††thanks: ORCID: https://orcid.org/0000-0002-3004-0930††thanks: ORCID:
http://orcid.org/0000-0001-8837-4127

# The 511 keV Excess and Primordial Black Holes

Celeste Keithb,c Dan Hoopera,b,c aFermi National Accelerator Laboratory,
Theoretical Astrophysics Group, Batavia, IL, USA  bUniversity of Chicago,
Kavli Institute for Cosmological Physics, Chicago IL, USA  cUniversity of
Chicago, Department of Astronomy and Astrophysics, Chicago IL, USA

###### Abstract

An excess of 511 keV photons has been detected from the central region of the
Milky Way. It has been suggested that the positrons responsible for this
signal might be produced through the Hawking evaporation of primordial black
holes. After evaluating the constraints from INTEGRAL, COMPTEL, and Voyager 1,
we find that black holes in mass range of $\sim$ $(1-4)\times 10^{16}$ g could
potentially produce this signal if they make up a small fraction of the total
dark matter density. Proposed MeV-scale gamma-ray telescopes such as AMEGO or
e-ASTROGAM should be able to test this class of scenarios by measuring the
diffuse gamma ray emission from the Milky Way’s inner halo.

††preprint: FERMILAB-PUB-21-088-T

The INTEGRAL satellite has detected an excess of 511 keV photons from the
inner Milky Way relative to astrophysical expectations. This signal consists
of a flux of (1.07 $\pm$ 0.03) $\times$10-3 photons cm-2 s-1, requiring the
injection of $\sim$ $2\times 10^{43}$ positrons per second Weidenspointner
_et al._ (2004); Churazov  _et al._ (2005); Weidenspointner  _et al._ (2007);
Jean  _et al._ (2006); Weidenspointner  _et al._ (2008); Kierans  _et al._
(2020); Prantzos (2006). A variety of potential astrophysical sources for this
signal have been proposed, including type Ia supernovae Kalemci  _et al._
(2006), gamma-ray bursts Casse  _et al._ (2004); Bertone  _et al._ (2006),
microquasars Guessoum  _et al._ (2006), low-mass X-ray binaries Bartels  _et
al._ (2018), and neutron star mergers Takhistov (2020); Fuller  _et al._
(2019). However, given the challenges involved in explaining the observed
characteristics of this signal (for a review, see Ref. Prantzos  _et al._
(2011)), a number of more exotic scenarios have also been put forth. In
particular, the annihilations of MeV-scale dark matter particles have been
considered in detail within this context Boehm  _et al._ (2004); Huh  _et al._
(2008); Hooper and Zurek (2008); Khalil and Seto (2008), although this
possibility is constrained by measurements of the damping tail of the cosmic
microwave background Wilkinson  _et al._ (2016) (see, however, Refs. Ema  _et
al._ (2021); Escudero (2019); Sabti  _et al._ (2020)). Explanations featuring
decaying Hooper and Wang (2004); Cembranos and Strigari (2008); Craig and Raby
(2009), or upscattering Finkbeiner and Weiner (2007); Pospelov and Ritz
(2007); Cline  _et al._ (2011); Cline and Frey (2012) dark matter particles
also remain potentially viable. Other exotic scenarios involving Q-balls
Kasuya and Takahashi (2005), pico-charged particles Farzan and Rajaee (2017,
2020), quark nuggets Lawson and Zhitnitsky (2017), or unstable MeV-scale
states produced in supernovae Davoudiasl and Perez (2010) have also been
discussed within this context.

Another possibility that we will consider here is that the excess positrons
could be produced through the Hawking evaporation Hawking (1975); Gibbons and
Hawking (1977) of a population of primordial black holes concentrated in the
Inner Galaxy Frampton and Kephart (2005); Bambi  _et al._ (2008); Cai  _et
al._ (2020) (see also, Refs. Laha (2019); DeRocco and Graham (2019)). As we
will show, black holes with masses in the range of $m_{\rm BH}\sim(1-4)\times
10^{16}$ g could produce the required flux of positrons while remaining
consistent with all existing constraints, including those from the COMPTEL,
INTEGRAL, and Voyager 1 satellites Laha  _et al._ (2020); Coogan  _et al._
(2020); Boudaud and Cirelli (2019). For other constraints, see Refs. Carr  _et
al._ (2010, 2020a); Dasgupta  _et al._ (2020).

In this letter, we explore the possibility that primordial black holes could
be the source of the positrons responsible for INTEGRAL’s 511 keV excess.
After identifying the regions of parameter space that can accommodate this
signal, we calculate the gamma-ray spectrum from these black holes (including
direct Hawking radiation, photons from the inflight annihilation of positrons,
and final-state radiation), and derive constraints based on data from the
gamma-ray telescopes INTEGRAL and COMPTEL. In scenarios in which black holes
can produce the observed 511 keV signal, we find that the number density of
black holes in the local halo is $\sim$ $10^{12}\,{\rm pc}^{-3}$,
corresponding to a median instantaneous distance of $\sim$ 10 AU to the
nearest such black hole, well within the boundaries of our Solar System. Given
the significant velocities of any primordial black holes in the Milky Way’s
halo, we should expect individual black holes to regularly pass through the
inner Solar System. Even at such close distances, however, it would be very
challenging to detect the Hawking radiation from an individual black hole in
this mass range. On the other hand, it should be possible to definitively test
this class of scenarios by characterizing the diffuse MeV-scale gamma-ray
emission from the halo of the Milky Way with proposed gamma-ray telescopes
such as AMEGO or e-ASTROGAM.

Figure 1: The predicted flux and angular profile of 511 keV photons, averaged
over $-8^{\circ}<l<+8^{\circ}$, and compared to the measurements of the
INTEGRAL satellite Bouchet  _et al._ (2010). Results are shown for four
choices of the density profile’s inner slope, $\gamma$. In each frame, we have
selected values of $m_{\rm BH}$ and $f_{\rm DM}$ which provide the best
possible normalization to this data. The solid curves represent an estimate
for the contribution from astrophysical sources in the Galactic Disk Robin
_et al._ (2003), while the dashed curves correspond to the total contribution
from disk sources and primordial black holes.

Black holes radiate all particle species lighter than or comparable to their
temperature, which is related to the mass of the black hole:

$\displaystyle T_{\rm BH}=\frac{M_{\text{Pl}}^{2}}{8\pi m_{\text{BH}}}\approx
1.05\,{\rm MeV}\,\bigg{(}\frac{10^{16}\,{\rm g}}{m_{\rm BH}}\bigg{)}.$ (1)

This radiation causes a black hole111For context, the Schwarzschild radius of
a black hole is given by $r_{s}=2m_{\rm BH}/M^{2}_{\rm Pl}\simeq 1.5\times
10^{-12}\,{\rm cm}\times(m_{\rm BH}/10^{16}{\rm g})$ to lose mass at the
following rate:

$\displaystyle\\!\begin{aligned}
\frac{dm_{\textrm{BH}}}{dt}&=-\frac{\mathcal{G}g_{*,H}(m_{\textrm{BH}})M_{\textrm{Pl}}^{2}}{30720\pi
m_{\textrm{BH}}^{2}}\\\ &\approx-8.2\times
10^{-7}\,\textrm{g/s}\left(\frac{g_{*,H}}{10.92}\right)\left(\frac{10^{16}\textrm{g}}{m_{\textrm{BH}}}\right)^{2},\\\
\end{aligned}$ (2)

where $\mathcal{G}\approx$ 3.8 is the appropriate greybody factor,
$M_{\text{Pl}}$ is the Planck mass, and $g_{*,H}$ counts the number of (spin-
weighted) degrees-of-freedom with $m\ll T_{\rm BH}$, receiving a contribution
of 6 from neutrinos, 4 from electrons, 0.82 from photons, and 0.1 from
gravitons MacGibbon and Webber (1990); MacGibbon (1991). Integrating this
expression, one finds that a black hole with an initial mass of $m_{\rm
BH}\sim 5\times 10^{14}$g will evaporate in a time equal to the age of the
universe.

