Figure 1.
Empirical evidence for the recurrent nature of crime, depicting the number of robberies per 100,000 population.
Next to the average for the United States (black), the figure shows averages of states with very high (blue), high (green), moderate (red) and low (grey) robbery rates. The four categories are obtained through grouping states by their average robbery rate over the period 1965–2010 and then assigning the 25% most affected states to the first category, the 25% next most affected states to the second category, etc. Examples of states with very high robbery rates are Washington D.C. and New York. States with low robbery rates are, for instance, North Dakota and Utah. In states with very high robbery rates, the number of robberies oscillates considerably over time. In states with high and moderate robbery rates numbers follow the same up and down trends but at significantly lower levels and with smaller variations. In the states with low robbery rates numbers are very low and remain more or less stable over time. Similar dependencies are found for other kinds of crime such as motor vehicle theft and property crime.
Figure 2.
Phase diagrams, demonstrating the spontaneous emergence and stability of the recurrent nature of crime and other possible outcomes of the evolutionary competition of criminals (), ordinary individuals (
) and punishing inspectors (
).
The diagrams show the strategies remaining on the square lattice after sufficiently long relaxation times as a function of the (relative) inspection costs and the (relative) temptation
, (A) for inspection incentive
and (B) for
. The overlayed color map encodes the crime rate, i.e. the stationary density of criminals in the system. (A) For small and intermediate values of
and
, cyclic dominance between the three strategies characterizes the evolutionary dynamics. Criminals outperform ordinary people, ordinary people outperform inspectors, and inspectors outperform criminals. This cyclic dominance leads to recurrent outbreaks of crime during the evolutionary process. If either
or
exceed a certain threshold, the cyclic phase ends with a continuous phase transition to a mixed
phase (lower solid line), where inspectors and criminals coexist. Further increasing the two parameters leads to another continuous transition (upper solid line) and an absorbing
phase, where criminals dominate. In other words, when a certain value of temptation
is exceeded, it cannot be compensated by larger inspection costs
anymore. A re-entry into the cyclic
phase is possible through a succession of two discontinuous phase transitions (dashed lines) occurring for sufficiently small
and decreasing inspection costs. First, the absorbing
phase changes abruptly to an absorbing
phase dominated by inspectors, which then changes abruptly to the cyclic phase. (B) Increasing the value of
increases the region of cyclic dominance, but also eliminates the possibility of complete dominance of inspectors. Qualitatively, however, the evolutionary dynamics within the different phases does not change compared to
, and remains the same also for
. Dash-dotted gray lines in A and B correspond to the condition
, i.e. where the probability for criminals to be detected is the same as the temptation to commit crime, and a transition to criminal behavior would be expected according to the rational choice equation (1).
Figure 3.
Representative cross-sections of phase diagrams, invalidating straightforward gain-loss principles (“linear thinking”) as proper description of the relationship between crime, inspection and punishment rates.
As the inspection costs increase, there are virtually no clear trends regarding the impact that such changes have on the outcome of the evolutionary process. (A) For
and
, increasing
initially leaves the stationary densities of strategies almost unaffected, while a discontinuous first-order phase transition to complete dominance of inspectors occurs at
. Another discontinuous first-order phase transitions follows at
, where the dominance of inspectors is replaced by the dominance of criminals. (B) For
and
, ordinary people first disappear in a second-order continuous phase transition at
, thereby terminating the cyclic
phase. In the following mixed
phase, the impact of increasing inspection costs
leads to an increase in the number of criminals, which finally gives rise to a complete dominance through another second-order continuous phase transition occurring at
. It is worth noting that the impact of increasing temptation
at a fixed value of
is analogous.
Figure 4.
Snapshots of three different realizations of the cyclic phase, revealing the microscopic dynamics behind the cycles of crime.
Invasion fronts can be either led by the criminals (red), by the inspectors (green), or by ordinary individuals (blue). (A–D) For small inspection costs , large temptation
and medium inspection incentives
, the situation is initially dominated by inspectors, after which clusters of ordinary individuals spread. Then, a front of criminals invades the area of ordinary individuals, but they are chased by inspectors. As a result, inspectors prevail, with clusters of ordinary individuals and a few criminals in between. (E–H) For medium inspection costs
, small temptation
and medium inspection incentives
, the situation is first dominated by ordinary individuals, then by criminals. Afterwards, inspectors form an invasion front entering the domain of criminals, followed by ordinary individuals. Eventually, the situation is dominated by ordinary individuals with some inspectors and criminal enclaves in between. (I–L) For even higher inspection costs
, moderate temptation
and medium inspection incentives
, the evolutionary process is dominated by criminals in the early stage and later by inspectors. Finally, however, inspectors and criminals prevail, with a few clusters of ordinary individuals in between. Supplementary videos showing all three evolutionary processes from the beginning until the convergence to the stationary distribution of strategies are available at the following links: (A–D) youtube.com/watch?v = pH9l-2h6PRo, (E–H) youtube.com/watch?v = gVnCN3a9ki8, and (I–L) youtube.com/watch?v = ehlDSde3BM4.
Figure 5.
Robustness of crime cycles against the variation of network topology.
The social interaction networks were constructed by rewiring links of a square lattice of size with probability
. For low values of
, small-world properties emerge, while for
we have a random regular graph. As
is small and increases, the stationary fractions of the three competing strategies remain almost the same. However, due to the increasing interconnectedness of the players, the amplitude of oscillations increases. When a critical threshold value of
is reached, the maxima become comparable to the system size and oscillations terminate abruptly. The winner is the strategy that mediates the evolutionary competition between the two other strategies. (A) For small inspection costs
, large temptation
and moderate inspection incentives
, ordinary people are the winners [see the evolution in panels (A–D) in Fig. 4, in particular panel (C)]. (B) For moderate inspection costs
, small temptation
, and medium inspection incentives
, criminals are the winners [see the evolution in panels (E–H) in Fig. 4, in particular panel (G)]. While cycles of crime are in general robust to variations of the network structure, the globalization by shortcut links adds another layer of complexity to the game that can result in the emergence of discontinuous phase transitions to absorbing states, for example, the prevalence of ordinary individuals (but not necessarily so). Note, however, that the evolutionary dynamics becomes more and more fragile as the cycles escalate [inset in (B)] shows the envelope of oscillations of
), until they eventually involve almost the whole population. A supplementary video depicting such an evolutionary process where criminals are the victors is available at youtube.com/watch?v = oGNOmLognOY. The final outcome of this dynamics may be hard to predict, especially if the population size is small and the strategies become subject to random extinction.