# mmWave Enabled Connected Autonomous Vehicles: A Use Case with V2V
Cooperative Perception

Muhammad Baqer Mollah, Honggang Wang,
Mohammad Ataul Karim, Hua Fang (Corresponding Author: Honggang Wang)Muhammad
Baqer Mollah and Mohammad Ataul Karim are with the Department of Electrical
and Computer Engineering, University of Massachusetts Dartmouth, MA 02747 USA
(Emails<EMAIL_ADDRESS>mkarim@umassd.edu).Honggang Wang is with the
Department of Graduate Computer Science and Engineering, Katz School of
Science and Health, Yeshiva University, New York, USA (Email:
honggang.wang@yu.edu).Hua Fang is with the Department of Computer and
Information Science, University of Massachusetts Dartmouth, MA 02747 USA
(Email: hfang2@umassd.edu).This work is supported by National Science
Foundation (NSF) under Grant No. # 2010366.This article has been accepted for
publication in IEEE Network. This is the author’s version which has not been
fully edited and content may change prior to final publication.DOI:
https://dx.doi.org/10.1109/MNET.2023.3321520©2023 IEEE. Personal use is
permitted, but republication/redistribution requires IEEE permission. See
https://www.ieee.org/publications/rights/index.html for more information.

###### Abstract

Connected and autonomous vehicles (CAVs) will revolutionize tomorrow’s
intelligent transportation systems, being considered promising to improve
transportation safety, traffic efficiency, and mobility. In fact, envisioned
use cases of CAVs demand very high throughput, lower latency, highly reliable
communications, and precise positioning capabilities. The availability of a
large spectrum at millimeter-wave (mmWave) band potentially promotes new
specifications in spectrum technologies capable of supporting such service
requirements. In this article, we specifically focus on how mmWave
communications are being approached in vehicular standardization activities,
CAVs use cases and deployment challenges in realizing the future fully
connected settings. Finally, we also present a detailed performance assessment
on mmWave-enabled vehicle-to-vehicle (V2V) cooperative perception as an
example case study to show the impact of different configurations.

###### Index Terms:

Autonomous vehicles, connected and autonomous vehicles, cooperative
perceptions, Internet of Things, millimeter-wave, vehicle-to-everything.

##  I Introduction

The successful deployment of future connected transportation is dependent on
the reliable and seamless operations of vehicular connectivity [1]. Employing
advanced technologies will always pave the way toward next generation
autonomous mobility by promoting connected and autonomous vehicles (CAVs). In
particular, the connectivity of envisioned CAVs use cases will be entirely
facilitated by communications vehicle-to-everything (V2X), enabling vehicle-
to-vehicle (V2V), vehicle-to-infrastructure (V2I), vehicle-to-network (V2N),
and vehicle-to-pedestrian (V2P), to name a few.

Once fully connected, autonomous vehicles are more likely to have the ability
to learn from surroundings (e.g., connected and non-connected vehicles,
infrastructures) and make decisions accordingly after compiling the sensory
data collected from own on-board units and neighbor vehicles. For instance,
according to the report in [2], it is estimated that a single connected
vehicle requires approximate total data rate for sensor data communications is
25 gigabytes per hour, and once fully connected and autonomous, it is expected
to reach up to 500 gigabytes per hour. Likewise, automotive manufacturers are
essentially focusing on extending their communications capability, while
meeting the ever-growing demands from CAVs. On the other hand, millimeter-wave
(mmWave), known also as the extremely high-frequency (EHF), band operates
roughly from 30 GHz (10 mm) to 300 GHz (1 mm), allowing itself as a key
enabler to facilitate this trend. Conversely, physical characteristics of
mmWave band is different from that of traditional sub-6 GHz. Thus, to move
forward with the progression of use cases of CAVs, it is essential to overcome
their associated deployment challenges. Accordingly, we present in this
article on how mmWave communications facilitate several opportunities and
efficiently utilize the development of envisioned use cases of CAVs.

Unlike [3, 4, 5], where the authors mainly focus on mobility supports, MIMO
beam selections, and algorithms for beam management on mmWave communications,
respectively, we discuss both details and specifics on functionalities, use
cases, and deployment challenges for mmWave enabled CAVs. More specifically,
we first present the inherent propagation and channel characteristics of
mmWave band before exploring other aspects. After that, we highlight most
recent vehicular standardization activities by IEEE and 3GPP 5G NR, and we
then focus on a number of prospective use cases enabled by mmWave assisted
CAVs. Followed by the use cases, we investigate the potential challenges while
CAVs are deployed at mmWave, including learning-based beamforming techniques,
efficient resource management, authentication during frequent handovers,
mmWave at 60 GHz, and real time deep learning in V2V. Before concluding, we
validate the performances of mmWave enabled V2V cooperative perception and
show the corresponding numerical insights.

##  II Background Features of mmWave

In this section, we outline the physical characteristics of mmWave and the
necessity of beamforming techniques.

###  II-A Physical Characteristics of mmWave Band

The mmWave signals have unfavorable propagation characteristics specifically
in outdoor environment. Compared to the signals at sub 6 GHz frequency bands,
mmWave signals have a much higher propagation loss while travelling through
the atmosphere. This is due to having small wavelength as comparable to the
size of air molecules, mmWave signals face atmospheric attenuation with
molecular absorption caused by oxygen ($\text{O}_{2}$), water vapor
($\text{H}_{2}\text{O}$), and other dust particles in the air. In general, the
wireless communications based on mmWave bands are mostly dependent on the
availability of line of sight (LoS) links and potentially viable for short
distance communications.

On the other hand, small wavelength of mmWave band essentially weaken the
diffraction capability of the signals. In fact, because of this limitation,
mmWave communications also suffer from a much higher sensitivity to blockage
by vehicles, human, buildings, and tree like physical objects compared to
lower carrier frequency band signals. Such blockage characteristics of mmWave
propagation drastically limit the signal strength, which, may result in
limiting the communications range. Nonetheless, there have been a number of
considerable channel measurement efforts on mmWave frequency bands. For
instance, the authors in [6, 7] conducted measurements at 28, 38, and 73 GHz
with a focus on fully understanding the large-scale physical characteristics
of these channels in urban outdoor environment. However, the results show that
the mmWave bands are particularly suitable for communications within 100-200 m
distance, even with non-line of sight (NLoS) links.

