# Complement Grover’s Search Algorithm: An Amplitude Suppression
Implementation

Andrew Vlasic<EMAIL_ADDRESS>Deloitte Consulting, LLP  Salvatore Certo
Deloitte Consulting, LLP  Anh Pham  Deloitte Consulting, LLP

###### Abstract

Grover’s search algorithm was a groundbreaking advancement in quantum
algorithms, displaying a quadratic speed-up of querying for items. Since the
creation of this algorithm it has been utilized in various ways, including in
preparing specific states for the general circuit. However, as the number of
desired items increases so does the gate complexity of the sub-process that
conducts the query. To counter this complexity, an extension of Grover’s
search algorithm is derived where the focus of the query is on the undesirable
items in order to suppress the amplitude of the queried items. To display the
efficacy the algorithm is implemented as a sub-process into QAOA and applied
to a traveling salesman problem. For a basis of comparison, the results are
compared against QAOA.

##  I Introduction

Grover’s search algorithm grover1996p  is a pivotal quantum routine that leads
to quadratic speedups in an unstructured search, counting, and amplitude
estimation procedures. Since the inception of this algorithm, it has been
utilized as a form of state preparation, an essential subroutine. Examples
include quantum topological analysis algorithm lloyd2016quantum  to prepare
the boundary of the homology group and Monte Carlo analysis
rebentrost2018quantum  preparing states with various probability
distributions. If only a subset of states are required for the circuit (see
lloyd2016quantum ) there is potential for the depth complexity of the Grover
subroutine to substantially increase, potentially eroding or eliminating the
quadratic advantage it provides.

This increase in complexity is found within the oracle query that searches for
the desirable states, and one may observe as the number of desirable states
increases the gate complexity of the oracle increases. In more detail, the
oracle is the first step in the algorithm and is constructed to add a negative
phase to the desired states. Depending on the task at hand and the complexity
of the oracle query, it may be more efficient (from a circuit requirements and
gate depth perspective) to invert this step and instead construct the oracle
to add a positive phase to the undesirable solutions and a negative phase to
the other states. Ergo, this oracle would search for fewer states thereby
decreasing complexity. This paper describes in detail how one may construct
such an oracle.

##  II An Overview of Grover’s Search Algorithm

The intuition behind Grover’s search algorithm is it locates items in an
unstructured data set that meet some criteria by increasing the amplitude of
the desirable items relative to the undesirable, so that when the circuit is
measured there is a higher probability of finding the desirable items.

The algorithm is comprised of a periodic function that can be applied multiple
times to iteratively change the amplitudes. In particular, the Grover
iteration is a two step process, where first we define the oracle operator $O$
defined as
$O:\mathinner{|{x}\rangle}\mathinner{|{q}\rangle}\to\mathinner{|{x}\rangle}\mathinner{|{q\oplus
f(x)}\rangle}$, where $\oplus$ is the modulo 2 operation and $f(x)=1$ if $x$
is the identified state and $0$ otherwise. One way to implement the oracle is
by utilizing a Hadamard state so that the evolution of the qubits is

$O\bigg{(}\mathinner{|{x}\rangle}\Big{(}\frac{\mathinner{|{0}\rangle}-\mathinner{|{1}\rangle}}{\sqrt{2}}\Big{)}\bigg{)}=(-1)^{f(x)}\mathinner{|{x}\rangle}\Big{(}\frac{\mathinner{|{0}\rangle}-\mathinner{|{1}\rangle}}{\sqrt{2}}\Big{)}.$

The Hadamard state is crucial in seamlessly adding the local $-1$ phase shift
to the desired items.

To understand the oracle more concretely consider a two qubit register and a
one qubit ancilla with an oracle that identifies the state
$\mathinner{|{11}\rangle}$. The oracle would be the Toffoli gate controlled on
the state $\mathinner{|{11}\rangle}$ which, targeting Hadamard state, would
add a negative phase.

Once the desirable states are identified there is an increase to their
amplitudes through the diffuser, defined by the operator
$2\mathinner{|{\psi}\rangle}\mathinner{\langle{\psi}|}-I$. Often referred to
as an inversion about the mean, the easiest way to implement is to apply a
phase shift of $-1$ to $\mathinner{|{0}\rangle}$. One may observe the diffuser
is of the form of a Householder matrix.

The subroutine with $\mathinner{|{0}\rangle}^{\otimes
n}\mathinner{|{0}\rangle}$ qubits searching for the string $x_{0}$ is
implemented as follows.

  1. 1.

Prepare the register qubits into the distribution state and prepare the
ancilla qubit with the gate $H^{\otimes n}\otimes HX$, where $X$ is the Pauli
operator which gives the mathematical representation
$\displaystyle\frac{1}{\sqrt{2^{n}}}\sum_{x=0}^{2^{n}-1}\mathinner{|{x}\rangle}\Big{(}\frac{\mathinner{|{0}\rangle}-\mathinner{|{1}\rangle}}{\sqrt{2}}\Big{)}$.

  2. 2.

Apply the oracle $O$ which yields
$\displaystyle\frac{1}{\sqrt{2^{n}}}\sum_{x=0}^{2^{n}-1}(-1)^{f(x)}\mathinner{|{x}\rangle}\Big{(}\frac{\mathinner{|{0}\rangle}-\mathinner{|{1}\rangle}}{\sqrt{2}}\Big{)}$.

  3. 3.

Apply the adjoints of the Hadamard gates $H^{\otimes n}$, recalling $H$ is a
self-adjoint operator.

  4. 4.

Apply conditional phase shift on the $\mathinner{|{0}\rangle}^{\otimes(n+1)}$
state to perform the phase shift of $-1$.

  5. 5.

Apply the Hadamard gates $H^{\otimes n}$.

  6. 6.

Repeat steps 2 through 5, when necessary, to approximate the final state
$\displaystyle\mathinner{|{x_{0}}\rangle}\Big{(}\frac{\mathinner{|{0}\rangle}-\mathinner{|{1}\rangle}}{\sqrt{2}}\Big{)}$.

  7. 7.

Finally, measure all the qubits except for the ancilla qubit in the
computational basis.

Figure 1 gives an explicit example of this algorithm in the form of a circuit.

Since the Grover iteration may be repeated for several iterations in the
circuit, for simplicity the operators in steps 2-5 are known as the Grover
Operator. One may see has the Grover operator has the form $H^{\otimes
n}\big{(}2\mathinner{|{0}\rangle}\mathinner{\langle{0}|}^{\otimes
n}-I_{2^{n}}\big{)}H^{\otimes
n}O=\Big{(}2\mathinner{|{\psi}\rangle}\mathinner{\langle{\psi}|}-I_{2^{n}}\Big{)}O$
where $\mathinner{|{\psi}\rangle}$ is the uniformly distributed states of
superpositions. Note that once operators are aggregated the final operator
$2\mathinner{|{\psi}\rangle}\mathinner{\langle{\psi}|}-I_{2^{n}}$, denoted
earlier as the diffuser.

The Hadamard gate $H^{\otimes n}$ assumes a uniform distribution, and this
algorithm has a natural extension for any sub-process that requires certain
states to be fed downstream in the algorithm. One may see that any
distribution may be considered calling the sub-process that loads an arbitrary
distribution $\mathcal{A}$. with this notation the Grover operator has the
general form
$\mathcal{A}\big{(}2\mathinner{|{0}\rangle}\mathinner{\langle{0}|}-I\big{)}\mathcal{A}^{\dagger}O$.
Other extensions of the algorithm include a creative extension of the Grover
operator in the quantum analog of derivative pricing rebentrost2018quantum .
The authors in suzuki2020amplitude  also give a general extension of Grover’s
algorithm. Lastly, an in-depth description of the Grover search algorithm can
be found in Chapter 6 in the book nielsen2002quantum .

Do to the intricate nature of Grover’s search algorithm implementation in the
NISQ-era has made algorithms with large qubit registers unusable. The authors
in zhang2022quantum  propose a method to bypass this shortcoming and enable a
version of Grover’s search algorithm in the NISQ-era.

