric topology upon this set of victims. The weighted
metric, similar to the case of offender space then describes a metric
topology characterizing the victim space where each victim is
represented by a unique point in the victim space. The possibility is
that those victims who live in particular socio-economic places,
share common characteristics or move in common circles are victimized
similarly. For example, it is known that domestic victims share many
common characteristics (Bell, 1985). Miethe and McCorkle (1998: 153)
report that "those who are between 30 and 59 years old, ethnic and
racial minorities, renters, and central-city residents are most at
risk of auto theft."

Finally, similar arguments apply in the case of physical targets. The geographical location, economic value, and physical characteristics like shape and accessibility can be used to develop a topology. For instance, for physical targets like motor vehicles we may use their make, year of manufacturing, color and market value to ascribe metric distances. For example, using the make of motor vehicle as the variable of interest we could define a metric as

d (t_{1}, t_{2}) = 0 if make of the vehicle is the same= 1 otherwise.

For the variables of year of manufacture and value, the choice of metric is the natural one. The weighted combination of these metrics can then be used to induce a topology on the set of stolen vehicles where each vehicle crime has a unique representation. In this topological space clusters or ‘hot targets’ suggest that the nature of vehicle is a criminogenic factor. For example, the attractiveness of the car in terms of its parts value, temporary usage or resale value poses a major risk of auto theft (Clarke and Harris 1992).

**Product Topology**

So far we have discussed the notion of metric spaces and topology separately for several dimensions of crime -- space, time, laws, offenders, victims and targets. We now introduce the definition of product metric spaces and product topology. This will allow us to consider the separate dimensions of crime in an integrated manner.

This can be done easily by defining the notion of distance between several dimensions as the sum of the distances on each individual component. For example, Manhattan metric is the product metric of the Euclidean metrics on the horizontal and vertical component of the spatial dimensions. This product metric indeed satisfies the three conditions -- positivity, symmetry and triangular inequality -- to qualify as a metric. The product metric helps create a product topology (X, T) in a similar manner.

More formally, given two metric
spaces (X_{1}, d_{1}) and (X_{2},
d_{2}), the product metric space is defined on the product
set X = X_{1} x X_{2}. Recall X_{1} x X_{2
}= {(x_{1}, x_{2}): x_{1} in
X_{1}, x_{2} in X_{2}}. The metric on X is
defined as d ((x_{1}, x_{2}), (y_{1},
y_{2})) = d_{1} (x_{1}, y_{1}) +
d_{2} (x_{2}, y_{2}).

If (X_{1}, T_{1})
and (X_{2}, T_{2}) are the topologies introduced by
the metric spaces (X_{1}, d_{1}) and (X_{2},
d_{2}) respectively, the product topology (X, T) is
introduced by defining

T = T_{1}x T_{2}= {U x V | U in T_{1}, V in T_{2}}.

In other words, the product
topology consists of products of all open sets in X_{1} and
all open sets in X_{2}. This definition of product metric
spaces and the product topology can be extended to any finite
number.

*T*opology of the
Criminal Event

We are now ready to describe the topological space for the criminal event. Let C be the set of all criminal incidents for a particular period of time, say a year, and from a region, which may be a city, police district or some neighborhood under consideration. An element of C, designated as c, is then a set of people and their behavior located in a space-time continuum that is marked by the violation of the criminal law. This set will also include crimes occurring in the region that may remain unreported to the police or other official agencies. Every such criminal incident involves a geographical site [the spatial dimension], a specific time of occurrence [the temporal dimension], a violation of the law [the legal dimension], an offender [the offender space] and the victim or target [the victim space] (Brantingham and Brantingham 1991). Each of these sub-components is a topological space formed by one or the other metric topologies described earlier. The product topology of all these topologies forms the topological space for the set of crimes where the metric topology is the sum of the individual metrics. We now present some concrete examples of product topologies associated with criminal events.

**Example 10: Euclidean Real-Time
Topology**

The Euclidean topology in one dimension is a straight line segment. Similarly a segment of the real-time topology is a straight line as well. Therefore, the product topology in this case will be rectangular. This is shown in the figure below as a joint space:

A limitation of this view is that the spatial clustering may not be so meaningful if the incidents are spread over a long period of time and therefore, appear visually scattered.