###  II-B Beamforming Techniques

Inherently, mmWave wireless links should be highly directional. The
transmission loss and short distance communications at this high frequency
band necessitate large-scale antenna arrays at both transmitter and receiver
ends for realizing high directive gains. In fact, it is possible to configure
many antenna arrays with small sizes due to their short wavelength. Using such
large-scale antenna arrays is significant for beamforming, which aids the
wireless links have an improved signal quality, i.e., signal-to-noise ratio
(SNR), by accommodating stricter alignment between transmitter and receiver
end beams guaranteeing a highly directional communication.

Radio frequency-based beamforming, such as analog, digital, and hybrid
techniques essentially have their own benefits and drawbacks depending on the
application scenarios. Hybrid techniques, among them, aim to attain a balance
between low-complexity analog and high-complexity fully digital techniques to
obtain the benefits, including hardware costs and processing times from both
analog and digital techniques. Perhaps, this hybrid beamforming, for example
[8], is particularly suitable for mmWave connected vehicular communications
involving high mobility scenarios.

Besides, the high directionality of mmWave communications imposes time
overhead in longer beam tracking, selections, and alignment processes, thereby
introducing higher latency. In fact, in order to address such latency issues,
it is essential to employ adaptive beamforming techniques so that robust
adaptation can be facilitated, which is one of the key steps to stabilize the
wireless link quality while the vehicles are moving. As such, the authors in
[9] developed an adaptive beamforming technique for mmWave so that a large
amount of sensory data can be disseminated among vehicles with multiple
gigabit data rates and lower latency. In summary, because of the
aforementioned unique characteristics, mmWave systems must rely heavily on
large-scale antenna arrays, highly directional communications with beamforming
techniques, and robust adaptive techniques to ensure reliable connections.

##  III Standardization Aspects

In this section, we highlight the standardization aspects in view of CAVs
deployment and mmWave V2X for CAVs use cases.

###  III-A CAVs Deployment

European Telecommunications Standards Institute (ETSI), the International
Organization for Standardization (ISO), the Society of Automative Engineers
(SAE), and the American Association of State Highway and Transportation
Officials (AASHTO) are actively developing standardization protocols related
to CAVs deployment and ecosystem. Specifically, efforts for communicating
standardized messages among the connected vehicles and surroundings are
defined by ETSI [10] into five types of messages, including (i) cooperative
awareness notifications, (ii) decentralized environmental safety alerts, (iii)
V2I, and vice versa, (iv) signal phase and timing information, and (v) map
data. In this regard, the ISO is also involved in defining the communications
for CAVs. For instance, the ISO/TS
19091111https://www.iso.org/standard/73781.html present a standardized
framework to establish the information exchange between roadside units (RSUs)
and vehicles, specifying the data elements and structures to facilitate
enhanced mobility and safety of the vehicles.

Further, the SAE International describes the functional approaches of message
structure, encoding and decoding while implementing the 5.9 GHz Dedicated
Short Range Communications (DSRC) standard in their J2735
Standard222https://www.sae.org/standards/content/j2735set_202211/. However,
this standard is designed particularly for DSRC messages; perhaps, it is
possible to extend it by deploying along with other technologies as described
in the J2735 standard as well. Nonetheless, SAE International also defines
vehicle driving automation levels jointly with
ISO333https://www.sae.org/standards/content/j3016_202104/, which comprises six
levels, such as no driving automation (level 0), assistance with human driver
(level 1), automation partially (level 2), automation conditionally (level 3),
automation highly (level 4), and automation fully (level 5).

Besides, the AASHTO specifies ten policy principles and various
recommendations under these policy
principles444https://cav.transportation.org/ for deployment as well as
operations of CAVs. In particular, the policy principles cover (i) national
strategy, (ii) safety, (iii) sustainability, (iv) future connectivity, (v)
investment, (vi) quality of life, (vii) state and federal roles, (viii)
uniform national policy, (ix) community engagement and collaboration with
industry, and (x) security and privacy-preserving data sharing.

Figure 1: Selective mmWave enabled use cases of CAVs.

###  III-B mmWave V2X Standardization Activities

3GPP 5G NR and IEEE 802.11 have already considered to integrate mmWave in V2X
communications. While cellular V2X by 3GPP and IEEE 802.11p (DSRC) may have
some limitations in terms of availability and reliability potentially demanded
by CAVs use cases, both 3GPP and IEEE are currently promoting the
standardization activities to address the limitations, such as 5G NR V2X [11]
and 802.11bd [12]. These two standardization activities are presented as
follows.

3GPP 5G NR: For the standardization activities, 3GPP 5G NR (New Radio) V2X and
IEEE 802.11 NGV (Next Generation V2X) are the two key groups promoting the
technical specifications and defining the advanced use cases. In general, 3GPP
5G NR has specified to operate in both sub-6 GHz as well as high band ($>$24
GHz, mmWave) frequencies. However, 3GPP standardization efforts in V2X arena
have come up with C-V2X (cellular V2X) in Release 14 with long-term evolution
(LTE) in two parts. The first part, referred as Direct Communications, has
been aimed to provide typical safety related services within V2X. Whereas, the
second part serves as cellular network to provide services which require wide
area coverage, thus, named as Vehicle-to-Network. Later, Releases 15 and 16
described the native supports for mmWave band by enabling improved
localization accuracy of the vehicles, employing large-scale antenna arrays,
and exploiting extra spatial and angular degrees of freedom within 5G New
Radio (NR) carrier. Further, 3GPP’s most recent release 17 [11] has focused on
supporting enhanced V2X communications within mmWave band by including
resource allocations techniques in multi-casting, user equipment (UE) power
savings, and improved sidelink for V2V operations. This release 17 has also
introduced a channel model at frequency range 2 (FR2), which, in principle,
covers from 24.25 GHz to 52.6 GHz. Clearly, the FR2 has a large amount of
spectrum, and consequently, it’s technical specifications will help to move
forward the mmWave V2X communications to enable ultra-low-latency services. In
fact, it is expected that the utilization of other bands will be considered to
include soon under other releases since the demands for more 5G NR
applications are becoming increasingly significant in autonomous driving
field.