The spectrum of Hawking radiation from an individual black hole is given by
Page (1976):

$\displaystyle\frac{dN}{dE}(m_{\text{BH}},E)=\frac{1}{2\pi^{2}}\frac{E^{2}\sigma(m_{\text{BH}},E)}{e^{E/T}\pm
1},$ (3)

where both the sign in the denominator and the absorption cross section,
$\sigma$, depend on the spin of the particle species being radiated. For
fermions (bosons), the sign is positive (negative). In the high-energy limit
($E\gg T$), the absorption cross section approaches $\sigma\simeq 27\pi
m^{2}_{\rm BH}/M^{4}_{\rm Pl}$ for all particle species. In the opposite
limit, this cross section depends both on the particle species in question,
and on the energy of those particles. In performing our calculations, we do
not rely on any limiting behavior, but instead use the full spectra as
presented in Ref. Page (1976).

Figure 2: In each frame, the orange band denotes the 2$\sigma$ region of
parameter space in which the 511 keV excess observed by INTEGRAL could be
produced through the Hawking evaporation of primordial black holes. Also shown
are the constraints on this parameter space derived from the measurements of
the INTEGRAL, COMPTEL, and Voyager 1 satellites. Black holes with masses of
$\sim(1-4)\times 10^{16}$ g could produce the observed excess if they
constitute a small fraction of the total dark matter and are distributed
according to a halo profile with a very steep inner slope, $\gamma\simeq
1.6-1.8$.

In order to identify the parameter space in which primordial black holes could
produce the observed 511 keV excess, we calculate the flux and spatial
distribution of the positrons injected from these objects and compare this to
the intensity and morphology of the 511 keV signal as reported in Ref. Bouchet
_et al._ (2010). The flux of 511 keV photons from black holes near the
Galactic Center observed over a region of solid angle, $\Delta\Omega$, is
given by:

$\displaystyle F_{511}(\Delta\Omega)$ $\displaystyle=\frac{L_{511}(m_{\rm
BH})}{4\pi}\int_{\Delta\Omega}\int_{los}n_{\textrm{BH}}(l,\Omega)\,dl\,d\Omega,$
(4) $\displaystyle\approx\frac{0.55\,L_{e^{+}}(m_{\rm BH})\,f_{\rm BH}}{4\pi
m_{\textrm{BH}}}\int_{\Delta\Omega}\int_{los}\rho_{\textrm{DM}}(l,\Omega)\,dl\,d\Omega,$

where $n_{\rm BH}$ is the number density of black holes, $L_{511}$ is the
number of 511 keV photons produced by a given black hole per unit time, and
the integrals are performed over solid angle and the line-of-sight, $l$. In
the second line, we have written the number density of black holes in terms of
the dark matter density, $n_{\rm BH}=f_{\rm BH}\,\rho_{\rm DM}/m_{\rm BH}$,
where $f_{\rm BH}$ is the fraction of the total dark matter that consists of
black holes. When a black hole emits a positron, it can either directly
annihilate to produce two 511 keV photons, or form a positronium bound state
with an electron, which results 25% of the time in two 511 keV photons, and
75% of the time in three photons, each with $E_{\gamma}<$ 511 keV. Given that
observations indicate that the positronium fraction in the interstellar medium
of the Milky Way is $f=0.967\pm 0.022$ Jean  _et al._ (2006), the number of
511 keV photons produced per positron is $2(1-f)+2f/4\approx 0.55$.

For the dark matter distribution, we adopt a generalized Navarro-Frenk-White
(NFW) halo profile Navarro  _et al._ (1996, 1997):

$\displaystyle\rho_{\rm
DM}=\frac{\rho_{0}}{(r/R_{s})^{\gamma}\,[1+(r/R_{s})]^{3-\gamma}},$ (5)

where $r$ is the distance from the Galactic Center. Throughout this study, we
have adopted a scale radius of $R_{s}=20$ kpc and have normalized $\rho_{0}$
such that the local density (at $r=8.25$ kpc) is 0.4 GeV/cm3. We take the
inner slope of this profile, $\gamma$, to be a free parameter.

In Fig. 1, we have plotted the 511 keV emission predicted from primordial
black holes as a function of galactic latitude (averaging over
$-8^{\circ}<l<+8^{\circ}$), for four choices of the density profile’s inner
slope, $\gamma$. In each frame, we compare the predicted profile of this
emission to the measurements by INTEGRAL Bouchet  _et al._ (2010). In addition
to the 511 keV emission resulting from positrons radiated from black holes, we
have included an estimate for the contribution from astrophysical positron
emission in the disk, as described in Ref. Robin  _et al._ (2003). Given the
uncertainties regarding the magnitude of the astrophysical contribution, we
have allowed its normalization to float from its default value by up to a
factor of two. From this figure, it is clear that a very steep density profile
is required if primordial black holes are to produce the observed intensity of
the 511 keV emission from the inner few degrees around the Galactic Center
(see also, Refs. Vincent  _et al._ (2012); Ascasibar  _et al._ (2006)).
Formally, our fit to this data favors $\gamma=2.2\pm 0.6$ (at $2\sigma$),
although a somewhat larger range could plausibly be accommodated if all of the
related uncertainties were taken into account. While this range is well above
that typically favored by numerical simulations of cold dark matter
($\gamma\sim 1-1.4$ Gnedin  _et al._ (2011a, 2004); Governato  _et al._
(2012); Kuhlen  _et al._ (2013); Weinberg and Katz (2002, 2007); Sellwood
(2003); Valenzuela and Klypin (2003); Colin  _et al._ (2006); Scannapieco  _et
al._ (2012); Calore  _et al._ (2015); Schaller  _et al._ (2015); Di Cintio
_et al._ (2014a, b); Schaller  _et al._ (2016); Bernal  _et al._ (2016)), the
lower portion of the range favored by our fit could be potentially viable if
adiabatic contraction is efficient (see, for example, Ref. Gnedin  _et al._
(2011b)). In light of this, we chose to focus on the lowest portion of the
range favored by our fit, $\gamma\sim 1.6-1.8$.

In each frame of Fig. 1, we have selected values of $m_{\rm BH}$ and $f_{\rm
DM}$ which provide the best possible normalization to the INTEGRAL data. The
orange bands shown in Fig. 2 represent the range of these parameters that lead
to the best-fit normalization for the 511 keV signal (For results using other
values of $\gamma$, or for non-monochromatic mass distributions, see the
Appendix.).

In addition to electrons and positrons, black holes produce gamma rays which
can be used to constrain this class of scenarios. Photons can be produced
directly as the products of Hawking evaporation (see Eq. 3), as well as
through the inflight annihilation of positrons, and as final state radiation.
This inflight annihilation, final state radiation, and the contribution
directly from black hole evaporation, make up our 511 keV photon emission from
black holes, and takes into account electron propagation in the Milky Way. The
flux of these gamma rays from a population of black holes observed over a
solid angle, $\Delta\Omega$, is given by:

$\displaystyle F_{\gamma}(\Delta\Omega)$ $\displaystyle=\frac{dN^{\rm
tot}_{\gamma}}{dE_{\gamma}}\frac{1}{4\pi}\int_{\Delta\Omega}\int_{los}n_{\textrm{BH}}(l,\Omega)\,dl\,d\Omega,$
(6) $\displaystyle=\frac{dN^{\rm tot}_{\gamma}}{dE_{\gamma}}\frac{f_{\rm
BH}}{4\pi m_{\rm
BH}}\int_{\Delta\Omega}\int_{los}\rho_{\textrm{DM}}(l,\Omega)\,dl\,d\Omega,$

where $dN^{\rm tot}_{\gamma}/dE_{\gamma}$ is the gamma-ray spectrum from an
individual black hole, including the contributions from direct Hawking
evaporation, inflight positron annihilation, and final state radiation. In
Fig. 3, we plot the spectrum of the gamma-ray emission from a black hole with
a mass of $m_{\rm BH}=2\times 10^{16}$ g, showing separately each of these
contributions. At the highest energies, the direct Hawking radiation
dominates, while inflight annihilation provides the largest contribution at
lower energies. For the calculation of the contributions from inflight
annihilation and final state radiation, see the Appendix.