[thin lines] Register 0 $\mathinner{|{0}\rangle}$
[wires=4,steps=3,style=rounded corners,fill=cyan!10, inner xsep=2pt,
background]Initial & H 1 [wires=4,steps=1,style=rounded corners,fill=cyan!25,
inner xsep=2pt, background]Oracle H [wires=4,steps=5,style=rounded
corners,fill=cyan!40, inner xsep=2pt, background]Diffuser X 1 X H [wires=3]
Register 1 $\mathinner{|{0}\rangle}$ H 1 H X 1 X H —[meter]—
Register 2 $\mathinner{|{0}\rangle}$ H 1 H X 1 X H
Ancilla $\mathinner{|{0}\rangle}$ X H X X H X H X

Figure 1: The circuit above displays a Grover’s search algorithm for the
string $\mathinner{|{001}\rangle}$. One may use the open source Qiskit
aleksandrowicz2019qiskit  for computational verification that this circuit
finds the state $\mathinner{|{001}\rangle}$.

##  III Grover’s Amplitude Suppression Algorithm

There have been many advances from different perspectives on Grover’s
algorithm yoder2014fixed ; byrnes2018generalized ; li2018complementary ;
cafaro2019continuous ; gilliam2020optimizing . In particular, the advancement
in li2018complementary  focuses on sets within the search space with the goal
of decreasing the number of gates required.

A potential method to decrease the complexity of this oracle is to focus on
finding the undesirable states. While at first this may seem intuitive, it is
not trivial to derive. Before explaining the algorithm in more detail,
notation of the sets of desirable and undesirable states will be needed for
simplicity. Denote the set $S$ as the collection of undesirable states,
$S^{\complement}$ as the complement of this set, and $|S|$ as the cardinality
of the set. Lastly, denote the oracle that finds the undesirable states as
$O_{S}$.

[thin lines] Register 0 $\mathinner{|{0}\rangle}$
[wires=4,steps=2,style=rounded corners,fill=cyan!10, inner xsep=2pt,
background]Initial & H [wires=4,steps=4,style=rounded corners,fill=cyan!25,
inner xsep=2pt, background]Amp. Suppr. Oracle 1 1 H
[wires=4,steps=5,style=rounded corners,fill=cyan!40, inner xsep=2pt,
background]Diffuser X 1 X H [wires=3]
Register 1 $\mathinner{|{0}\rangle}$ H 1 1 H X 1 X H —[meter]—
Register 2 $\mathinner{|{0}\rangle}$ H 1 1 H X 1 X H
Ancilla $\mathinner{|{0}\rangle}$ X X X Z X H X H X

[thin lines] Register 0 $\mathinner{|{0}\rangle}$
[wires=4,steps=3,style=rounded corners,fill=cyan!10, inner xsep=2pt,
background]Initial & H 1 [wires=4,steps=6,style=rounded corners,fill=cyan!25,
inner xsep=2pt, background]Oracle 1 1 1 1 1 H [wires=4,steps=5,style=rounded
corners,fill=cyan!40, inner xsep=2pt, background]Diffuser X 1 X H [wires=3]
Register 1 $\mathinner{|{0}\rangle}$ H 1 1 1 1 1 1 H X 1 X H —[meter]—
Register 2 $\mathinner{|{0}\rangle}$ H 1 1 1 1 1 1 H X 1 X H
Ancilla $\mathinner{|{0}\rangle}$ X H X X X X X X X H X H X

Figure 2: (a) This circuit displays an amplitude suppression of Grover’s
search algorithm for all states except for $\mathinner{|{000}\rangle}$ and
$\mathinner{|{111}\rangle}$. For the six desirable states, instead of six
oracles there are two oracles and two single qubit gates. (b) This is the
classical implementation searching for all of the six desirable states.

To obtain the goal of adding the negative phase to the desirable states by
acting on the undesirable states, which is performed within the oracle, an
ancilla qubit is introduced and flipped to the $\mathinner{|{1}\rangle}$ state
with the Pauli $X$ gate, which allows the Pauli $Z$ gate to give the negative
phase to $S^{\complement}$. Unlike the classical oracle, the ancilla qubit is
not put into a Hadamard state but left in the $\mathinner{|{0}\rangle}$ state.
To understand this process mathematically, after setting the qubits in the
register into a uniform superposition and applying the amplitude suppression
oracle gives the superpositions of

$\displaystyle
O_{S}\bigg{(}\frac{1}{\sqrt{2^{n}}}\sum_{i=1}^{2^{n}-1}\mathinner{|{i}\rangle}\mathinner{|{0}\rangle}\bigg{)}=\frac{1}{\sqrt{2^{n}}}\sum_{i\in
S^{\complement}}-\mathinner{|{i}\rangle}\mathinner{|{1}\rangle}+\frac{1}{\sqrt{2^{n}}}\sum_{i^{\prime}\in
S}\mathinner{|{i^{\prime}}\rangle}\mathinner{|{0}\rangle}.$

The general process of the algorithm is as follows.

  1. 1.

Prepare the register qubits into the distribution state with the gate
$H^{\otimes n}\otimes I$ and add an ancilla qubit initialized to the
$\mathinner{|{0}\rangle}$ state, leading to
$\displaystyle\frac{1}{\sqrt{2^{n}}}\sum_{i=0}^{2^{n}-1}\mathinner{|{i}\rangle}\mathinner{|{0}\rangle}$.

  2. 2.

Apply the oracle $O_{S}$ which yields
$\displaystyle\frac{1}{\sqrt{2^{n}}}\sum_{i\in
S^{\complement}}-\mathinner{|{i}\rangle}\mathinner{|{1}\rangle}+\frac{1}{\sqrt{2^{n}}}\sum_{i^{\prime}\in
S}\mathinner{|{i^{\prime}}\rangle}\mathinner{|{0}\rangle}$

  3. 3.

Apply Hadamard gates $H^{\otimes n}$, recalling $H$ is a self-adjoint
operator.

  4. 4.

Apply conditional phase shift on the $\mathinner{|{0}\rangle}^{\otimes(n+1)}$
state to perform the phase shift of $-1$.

  5. 5.

Apply the Hadamard $H^{\otimes n}$.

  6. 6.

Repeat steps 2 through 5, when necessary, to approximate the final state
$\displaystyle\frac{1}{\sqrt{|S^{\complement}|}}\sum_{i\in
S^{\complement}}-\mathinner{|{i}\rangle}\mathinner{|{1}\rangle}$.

  7. 7.

Finally, measure all the qubits except for the ancilla qubit in the
computational basis.

To make this algorithm more tangible, Figure 2(a) shows a circuit of preparing
the states $\mathinner{|{001}\rangle},\ldots,\mathinner{|{110}\rangle}$.
Figure 3 displays results of this algorithm with registers of size 3 qubits
and 2 qubits.

While there are creative ways to prepare states huang2018demonstration , the
algorithm presented is scalable, hence, agnostic to the number of qubits
required. Moreover, since the only parts of the algorithm that changed from
the original Grover’s search algorithm are the oracle and the initial step of
the ancilla, original analysis on the number of iterations needed still holds
true; see nielsen2002quantum  for further information. In particular, for $R$
the number of iterations, $N$ the total number of states, and $M$ the number
of undesirable states the inequality holds $\displaystyle
R\leq\Big{\lceil}\frac{N}{M}\Big{\rceil}$, where $\lceil\cdot\rceil$ is the
ceiling function.

Although this algorithm does not set the ancilla qubit in a superposition,
which may be perceived as not utilizing quantum and hence losing an advantage,
there are instances of applying control gates to ancilla vazquez2021efficient
; suzuki2020amplitude . Furthermore, the addition of the Pauli $X$ gate and
Pauli $Z$ gate stay constant as the number of qubits in the register grows.
Therefore, with at least a two qubit register, the subtraction of one control
gate in the oracle yields a more efficient circuit.

Figure 3: The bar graphs are results from running the algorithm on a simulator
and a QPU. Figure (a) displays the results from the circuit in Figure 2 ran on
Qiskit aleksandrowicz2019qiskit  using a simulator backend after 3 Grover
Iterations. Figure (b) are the results of running the circuit in Figure 2 but
with a two qubit register and ran on an IBM QPU.

Figure 4: The figures show the increase in the number of gates required for
the oracle in the circuit in Figure 2 and the classical analog this this
circuit. Figure (a) is the classical oracle. Figure (b) is oracle size in
Figure 2. Finally, Figure (c) compares these different methods.

##  IV Analysis of the Oracle Depth

[thin lines] Register 0 $\mathinner{|{0}\rangle}$ & 1
Register 1 $\mathinner{|{0}\rangle}$ X 1 X
Register 2 $\mathinner{|{0}\rangle}$ 1
Ancilla $\mathinner{|{0}\rangle}$ X

Figure 5: The circuit displays an oracle identifying the state
$\mathinner{|{101}\rangle}$.