IEEE 802.11 NGV: Similarly does, IEEE 802.11 NGV was initiated to develop IEEE
802.11 specifications in physical (PHY) and medium access control (MAC) layers
particularly for V2X communications. Thereby, the first amendment building
upon IEEE 802.11 was IEEE 802.22p to offer wireless access in vehicular
environment (WAVE), which presented new functionalities to provide inter-
vehicle connectivity to the vehicles in dynamic scenarios within DSRC. While
enabling inter-vehicles communication was a significant achievement, IEEE
802.11p WAVE is unable to fulfill the low latency and high throughput
requirements of recent use cases in semi or fully autonomous driving, such as
cooperative driving, High-Definition (HD) map updating in real-time, and
vehicular platooning. For this reason, inherited from IEEE 802.11p, the IEEE
802.11 NGV has recently developed IEEE 802.11bd PHY and MAC specifications in
[12] with a focus on 5.9 GHz band, but also support from 57 GHz to 71 GHz,
which is considered the first standard using new spectrum at mmWave band for
V2X communications. Basically, this IEEE 802.11bd standard has redefined
fundamental principles of PHY and MAC layers through incorporating innovative
techniques to take benefits of beamforming gains and cope with mmWave
propagation losses. Besides, this standard has also defined to support
increased throughput, reduced end-to-end latency, and higher channel bandwidth
up to 40 MHz, while ensuring coexistence and backward compatibility with IEEE
802.11p, among other advancements. However, apart from this IEEE 802.11bd,
IEEE 802.11 has two more specifications on mmWave, namely IEEE 802.11ad and
IEEE 802.11ay, but these two standards are targeted to serve indoor wireless
communications applications.

To sum up, both 3GPP 5G NR-V2X and IEEE 802.11bd represent the latest mmWave
enabled specifications for V2X employing innovative techniques and procedures,
which could be considered as basis of opening up new possibilities for V2X
communications in the upcoming beyond 5G and 6G era, thereby, have a vision on
fulfilling the requirements of advanced use cases of future fully connected
vehicles.

##  IV mmWave in CAVs Use Cases

Once CAVs are fully deployed, mmWave communications is expected to support a
number of use cases. In the following, we discuss such CAVs use cases,
including but not limited to.

###  IV-A Cooperative Perception

It is possible to share the perceptual data in cooperative driving, for
example, LiDAR sensor data between vehicles and process them for the purpose
of avoiding potential collisions. In the context of CAVs, referred to as
cooperative perception, and it is particularly suitable for use in complex
urban areas. In fact, cooperative perception techniques facilitate autonomous
vehicles to enlarge their visibility areas by developing a layout of
surrounding objects and traffic, such as pedestrians, bicycles, buildings, and
non-connected vehicles. With this functionality, indeed, the vehicles can
collect the perceptual data from connected neighboring vehicles through V2V
communications in order to manage blind spots, detect hidden objects, and
recognize NLoS traffic ahead. However, for this use case, leveraging mmWave is
necessary, specifically for real-time integration of multimedia data sharing
and processing. Both high data rate, lower latency, and highly reliable
communications are desired, thereby ensuring safe automated driving.

###  IV-B Vehicular Platooning

Instead of driving individually, a group of vehicles may consider driving
together by making a platoon on highways, which results in potentially lower
fuel consumption, reduced congestion, increased lane capacity as well as
improved safety, eventually leading to optimized transportation. In this
platoon formation, the lead vehicle may have a driver, whereas the other
following vehicles can be driverless, and the lead vehicle basically
represents the speed and trajectory reference for the followers. However, in
such a connected driving concept, V2V communications are expected to play an
important role in maintaining connectivity and exchanging the high-volume of
onboard sensor data among the vehicles. Besides, the vehicles in platoon
travel maintain relatively shorter spacing in-between. Essentially, compared
to DSRC, V2V enabled by mmWave has become one of the most suitable
communication technologies to enable vehicle platooning.

###  IV-C Precise Positioning

The unique characteristics of mmWave band, such as having large arrays of
small sized antennas for directional communications, can be employed to
provide highly accurate positioning for automated driving, which is considered
as one of the key requirements for future CAVs. Therefore, mmWave enabled CAVs
is expected to possess the capability of precise positioning, which is able to
offer environmental perception in real-time or near-instantly. However, mmWave
along with global navigation satellite system (GNSS) and time of arrival (ToA)
type localization techniques are utilized to improve the positioning accuracy,
perhaps most importantly, by employing multiple techniques to allow for
centimeter level accuracy positioning. Such capabilities lead to more
efficient and safe automated driving, particularly in urban vehicular
scenarios. In fact, native support for mmWave applicable positioning has been
introduced in Release 16 of 3GPP 5G NR.

###  IV-D Multimodal Beamforming

To increase the performance of beamforming in terms of reduced processing time
and improved directional accuracy, an enhanced beamforming method that uses
GPS, Camera, Radar, and LiDAR like multiple modalities from both non-RF data
sources can be utilized along with RF-only mmWave can be employed. This
beamforming method is referred as multimodal beamforming. It can be used to
detect pedestrians and any other objects either between two connected vehicles
or between a vehicle and a roadside unit. Accordingly, deep learning-assisted
object detection techniques are applied after fusing the non-RF visual sensor
data to develop a surrounding visualization of the vehicle. For instance, the
work in [13] presents a beam selection procedure utilizing deep learning on
distributed multimodal data. The technique when employed to process synthetic
and home-grown real-world data sets has shown to yield 95% and 96% enhancement
respectively beam selection speed as well as predicting the top-10 best beam
pairs.

###  IV-E RSU V2I Communications

Having a dynamic high-definition 3D map is essential in automated driving to
maneuver autonomous vehicles efficiently and safely. The onboard visual
sensors are used to build crowdsourced maps presented as real-time point
clouds and high-resolution multimedia data. However, due to the necessity of
keeping the map updated in real-time, the vehicles are essentially dependent
on the RSUs by considering edge computing nodes, which may introduce a
significant delay while collecting and processing the necessary information.
Ensuring the required high throughput as well as stringent latency with mmWave
V2I can help to enhance the capability of serving on-demand requests on
location-aware and time-sensitive updates, navigation and control by obtaining
the perception of the surroundings instantly, and providing, in turn, safe
operation in automated driving.

In summary, these aforementioned CAVs use cases, shown in Fig. 1, necessitate
adherence to the strict requirements given very high data rate, lower latency,
precise positioning, and highly reliable communications. And to satisfy these
demands, CAVs’ use cases promise to be highly beneficial in mmWave
communications. Accordingly, connectivity with mmWave communications is
becoming an increasingly important topic in the CAVs areas.

Figure 2: A summary of key deployment challenges of mmWave enabled CAVs.

##  V Deployment Challenges

In this section, we discuss key challenges that deserve further attention
while deploying mmWave enabled advanced CAVs use cases as follows with Fig. 2
representing a summary of such challenges.