Figure 3: The gamma-ray emission from a black hole with a mass of $m_{\rm
BH}=2\times 10^{16}$ g, showing separately the contributions from direct
Hawking radiation, inflight positron annihilation, and final state radiation.

To constrain the abundance of primordial black holes in the Inner Galaxy, we
make use of gamma-ray measurements from the INTEGRAL Bouchet  _et al._ (2011)
and COMPTEL Strong  _et al._ (1999) satellites (for lower mass black holes,
one would instead derive constraints from gamma-ray measurements at higher
energies, such as those from EGRET Lehoucq  _et al._ (2009) or Fermi Ackermann
_et al._ (2018)). We utilize the gamma-ray fluxes as measured in the 0.1-0.2
MeV, 0.2-0.6 MeV and 0.6-1.8 MeV bands by INTEGRAL, and in the 1-3 MeV, 3-10
MeV, and 10-30 MeV bands by COMPTEL. For each INTEGRAL band, we integrate the
predicted gamma-ray flux over each of the angular bins presented. We then vary
$f_{\rm BH}$ until we identify the value at which the total $\chi^{2}$ has
increased from its best fit at the 2$\sigma$ level (for similar analyses, see
Refs. Laha  _et al._ (2020); Cirelli  _et al._ (2020)). For our comparison to
the COMPTEL data, the large systematic (and highly correlated) error bars make
it inappropriate to conduct a $\chi^{2}$ analysis, so we simply require that
our predicted flux does not exceed any of the measurements by more than
$2\sigma$ (see also, Ref. Coogan  _et al._ (2020)). We also apply constraints
derived from the local electron/positron flux as measured by Voyager 1 Boudaud
and Cirelli (2019).

In Fig. 2, we plot the INTEGRAL, COMPTEL, and Voyager 1 constraints on the
value of $f_{\rm BH}$, as a function of $m_{\rm BH}$, for $\gamma=1.6$ or 1.8.
After taking these constraints into account, we conclude that $\sim(1-4)\times
10^{16}$ g black holes could produce the observed 511 keV excess if they
constitute a small fraction of the total dark matter halo, $f_{\rm BH}\sim
0.0001-0.004$, and are distributed according to a halo profile with a very
steep inner slope. For this range of parameters, there are approximately $\sim
10^{22}-10^{23}$ black holes in the innermost kpc of the Milky Way. It is
plausible that black holes in this mass range could have been formed in the
very early universe. In particular, the horizon enclosed a total mass of $\sim
10^{16}$ g when the universe was at a temperature of $\sim 10^{8}$ GeV Hawking
_et al._ (1982); Garcia-Bellido  _et al._ (1996); Kawasaki  _et al._ (2016);
Clesse and García-Bellido (2017); Kannike  _et al._ (2017); Kawasaki  _et al._
(1998); Cai  _et al._ (2018); Yoo  _et al._ (2018); Young and Byrnes (2015);
Clesse and García-Bellido (2015); Hsu (1990); La and Steinhardt (1989a, b); La
_et al._ (1989); Weinberg (1989); Steinhardt and Accetta (1990); Holman  _et
al._ (1990); Khlopov and Polnarev (1980). Note that although we have adopted a
monochromatic mass distribution in our calculations, a distributions of masses
with a variance as large as roughly an order of magnitude could plausibly
produce the observed 511 keV signal without conflicting with existing
constraints. We also note that in the presence of a sizable population of
primordial black holes, the remaining dark matter cannot consist of particles
that are capable of producing photons or other readily detectable particles
through their self-annihilations Carr  _et al._ (2020b); Adamek  _et al._
(2019).

In the regions of parameter space in which primordial black holes could
generate the observed 511 keV excess, the number density of black holes in the
vicinity of the Solar System is given by:

$\displaystyle n^{\rm local}_{\rm BH}$ $\displaystyle=\frac{f_{\rm
BH}\rho^{\rm local}_{\rm DM}}{m_{\rm BH}}$ (7) $\displaystyle\simeq 1.0\times
10^{12}\,{\rm pc}^{-3}\times\bigg{(}\frac{f_{\rm
BH}}{10^{-3}}\bigg{)}\,\bigg{(}\frac{2\times 10^{16}\,{\rm g}}{m_{\rm
BH}}\bigg{)}$ $\displaystyle\simeq 1.2\times 10^{-4}\,{\rm
AU}^{-3}\times\bigg{(}\frac{f_{\rm
BH}}{10^{-3}}\bigg{)}\,\bigg{(}\frac{2\times 10^{16}\,{\rm g}}{m_{\rm
BH}}\bigg{)},$

where we have again adopted a local dark matter density of $\rho^{\rm
local}_{\rm DM}=0.4\,{\rm GeV}/{\rm cm}^{3}$. For this number density, we
should expect the closest black hole to be located at a distance of only
$d\sim(3/4\pi n_{\rm BH})^{1/3}\sim\mathcal{O}(10\,{\rm AU})$, and for the
Solar System to contain several hundred black holes at any given moment.
Considering the significant velocities expected of any black holes that are
part of the Milky Way’s dark matter halo, we conclude that black holes should
regularly pass through the inner Solar System in this class of scenarios. Over
a ten year window, for example, and adopting a representative velocity of 300
km/s, we predict a closest approach of a few AU or less. At such close
proximity, one might think that the Hawking radiation from an individual black
hole could be detectable, especially in light of proposed satellite-based
gamma-ray telescopes optimized for sensitivity to MeV-scale photons, such as
AMEGO Caputo  _et al._ (2019) and e-ASTROGAM De Angelis  _et al._ (2017) (see,
for example, Ref. Sobrinho and Augusto (2014)). Even for these future
missions, however, the design sensitivity to point sources of MeV-scale
photons is only $\sim$ $10^{-6}\,{\rm MeV}\,{\rm cm}^{-2}\,{\rm s}^{-1}$. In
comparison, the gamma-ray flux from an individual black hole is predicted to
be $\sim$ $10^{-10}\,{\rm MeV}\,{\rm cm}^{-2}\,{\rm s}^{-1}\times(10\,{\rm
AU}/d)^{2}\,(2\times 10^{16}\,{\rm g}/m_{\rm BH})^{2}$, leading to a
detectable signal only for a black hole closer $d\mathrel{\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}1{\rm AU}$. Even
if we were to be lucky enough to have a black hole at such a close proximity,
its proper motion would significantly complicate the search for its Hawking
radiation.

A more promising way to test this class of scenarios is to use a telescope
such as AMEGO Caputo  _et al._ (2019) or e-ASTROGAM De Angelis  _et al._
(2017) to detect and characterize the diffuse gamma-ray emission generated by
black holes in the Milky Way’s inner halo, or as they contribute to the
isotropic gamma-ray background Ray  _et al._ (2021). Instruments such as AMEGO
or e-ASTROGAM should be able to improve substantially (by approximately an
order of magnitude) on the limits derived in study using COMPTEL and INTEGRAL
data, making it possible to definitively test the range of parameter space in
which primordial black holes could produce the observed 511 keV signal.