For a deeper understanding of the potential increased efficiency of this new
algorithm the circuit depth of the classical oracle and amplitude suppression
oracle will be compared. The oracles from the classical Grover’s search
algorithm are assumed to be scalable, bypassing creative oracles that are
qubit-number specific. To be scalable for each string in the desirable state
there is a general Toffoli gate identifying a particular state with
appropriate Pauli $X$ gates to flip the qubit then flip back to the original
state; see Figure 5 for an example. This particular implementation of flagging
states should be hardware agnostic.

Each Pauli $X$ gate and each general Toffoli gate are counted as one. General
Toffoli gates are only counted as one since these gates are the same in either
oracle, and moreover, each hardware system implements general Toffoli gates
differently which would lead to different growths.

The analysis is of the circuit in Figure 4. Taking the register size of 2 to
20 qubits, the number of qubits in each oracle is counted. Since Grover’s
algorithm prepares all states except for $\mathinner{|{0\ldots 0}\rangle}$ and
$\mathinner{|{1\ldots 1}\rangle}$ the growth of the required state
preparations in the classical implementation is $2^{n}-2$ for $n$ the number
of qubits, while the amplitude suppression implementation always requires $2$
states to prepare. In particular, the classical case grows with $n$ the
following formula
$\displaystyle(2^{n}-2)+2\cdot\sum_{i=1}^{n}\left(\begin{array}[]{c}k\\\
i\end{array}\right)$, and the amplitude suppression case grows as $4+2\cdot
n$. From this one may observe the classical implementation grows exponentially
and amplitude suppression grows linearly. Figure 2 shows and compares these
different growths.

While this is an extreme case to explore there are examples which such an
algorithm is not required. For example, if the first of position needs to be
in the state $\mathinner{|{1}\rangle}$ for a number of particular states. In
this instance one may implement Grover around this position with a simple
oracle for state preparation. However, this analysis displays the potential
increase in efficiency when intricate states are required in the general
algorithm.

##  V Application to the Quantum Approximate Optimization Algorithm

On application of this algorithm is to use it as state preparate for the
Quantum Approximate Optimization Algorithm (QAOA) farhi2014quantum , which was
derived as a means to find the lowest energy within in a system. This was
achieved by taking the adiabatic quantum evolution and applying the Trotter
product formula to approximate the Hamiltonian. The ground state is then
mapped to the solution of the mathematical problem that was encoded into the
circuit. From the derivation one may see how numerous combinatorial
optimization problems can be translated into a QAOA circuit, including various
graph problems and various mathematical programming problems.

QAOA requires an ansatz for the initial condition and typically requires
application of a classical optimizer to derive the parameters that yield the
optimal or near-optimal solution. A uniform distribution of all the states was
usually taken as the initial sub-process, however, there have been recent
advancements that utilize the mixer Hamiltonian to guide the derivation of
superpositions closer to the optimal solution. egger2021warm . This algorithm
is noted as the warm start quantum approximate optimization algorithm (WS-
QAOA). For completeness, other derivations using custom mixer Hamiltonian for
QAOA have been considered tate2021classically .

WS-QAOA works by relaxing the QUBO and solving it classically to get an
approximate solution. This approximate solution is then given as an initial
state to QAOA. For problems where the gap of the relaxation to the QUBO
solution is minimal, this method is effective and efficient. When this gap is
adequately large however, its efficacy is drastically reduced. Some instances
where a relaxed problem solution may be far from the true QUBO minimum is in
cases like the Traveling Salesman Problem.

To show the efficacy the algorithm in Figure 2 (a) is implemented as the first
sub-process in QAOA and applied to a toy traveling salesman problem. For a
basis of comparison, the same traveling salesman problem is applied to QAOA
and the results are compared. Comparison against this initialization method
against others methods, like WS-QAOA, is left for future research. The
circuits are written in Qiskit aleksandrowicz2019qiskit , and as noted in
egger2021warm , there are packages to run the original QAOA, (as well as WS-
QAOA). The results are given in Figure 6 where the original QAOA using
amplitude suppression Grover’s algorithm out-performs Qiskit’s QAOA package.
The reason for the performance boost is the amplitude suppression Grover’s
algorithm decreases the amplitudes of the infeasible states, allowing for
initial higher probabilities of feasible states, in which the solution can be
found faster.

Figure 6: Comparison of the QAOA and QAOA using amplitude suppression Grover
as the initial start of the circuit.

##  VI Discussion and Conclusion

In the case where the desirable states $S^{C}$ outnumber the undesirable
states $S$, the algorithm presented above represents an efficient way to
structure the oracle. While this structure normally depends entirely on the
domain and query involved in the algorithm, certain applications of state
preparation include specific states to be omitted, which prompts a different
approach to efficiently construct the Grover operator. The benefit proposed
here relies on the fact that we are only adding an ancilla qubit within the
context of decreasing the amplitudes of certain states, rather than during the
initial preparation of the state. In fact, some tasks that require the Grover
Operator will see efficiency gains through this method structuring the oracle
on the undesirable states, rather than the desired solutions themselves.

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# On the Absoluteness of Rotation.

###### Abstract

We argue that in the general relativistic calculation of planetary orbits, the
choice of a reference frame which is an obligatory condition in the Newtonian
approach is replaced by an appropriate boundary condition on the solution of
Einstein equation. Implications of this observation on the nature of rotation
and the physical interpretation of the metric tensor are discussed.

Péter Hraskó

Department of Theoretical Physics, University of Pécs, Hungary

Dávid Szepessy

Student of the Eötvös Loránd University (ELTE), Budapest, Hungary

In Principia [1] we read the following lines about Newton’s famous bucket
experiment:

> If a vessel, hung by a long cord, is so often turned about that the cord is
> strongly twisted, then filled with water, and held at rest together with the
> water; after, by the sudden action of another force, it is whirled about in
> the contrary way, and while the cord is untwisting itself, the vessel
> continues for some time this motion; the surface of the water will at first
> be plain, as before the vessel began to move; but the vessel by gradually
> communicating its motion to the water, will make it begin sensibly to
> revolve, and recede by little and little, and ascend to the sides of the
> vessel, forming itself into a concave figure… This ascent of the water shows
> its endeavour to recede from the axis of its motion; and the true and
> absolute circular motion of the water, which is here directly contrary to
> the relative, discovers itself, and may be measured by this endeavour. … And
> therefore, this endeavour does not depend upon any translation of the water
> in respect to ambient bodies, nor can true circular motion be defined by
> such translation. …; but relative motions…are altogether destitute of any
> real effect. …

In the 19th century, this experiment could have been rephrased in the
following way: When calculating planetary orbits, only the law of universal
gravitation is taken into account on the right-hand side of Newton’s equation,
no inertial forces are included. With this, we automatically assume that we
are in an inertial (specifically non-rotating) frame of reference. However,
such an inertial system of the size of the Solar System cannot be realized, so
instead, we check the correctness of the calculations in relation to distant
stars and find that the observations confirm the calculations very accurately.
Why?

For the sake of brevity, we will refer to the question formulated in the
original bucket experiment and in its form outlined in the previous paragraph
as Newton’s problem.

Newton’s problem is logically similar to the problem of the equality of the
inertial and gravitational mass (which, of course, is also a ”Newton’s
problem”). Knowing the genesis of the general theory of relativity, however,
we are inclined to consider the latter to be much more fundamental than the
former. But it is by no means certain that this is no more than hindsight.
Perhaps it is enough to refer to Lorentz’s electromagnetic theory of gravity
[2], which he published in 1900. According to the priorities in physics of
those times, Lorentz considered the elimination of the action at a distance
from Newton’s theory to be much more important than explaining the equality of
inertial and gravitational mass. His ambitious theory built on the analogy of
electrodynamics transformed gravitation into retarded interaction between
celestial bodies, but the equality of the two kinds of mass in his theory
remained a mystery to be solved in the future. We believe that within the
framework of Newtonian dynamics, Newton’s problem is as important as the
equality of the inertial and gravitational masses. Below, however, we argue
that the general theory of relativity solves the former problem too in a
natural way.

Newton’s answer to Newton’s problem was that rotation is absolute [1]:

> It is indeed a matter of great difficulty to discover, and effectually to
> distinguish, the true motions of particular bodies from the apparent;
> because the parts of that immovable space in which these motions are
> performed, do by no means come under the observations of our senses.