AI-Enabled Beamforming: Blockage-prone mmWave communications introduce
significant challenges in beam alignment algorithms while maintaining reliable
transmitter and receiver links in high mobility and dynamic environments,
which can severely impact delay-sensitive CAVs. Consequently, such beam
alignment algorithms are required to perform adaptive and frequent updates of
large beamforming arrays, perhaps adding further processing overheads.
However, facilitated with the capability of building the trained model from
calibrated beam information and RF signals, it is possible to utilize the
trained model later to predict the optimal beam selection, eventually,
realizing higher beamforming gain. Likewise, learning-based techniques can
also be utilized for the purpose of blockage classification and afterward,
predict the beam pairs to bypass the blockage, resulting in maximizing the
mmWave network capacity. For example, the authors in [14] present to equip the
mmWave base stations with RGB cameras to get the surrounding sensing
information, such as moving vehicles, pedestrians, etc. Through utilizing the
multivariate regression and vision-based algorithms on the surrounding sensing
information, the base stations can proactively predict the potential LoS link
blockages, thereby avoiding sudden link failures by doing timely handover.

Efficient Resource Management: Doppler effect caused by vehicular mobility
impacts the energy efficiency and network capacity of mmWave enabled CAVs.
Accordingly, the network coordination overhead costs increase essentially with
the increase of desired high data rates. These in turn, however, demand to
reconsider developing effective resource management as well as optimization
techniques, such as network planning, optimized allocation of resources (e.g.,
bandwidth, channel, power), admission control, and scheduling. Moreover,
promising self-organizing networks could support developing dynamic network
configuration and planning. In particular, such self-organizing capabilities
should also consider the sensitivity of specific CAVs use cases to include
complex interference, routing, and time delay caused by vehicular mobility and
mmWave communications.

Authentication during Frequent Handovers: Due to short range communication
capability, connected vehicles enabled by mmWave communications will require
frequent handover while moving. In mmWave network, a part of security mainly
relies on authentication techniques for both messages and involved parties.
However, frequent handovers essentially demand multiple authentications, which
may result in unexpected communications overheads. To be effective, fast and
cooperative authentication techniques need to be further considered in the
future while realizing successful authentications with moderate communication
costs.

(a) Throughput performances in 28 GHz with highway and urban scenarios

(b) Throughput performances in 60 GHz with highway and urban scenarios

(c) Latency performances in 28 GHz and 60 GHz with highway and urban scenarios

Figure 3: (a)&(b) The impacts of V2V distances on achievable throughput and
(c) Results of latency performances with respect of V2V distances in mmWave
communications at 28 GHz and 60 GHz with two propagation scenarios.

mmWave at 60 GHz: In recent years, mostly lower bands of mmWave have been
studied well, for example, at 28 GHz and proven the feasibility of employing
V2X communications. However, mmWave at unlicensed 60 GHz (57 – 64 GHz) has
also been considered as a promising frequency band for the upcoming paradigm
shift on the procedures of communications technologies utilized to support
specifically the growing high data rates demands from V2X for CAVs. On the
other hand, to make highly directional communications through employing
stricter alignment between receiver and transmitter end beams inherently
brings its own unique limitations. Thus, it is necessary to do feasibility
studies of CAVs in 60 GHz communications, including investigating the
intermittent connectivity among vehicles, RSUs, and other entities. We believe
that reducing the impact of the Doppler effect is necessary for high data rate
(e.g., gigabyte) applications. Besides, incorporating sensing, communications,
and localization like different functionalities along with other mmWave and
below 6 GHz spectrum can be a promising direction to facilitate development of
realistic use cases.

Real Time Deep Learning in V2V: It is worth mentioning that deep learning
techniques have attracted growing interest in mmWave enabled vehicular
communications because of having capability of enhancing the performances of
communications. Through training the ray-tracing (e.g., angle of arrival,
angle of departure, time of arrival, received power), channel state
information, LiDAR, RADAR, camera, and location data, deep learning techniques
have made possible to extract the features of the system, which eventually
result improved vision-based applications for connected vehicles. On the other
hand, when it comes to CAVs uses cases having V2V communications, such as
collaborative perception and platooning, inefficient resource allocation
(e.g., channel, energy budget, bandwidth) techniques and dynamic V2V links may
lead performance degradation substantially while deploying real-time deep
learning algorithms. Besides, it becomes more challenging when the vehicles
have to handle a large amounts of data for training. However, edge
intelligence algorithms can be utilized to alleviate the high communications
overhead, latency, and complexity, and also, transfer learning techniques can
be applied so that the trained models or knowledge can be effectively reused.

(a) 28 GHz and 60 GHz with highway and urban scenarios in 100-meter distance

(b) 28 GHz and 60 GHz with highway and urban scenarios in 500-meter distance

Figure 4: The impact of packet reception ratio over source data rate. The
packet reception ratio is significantly becoming smaller with source data rate
decreasing.

##  VI A Use Case: V2V mmWave Communications

In this section, we present a detailed performance study of a practical CAVs
use case to realize how high frequency mmWave bands impact various aspects of
vehicular scenarios. However, due to the costly mmWave hardware and the
complexity of developing as well as testing the network components, it might
be impractical to validate the performance matrices with a fully implemented
testbed. For this reason, we choose MilliCar presented in [15] to perform the
end-to-end simulations, which is a Network Simulator-3 (NS-3) mmWave module
followed by the NS-3 mmWave module by NYU-
Wireless555https://github.com/nyuwireless-unipd/ns3-mmwave. On the other hand,
we choose V2V cooperative perception communications at mmWave bands as an
example use case to implement and analyze the corresponding performances. In
particular, cooperative perception requires perceptual data sharing with very
high throughput within short distance, which makes mmWave V2V communications
as the most suitable use case. While the best mmWave in CAVs use cases may not
be just limited to V2V cooperative perception, the parameters in the
simulation environment are chosen in accordance with the V2V cooperative
perception use case.

The simulated settings are comprised of two representative scenarios, namely
vehicular highway and urban scenarios having medium traffic density, with
highway covering 1500 vehicles/hour/direction and urban area covering 316
vehicles/$\text{mile}^{2}$. Both scenarios may include LoS, Non-LoS (NLoS),
and Vehicle NLoS (NLoSv) types of states depending on the vehicle moving
directions and surroundings. We perform the simulations on V2V links operating
at $f_{c}$ = 28 GHz (with 400 MHz bandwidth) and 60 GHz (with 800 MHz
bandwidth) frequencies, with numerology index 3 in the physical layer (sub-
carrier spacing = 120 kHz and slots per frame = 8) and the vehicles exchange
1024 bytes of User Datagram Protocol (UDP) packets with each other by NR
sidelink interface in order to share the perceptual data unless otherwise
mentioned. Accordingly, the obtained simulation results are illustrated in
Fig. 3 and Fig. 4.