In summary, we have shown in this letter that a population of primordial black
holes could potentially produce the excess of 511 keV photons observed from
the Inner Galaxy by the INTEGRAL satellite. To reproduce the angular
distribution of this signal, the black holes must be distributed with a very
cuspy halo profile, featuring an inner slope of $\gamma\sim 1.6$ or higher.
Furthermore, to be consistent with gamma-ray constraints from COMPTEL and
INTEGRAL, and cosmic-ray electron constraints from Voyager 1, the bulk of
these black holes must have masses in the range of $m_{\rm BH}\sim(1-4)\times
10^{16}\,{\rm g}$. To provide the observed normalization of this signal, these
black holes must constitute a relatively small fraction of the total dark
matter abundance, $f_{\rm DM}\sim 0.0001-0.004$.

In our calculations, we have included the photons, electrons and positrons
that are directly produced through Hawking radiation, as well as the gamma-
rays that are generated through the inflight annihilation of positrons, and as
final state radiation. For the range of $m_{\rm BH}$ and $f_{\rm DM}$ required
in this scenario, we predict that the local halo of the Milky Way should
contain a considerable number density of black holes, $n^{\rm local}_{\rm
BH}\sim 10^{12}\,{\rm pc}^{-3}$. Although such black holes would be very
challenging to detect individually, future satellite-based gamma-ray
telescopes such as AMEGO or e-ASTROGAM will be able to definitively test this
class of scenarios by measuring the spectrum and morphology of the MeV-scale
emission from the Inner Galaxy.

###### Acknowledgements.

CK would like to thank Cory Cotter for helpful discussions. CK is supported by
the National Science Foundation Graduate Research Fellowship Program under
Grant No. DGE-1746045. DH is supported by the Fermi Research Alliance, LLC
under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy,
Office of Science, Office of High Energy Physics.

##  Appendix A Inflight Annihilation

In the energy range of interest for this study, most positrons lose energy via
ionization and become non-relativistic before annihilating. A small fraction
of such positrons, however, will annihilate inflight prior to reaching non-
relativistic velocities. Such annihilations can produce photons with energies
greater than 511 keV, thus contributing to the continuum spectrum of diffuse
gamma rays.

Following Ref. Beacom  _et al._ (2005), the spectrum of gamma rays from the
inflight annihilation of positrons is given by:

$\displaystyle\frac{dN^{\rm IA}_{\gamma}}{dE_{\gamma}}$
$\displaystyle=\frac{\pi\alpha^{2}n_{H}}{m_{e}}\int^{\infty}_{m_{e}}dE_{e^{+}}\frac{dN_{e^{+}}}{dE_{e^{+}}}\int^{E_{e^{+}}}_{E_{\rm
min}}\frac{dE}{dE/dx}\,\frac{P_{E_{e^{+}}\rightarrow E}}{(E^{2}-m_{e}^{2})}$
(8)
$\displaystyle\times\bigg{(}-2-\frac{(E+m_{e})(m_{e}^{2}(E+m_{e})+E_{\gamma}^{2}(E+3m_{e})-E_{\gamma}(E+m_{e})(E+3m_{e}))}{E_{\gamma}^{2}(E-E_{\gamma}+m_{e})^{2}}\bigg{)},$

where $\alpha$ is the fine structure constant, $n_{H}$ is the number density
of neutral hydrogen atoms, and $dN_{e^{+}}/dE_{e^{+}}$ is the spectrum of
positrons radiated from the black hole. $dE/dx$ is the energy loss rate of a
positron due to ionization in the presence of neutral hydrogen, as described
by the Bethe-Bloch formula. Since $dE/dx$ is proportional to $n_{H}$, the flux
of gamma rays from inflight annihilation is not sensitive to the value of the
gas density.

The probability that a positron of energy $E_{e^{+}}$ will survive until its
energy has been reduced to $E$ is given by

$\displaystyle P_{E_{e^{+}}\rightarrow
E}=\exp\bigg{(}-n_{H}\int^{E_{e^{+}}}_{E}\sigma_{\rm
ann}(E^{\prime})\,\frac{dE^{\prime}}{|dE^{\prime}/dx|}\bigg{)},$ (9)

where $\sigma_{\rm ann}$ is the cross section for a positron annihilating with
an electron at rest. For positrons with an energy below a few MeV,
$P_{E_{e^{+}}\rightarrow m_{e}}$ is always between $\sim$ 0.95 and unity. In
other words, only a few percent of the positrons annihilate before becoming
non-relativistic.

##  Appendix B Final State Radiation

Electrons, positrons, and other charged particles emitted from a black hole
can radiate photons as final state radiation. This process contributes as
follows to the gamma-ray spectrum:

$\displaystyle\frac{dN^{\rm FSR}_{\gamma}}{dE_{\gamma}}$
$\displaystyle=\frac{\alpha}{2\pi}\int
dE_{e}\frac{dN_{e}}{dE_{e}}\,\bigg{(}\frac{2}{E_{\gamma}}+\frac{E_{\gamma}}{E^{2}_{e}}-\frac{2}{E_{e}}\bigg{)}\,\bigg{[}\ln\bigg{(}\frac{2E_{e}(E_{e}-E_{\gamma})}{m^{2}_{e}}\bigg{)}-1\bigg{]},$
(10)

where $dN_{e}/dE_{e}$ is the spectrum of electrons and positrons radiated from
the black hole.

Figure 4: As in Fig. 2, but for black hole distributions with an inner slope
of $\gamma=1$ (left) or $\gamma=2$ (right). In the $\gamma=1$ case, we do not
show any region for the 511 keV excess, as the angular distribution of this
signal cannot be accommodated for this choice of halo profile. In the
$\gamma=2$ case, the region favored by the 511 keV excess is ruled out by a
combination of the constraints from COMPTEL, INTEGRAL, and Voyager 1.

##  Appendix C Results For Other Values of $\gamma$

In Fig. 2, we showed the regions of black hole parameter space that can
accommodate the observed characteristics of the 511 keV excess, along with the
constraints from gamma-ray and cosmic-ray electron measurements, for halo
profiles with an inner slope of $\gamma=1.6$ or 1.8. In Fig. 4, we show the
corresponding results for the cases of $\gamma=1$ or $\gamma=2$. In the
$\gamma=1$ frame, we do not show any region for the 511 keV excess, as the
angular distribution of this signal cannot be accommodated for this choice of
halo profile. In the $\gamma=2$ case, the region favored by the 511 keV excess
is ruled out by a combination of the constraints from COMPTEL, INTEGRAL, and
Voyager 1.

Figure 5: the constraints on the black hole parameter space from individual
energy bins of data from the INTEGRAL (dashed) and COMPTEL (solid) satellites,
for the case of $\gamma=1.8$.

##  Appendix D Bin-by-Bin Gamma-Ray Constraints

In Fig. 5, we show the constraints on the black hole parameter space as
derived from individual energy bins of INTEGRAL and COMPTEL data, for the case
of $\gamma=1.8$. INTEGRAL provides the most stringent constraints on black
holes more massive than $\sim 2\times 10^{16}\,{\rm g}$, while COMPTEL (and
Voyager 1) are most restrictive for smaller values of $m_{\rm BH}$.