However, this concept was and presumably is still rejected by most physicists.
Hans Reichenbach, for example, comments it as follows [3]:

> Newton’s argument was severely criticized by Ernst Mach, who showed that it
> involved a serious non sequitur. Newton noted quite correctly that the
> variations in the shape of the surface of the water are not connected with
> the rotation of the water relative to the sides of the bucket. But he
> concluded that the deformations of the surface must therefore be attributed
> to a rotation relative to absolute space. However, this conclusion does not
> follow from the experimental data and Newton’s other assumptions, for there
> are in fact two alternative ways of interpreting those data: the change in
> the shape of the water’s surface is a consequence of either of a rotation
> relative to absolute space or of a rotation relative to some system of
> bodies different from the bucket.

As a matter of fact, intuition seems indeed to suggest that if uniform
translation is a relative phenomenon then uniform rotation should be a
relative process either. This expectation, however, is not justified at all,
because there is a very significant difference between uniform translation and
uniform rotation even at the level of pure kinematics. The relativity of
uniform translation is related to the existence of a velocity addition law in
both Newtonian and relativistic physics, which endows uniform translations
with a Lie group structure. However, there is no analogous ”angular velocity
addition law” for rotations around axes intersecting at some common point
because two consecutive uniform rotations in general lead to nonuniform
rotation. As a result, uniform rotations around different axes do not form a
Lie group, even though time-independent translations and rotations both have a
Lie group structure. This fundamental difference between uniform rotations and
uniform translations makes ideas based on the relativity of rotation
untenable.

In the 20th century, planetary orbits are interpreted as geodesics in space-
time deformed by the huge mass of the Sun. To calculate them one is faced with
the problem to choose the coordinate system best suited to the physical
situation under study. In sharp difference with Newtonian treatment, there
seems to be no place for choosing an inertial reference frame too in addition
to the coordinate system because in general relativity no global inertial
frames exist. In spite of these fundamental differences results of general
relativistic calculations turn out very similar to those of Newton’s theory of
gravity, though they correct and implement them in several important respects.

The subject of the present note concerns this difference between Newtonian and
general relativistic treatment of planetary motion and can be summarized in
the following statement which justifies the established practice: In general
relativity the choice of inertial reference frame is replaced by the
requirement that the solution be asymptotically Minkowski up to a time-
independent transformation of space coordinates. The Schwarzschild and the
Kerr solutions obviously obey this criterion. Notice that it refers to
coordinates rather than the geometric property of asymptotic flatness which is
supposed to be the case.

These solutions can of course be rewritten in rotating coordinates — there are
problems that are easier to solve in a rotating coordinate system both in
Newtonian gravity and general relativity. For example, when a point mass
revolves uniformly in a circular orbit around a spherically symmetric star of
mass $M$, we obtain in rotating Schwarzschild coordinates

$\begin{split}ds^{2}=&\left[1-\frac{r_{g}}{r}-\left(\frac{r\omega}{c}\right)^{2}\cdot\sin^{2}\vartheta\right]c^{2}dt^{2}-\\\
&-2\omega
r^{2}\sin^{2}\vartheta\>dtd\varphi-\frac{dr^{2}}{1-r_{g}/r}-r^{2}(d\vartheta^{2}+\sin^{2}\vartheta\>d\varphi^{2}),\end{split}$

the following answer for the relation between the radius $r$ of the rotation
and its angular velocity $\omega$:

$\Gamma_{tt}^{r}=\frac{r-r_{g}}{r}\left(\frac{MG}{r^{2}}-r\omega^{2}\right)=0.$

In the language of Newtonian physics, formally (because $r$ is not a distance,
but a coordinate), this condition exactly corresponds to the equality of
gravitational attraction and centrifugal force. However, in the theory of
general relativity, the criterion of non-rotation cannot consist in the
absence of the centrifugal force (a concept which is meaningless outside local
frames), but rather in the smooth, continuous, singularity-free transition of
the metric around the star into the asymptotic Minkowski metric. The rotating
coordinate system does not meet this condition because it has a characteristic
coordinate singularity: The component $g_{00}$ changes sign at finite $r$
while $g_{t\varphi}\longrightarrow\infty$ as $r\longrightarrow\infty$. It is
for this reason that the line element above is related to a nonrotating object
in rotating coordinates rather than the other way around.

In general relativity, considering an isolated star is an accepted
idealization [4], although strictly speaking such objects don’t exist in the
Universe. According to this assumption, the space-time of a solitary star
${\cal S}_{1}$ can be considered asymptotically flat, because the influence of
other stars and the cosmological curvature of space-time can be disregarded
with high accuracy.

Let us now consider another star ${\cal S}_{2}$ in this asymptotic domain.
This star will usually move with some constant velocity $\vec{v}$ in the
asymptotically Minkowski coordinates ${\cal K}_{1}$ of the star ${\cal
S}_{1}$. The asymptotic Minkowski coordinates ${\cal K}_{2}$ belonging to
${\cal S}_{2}$ will therefore differ from the asymptotic coordinates of ${\cal
K}_{1}$ in a Lorentz transformation corresponding to the stellar velocity. In
the region between the two stars, distant from both, the homeomorphism between
the two maps ${\cal K}_{1}$ and ${\cal K}_{2}$ will, therefore, be the Lorentz
transformation corresponding to the speed of ${\cal S}_{2}$111Relation of this
kind of description to the Standard Model of Cosmology is unclear because the
Einstein-Straus problem [5] has yet no universally accepted solution. The
description of the Universe as the multitude of overlapping coordinate patches
of individual stars is in a sense dual to the Standard Model which is based on
the notion of the cosmological fluid of smoothed-out matter. From this point
of view the homeomorphisms we are speaking about may be slightly different
from Lorentz-transformation but we hope that this quantitative uncertainty
does not invalidate our argumentation.. The roles of the two stars are
obviously interchangeable, so their translational motion is relative222If
${\cal S}_{2}$ has an angular momentum, this homeomorphism must also contain a
spatial rotation, which is determined by the direction of the $z$-axis of
${\cal K}_{2}$ relative to the axes of ${\cal K}_{1}$..

Their rotation is, however, absolute. In the common domain, the two
overlapping coordinate systems are Minkowski, so they rotate neither by
themselves nor relative to each other. They are the asymptotics of the
coordinate systems in the near stellar regions which on their part are
continuations of the inner solution of the Einstein equations. The metric of
the internal coordinate system fixes the energy-momentum tensor and from the
components of this tensor, we can calculate the motion of the material the
star is made of relative to the overlapping common nonrotating coordinate
system described at the beginning of this paragraph. If, for example, the
outer solution is Kerr, then the star’s material is generally rotating in an
absolute sense.

If this argumentation is correct, then Newton’s problem ceases to be a
problem. To illustrate this, consider the calculation and measurement of the
geodetic precession of a gyroscope orbiting ${\cal S}_{1}$ as the 21st-century
version of the bucket experiment. The calculation is performed in the
Schwarzschild or Kerr coordinates of ${\cal S}_{1}$. But since these
coordinate systems only exist in our imagination, we relate the gyroscope’s
orientation to the direction of the very distant guide star ${\cal S}_{2}$.
According to experience, the precession rate we measure corresponds to the
calculation within experimental error. Why?

Because the coordinate system ${\cal K}_{1}$ used for the calculation is
actually not independent from that of the guide star, since it is part of an
overlapping singularity-free coordinate system consisting of two maps
(coordinate patches) that contains both objects and does not rotate. As for
the requirement of regularity, it is made possible by the fact that the
general theory of relativity is a field theory, in which this is a meaningful
prescription for the field quantity $g_{ij}(x)$. In Newtonian physics, which
is a particle theory, no such possibility exists, and it is this fundamental
difference which makes connecting ${\cal S}_{1}$ with the very distant ${\cal
S}_{2}$ a much more difficult task.

In the solution of electrodynamic problems, the lack of singularity in regions
free of charges is a natural requirement because Maxwell’s equations describe
the dynamics of a special type of continuous substance, the electromagnetic
field. Today in general relativity no such background is felt behind
$g_{ij}(x)$. Moreover, the equivalence of coordinate systems seems even to
prohibit this kind of interpretation of the metric tensor.

Coordinate systems are, however, equivalent to each other only in that either
they all satisfy Einstein’s equations, or none of them do. In other respects
they may be distinguished from each other and in calculations this possibility
is systematically exploited. For example, we can require that the coordinate
system reflect the symmetries of the physical object whose space-time is being
examined. Since coordinate systems, which are equivalent from the point of
view of the Einstein equations, may differ from each other in coordinate
singularities, we are allowed, if necessary, to require that they do not
contain certain types of coordinate singularities.