Figs. 3a and 3b highlight the achievable average throughput and latency,
respectively, over different V2V distances ($\mathcal{D}$ = 10 meters to 500
meters) while two vehicles are moving one behind the other in the same lane at
approximate 45 mph speed. From the results, we can observe the following
points.

  * •

As can be noticed, mmWave at 28 GHz and 60 GHz highway scenarios have higher
throughput and lower end-to-end latency than the urban scenarios up to a
certain V2V distance ($\sim$300 m). This is due to the obstacle free signal
propagation in highways, resulting in the high probability of LoS.

  * •

Because of significant path loss, the achievable throughput of mmWave links
significantly drops as the V2V distance increases. At the 350 meters, even the
LoS V2V links is very likely to be blocked in highway scenarios. Thereby the
throughput starts becoming worse.

  * •

Presence of many obstacles leads to high penetration loss in urban scenarios
can be drastic and subsequently provide the low performance of achievable
throughput. We can observe it specifically in urban cases by comparing the bar
lines from V2V distances at 300m and 200m corresponding to the 28 GHz and 60
GHz, respectively. However, proper beamforming management techniques may
compensate partially or fully for it.

  * •

The size of the buffer in this simulation setting is fixed at 500 packets.
Although getting high throughput depends on the buffering, large buffer size
may result in increased end-to-end delay because of buffering at the queue.

Figs. 4a and 4b depict the impact of Packet Reception Ratio (PRR) over
different source data rate values at 28 GHz and 60 GHz. In this simulation
setting, we consider there are four vehicles divided into two groups moving on
the same lane at approximately 45 mph speed. We also consider each link end is
equipped with 4 antenna elements, the resources are shared orthogonally among
the vehicles, and 64-QAM modulation technique is employed. With these
settings, we can get a notable increase in PRR values over a lower source data
rate by comparing the curves. Specifically, by comparing the curves, the high
frequency at 60 GHz has slightly lower PRR values than 28 GHz. This is again
due to mmWave signal attenuation problems. Indeed, with a lower source data
rate, the PRR value is close to 1, which is very suitable to safety-critical
services. However, robust forward error correction (FEC) techniques might be
required to maintain high PRR.

##  VII Conclusion

In this article, we have presented a detailed discussion on the prospects of
mmWave on envisioned CAVs use cases. Specifically, we have detailed the
physical characteristics of mmWave signals and the necessity of beamforming
techniques. We also have discussed the mmWave standardization activities,
which apply to CAVs specified by IEEE and 3GPP 5G NR. After that, we have
highlighted the selected use cases of CAVs and discussed how and to what
extent mmWave can support them. Further, we have investigated several specific
deployment challenges. Finally, we have evaluated the performance of a V2V
cooperative perception scenario as an example use case. However, we plan to
extend the performance evaluation part to prototype development and then
evaluate the performance in our future works. We expect this article will
facilitate further exploration in this crucial research area.

## Acknowledgments

The authors would like to thank the editor and anonymous reviewers for their
constructive suggestions and valuable comments.

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Biographies

Muhammad Baqer Mollah is currently pursuing a PhD degree in the Department of
Electrical and Computer Engineering at the University of Massachusetts
Dartmouth. His research interests include advanced communications, security,
and edge intelligence techniques for Internet of Things (IoT) and connected
vehicles.

Honggang Wang is a Professor in Department of Graduate Computer Science and
Engineering at Katz School of Science and Health, Yeshiva University. His
research interests include AI and Internet of Things, connected vehicles,
smart health, and cyber security. He served as the Editor-in-Chief of IEEE
Internet of Things Journal (2020 - 2022).

Mohammad Ataul Karim is a Professor in the Department of Electrical and
Computer Engineering at the University of Massachusetts Dartmouth. His areas
of research encompass optical computing, information processing,
pattern/target recognition, computer vision, and Internet of Things.

Hua Fang is a Professor in the Department of Computer and Information Science
at the University of Massachusetts Dartmouth. Her current research interests
involve real-time machine or statistical learning, and visual analytics of
wearable biosensor data streams, broadly in
E-/M-/Digital/Tele-/Virtual/Precision health, and Internet of Things.


11institutetext:  US Naval Research Laboratory, Code 6792, Nonlinear System
Dynamics Section, Plasma Physics Division, Washington, DC 20375
College of William and Mary, Department of Applied Science, Williamsurg, VA
23187-8795

#  Rare Event Extinction on Stochastic Networks

Brandon S. Lindley 11 Leah B. Shaw 22 and Ira B. Schwartz 111122

###### Abstract

We consider the problem of extinction processes on random networks with a
given structure. For sufficiently large well-mixed populations, the process of
extinction of one or more state variable components occurs in the tail of the
quasi-stationary probability distribution, thereby making it a rare event.
Here we show how to extend the theory of large deviations to random networks
to predict extinction times. In particular, we use the theory to find the most
probable path leading to extinction. We apply the methodology to epidemic
models and discover how mean extinction times scale with epidemiological and
network parameters in Erdős-Rényi networks. The results are shown to compare
quite well with Monte Carlo simulations of the network in predicting both the
most probable paths to extinction and mean extinction times.

###### pacs:

05.50.-a

###### pacs:

87.23.Cc

Fluctuation phenomena, random processes, noise, and Brownian motion Population
dynamics

In many models of finite populations, random fluctuations occur due to
internal interactions between individuals or agents, and/or external
stochastic forces. Such fluctuations are evident in the modeling of well mixed
populations that support epidemics of disease spread [1], as well as
ecological bio-diversity of species [2, 3], among others. Typically, the
dynamics performs small random fluctuations about an attracting state.
However, in almost all populations of finite size, fluctuations may organize
in such a way to drive one or more components of the population to extinction.
Mechanisms conjectured to play a role in extinction processes include small
population size, low contact frequency for frequency-dependent transmission,
competition for resources, and evolutionary pressure [4], as well as
heterogeneity in populations and transmission in coupled population models [5,
6].