##  Appendix E Non-Monocromatic Mass Spectrum

In addition, we show the constraints in the black hole parameter space for a
non-monochromatic mass spectrum in Fig. 6. For this mass spectrum, we follow a
log-normal distribution and plot the parameter space as a function of median
black hole mass, with $\sigma=1$ (left) and $\sigma=2$ (right). We consider a
value of $\gamma=1.8$. For a Gaussian distribution of black holes masses, the
constraints from COMPTEL and/or INTEGRAL are approximately saturated across a
wide range of median mass values. In such a scenario, a large fraction of the
gamma-ray emission observed from the Inner Galaxy must originate from PBHs.

Figure 6: As in Fig. 2, but for black hole distributions with an inner slope
of $\gamma=1.8$, with a non-monochromatic mass distribution. We do not show
the Voyager constraints, as they are specific to a monochromatic mass
spectrum. The non-monochromatic mass spectrum is calculated using a log normal
distribution, with the median black hole mass being plotted on the $x$ axis,
with $\sigma=1$ in the left plot, and $\sigma=2$ on the right.

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All authors contributed equally to this work.

All authors contributed equally to this work.

All authors contributed equally to this work.

All authors contributed equally to this work.

[1]Computer Science Department, Stanford University, Stanford, USA

2]Mathematics Department, Stanford University, Stanford, USA

3]Electrical Engineering Department, Stanford University, Stanford, USA

# Actionable Insights on Philadelphia Crime Hot-Spots: Clustering and
Statistical Analysis to Inform Future Crime Legislation

Ishan S. Khare  Tarun Martheswaran  Rahul Thomas  Aditya Bora  *  [  [

##  1 Abstract

Philadelphia’s problem with high crime rates continues to be exacerbated as
Philadelphia’s residents, community leaders, and law enforcement officials
struggle to address the root causes of the problem and make the city safer for
all. In this work, we deeply understand crime in Philadelphia and offer novel
insights for crime mitigation within the city. Open source crime data from
2012-2022 was obtained from OpenDataPhilly. Density-Based Spatial Clustering
of Applications with Noise (DBSCAN) was used to cluster geographic locations
of crimes. Clustering of crimes within each of 21 police districts was
performed, and temporal changes in cluster distributions were analyzed to
develop a Non-Systemic Index (NSI). Home Owners’ Loan Corporation (HOLC)
grades were tested for associations with clusters in police districts labeled
‘systemic.’ Crimes within each district were highly clusterable, according to
Hopkins’ Mean Statistics. NSI proved to be a good measure of differentiating
systemic ($<$ 0.06) and non-systemic ($\geq$ 0.06) districts. Two systemic
districts, 19 and 25, were found to be significantly correlated with HOLC
grade (p $=2.02\times 10^{-19}$, p $=1.52\times 10^{-13}$). Philadelphia crime
data shows a high level of heterogeneity between districts. Classification of
districts with NSI allows for targeted crime mitigation strategies.
Policymakers can interpret this work as a guide to interventions.

Keywords: Patterns, Redlining, Clustering, Philadelphia

##  2 Background and Introduction

Philadelphia is one of the most populated cities in the United States, with a
rich history and a diverse population. However, like many large cities,
Philadelphia has struggled with high crime rates for decades. It has long been
known for its high crime rates, particularly violent crimes such as homicide
and aggravated assault. In recent years the city continues to experience
higher crime rates compared to many other large cities in the United States
[2]. According to the Philadelphia Police Department, in 2021 there were 499
homicides, which is the highest number in the city in 30 years [18].

Over the past 10 years, Philadelphia has implemented various policing
strategies aimed at reducing crime rates in hotspot districts. These include
community policing, which emphasizes collaboration and communication between
the police and community members, as well as hot-spot policing, which focuses
on targeting areas with high levels of criminal activity [3]. Despite these
efforts, crime rates in some districts of Philadelphia remain stubbornly high:
policing these districts requires a nuanced understanding of the local
community, its needs, and its priorities [19].

Indeed, one of the key challenges in addressing crime in Philadelphia is the
issue of policing districts. Philadelphia is divided into 21 police districts,
each of which has its own unique characteristics and challenges. Some
districts, such as Center City and University City, are densely populated and
experience a high volume of tourism and commerce, while others are more
residential and may be marked by poverty and disinvestment [7]. Any crime
prevention strategy that neglects these differences will ultimately be
ineffective.

Furthermore, it is critical that community leaders understand the policing
alone is very unlikely to reduce crime [5]. Especially given the recent events
regarding police brutality, is it paramount that policy makers and community
leaders use different methods to try and reduce crime. For instance, tools
such as educational investments and infrastructure investments have proven to
be effective ways of solving root causes of crime [12]. These among other
techniques should be heavily considered and appropriately implemented when
creating crime mitigation strategies [15].

Our project aims to explore and utilize the temporal and geospatial trends in
Philadelphia’s crime rates in order to better inform the policing,
infrastructure, and educational decisions within particular districts.
Ultimately, we seek to provide insight into the complex and multifaceted
nature of crime and policing in Philadelphia. By examining the data collected
regarding previous crimes, demographics, systemic discriminatory practices
like redlining, among other factors, we formed a deeper understanding of the
issues exacerbating crime rates in Philadelphia.

Ultimately, we discovered that crime can be classified based on location and
temporal features which can assist policy makers in creating more effective
targeted legislation.

##  3 Technical Exposition

###  3.1 Data Cleaning and Feature Engineering

We extensively use crime data from OpenDataPhilly, providing a comprehensive
table of data of crimes reported in Philadelphia from 2012 to 2022 [17]. The
particularly important features are the time of dispatch, geospatial data
represented in terms of latitude and longitude, the street address of the
crime, and associated police district. The main feature we use is the
geospatial data, in terms of latitude and longitude.

Data cleaning of crime data is kept relatively simple, as the main data we
wish to use is the geospatial information of crimes. First, we delete rows
with ‘NA’ listed under the location. Second, we remove rows with invalid
dispatch times and dates, and street addresses. These invalid entries account
for $0.7\%$ of all rows. In addition, we remove data points belonging to
districts 4 and 23 as there is only one datapoint for each of the districts.

Finally, using the district feature, we computed aggregate counts of crime in
each district from 2012 to 2022. We found that in districts $4$ and $23$,
there had only been one crime in $10$ years, so we removed these entries. This
completes the cleaning and feature engineering for data obtained and compiled
from the OpenDataPhilly dataset.

###  3.2 Exploratory Data Analysis

####  3.2.1 Understanding Police Districts

In Philadelphia, a police district is a geographic area that is patrolled by
officers from the Philadelphia Police Department. According to the City of
Philadelphia, there are a total of 21 police districts in the city.

Each police district covers a specific area of the city and is responsible for
responding to emergency calls, investigating crimes, and maintaining public
safety in that area. The boundaries of the police districts are based on
factors such as population density, crime rates, and geographic features. Each
district is staffed by police officers, detectives, and support personnel and
is responsible for responding to calls for service, conducting investigations,
and engaging with the community [4].

Given that police resources are allocated within these districts, we begin our
analysis by analyzing trends with respect to these district boundaries. We
also choose to analyze crime distributions within Philadelphia without the
context of the police district boundaries to see if there are larger
conclusions/trends which cross district borders.

####  3.2.2 Temporal Analysis By Police District

To better understand the city of Philadelphia’s crime incidence we wish to
granularize our analysis. An obvious point of exploration is a location based
exploration, specifically by police district. With the upcoming plots we wish
to reveal characteristics about the police district level crime rate.
Specifically we want to determine the relative frequency of the crime per
district. Here we create heatmaps of the crime distribution by police district
across the city of Philadelphia over a 9-year time frame with 4-year time
increments (Figure 1).

Figure 1: Heat map of crime percentages for each Philadelphia police district
from 2013 - 2019. Dark red regions indicate high crime frequency whereas light
red regions indicate lower crime frequency.