If we perceived vacuum as a continuous substance rather than pure void or
emptiness then the absence of singularity in regions of space without masses
would become as natural property of the metric as the continuity of
${\vec{E}}(x)$ and ${\vec{B}}(x)$ in charge free regions.

In 1920, in his now little-known Leyden lecture [6], Einstein argued that
general relativity actually requires the rehabilitation of the ether
hypothesis, which would play the role of such a substance333L. Kostro
published several articles and a book about the evolution of Einstein’s
conception of the ether. In [7] one can find references to these works.:

> [T]here is a weighty argument to be adduced in favour of the ether
> hypothesis. To deny the ether is ultimately to assume that empty space has
> no physical qualities whatever. The fundamental facts of mechanics do not
> harmonize with this view. For the mechanical behaviour of a corporeal system
> hovering freely in empty space depends not only on relative positions
> (distances) and relative velocities but also on its state of rotation, which
> physically may be taken as a characteristic not appertaining to the system
> itself. In order to be able to look upon the rotation of the system, at
> least formally, as something real, Newton objectiveses space. Since he
> classes his absolute space together with real things, for him rotation
> relative to an absolute space is also something real. Newton might no less
> well have called his absolute space ”ether”; what is essential is merely
> that besides observable objects, another thing, which is not perceptible,
> must be looked upon as real, to enable acceleration or rotation to be looked
> upon as something real.

As a consequence of the absence of privileged coordinate systems (rest system,
for example), this ether, whose existence is compatible with both special and
general relativity, must differ from the pre-relativistic concept of ether in
that no state of motion or rest can be attributed to it [5]:

> Not every extended conformation in the four-dimensional world can be
> regarded as composed of world threads. The special theory of relativity
> forbids us to assume the ether to consist of particles observable through
> time, but the hypothesis of ether in itself is not in conflict with the
> special theory of relativity. Only we must be on our guard against ascribing
> a state of motion to the ether.

Developments in physics in the last hundred years support Einstein’s concept
of the ether: The vacuum cannot be mere emptiness, because according to
quantum field theory, it has physical properties. In Einstein’s equation, the
term containing the cosmological constant can be interpreted as the energy-
momentum tensor of the vacuum [8]:

$T^{(vac)}_{ij}(x)=\frac{c^{4}}{8\pi G}\Lambda g_{ij}(x).$

This conception, which today is only a plausible hypothesis, might be the
indication of a connection between the physical properties of the vacuum and
the metric tensor. Since $\nabla_{i}g_{jk}=0$ no four-vector can be formed
from $T^{(vac)}_{ij}$ by differentiation and so its existence is compatible
with the requirement that no state of motion be attributable to the ether of
general relativity.

References

[1] Newton, I.: Principia, Book 1, Scholium, English translation by I. Bernard
Cohen and Anne Whitman, University of California Press (2016)

[2] https://en.wikipedia.org/wiki/Lorentz_ether_theory

[3] Reichenbach, H.: Problems in the Logic of Scientific Explanation, p. 209.
New York: Harcourt, Brace and World (1961)

[4] Wald, R. M.: General Relativity, ch. 11. The University of Chicago Press
Chicago and London (1984)

[5] Einstein, A. and Straus E. G., Rev. Mod. Phys. 17, 120:

https://doi.org/10.1103/RevModPhys.17.120

[6] Einstein, A.: Äther und relativitätstheorie. Springer (1920). English
translation: https://einsteinpapers.press.princeton.edu/vol7-trans/17

[7] Kostro, L.: https://link.springer.com/chapter/10.1007/978-1-4615-2560-8_22

[8] Weinberg, S: Cosmology, Oxford University Press (2008)


# A structural study of Big Tech firm-switching of inventors in the post-
recession era

Yidan Sun  Information Sciences Institute
University of Southern California
Marina del Rey, California, United States 90292
Email<EMAIL_ADDRESS>Mayank Kejriwal  Information Sciences Institute
University of Southern California
Marina del Rey, California, United States 90292
Email<EMAIL_ADDRESS>
###### Abstract

Complex systems research and network science have recently been used to
provide novel insights into economic phenomena such as patenting behavior and
innovation in firms. Several studies have found that increased mobility of
inventors, manifested through firm switching or _transitioning_ , is
associated with increased overall productivity. This paper proposes a novel
structural study of such transitioning inventors, and the role they play in
patent co-authorship networks, in a cohort of highly innovative and
economically influential companies such as the five Big Tech firms (Apple,
Microsoft, Google, Amazon and Meta) in the post-recession period (2010-2022).
We formulate and empirically investigate three research questions using Big
Tech patent data. Our results show that transitioning inventors tend to have
higher degree centrality than the average Big Tech inventor, and that their
removal can lead to greater network fragmentation than would be expected by
chance. The rate of transition over the 12-year period of study was found to
be highest between 2015-2017, suggesting that the Big Tech innovation
ecosystem underwent non-trivial shifts during this time. Finally, transition
was associated with higher estimated impact of co-authored patents post-
transition.

##  1 Introduction

Innovation has long been recognized in the economics and social sciences for
driving long-term productivity and economic measures of aggregate income, such
as the Gross Domestic Product (GDP) [1, 2, 3, 4]. For private organizations,
especially in a knowledge-based economy [5, 6] such as is prevalent in much of
the industrialized world, innovative products and services developed and
commercialized by the firm can be an important driver111As with any complex
system, it bears noting that the relationship between innovation and valuation
is itself a complex one, and not always significant (some instances of
negative association for specific sectors and types of innovations may be
found in, for example, [7]). Both the sector and geography can play a role.
Most studies have found support for the claim that for technology and
‘science-based’ firms (to quote the terminology used in [8]), there is a
positive relationship. of the firm’s valuation [8, 4]. Innovation can also
have positive societal benefits in the form of higher corporate social
responsibility (CSR) by more innovative forms. For example, in an influential
recent work, Mishra analyzed a sample of more than 13,000 US ‘firm-years’ over
a 15 year period starting from the early 1990s and found that, mediated by
high CSR, more innovative firms ended up achieving significantly higher
valuation following a period of innovation [9].

The positive effects of innovation on valuation are especially strong for
technology sectors (e.g., fintech [10]), and for early-stage companies such as
start-ups [11]. Although not the only way of measuring _innovation
productivity_ in an organization, the number of patents filed by the
organization is an objective measure that has been widely studied by
researchers and policymakers alike[12, 13, 14]. At the time of writing, there
is an enormous body of literature on constructing and studying patent
networks, estimation of valuation from patent analysis (including network
analysis), textual analysis of patents, as well as a range of qualitative
studies on patents[15, 16, 17, 18].

In this paper, we use patent analysis, including construction and study of a
patent co-authorship network, to understand the structural properties of
_firm-switching_ or _transitioning_ inventors. We focus our study on a cohort
of five ‘Big Tech’ firms (Apple, Google, Meta, Microsoft, and Amazon) that, in
the aftermath of the 2008 financial crisis, have ended up outperforming the
broader market by a considerable margin and are among the most valuable
companies in the world222https://www.cnbc.com/2020/01/28/sp-500-dominated-by-
apple- microsoft-alphabet-amazon-facebook.html. Here we define an inventor as
co-author on a patent where one of these five organizations serves as the
‘assignee’ organization i.e., the organization that files the patent, and to
whom the intellectual property legally belongs. A transitioning inventor is
one who is originally employed by one of the five Big Tech firms (e.g.,
Apple), and is co-author on patents filed by Apple, but subsequently switches
to a different firm (e.g., Amazon) and starts co-authoring patents filed by
that firm.

There is increasing evidence that such firm-switching or ‘transitioning’
inventors may play an under-appreciated role in boosting innovation potential
within the technology ecosystem [19]. Intuitively, the ability to transition
represents inventor ‘mobility’ [20], and even for general workers (not just
inventors), evidence suggests that mobility can lead to productivity gains and
better incentive alignment in the overall economy [21]. Highly productive
inventors are also more likely to be ‘poached’ by a rival company that is
looking to innovate in a similar area. The rival company may be incentivized
to offer greater benefits, including higher pay and more freedom to the
inventor in developing and publishing their ideas, many of which the
organization can potentially look to patent. Because poaching is usually
targeted, in theory, this can motivate the inventor further due to greater
alignment between the company’s incentives and the inventor’s goals. While
firm switching behavior has received some theoretical and empirical attention
in the economics literature [19, 21, 22], it has never before been studied
from a structural perspective (i.e., from the perspective of _economic
complexity_ [23]), in the five Big Tech organizations in the post-recessionary
era (2010-2022).