A more useful representation is the Euclidean topology in two dimensions, where the spatial dimension topology is simply shown by the city map. In this case, the Euclidean real-time topology is three-dimensional with two spatial dimensions and a time dimension that can be shown vertically (Lodha and Verma 1999).

The shape is simply the city map
"grown" vertically along the time dimension. In this view crime
clusters that appear together both in space and time demand attention
by appearing together. Although Euclidean real-time topology is
common, even this simple topology has not been investigated or mapped
in-depth.

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**Example 11: One-Dimensional
Euclidean Daily Cycle Topology**

This is the product topology on the space-time by taking the product of Euclidean topology on the spatial dimension as in Example 1 with the daily cycle topology on the time dimension as in Example 5. Since the shape of the daily cycle topology is circular, this topology introduces a cylindrical shape when the space is one-dimensional. This is shown in Figure 7, where the space is shown along the x-axis and the time is "rolled over" to form a circular ring. This cylindrical view of the world is useful in identifying criminal patterns that are cyclic with respect to a 24-hour routine.

**Example 12: Two-Dimensional
Euclidean Daily Cycle Topology**

This is the product topology by taking the product of two-dimensional Euclidean topology on space and the daily cycle topology on time as discussed in Example 5. For the purpose of illustration, we will assume that the city is like a circular disk. Then, each of the two components in this product topology is circular shaped. One way to think about it is to approach this shape in two steps. In the first step, one stretches the circular city in time dimension producing a cylindrical shape. In the second step, one folds over the time dimension at the two ends of the cylinder to join them, which then introduces the shape of a solid torus. This is shown in Figure 8.

In case the city is not a circular disk, one can still construct a topological torus in two steps. In the first step, one needs to extrude the city map in the time dimension creating a prism with the city map as the base. In the next step, one needs to roll over the two ends of the prism to join them together, and identify that, in the daily cycle topology, "0000" hour is the same as the "2400" hour. This gives a topological torus, where each section of the torus is not a circular disk, but simply the shape of the city. The concept of torus has been introduced in criminological literature (Milovanovic 1997) and becomes a useful model of criminal behavior when associated with the notion of metric space.

**Example 13: Location
Analysis**

This is the product topology in the space-law dimensions of crime. This mapping has been found to be useful by Brantingham and Brantingham (1995), where they showed different hot spots for different crime types in a city. For example, certain spots are hot with respect to auto theft, while other spots may be hot with respect to burglary. This analysis has been referred to as Location Analysis (Brantingham and Brantingham, 1995), but this is a special case of the product topology of space and law.

**Projection
Mapping**

We now discuss the concept of
projection mapping and its relevance to criminology. The projection
mapping P from a product metric space (X_{1} x X_{2},
d_{1} x d_{2}) or a product topological space
(X_{1} x X_{2}, T_{1} x T_{2}) into
one of its component, say the first component X_{1}, is
defined simply as follows:

P: (x_{1}, x_{2}) -> x_{1 .}

In other words, all other components are ignored or dropped except the ones that are of interest. Projection mapping into the spatial component is a popular mapping in the current crime mapping literature (Eck 1997; LeBeau and Vincent 1997; Ratcliffe and McCullagh 1998). If the space is two-dimensional as in a city, this topology can be viewed as a solid cylinder, the base of which is a city map. This view is also useful in identifying the hot spots and burning times together.

**Example 14: Projection Mapping
into Space**

As described in the introductory paragraph, a criminal event has five dimensions. A criminal event should therefore be truly viewed as a point in a five-dimensional space involving space-time, law, offender and target-victim dimensions as suggested before. For each of these dimensions, the notion of metric helps us to define when two criminal occurrences may be considered to be close together in this multi-dimensional space. In practice, however, one usually projects criminal events into the spatial component and maps crime occurrence on a city map disregarding the other components of crime.

Although spatial projection has been very useful, there are many other projection mappings, such as the projection into the time component or the law component that helps us to identify or focus on ‘burning’ times and ‘stinging’ laws. Furthermore, the examples of product topologies described before can also be viewed as projection mappings from the multi-dimensional product topology to two dimensions. As an example, location analysis is a projection map from the five dimensional topological space of criminology into the spatio-legal two-dimensional space of crime.