In epidemic models where the population is well-mixed, extinction of
infectious individuals has been shown to be affected by noise intensity [7],
peak infectious population size [8], and seasonal phase occurrence [9].
Moreover, since the extinct state is typically unstable in the deterministic
mean field and is an absorbing state of a stochastic process, time scales for
extinction may be exponentially long [10]; i.e., the probability of extinction
is a decreasing exponential function [11]. Vaccination and treatment programs
have been studied to speed up the extinction of disease in well-mixed
populations [12, 10]. For example, although most vaccination schedules are
designed to be administered periodically (deterministic) [13, 14, 15], Poisson
distributed scheduling was recently shown to be more efficient than regular
treatment schedules [16].

In network populations, outbreak extinction probabilities have been predicted
for early times when an infection has just been introduced [17, 18]. Other
studies of extinction on networks attempt to predict whether a persistent non-
extinct state exists, such as for computer viruses in growing networks [19]
and epidemics in various network geometries (e.g., [20]). This is equivalent
to finding the bifurcation point where the extinct state becomes unstable and
thus can frequently be predicted using deterministic approximations (e.g.,
[19]).

Extinction times for an endemic disease in a network have occasionally been
studied, mostly numerically. In [21], extinction times were measured in
simulations of an epidemic on an adaptive network, and the log of the
extinction time was observed to increase with a power of the distance from a
bifurcation point. In [22], lifetimes were measured in a similar model with
the addition of pulsed vaccination. Similar lifetimes were obtained in
simulations by adding stochastic pulsed vaccination to a deterministic mean-
field model based on a pair approximation, suggesting that it is not necessary
to model the full stochastic network dynamics in detail to capture the
extinction time. Here we will present a method for finding the most probable
path to extinction and the extinction time in networks that can be described
using a pair approximation.

Here we consider the problem of epidemic extinction in stochastic networks,
and we find that the extinction process depends not just on the nodes of the
network, but also on how the links change as the system evolves. That is,
along the most probable path to extinction, we have derived a new approximate
model showing that both nodes and links play an important role in the mean
time to disease eradication. We have introduced a novel mathematical tool so
that the path is derived constructively.

The specific example we will consider here is a network of $N$ nodes and $K$
links, with an average degree of $2K/N$. The dynamics on the network is an SIS
epidemic model, where susceptibles capable of acquiring the disease become
infectious through a contact with an infective individual, and become
susceptible again after a recovery period. Here we divide the population into
two groups, with $S$ as the number of susceptible individuals and $I$ as the
number of infected individuals, such that $S+I=N$. The population is closed,
and there are no births and deaths.

To quantify how the network topology compares with a well-mixed global, or
all-to-all, coupling structure, we allow for two modes of disease spread.
Specifically, global disease transmission occurs in a well-mixed population
when the disease transmits from any infected person to any susceptible person.
In contrast to global transmission models, local disease transmission occurs
when a network of connections between individual persons is considered, and
diseases can only spread along links between infected and susceptible
individuals. In the case of transmission via only the network, we let $p$ be
the infection rate per SI link. When transmission is only global, we define a
global contact rate of $p2K/N$, selected so that the epidemic threshold occurs
at $p^{*}=\frac{rN}{2K}$ in both the globally coupled and network transmission
cases. We will introduce a homotopy parameter $\epsilon$ which continuously
transforms the system between the local and global transmission models.
Finally, $r$ is a recovery rate.

We approximate the dynamics of the network by considering transitions in nodes
and links, similar to the pair-based proxy model of [23]. We let
$\boldsymbol{X}$ denote the vector with components containing both the number
of nodes and number of links of the network. Specifically, we let
$\boldsymbol{X}=[S,I,N_{SS},N_{SI},N_{II}]$, where $S,I$ denotes the numbers
of susceptibles and infectives, respectively, and $N_{AB}$ denotes the number
of connections between nodes of type A and B. We assume large but finite
population size, $N$, and we suppose the dynamics proceeds as a Markov
process. To complete the formulation for the network dynamics, we suppose
there exist $M$ transition events with transition rates
$W(\boldsymbol{X},\boldsymbol{\nu}_{k})$ having increments
$\boldsymbol{\nu}_{k}$. The dynamics of the probability density,
$\rho(\boldsymbol{X},t)$, can now be modeled as a master equation [11, 24,
25]:

$\begin{split}\frac{\partial\rho}{\partial
t}(\boldsymbol{X},t)&=\sum_{k=1}^{M}[W(\boldsymbol{X}-\boldsymbol{\nu}_{k},\boldsymbol{\nu}_{k})\rho(\boldsymbol{X}-\boldsymbol{\nu}_{k},t)\\\
&\quad-W(\boldsymbol{X},\boldsymbol{\nu}_{k})\rho(\boldsymbol{X},t)],\end{split}$
(1)

where $\rho(\boldsymbol{X},t)$ represents the probability of finding the
system in state $\boldsymbol{X}$ at time $t$.

The rare events are characterized by observing the targeted event in the tail
of the probability distribution. Typically, when observing the times for the
event to occur, one sees that the distribution of times is exponential [26].
When the population is sufficiently large, we may assume the distribution of
times possesses such an exponential tail [10].

Making a homogeneity assumption that each node’s local neighborhood is
identical to that of all other nodes of the same type, we can find its
neighbors from the system average. For example, a node of type S would have an
expected degree $d_{S}=(2N_{SS}+N_{SI})/S$ and has, on average, $2N_{SS}/S$
neighbors of type S.

We next specify the possible transitions and associated transition rates for
the pair-based proxy model, so that the approximate stochastic process can be
described by a master equation that captures fluctuations in the probability
density. There are three types of state transitions to consider:
S$\rightarrow$I by transmission along the network, S$\rightarrow$I by global
disease transmission, and I$\rightarrow$S by disease recovery.