We formulate and investigate three specific research questions (RQs) to better
understand the structural role and other properties of these transitioning
inventors in the broader Big Tech ecosystem during this period:

RQ1: How does the degree distribution and other structural properties of the
transitioning inventors in the patent co-authorship network compared with
those of the other inventors, and does the removal of these inventors from the
network lead to lower connectivity and more fragmentation than would be
expected through chance?

RQ2: Does the rate of inventor transition remain constant (or a show a
monotonic trend) over time, or is there a specific period during which the
majority of transitions occur?

RQ3: Is transition associated with an increase in estimated impact of co-
authored patents (reflected through a subsequently defined measure, such as
the normalized number of citations received by their patents) _after_
transition, as compared to the impact of the patents co-authored _before_
transition?

The first research question explores the structural properties of
transitioning inventors using the usual tools of network science. We consider
these properties both with respect to the overall co-authorship network (that
includes all patent authors within Big Tech in the period under study,
including the vast majority of patent authors in Big Tech who have not
transitioned) and also the ‘sub-network’ where only the transitioning
inventors are represented as nodes. The second research question is instead
considering what happens to the rate of transition over the period under
study; do we, for example, see relatively stable and constant rates, steadily
increasing or decreasing rates, or a trend that is more complex? We also
consider whether some organizations experience greater transitions (whether
incoming or outgoing) than others or if the transition-mixture is relatively
even among the five Big Tech firms. Finally, the motivation behind studying
the third research question is to understand whether transition is associated
with an estimated measure of ‘excess’ impact due to transition. While we only
consider one measure of excess impact, our definition attempts to control for
potential biases, such as the length of time elapsed since the patent was
granted.

##  2 Materials and Methods

Our research primarily focuses on analyzing patents filed by five major
technology corporations: Google, Meta (formerly known as Facebook), Apple,
Amazon, and Microsoft. To gather the relevant patent data, we utilized the
Query Builder feature in the PatentsView platform [24]. PatentsView is a
comprehensive visualization, data dissemination, and analysis tool
specifically designed for intellectual property (IP) data. The platform
receives support from the Office of the Chief Economist at the U.S. Patent &
Trademark Office (USPTO). The Query Builder feature within PatentsView allows
researchers to identify and retrieve specific subsets of patents of interest
from the vast collection of all U.S. patents. To obtain patent data for this
study, we issued the following query to Query Builder: ‘ _Assignee
Organization_ contains333The special query keyword ‘contains’ will return any
patent with an assignee organization that has the specified organization name
as a substring. For example, a patent filed by ‘Google LLC’ as the assignee
organization will be returned if the organization name is specified as
‘Google’. While we could use the special keyword ‘equals’ rather than
contains, it is less robust than the former and does not capture subsidiary
organizations, as discussed subsequently. _[Organization Name]_ ’. Here, the
_[Organization Name]_ is a placeholder for each of the five Big Tech companies
we targeted and the _Assignee Organization_ is a field in the PatentsView
dataset. We also specified a time constraint as we wanted to focus on patents
filed by these companies between 2010 and 2022 (inclusive), and their patents
granted within this period.

During the query process, we noticed that there were instances where other
irrelevant assignees included similar keywords. For example, when searching
for ‘ _Assignee Organization_ contains _Apple_ ’, entries such as ‘Appleton
Papers Inc’ would also be retrieved, which is unrelated to the company that we
intended to study. To maintain the accuracy of our analysis, we cross-verified
the patents and inventors associated with each retrieved assignee, using
external sources, by identifying the assignees that are genuinely related to
the companies we intended to study.

Furthermore, these technology corporations often have numerous branches,
alternate names, and variations in spelling. For instance, Google may be
referred to as Google LLC, Google Technology Holdings LLC, or even encounter
misspellings like Google LCC. To ensure a comprehensive analysis, we
consolidated all subsidiary branches and alternate names under one unified
name. In this example, we merged all variations into ‘Google’. The same
approach is also applied to the other four companies being studied, including
Meta, Apple, Amazon, and Microsoft.

The complete set of fields444A data dictionary describing all of these fields
may be accessed at https://patentsview.org/query/data-dictionary. available
for each patent in the PatentsView Query Builder can be quite extensive
(including the patent text). Our analysis is restricted to a selected subset
of these fields. These fields of interest include:

  * •

app_date: The date when the patent application was filed.

  * •

assignee_organization: The organization or company to which the patent rights
have been assigned. In our study, these organizations refer to the five
companies under investigation: Google, Meta, Apple, Amazon, and Microsoft.
Each patent can only have one assignee organization.

  * •

inventor_key_id: A unique identifier for each inventor.

  * •

inventor_first_name: The first name of the inventor(s) listed on the patent.

  * •

inventor_last_name: The last name of the inventor(s) listed on the patent.

  * •

patent_title: The title or name given to the invention covered by the patent.

  * •

patent_date: The date when the patent was granted.

  * •

patent_number: The unique identification number assigned to the patent.

  * •

citedby_patent_number: The number of other patents that have cited this
particular patent.

Figure 1: (a) The distribution of granted patents and filed patents; (b)
distribution of patents granted per year, by company; (c) distribution of
patents applied for by each company per year.

Basic descriptive statistics. After retrieving the data, we found that, since
2010 (and up until 2022), 2,329 individuals have been granted patents with
multiple assignee organizations. More details on the patents filed and
granted, distributed across the five organizations, are illustrated in Figure
1. Among the 2,329 inventors with multiple assignee organizations, 2,213
inventors have patents associated with exactly two different assignee
organizations. In relative terms, the number of individuals in these five
companies (between 2010 and 2022) authoring patents with multiple assignee
organizations is a small, but non-trivial, fraction of the total number of
individuals authoring patents with these five organizations (74,637
inventors).

We ordered the patents submitted by each of these 2,213 inventors according to
the application date of the patent. Within this group, 873 inventors
alternated between two different assignee organizations multiple times, filing
patents under one assignee and then switching to the other, and vice versa. On
the other hand, 1,340 of these inventors transitioned only once, initially
filing patents for one assignee organization and subsequently moving to
another.

Qualitative spot-checks. To gain preliminary qualitative insights into these
1,340 inventors’ professional trajectories, we randomly selected a subset of
30 inventors from the group of 1,340 and conducted a preliminary investigation
of their profiles using external sources such as LinkedIn. Our findings
revealed that all 30 inventors had undergone transitions from one company to
another among the five companies under investigation throughout their careers.
Importantly, we observed a correlation between the timing of these job
transitions and the periods when they started filing patents with their
subsequent company, according to their patent application dates (app_date).
Furthermore, the companies they were employed by corresponded with the
companies they assigned their patents to. This suggests that a change in the
patent assignee often indicates a job transition, rather than holding
concurrent positions at two different companies. Therefore, in this study, we
assumed that for the 1,340 inventors who changed their patent assignees once,
this shift likely represented a move from one company to another among the
five companies being investigated.

Co-inventor patent network (CPN) construction. We constructed the co-inventor
patent network (CPN), denoted here as $C_{1}=(V,E)$, by using authorship data
from all of the patents collected ($|V|=74,637$), as described in the previous
section. In other words, each unique inventor was assigned a unique node in
the CPN, and an (undirected, unweighted) edge was created between two
inventors if they co-authored a patent together as employees in a Big Tech
firm during the period of interest (2010-2022).

Recall that the subset of these inventors who transitioned just once between
different organizations from 2010 to 2022 consists of 1,340 inventors. To
understand the structural properties of these inventors with respect to the
overall co-inventor network, we also constructed the subgraph $C_{2}\subset
C_{1}$, of which the nodes consists of these 1,340 inventors, and the edges
are the subset of edges in $C_{1}$ that can only exist between these nodes.
The goal behind constructing $C_{2}$ is to study the unique collaboration
dynamics of inventors who have experienced a transition in their
organizational affiliation. Finally, for investigating the fragmentation
aspect of RQ1, we also consider the network $C_{r}$, which is obtained by
removing all nodes in $C_{2}$ from $C_{1}$ (in other words, the 1,340
transitioning inventors), and all edges in $C_{1}$ that were incident on at
least one of the 1,340 inventors.

##  3 Results and Discussion

###  3.1 RQ1

Figure 2: (a) The degree distribution of all nodes in the CPN $C_{1}$; (b)
degree distribution of the subset of 1,340 transitioning inventor nodes in
$C_{1}$; (c) degree distribution of all nodes in $C_{2}$.