At first glance, we see that there are no changes of greater than 0.5 with
respect to the distribution over time within certain police districts.
However, we do observe that there are general shifts in distribution within
the districts which motivates a more narrowly-focused exploration into
individual geographic regions (i.e. districts). This narrow-focus would allow
for the discovery of any potential temporal changes that are not otherwise
recognizable. In particular, Figure 1 and Figure 2 help us identify and
segment out districts where the crime frequency remained consistent over time
versus districts where the crime frequency changed. Examining districts
individually can help us determine whether there was uniformity in crime
frequency within a given district or not. Such exploration may be useful in
determining if the general trend for each district casts a shadow upon or
hides meaningful insight at a finer level.

Figure 2: For each Philadelphia police district, we plot the crime frequencies
for every year from 2013 - 2019.

####  3.2.3 DBSCAN Clustering

From the previous heat map visualizations in Figure 1, we infer that we should
cluster crimes to extrapolate geospatial hotspots with similar levels of
crime. One common clustering algorithm is $k$-means clustering, where we group
all points into $k$ clusters of close points. This is done by minimizing an
objective function, based on sums of distances squared of points to the
centroids of their clusters. However, there are a few downsides to this
algorithm:

  * •

First, running $k$-means requires prior knowledge of the number of clusters,
which is inconvenient for a large of clusters.

  * •

Furthermore $k$-means has difficulty clustering data which has pockets of
different sizes and densities, which is something we observe in our heat map
visualizations.

  * •

Finally, $k$-means performs poorly when a large number of outliers are
present.

Given these obstacles, we instead opt for a clustering algorithm called DBSCAN
(density based spatial clustering) [8]. This is a non-parametric algorithm,
which groups together points which are packed together; it does not required
knowledge of the cluster count beforehand. It is also one of the most
commonly-used clustering algorithms, and has been proven to be particularly
well-suting in dealing with latitudes and longitudes. Thus, it is well-suited
to the task of clustering crimes based on geospatial proximity. Another
advantage of DBSCAN is its ability to separate high density areas from low
density areas. As seen in the heat map, crime percentages range from $0.4\%$
to over $1\%$, so we can infer DBSCAN will perform well in this regard.

The main idea behind DBSCAN is that a point belongs to a cluster if it is
close to many points from that cluster. There are two parameters used in
DBSCAN:

$\epsilon:\text{minimum distance away for a new point to be added to a
cluster,}$

and

$p_{min}:\text{minimum number of data points to define a cluster.}$

To choose $\epsilon$, we use KNN ($k$ nearest neighbors) [11]. By plotting the
geospatial distances from one data point to the others, sorted in ascending
order, we form an “elbow” curve. This curve has a sharp change in concavity,
which is where we choose $\epsilon$. The value of $p_{min}$ is traditionally
chosen to be twice the number of features, so in this case, we set it to
$2\cdot 2=4$.

Once DBSCAN is finished running, it classifies all points into three levels of
“density”.

  * •

A core point is one where at least $p_{min}$ points within distance
$\epsilon$.

  * •

A border point is one that lies at most $\epsilon$ away from a core point, but
less than $p_{min}$ points within distance $\epsilon$.

  * •

An outlier (noise) is anything that is not a core or border point.

The clusters we obtain will consist of the core points: these represent crime
hotspots, areas where the geospatial distribution of crimes for a particular
year is particularly dense.

####  3.2.4 Clustering Within Philadelphia Over Time

Our preliminary motivation for this section is to analyze crime within
Philadelphia without the context of the police borders. The reason for this is
to better understand trends in crime that cross the boundaries. Initially, we
plotted a heatmap of crime frequency over all of Philadelphia for the years
2012-2022, as displayed in Figure 3. We can visually observe that there seem
to be pockets of higher crime frequency, which motives the need for
clustering. Clustering can help identify spatial patterns and groupings in
geographic data, allowing for more efficient analysis and decision-making.

Figure 3: This figure shows a heatmap of crime frequency over all of
Philadelphia for the years 2012-2022.

Here, we have used DBSCAN clustering over a $9$-year time frame. Above, we
have selected clustering graphs for every $3$ years, starting at $2013$ and
ending at $2022$. Observe that the the locations of particular clusters change
significantly over time. Also, notably, the number of clusters has decreased
over time. Furthermore, when we overlay one such cluster plot with the
district lines, we observe that clusters overlap between different districts.
This is actually a reason to consider clustering within each district, rather
than on all Philadelphia: this overlap suggests an extra level of complexity
that cannot be resolved by solely district or solely proximity of latitude and
longitude, so we are motivated to use a combination of both.

Figure 4: DBSCAN’s Clustering over 9-year time frame

Now, there are a few other reasons that we want to run DBSCAN clustering
within each district. First, from our qualitative observations in Figure 4,
the density is different within each district, so it would make sense to
represent this non-uniformity through district-focused clustering. Second,
aggregate clustering gives at most $13$ clusters in all cases, which is
significantly less than the number of actual police districts: so, we might
gain better insight into geospatial trends of crime by specifying the
algorithm runs within each district. Finally, with regards to politics, we
would expect districts to separate their police officers and resources: so any
sort of policing strategy should not operate based on clusters which lie in
multiple districts.

####  3.2.5 Hopkins statistic testing

Before we run clustering on each district, we introduce a statistical testing
method which evaluates the success of clustering. The Hopkins statistic
provides a measure of how ”uniform” a collection $\mathcal{P}$ of $N$ points
is [13]. When the Hopkins value is close to $0$, the data is largely uniform,
and when it is close to $1$, the data is highly clustered.

We compute the Hopkins statistic as follows. We first select $n<<N$, which
will be our subcollection size for the testing. Then, we choose a random
subset $\mathcal{Q}$ of $n$ points from $\mathcal{P}$. Next, we randomly and
uniformly generate a set $\overline{\mathcal{Q}}$ of $n$ random points.
Finally, as our data is $2$-dimensional, the Hopkins statistics for these
random sets $\mathcal{Q},\overline{\mathcal{Q}}$ are given by

$H(\mathcal{Q},\overline{\mathcal{Q}})=\frac{\sum_{q\in\mathcal{Q}}\min_{p\neq
q\in\mathcal{P}}d(p,q)^{2}}{\sum_{q\in\mathcal{Q}}\min_{p\neq
q\in\mathcal{P}}d(p,q)^{2}+\sum_{\overline{q}\in\overline{\mathcal{Q}}}\min_{p\in\mathcal{P}}d(p,\overline{q})^{2}}.$
(1)

Here, $d(p,q)$ is the distance between $p$ and $q$.

####  3.2.6 Clustering within districts over time

Figure 5: Mean Hopkins Statistic Per District

The results of the Hopkins test, as displayed in Figure 5 show that within
each district the data is highly cluster-able. This justifies the use of
DBScan to visualize these clusters and determine their geographic
displacements over time which is critical as explained in methodology.

###  3.3 Methodology

The Hopkins test showed us that the crime distribution complexity could be
well-modeled within each of the police districts. Given this result, we
propose the following methodology to help classify the type of crime occurring
in a district in order to implement prevention and mitigation measures which
are most likely to be successful.

The overarching goal is to be able to create a model which helps policy makers
develop the most targeted interventions to help prevent crime.

###  3.4 Defining ”systemic” vs ”non-systemic” crime

After visual analysis of the DBSCAN output on the 21 police districts, a clear
theme could be discerned: certain districts would have clusters which don’t
change geographically over time (Figure 7) whereas other districts do (Figure
6). We believe that differentiating these districts can be extremely useful
for policy makers to implement targeted solutions to try and reduce crime.