To determine the increments for each transition, notice that whenever a node
changes from one state to another, all of its associated links change. For
example, if a susceptible node becomes infected via local disease
transmission, then S$\rightarrow$I, and therefore all its SS links become SI
links. Further, $1+N_{SI}/S$ links of type SI will be lost, creating as many
II links. Using this logic, we can write the following transition increments
for transmission along the network, global transmission, and recovery,
respectively:

$\displaystyle\boldsymbol{\nu}_{1}$
$\displaystyle=\left[-1,1,-\frac{2N_{SS}}{S},\frac{2N_{SS}}{S}-\left(1+\frac{N_{SI}}{S}\right),\left(1+\frac{N_{SI}}{S}\right)\right]$
(2) $\displaystyle\boldsymbol{\nu}_{2}$
$\displaystyle=\left[-1,1,-\frac{2N_{SS}}{S},\frac{2N_{SS}}{S}-\frac{N_{SI}}{S},\frac{N_{SI}}{S}\right]$
$\displaystyle\boldsymbol{\nu}_{3}$
$\displaystyle=\left[1,-1,\frac{2N_{SI}}{I},-\frac{N_{SI}}{I}+\frac{2N_{II}}{I},-\frac{2N_{II}}{I}\right].$

For each transition $\boldsymbol{\nu}_{k}$ we have an associated transition
rate $W(\boldsymbol{X},\boldsymbol{\nu}_{k})$ given by

$\displaystyle W(\boldsymbol{X},\boldsymbol{\nu}_{1})$ $\displaystyle=\epsilon
pN_{SI}$ (3) $\displaystyle W(\boldsymbol{X},\boldsymbol{\nu}_{2})$
$\displaystyle=(1-\epsilon)p\frac{2K}{N}\frac{{\color[rgb]{0,0,1}SI}}{N}$
$\displaystyle W(\boldsymbol{X},\boldsymbol{\nu}_{3})$
$\displaystyle=r{\color[rgb]{0,0,1}I},$

for transmission along the network, global transmission, and recovery,
respectively. We have introduced the homotopy parameter $\epsilon$ which
continuously transforms the system between the global and network transmission
models. Here $\epsilon=0$ returns the standard equations for the SIS model in
a globally-coupled (well mixed) population, and $\epsilon=1$ is a locally-
coupled network SIS model. Even the pair-based proxy system is sufficiently
high dimensional that finding the most probable path to extinction will be
difficult a priori, but continuously varying $\epsilon$ will allow us to track
the path from the known case of global coupling to the network case. This
approach to tracking the path to extinction by continuously varying a
parameter is new. However, we note that far from the extinction bifurcation
point, if the well mixed population is constrained in fluctuation so that
$S+I=N$, the limit of $I(\epsilon)$ as $\epsilon$ approaches zero may not
exist since the SIS model has been shown to be fragile [27].

The rare events we are interested in are those of extinction where the number
of infected nodes goes to zero. As in the globally coupled case
($\epsilon=0$), the Monte Carlo dynamics of the network ($\epsilon=1$)
exhibits random fluctuations about an attracting state, and then the
fluctuations organize in such a way as to drive the infected nodes to
extinction. We wish to characterize the probability of an extinction event in
the large population limit and compare it to the globally coupled case. In
particular, we are interested in the most probable path and mean times to
extinction. To understand the scaling of extinction times in terms of
epidemiological parameters and network topology, we employ large deviation
theory techniques for finite populations [28, 25].

We introduce the scaled variables $\boldsymbol{x}$ with components defined by
$x_{1}=I/N$, $x_{2}=N_{SI}/N$, and $x_{3}=N_{II}/N$, where the remaining
variables $S$ and $N_{SS}$ can be eliminated by conservation of nodes and
links. We define the scaled transition rates
$w(\boldsymbol{x},\boldsymbol{\nu}_{k})=W(\boldsymbol{X},\boldsymbol{\nu}_{k})/N$
on these variables. Henceforth, we will refer to the fraction of infecteds and
susceptibles as $P_{I}=I/N$ and $P_{S}=S/N$. The link variables will be given
as $L_{AB}=N_{AB}/N$. Given that the probability current at the extinct state
is sufficiently small for large enough $N$, and that the increments are
sufficiently small with respect to $N$, there will exist a quasi-stationary
probability distribution about a state with a non-zero number of infected
individuals that decays into the stationary (extinct) solution over
exponentially long times [16]. The extinction rate of infected individuals can
be calculated from the tail of the quasi-stationary distribution. It has been
demonstrated that a WKB approximation to the quasi-stationary distribution
allows one to approximate the mean extinction time provided the population is
sufficiently large when $\epsilon=0$ [28].

Employing the WKB approximation for the probability,
$\rho(\boldsymbol{x},t)=A(\boldsymbol{x})\text{exp}(-NR(\boldsymbol{x},t))$,
from the first order expansion with respect to $N^{-1}$, we can write the
Hamilton-Jacobi equation,

$\frac{\partial R}{\partial t}+H\left(\boldsymbol{x},\frac{\partial
R}{\partial\boldsymbol{x}}\right)=0.$ (4)

Here the function $H$ is called the Hamiltonian of the system, and $R$ is
known as the action. The variable $\boldsymbol{p}=\partial_{x}R$ is the
conjugate momenta of the Hamilton-Jacobi equation, and the Hamiltonian can be
written as

$H(\boldsymbol{x},\boldsymbol{p})=\sum_{k=1}^{3}w(\boldsymbol{x},\boldsymbol{\nu}_{k})(e^{\boldsymbol{p}\cdot\boldsymbol{\nu}_{k}}-1).$
(5)

We analyze the system by solving the characteristic equations
$\dot{\boldsymbol{x}}=\partial_{\boldsymbol{p}}H(\boldsymbol{x},\boldsymbol{p}),$
$\dot{\boldsymbol{p}}=-\partial_{\boldsymbol{x}}H(\boldsymbol{x},\boldsymbol{p})$.

Since we are interested in the probability of extinction events, we must
compute the action $R$ that maximizes this probability. This will, naturally,
be the one that satisfies the Hamilton-Jacobi Eq. 4. This implies that there
will be some trajectory along which $R$ is minimized, which represents the
maximal probability of such an extinction event, and that this trajectory,
which we will call the “most probable path,” satisfies the characteristic
equations for the Hamiltonian plus boundary conditions [28, 25].

The boundary conditions for the characteristic equations are given as steady
state solutions of $\dot{\boldsymbol{x}}=\dot{\boldsymbol{p}}=0$. In fact, for
epidemic models, the quasi-stationary probability distribution peaks at an
endemic state, where the number of new infections equals the number of
recoveries per unit time, and corresponds to the zero conjugate momenta case,
i.e., the steady state is at
$(\boldsymbol{x},\boldsymbol{p})=(\boldsymbol{x}_{e},\mathbb{0}$). The other
steady state is the extinct state, $P_{S}=1,$ $P_{I}=0$, which is saddle
point. However, here we find that the conjugate momenta at this state,
$\boldsymbol{p}_{0}$, is non-zero, since there is a non-zero probability
current at that point. We label the endemic state as $(\boldsymbol{x}_{e},0)$,
and the non-trivial extinct state as
$(\boldsymbol{x}_{0},\boldsymbol{p}_{0})$. The steady states of the
characteristic equation corresponding to zero conjugate momenta are those
which satisfy the deterministic epidemic model.