The original CPN ($C_{1}$) consisting of 74,637 inventors (as nodes) is
connected by 372,245 edges or unique co-authorships555Because the CPN is
undirected, a co-authorship relation is counted only once. In the future, we
will also consider the weighted version of this network.. The degree
distribution of $C_{1}$ follows a power-law distribution with a gamma value of
-2.32. Despite the large number of inventors, only a small fraction
($\sim$2.5% or 1,900 nodes) are singletons, indicating a relatively well-
connected network. The average inventor in this network has co-authored
patents with approximately 10 other inventors, with the most prolific inventor
having co-authored with 460 others. The network is divided into 3,598
connected components (including singletons), with the largest connected
component consisting of 66,818 inventors. The density of the network is
relatively low ($\sim$0.000134), implying that it is still quite sparse,
despite the inventors being divided only among five unique organizations.
However, as expected from the social nature of such collaborations, the
average clustering coefficient is moderately high ($\sim$0.675), which
indicates substantial clustering and transitivity among the inventors.

We also computed the degree distribution of the nodes corresponding to the
1,340 inventors who transitioned only once from 2010 to 2022 within the
network graph $C_{1}$. Figure 2 (a) and (b) illustrate the overall degree
distribution (all nodes) in $C_{1}$ as well as the degree distribution of the
subset of 1,340 nodes in $C_{1}$. We then ranked these degrees and compared
the ranks with respect to the overall network. We found that 80.61% of the
transitioning inventors are in the top quartile (25%) of all nodes, and 95.07%
of the transitioning inventors are within the top half (50%) of all nodes.
These are significantly higher than would be expected for a random sample of
1,340 nodes selected from the network. These numbers suggest that the majority
of transitioning inventors have high degree centrality, indicating some degree
of influence (at least in a local sense) within the network. However, by
calculating the clustering coefficient for each of the transitioning inventors
in $C_{1}$ and subsequently averaging these coefficients, we derive an average
local clustering coefficient of approximately 0.379 for the these
transitioning inventors, which is lower than the network average of 0.675.
This suggests that transitioning inventors are structurally embedded in a more
star-like structure: all collaborators of transitioning inventors do not
themselves collaborate on the same patents as often. In other words,
transitioning inventors are co-authoring patents with groups of inventors
between whom the only connection is the transitioning inventor themselves.

Recall that we had also constructed the subgraph $C_{2}$ that only contains as
nodes the 1,340 inventors who transitioned just once from 2010 to 2022. We
found that $C_{2}$ is denser ($\sim$0.000791) than $C_{1}$ while having lower
average clustering coefficient (0.0986) than $C_{1}$. Among the 1,340
inventors, there are 710 co-authorship connections, but nearly half of the
1,340 inventors have not co-authored patents with any others in $C_{2}$
(although they have co-authored patents with inventors in the larger network
$C_{1}$). These transitioning inventors therefore have fewer co-authorship
connections on average, as evidenced by an average degree of approximately
1.06 and a maximum degree of 13. The shape of this degree distribution is
illustrated in Figure 2 (c). While the data is too small to conclusively
interpret the shape as that of a scale-free distribution, it is also not
inconsistent with such a distribution.

The ‘remaining’ network, $C_{r}$, which mainly consists of inventors who have
not transitioned, has a slightly lower average number of co-authorship
connections per inventor compared to $C_{1}$. Additionally, the number of
inventors without any co-authorship increases, suggesting that the removal of
transitioning inventors led to some inventors in $C_{1}$ becoming isolated. We
explore fragmentation due to the removal of these nodes in more detail
subsequently. As expected, the density of $C_{r}$ is lower than that of
$C_{1}$, indicating less intense collaboration among the remaining inventors.

We evaluated the fragmentation caused by the removal of these 1,340 inventors
on the CPN by comparing the number of connected components and the average
clustering coefficient between the original network ($C_{1}$) and $C_{r}$ (the
network after removing the transitioning inventors), as well as a network
generated by randomly removing 1,340 inventors from the original network.
Compared to $C_{1}$, the average number of connected components in $C_{r}$
increases to 3,952. This indicates that transitioning inventors play a
bridging role in the innovation ecosystem, which was also suggested earlier by
their lower clustering coefficient (but higher degree centrality) compared
with the overall network. There are now (i.e., after removing the
transitioning inventors) more isolated groups of inventors who are
interconnected within their groups but not connected to other groups in the
network.

Note that this fragmentation is greater than would be expected by chance,
lending an affirmative answer to the second part of RQ1. To quantify this, we
performed an experiment where we _randomly_ removed an equivalent number of
inventors (1,340) from the network $C_{1}$, as noted above. We repeated this
process independently 100 times, and averaged the number of connected
components across these 100 iterations. This average (3,897.1 connected
components) is found to be significantly lower than 3,952 (N=100, p =
$1.46\times 10^{-9}$, one-sided Student’s t-test). This suggests that
transitioning inventors play a greater role in maintaining the network’s
global connectivity than a random group of inventors embedded in the same
environment (the five Big Tech firms), as their removal results in slightly
increased fragmentation compared to the removal of a random set of inventors.

Similarly, the average clustering coefficient exhibits interesting behavior
upon the removal of inventors. The average clustering coefficient in the
remaining network ($C_{r}$) is 0.674. This suggests that, on average, the
remaining inventors in $C_{r}$ still tend to form moderately interconnected
groups. Despite the network fragmentation caused by the removal of the 1,340
inventors, collaboration among groups of inventors remains largely unaffected,
as indicated by the similar average clustering coefficient in $C_{r}$ and
$C_{1}$.

In contrast, when we repeatedly remove a random set of 1,340 inventors from
the network, the average clustering coefficient of the nodes in the remaining
network is slightly lower, at 0.660. This value is also found to be
significantly lower than 0.674 (N=100, p = $4.074\times 10^{-20}$, one-sided
Student’s t-test). This indicates that the removal of the specific group of
transitioning inventors results in a network ($C_{r}$) that maintains slightly
higher transitivity as compared to the network resulting from the random
removal of inventors.

###  3.2 RQ2

The primary focus of RQ2 is to examine if the rate of transition of the 1,340
inventors from 2010 to 2022 is relatively stable over the period of study, or
if it peaks (or shows other irregularities) during some periods. We also seek
to investigate if transitioning occurred uniformly often between the five
organizations, or if some organizations are, in fact, over-represented in
terms of out-transitions or in-transitions than would be expected through
chance alone. As a first step toward these investigations, we determined an
estimator for the transition time $T_{t}$ when an inventor transitioned from
the first organization to the second organization. To do so, we identified
when a transitioning inventor filed their _last_ patent with the first
assignee organization and when they filed their _first_ patent with the second
assignee organization. The estimate of $T_{t}$ was then calculated the
midpoint between these two points in time666In a slight abuse of terminology,
we continue to refer to this estimator as $T_{t}$, rather than the unknown
variable itself (mathematically representing a more ‘exact’ time when the
inventor transitioned from the first to the second organization) that is being
estimated.

Figure 3: The number of inventor-transitions from their first assignee
organization to their second assignee organization each year from 2010 to
2022. (a) The number of inventor-transitions outflow by company, with the
color in each bar indicating the destination company the inventor transitioned
to. (b) The number of inventor-transitions inflow by company, with the color
in each bar indicating the source company the inventor transitioned from.

Figure 3 illustrates the number of inventors transitioning from their initial
assignee organization to their second assignee organization each year. We show
company-segregated distributions for both incoming and outgoing transitions
(and for each year) to gain deeper insights into the trend. Among the 1,340
inventors who transitioned just once from 2010 to 2022, Microsoft, on average,
had the most significant outflow of inventors, while Meta had the lowest.
Regarding transitions into the companies, Google, Amazon, and Meta have the
highest average influx of inventors. Therefore, outflowing transitions were
significantly more skewed than incoming transitions, which were more evenly
divided. Furthermore, the annual rate of inventor transitions suggests that
there is a peak from 2015 to 2017 for both outgoing and incoming transitions.
Together, the figure shows a concentrated transitioning period between the
assignee organizations, which further implies that the innovation ecosystem
within these organizations underwent a non-trivial shift (with the greatest
changes suggested for Microsoft) during a relatively brief 3-year period.