Figure 6: District 7: Clusters can be seen to change geographic location over
time Figure 7: District 25: Clusters can be seen to stay in the same
geographic location over time

Specifically, we propose a method for classifying districts within two
categories, systemic and non-systemic, which we define below. This
classification allows for targeted implementation of interventions measures.

####  3.4.1 High Level Overview Of Systemic Crime

We have defined systemic crime to be a district in which the crime clusters
have not changed in geography over time (statistical procedure for this
determination is below). Given this, we suggest that interventions which
address the root causes of crime will be more effective than increased
policing. For instance, a tool like education and infrastructure investments
would be more fruitful interventions in these areas. We attempt to discern
factors which drive the ”systemic-ness” of the crime such as comparing against
historic spatial racism which refers to the use of geographic or spatial
factors to perpetuate discriminatory practices.

####  3.4.2 High Level Overview Of Non-Systemic Crime

We define a Non-systemic crime district to be one in which the clusters of
crime changed geographic locations over time. A real world example of a non-
systemic crime district could be one in which crime spiked as a result of
recent civil unrest. In a situation such as this, data suggests that increased
policing would be more effective in decreasing the rate of crime given that
the crime is ”non-systemic” in nature [20] [22].

Now, we introduce our methodology for differentiating clusters within
districts as being systemic vs non-systemic.

###  3.5 Designing a Systemic Index: Procedure for defining a District with
Systemic Crime vs Non-Systemic Crime

We want to differentiate systemic crime, where crime hotspots’ geographic
locations have not changed over time, from non-systemic crime, where they have
changed significantly over time. In this sense, how ’systemic’ the crime is
corresponds to how much variation there is in the locations of clusters over
the years. As it is quite complicated to determine variations in clusters, it
would be prudent to instead represent these clusters by their cluster
representative, the average of their core points in DBSCAN. Formally, for a
cluster $\mathcal{C}$, with $\mathcal{C}_{core}$ being the set of core points
in $\mathcal{C}$, the representative is

$\tilde{x}=\underset{x\in\mathcal{C}_{core}}{\text{avg}}x.$ (2)

Then, for each district, we have temporal data of these collections of cluster
representatives, and want to determine whether these sets change geospatially
over time. The first step is to define a distance between two sets of points.

Suppose we have two sets $S_{1},S_{2}$ of points in the $2$D plane, and we
want to see how ’far’ apart they are. The traditional idea would be to use a
permutation method. Here, we randomly permute the order of $S_{2}$, pair off
points in $S_{1}$ and $S_{2}$, and get the sum of the distances between paired
points. We average these sums over a large number of random permutations, and
this will give a sort of distance metric on sets of points.

One major disadvantage of this method is its bias in classifying areas with
multiple clusters as systemic crime, because the sum of distances will be
larger. Also, it is not clear which points to pair off if the sizes do not
match up. Furthermore, this metric is fairly inaccurate when two sets of
points are close together: in fact, for two equal sets of sparse points, the
computed random permutation metric would be quite large, but the actual
distance should be zero.

Thus, we propose a novel metric on two sets of points $S_{1},S_{2}$. This is
defined as follows, where $d$ represents the usual Euclidean metric between
two points on a plane, with latitude and longitude being the coordinates:

$d_{set}(S_{1},S_{2})=\underset{p_{1}\in S_{1}}{\text{avg}}\underset{p_{2}\in
S_{2}}{\text{min}}d(p_{1},p_{2})+\underset{p_{2}\in
S_{2}}{\text{avg}}\underset{p_{1}\in S_{1}}{\text{min}}d(p_{2},p_{1}).$ (3)

Essentially, for each point in $S_{1}$, we find the closest distance to a
point in $S_{2}$, and then average all these minimal distances; we do the same
process from each point in $S_{2}$ to $S_{1}$; and then finally sum these two
averages. This avoids the bias of the random permutation metric, because it
takes averages rather than sums. Furthermore, this handles the case where
$|S_{1}|\neq|S_{2}|$ well, as it computes the average for each of the two sets
and adds them. Finally, if two sets of points are close together, because we
are taking an average of minimum distances, then the metric will return a
small value, which is fairly accurate.

With the above metric, we define the non-systemic index $\mathcal{NSI}(D)$ for
each district $D$. Suppose we have an array of $n$ sets of points for the
district $D$, say $[S_{1},\ldots,S_{n}]$, where $S_{i}$ is the set of cluster
representatives for year $i$. Our goal is to quantify the overall physical
proximity of these sets. Indeed, by using the distance metric defined in
Equation 3, we can form an $n\times n$ distance matrix $\mathcal{M}(D)$ for
these sets, with entries

$(\mathcal{M}(D))_{i,j}=d_{set}(S_{i},S_{j}),\;\;\;\forall\;1\leq i,j\leq n.$
(4)

Then, the $\mathcal{NSI}$ is computed as the Frobenius norm ($2$-norm) of this
matrix:

$\mathcal{NSI}(D)=\|\mathcal{M}\|_{2}.$ (5)

Note that a higher $\mathcal{NSI}$ indicates that the distance matrix is
larger in magnitude, meaning the cluster representatives physically move more
over time, and thus the crime hotspots change significantly in that district.
That is, a high $\mathcal{NSI}$ indicates that the district is non-systemic.

Figure 8: Police districts are ranked in ascending order by non-systemic
indices over the years 2012-2022.

From the bar graph in Figure 8, we see that it will be interesting to study
the underlying factors for the more systemic and non-systemic districts. Based
on the $\mathcal{NSI}$, we will use District 25 and District 19 in our
analysis of systemic districts, as they have extremely low scores. Also, we
will study District 7 and District 8 in our non-systemic analysis, as they
have the highest $\mathcal{NSI}$ values.

###  3.6 Systemic Analysis

We have defined systemic crime as crime in which clusters do not significantly
change in geographic position over time. The goal is to now determine what
underlying factor has caused the crime to remain in the same location. We
propose and hypothesize that one of the major causes for this crime is
attributed to systemic racism and the historical redlining that occurred in
the city.

####  3.6.1 What is Redlining?

Redlining, the practice of denying financial services such as loans or
insurance to residents of certain neighborhoods based on their race or
ethnicity, has had a significant impact on crime rates in affected
communities. Redlining occurred primarily in the United States from the 1930s
through the 1960s, although its legacy continues to affect many communities
today [23]. One of the ways in which redlining has impacted crime is by
exacerbating poverty and economic inequality in redlined neighborhoods. By
denying residents access to credit and other financial resources, redlining
has made it more difficult for people in these neighborhoods to start
businesses, buy homes, and otherwise build wealth [10]. This has led to higher
levels of poverty and unemployment, which are both risk factors for criminal
behavior [1].

Figure 9: Redlining Data from 1930s with Current Day Police District Map
Overlayed

In Figure 9, we overlay the red-lined neighborhood geography on top of our
clusters to visually examine whether there is an overlap in the red-lined
neighborhoods and the clusters from DBScan. To support our hypothesis, we are
correlating the presence of redlining and institutionalized racism. We map
each latitude-longitude pair to redlining Home Owner’s Loan Corporation (HOLC)
grade (A is the best whereas D is the worst)- with Mapping Inequality data
from The University of Richmond [16].

We elect to use police districts 19 ($\mathcal{NSI}=0.0332$) and 25
($\mathcal{NSI}=0.0273$), as these are classified as systemic, falling below
the median $\mathcal{NSI}$ for districts of $\approx 0.06$. For our analyses,
we re-cluster each district with data across all years, as $\mathcal{NSI}$
established temporal homogeneity. We then compute the breakdown of HOLC grade
for each cluster and observe notable differences and inequities. Both police
districts are characterized by one far larger cluster and several relatively
small clusters. We chose to combine the HOLC grade counts for these smaller
clusters. To assure mitigation of introduced bias, we visually observe the
HOLC grade breakdown from 9. By cross-referencing these with our earlier made
DBSCAN clusters, we see, geographically, that the grades in smaller clusters
are primarily homogeneous. We then use Chi Square Tests for Independence to
determine if there is a statistically significant association between the two
variables [21].