**The Advantages of Using
Topology**

A mathematical representation of the criminal event as outlined above is likely to present several advantages to the crime researchers. Rather than concentrating only upon one of the dimensions, the criminologist can deal with all the five dimensions simultaneously. If any one dimension appears to show certain specific characteristics, like spatial criminogeneity, then the possibility of similar characterization in the other dimensions can also be explored. The researcher is not constrained to focus only upon the "hot spots" but can as easily examine the "burning times and the stinging laws," as well as the set of people whose similarities cause interaction as offender and victim. Rather than labeling only the place as being criminogenic, the incidents could be seen as the manifestation of movement patterns, routine activities, statutory and administrative laws and socio-economic and demographic characteristics associated with those place-time-law and people dimensions. Criminogeneity is not in any one dimension but in combination with other dimensions. Any police strategy must therefore not only seek to identify the hot spots to develop preventive measures but obviously should consider the five dimensions simultaneously. An all-comprehensive crime control policy is required that would involve integrated measures by the police, community, urban planners, law makers and others to address the crime problem in its totality rather than in isolation. Perhaps this is the mathematical representation of what has been described as ‘problem-oriented’ policing (Goldstein, 1990).

There do remain several limitations in this approach. The concept of crime itself is controversial and any attempt to measure its dimensions can only provide a restricted picture (Coleman and Moynihan 1996). The vast number of laws and regulations makes any attempt to classify criminal events a very difficult task. There is also the problem of differential law enforcement that comes from police practices and discretion exercised by the officers. In many cases, crimes are not even registered or treated outside the purview of the law. Crime is a social construct (Maguire 1994), and even the dark figure of crime "expands and contracts with the values that we bring to our study" (Young J. 1992: 58). As Coleman and Moynihan state (1996: 20):

Crimes and dark figures are simply ‘not there’ waiting to be counted by the application of a simple rule to unambiguous events in a laboratory setting by neutral observers. Instead, any crime rate is produced in the sense that people with particular interests, concerns and objectives use a set of definitions, rules and procedures in a complex environment to arrive at that product.

Nevertheless, once a criminal
event is perceived (by whatever method or inquiry) topology can
provide the means to address all its dimensions in a simultaneous
manner. This "qualitative" mathematics provides a powerful tool for
analyzing the relationships between the different aspects of crime
phenomenon. It injects new perspective into the way we view and deal
with criminal behavior. The advantages of using mathematics are many.
By its symbolic construction mathematics has built a more efficient
system of communication than the modern languages. As Weyl (1956)
suggests, a mathematician is free to forget what the symbols stand
for and concentrate, like the librarian, only on the catalogue alone.
The details are unimportant; what matters is that once the initial
symbolic scheme S_{0} is given, further work can be carried
along by an absolutely rigid construction that leads from
S_{0} to S_{1} to S_{2} and so on. The idea
of iteration, familiarly encountered with natural numbers, can be
extended in a purely symbolic manner: the construction of not only 1
or 2 but 3, 4, 5, ... and even to manifold dimensions. The
characteristic of being completely precise is the tool that makes
long chains of reasoning possible and exciting. Following this
analogy, once its five dimensions describe a criminal incident, we
can associate a unique mathematical object in a five-dimensional
space. Thereafter, various mathematical techniques may be applied to
examine the nature, location and relations of these
objects.

The topological representation presented in this paper provides a unifying view of hot spots, burning times and stinging laws by labeling them as clusters in space, time and the legal dimension. Furthermore, projection into space takes us to the usual spatial techniques, and projection into space-law takes us into location analysis. By projecting into other spaces such as time-law, one can gain insight about the association between time and law, or, by projecting into space-time, when crimes are most likely to occur at certain spots. More significantly, the unified approach presented in this work opens up the possibility of considering the different dimensions of crime singly or simultaneously to analyze criminal patterns in a realistic manner.

The notion of metric space can be used to explore the properties of the criminal events and their multifaceted nature in innovative and novel ways. Many real or perceptual distance functions can be conjectured that will have the potential of opening up a new world of visualizing and analyzing criminal events. Beyond the familiar number crunching type of techniques, the mathematics of structures, relationships and abstractness is likely to reveal to crime analysts deeper insights into the subject matter of crime and its control mechanism. Simple reasoning and conceptual mathematical relationships will help develop knowledge of an abstract non-empirical domain that nonetheless would still provide practical utility in the empirical world.

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