While the results in Figure 3 are illustrative, they are tabulated at the
granularity of years. To gain further insight into these transitions at a
finer granularity (which is also useful because the rate of transition
increases during a relatively narrow time period of 2-3 years), we compute two
additional variables for each transitioning inventor: first, we record the
time when they first filed a patent with their first assignee organization as
their ‘innovation start’ time $T_{s}$. Similarly, we marked the time when they
last filed a patent with their second assignee organization as their
‘innovation end’ time $T_{e}$. This period between the start and end times is
referred to as the ‘innovation period’777It bears noting that these terms
should only be interpreted in the context of this study and time period e.g.,
some inventors may have been innovating in other organizations (e.g., startups
acquired by some of these organizations, or research labs) before or after the
time period of study. The ‘innovation end’ time therefore should not be
‘literally’ taken to imply that the inventor has stopped innovating or filing
patents.. Given these additional variables, we sought to identify the periods
during which transitions occurred most frequently at a finer granularity
(quarters or 3-month periods, rather than years). To achieve this, we utilized
a moving three-month window that starts from January 2010 and is incrementally
rolled (in increments of one day) all the way until December 2022. Within each
such window, we determined how _long_ each inventor was associated with either
the first or second assignee.

Formally, this calculation may be described as follows: first, for every
inventor $i$ in each window, we calculated the length of time spent innovating
in the first assignee organization. We do so by measuring the time from the
window’s start (or from $T_{s}$ if it commenced after the window began) to
either the window’s end or $T_{t}$ (the time of transition), whichever was
sooner. This quantity represents the duration that $i$ filed patents with,
when they were employed by their first assignee organization. We denote this
duration as $D^{1}_{i}$. Similarly, we can compute such a duration
($D^{2}_{i}$) where $i$ filed patents with their second assignee organization,
by measuring the length of time from the later of the window’s start, or
$T_{t}$, and ending at the earlier of the window’s end or $T_{e}$. An example
of this procedure using actual data is illustrated for a small set of
transitioning inventors in Figure 4 (a).

Figure 4: (a) An illustration of the 3-month moving window for a set of 20
inventors. The start of the blue bar ($T_{s}$) marks the beginning of the
inventor’s ‘innovation start’ time. The transition time to the second
organization is denoted by $T_{t}$, where the blue bar ends and the green bar
begins. The end of the green bar ($T_{e}$) marks the ‘innovation end’ time;
(b) The trend of the ratio $\frac{D2}{D1}$ (‘proportion of the job 2
duration’) between 2010 and 2022, and its first derivative.

Using this procedure, we calculate the total ‘duration’ of innovation in the
first organization within a window, by summing all individual durations
(namely., the ‘blue’ bars in Figure 4 (a) within a ‘red’ window, but for all
transitioning inventors rather than just the ones shown in the figure) for the
first organization. We denote this total duration as $D1$. Similarly, we can
compute the total duration for the second job (the sum of all the green bars
in a red window) for all inventors within a window, as $D2$. Denoting the
number of all transitioning inventors as $I=1340$, we can now compute the
proportion $\frac{D2}{D1}$ within each window, and plot this proportion
against the mid-point of the window (which spans 3 months).

The results are illustrated in Figure 4 (b). We can use the plot to determine
when transition, as a measure of this relative proportion of innovation
activity in the first versus the second assignee organization, exhibits the
most significant change. We do so by examining the slope of this curve. We are
specifically interested in identifying a two-year span with the steepest
slope. A steeper slope indicates a more rapid change in the ratio
$\frac{D2}{D1}$. Our analysis indicated that the derivative of the segment
from December 2014 to November 2016 remains high (with a short blip in late
summer 2016) before declining secularly afterward. This supports the previous
result that the most pronounced transition happened in this period, but it
also shows that rates had been increasing leading up to this period, before
declining continuously after this period. Further investigation of this
period, as well as hypothetical causes for a sustained high transition rate,
could be an interesting area for further sociological research exploring
innovation patterns in Big Tech.

###  3.3 RQ3

RQ3 aims to investigate whether the transition between assignee organizations
has had a positive impact on the inventors’ ability to produce more high-
impact inventions. We note at the outset that determining impact of an
innovation, especially in the technology industry, is a controversial topic
[25], because the impact could occur over long time horizons, be an indirect
enabler of other technologies, and is not always measurable using simple
metrics. Nevertheless, one way of examining the impact is by considering the
citation count of an inventor’s patents before and after an inventor
transitions. Done in aggregate across all patents filed by the 1,340
transitioning inventors from 2010 through 2022, this count allows us to gauge
(in a limited, but still reasonable, manner) the potential association between
transition and innovation impact.

One bias that must be accounted for before computing such an association is
that patents filed earlier will obviously be expected to receive higher
citation counts. Hence, we standardize or ‘normalize’ the citation count
$P_{t,o}$ for a patent filed in year $t$ and by an inventor currently in
organization $o$ by adopting the following methodology: for each unique
combination of year ($t$) and organization ($o$), we calculate the mean
($\bar{P}_{t,o}$) and standard deviation ($s_{t,o}$) of citation counts for
all the patents filed within that timeframe and in that organization. We can
then normalize $P_{t_{o}}$ and represent it as a new variable $Z_{t,o}$ using
standard Z-score normalization:

$Z_{t,o}=\frac{{P_{t,o}-\bar{P}_{t,o}}}{s_{t,o}}$ (1)

By implementing such a Z-score normalization, we are now able to derive a
normalized citation count, or estimated impact, for each patent within our
dataset. Similarly, we can calculate the estimated impact for an _inventor_
over a given time period by averaging the estimated impacts of all patents
filed in that time period where that inventor was a co-author. With these
measures of estimated impact, we investigated RQ3 by first calculating the
average estimated impact for each inventor using the patents filed in the
entire time period _before_ the inventor transitioned888Using the terminology
introduced earlier for RQ2, this would be the average estimates impact of all
patents where the inventor was a co-author in the period $T_{t}-T_{s}$; recall
that both of these estimators apply individually to each inventor., as well as
the inventor’s average estimated impact _after_ the transition. Because these
two estimates are obtained for each of the 1,340 transitioning inventors, we
obtain two ‘paired lists’ (representing before and after impacts for the same
inventor). Subtracting the ‘before transition’ estimated impact from the
‘after transition’ impact, we obtain a single vector containing the estimated
_excess_ impact (due to transition) for each of the 1,340 inventors. We plot
the distribution of this estimated excess impact as a histogram in Figure 5. A
paired Student’s t-test showed evidence in favor of the research hypothesis
that the after-transition impact is greater than the before-transition impact
(i.e., we were able to reject the null hypothesis convincingly; N=1,340; p=
0.00025).

Figure 5: Distribution of estimated excess impact, using the methodology
described in the main text, for the 1,340 transitioning inventors. Note that
the y-axis is on the log scale.

While this result would need to be replicated for other measures of estimated
impact, and other normalization procedures, it does suggest that the
innovation potential of Big Tech has likely been improved, rather than
diminished, due to such switching activity (whether voluntarily
intended999Transitioning or switching does not always imply volition on the
worker’s part. In some cases, the first assignee organization may have
terminated the worker, while in other cases, the worker may have been induced
to leave and join the second organization. Controlling for volition in such
analyses is an important, but difficult, study that future sociological
research may want to consider. by the worker or not).

##  4 Conclusion and Future Work

In investigating RQ1, we found that transitioning inventors can play key
structural roles in patent co-authorship networks in Big Tech firms in the
period of study, much greater than would be predicted by chance. For example,
their removal leads to greater fragmentation, as measured by the change in the
number of connected components, compared to the chance removal of an
equivalent number of nodes. Results for RQ2 showed that transitioning over
this period did not exhibit a regular or ‘secular’ trend: rather, there was a
significant outflow of inventors from Microsoft and a prominent inflow into
Google, Amazon, and Meta. The rate of transition peaked between 2015 and 2017
and stayed high consistently during this period, suggesting a marked shift in
the innovation ecosystem of these organizations during this period. Following
this period, the rate of transition declined and ultimately reached a low
level by the time of the pandemic. Finally, in investigating RQ3, we found
that estimated excess impact, as measured using a normalized citation score of
patents that a transitioning inventor co-authored (before and after
transition), was positive and statistically significant. This finding suggests
that the productivity of transitioning innovators, on average, across Big
Tech, was not diminished by transition.

One interesting area of future research is to investigate some of the same
research questions but through a causal lens, using additional data (e.g.,
survey and employment records) and causal inference techniques. Investigating
questions like RQ3 by considering a wider body of estimated measure impacts is
also a fruitful area for further investigation. Finally, using similar
techniques to investigate post-pandemic effects on some of these measures
(which will likely only manifest a few years from now) is also of interest.

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