HOLC Grade | Cluster 1 | Other Clusters
---|---|---
A | 0.4% | 7%
B | 20% | 39%
C | 16% | 52%
D | 63.6% | 2%
Table 1: Percentage breakdown of HOLC grade by Cluster in District 19. ($\chi^{2}=90.2$, $p=2.02\times 10^{-19}$) HOLC Grade | Cluster 1 | Other Clusters
---|---|---
A | 0% | 0%
B | 17% | 0%
C | 54% | 100%
D | 29% | 0%
Table 2: Percentage breakdown of HOLC grade by Cluster in District 25.
($\chi^{2}=59.0$, $p=1.52\times 10^{-13}$)

Both Districts 19 and 25 displayed significant associations with HOLC grade,
implying differences between the two cluster groups. It is important to note
that District 25 contained no ”A” HOLC grade-regions and combined clusters
were only in ”C” grade regions, as detailed in Table 2. However, this still
leads us to an important discussion surrounding the lasting effects of
redlining.

###  3.7 Non-Systemic Analysis

We base our non-systemic analysis on police districts that displayed a high
non-systemic index. This implies low variance in the geographic distribution
of crime clusters from year to year. We attempt to determine if there is a
present association between the clustered crime regions and the distribution
of historical redlining grades.

As shown in Figure 8, the two districts that exhibited the most non-systemic
behavior from 2012-2022 were district 7 and district 8. Let us now attempt to
understand potential reasons why this observation may be arising. From
district map data and online reading, we know that both districts 7 and 8 have
large areas of farmland and are more agricultural/rural communities. Let us
first turn to district 7, which, on average, has the lowest frequency of crime
among all the districts (Figure 2). As defined earlier, being non-systemic
means having clusters of crime that change geographic locations over time. In
general, given the smaller sample size (total number of crimes committed) in
district 7, small changes in the number of crimes committed can have a
significant impact on the overall crime rate and consequently the clustering.

Now, we can turn to district 8. Figure 2 shows that district 8 has the highest
incidences of crime during the years 2014-2015. It is well known and well-
documented that economic conditions can be a causal factor for crime [9]. In
particular, rural areas may be more vulnerable to changes in economic
conditions, such as a downturn in the agricultural sector. When people lose
their jobs or experience financial hardship, they may be more likely to engage
in criminal behavior. News articles from 2014-2015 announced that the “Average
household farm income in 2015 is projected to be $19,121, down from last
year’s high of $28,687, according to the USDA” [6]. Thus, the economic
pressure in the agricultural communities of districts 7 and 8 may be leading
to the non-system patterns we observe. The authors acknowledge that sometimes
it can be easy to ”explain-away” certain observations so further investigation
into districts 7 and 8 would be necessary before making strong claims.

###  3.8 Interpretation of Results: Targeted Intervention

In our analysis, we have identified that police districts can be classified as
either being hosts to systemic or non-systemic crime. This determination is
extremely useful for legislators as data shows that different crime mitigation
techniques are more effective within systemic vs non-systemic districts.
Overall we can say that we have taken a major stride towards achieving our
goal of creating actionable insights for effective/targeted crime mitigation
strategies.

While our model is capable of analyzing any particular district within the
city of Philadelphia, we chose to analyze particular districts due to their
more extreme $\mathcal{NSI}$ scores. As a recap, the $\mathcal{NSI}$ score
allows us to determine the ”systemic-ness” of a particular district. For the
following districts, our model along with current event information could
result in suggestions similar to the following.

Districts 7 and 8 (Northeast Philadelphia): The high $\mathcal{NSI}$ score
indicates these districts as non-systemic. Poverty within this district has
also recently been on the rise with neighborhoods such as the Northeast
neighborhood of Holmesburg, which went from a 2 percent poverty rate to 19
percent [14].

Given these two factors, we suggest more reactionary measures. Specifically, a
potential policy suggestion would be to increase police presence within this
area as well as implementing economic stimulus legislation.

Our analyzed systemic Districts (19 and 25), both displayed significant
statistical associations with HOLC grade. For District 19, we see that 63% of
crimes took place in regions historically rated as $D$. This is clearly vastly
different from the 2% in the other combined clusters. This implies that
historical and institutionalized bias still plays an important role in
determining the geographic distributions of crime. In District 25, the
statistical results are not as reliable, as we see all smaller clusters reside
in $C$ grade regions. However, the lack of $A$, and small amount of $B$ graded
regions overall in District 25 leads us to a similar conclusion as District
19. In these cases, a push for such interventions as targeted economic
development can attract investing and ultimately reduce poverty rates in this
area. Another initiative that can combat the historical impacts of redlining,
specifically in the context of crime, is community policing. Using such
methods as DBSCAN, we have reliably identified crime hotspots within each
District. With our research, police forces should formulate plans to increase
attention towards these hotspots in the future.

###  3.9 Limitations

While the Hopkins statistic can be a useful tool in determining the clustering
tendency of a data set, there are some assumptions and limitations in its use.
These include:

  1. 1.

Sensitivity to data distribution: the Hopkins statistic assumes that the data
is uniformly distributed, and it may not work well if the data has a skewed
distribution or contains outliers. It is unclear if the clustered data is
normally distributed.

  2. 2.

The dataset has no noise: The Hopkins statistic assumes that the dataset does
not contain any noise or outliers that could bias the clustering tendency, but
in the crime dataset there may be a large number of outliers in certain
districts.

  3. 3.

The dataset has a moderate to high number of dimensions: The Hopkins statistic
assumes that the dataset has a moderate to high number of dimensions, as
otherwise, it might not be able to distinguish between a clustered and non-
clustered dataset. Here, this is potentially problematic, as we only use two
dimensions (latitude, longitude). Nevertheless, a counterpoint is that because
the dataset had many clusters and a complex structure, low dimensionality was
not an issue.

Similarly, the $\chi^{2}$-model has a few assumptions and limitations to
consider:

  1. 1.

Independence: The observations should be ideally independent of each other
here in the $\chi^{2}$-test. However, here, the HOLC grade observations may be
dependent on each other, based on historical and demographic factors that have
affected redlining.

  2. 2.

Sample size: The sample size should be sufficiently large to ensure that the
expected frequencies are not too low. This was not an issue in District 19,
but was potentially problematic in District 25, where some clusters had zeroes
in their percentage breakdown.

  3. 3.

Association, not causation: The $\chi^{2}$-model can detect an association
between two variables, but it cannot determine whether this is causal.

Violation of these assumptions can lead to incorrect results and conclusions.
However, violating one or more of these assumptions does not necessarily mean
that the Hopkins statistic or $\chi^{2}$-model cannot be used. Rather, it is
generally important to be aware of these assumptions and their potential
impact on the results of the analysis.

##  4 Conclusion and Future Work

In this paper, we began by exploring crime distributions within the city of
Philadelphia and ultimately decided upon temporal and geospatial analysis
within set police district boundaries. We then proposed a method for
classifying the type of crime within these districts and began to explore the
potential underlying causes including factors such as redlining. This research
provides evidence that crime is not necessarily a uniform phenomenon. In this
study, we found that crime can be differentiated by its temporal and
geographical properties. Our results could allow policy makers to make more
targeted interventions for the different types of crime. In order to further
this work, we would like to explore different underlying factors and their
relationships to the types of crime within the city.

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