For the case $\epsilon=0$, analytic solutions are available for the most
probable path to extinction [29], and hence the action may be computed
directly. However, for $\epsilon=1$, the additional equations for the mean
link fractions complicate the analysis, and the most probable path must be
computed numerically. We therefore employ the iterative action minimizing
method (IAMM) [30] to compute the most probable path between
$\boldsymbol{x}_{e}$ and $\boldsymbol{x}_{0}$, and the action will be given by
the path integral
$\int_{\boldsymbol{x}_{e}}^{\boldsymbol{x}_{0}}\boldsymbol{p}\cdot
d\boldsymbol{x}$ along this path since $H(\boldsymbol{x},\boldsymbol{p})=0$
along the path.

Using the IAMM, we perturb off the $\epsilon=0$ solution, where the path can
be analytically defined. For an all-to-all connected graph,
$L_{SI}=\frac{K}{N}2P_{S}P_{I}$ and $L_{II}=\frac{K}{N}P_{I}^{2}$ and momenta
conjugate to the link variables are 0. We then use continuation as a function
of $\epsilon$ to get to the locally-coupled $\epsilon=1$ case. We demonstrate
the robustness of this approach in Fig. 1, where the computed most probable
path is plotted for several values of $\epsilon$. It is clear from Fig. 1a
that the path predicted for the discretely coupled network ($\epsilon=1$) is
quite different from the globally coupled disease dynamics ($\epsilon=0$).
Note that as $\epsilon$ increases and infection spreads primarily along the
network, II connections become more prominent because infected nodes arise
next to other infected nodes. Also, as shown in Fig. 1b, the action decreases,
corresponding to a decrease in the extinction time, presumably due to
stochastic effects in local neighborhoods of nodes.

Figure 1: (a) The most probable path projected onto the $I/N$ versus $L_{II}$
axis for various $\epsilon$ values up to $\epsilon=1$. (b) The action
predicted by the IAMM along the most probable path as a function of the
continuation parameter $\epsilon$. Parameters: $p=1.03\times 10^{-4}$,
$r=0.002$ and $K/N=10$.

To validate the predicted paths and extinction times, we compare the numerical
results for our approximate system to Monte Carlo simulations of an SIS
epidemic model on an Erdős-Rényi random network. Here, the Monte Carlo
simulations are completed using the Gillespie algorithm [31] with an initial
condition at the endemic steady state. For the networks, 1000 trajectories are
run to extinction, 100 for each of 10 randomly generated network geometries.
From the set of paths that go extinct, a density function is created from the
prehistory of these paths, and a clear local maximum can be identified. This
maximum corresponds to the most likely trajectory connecting the endemic and
extinct points, and is shown in the density plots of Fig. 2. Using the IAMM to
compute the most probable path and comparing it to the prehistory of
extinction events on stochastic networks, as shown by the dashed curve in Fig.
2, demonstrates that our proxy model approximates well the path to extinction.

Figure 2: (a) The position density function of extinction paths computed from
stochastic simulations on Erdős-Rényi random networks, projected onto the
$I/N$ versus $L_{II}$ axis, and overlaid with the predicted most probable path
(dashed curve). (b) The same graph as (a) but projected onto $I/N$ versus
$L_{SI}$. Parameters: $p=1.03\times 10^{-4}$, $r=0.002$, $N=10^{4}$,
$K=10^{5}$, and $\epsilon=1$.

From Monte-Carlo simulations, we can also approximate the mean lifetime of the
disease from endemic state until extinction. We assume that the mean lifetime
$\tau$ is inversely proportional to the probability of the event, and thus can
be related to our action by the formula $\tau=B(\boldsymbol{x})e^{NR}$. Note
the appearance of the prefactor $B(\boldsymbol{x})$ in this calculation. The
prefactor may depend on all parameters for a given problem, but in general
scales as $\frac{1}{\sqrt{N}}$ for sufficiently large $N$. For our purposes,
the prefactor can be found analytically for $\epsilon=0$,

$B=\frac{\sqrt{2\pi\frac{R_{0}^{\text{eff}}}{N}}}{r(R_{0}^{\text{eff}}-1)^{2}},$
(6)

where the effective reproductive number is given as
$R_{0}^{\text{eff}}=2pK/(Nr)$ [32, 16].

In Fig. 3, we plot the log of the mean lifetimes of the disease compared to
our numerical predictions of the action, incorporating the prefactor in Eq. 6.
Because it is analytically intractible to find the prefactor for $\epsilon>0$,
we assume that, since the action does not vary greatly with respect to
$\epsilon$, the change to the prefactor will be negligible, and thus we can
use the same prefactor for the case when $\epsilon=1$.

Figure 3a shows the log of the mean lifetime versus population for Monte Carlo
simulations of the disease on discrete networks ($\epsilon=1$) and the mean-
field prediction generated by the IAMM, and scaled by the analytical
prefactor, i.e., $\ln{\tau}/N\approx R+\ln{B}/N$. Because the action $R$ is
invariant with respect to $N$, the scaling depends entirely on the prefactor,
and the good agreement between our analytical approximation and stochastic
simulations shows that our approximation of the prefactor is sufficient to
capture the dynamics. Figure 3b varies a different parameter, the infection
rate $p$, and shows the lifetime scaling predicted by our model and the
stochastic simulations as the probability of disease propagation along the
network is increased.

Figure 3: a) $\ln(\tau)/N$ versus $N$ predictions (solid curve) compared to
mean over 1000 realizations of extinction on a random network (circles) for
$r=0.002$, $p=1.03\times 10^{-4}$, $K/N=10$, and $\epsilon=1$. b)
$\ln(\tau)/N$ versus $p$ predictions (solid curve) compared to mean over 1000
network extinction events (circles) for $r=0.002$, $K=10^{5}$, $N=10^{4}$, and
$\epsilon=1$. Note there is no fitting parameter in the theoretical plots.

We have presented a method for predicting extinction in stochastic network
systems by analyzing a pair-based proxy model. Both extinction times and paths
to extinction were obtained. Tracking the path to extinction was aided by
perturbing from the known path in a well-mixed system. In the future, the
pair-based proxy method will be extended to systems such as epidemics on
adaptive networks (e.g., [33]) by adding network adaptation to the list of
transitions (Eqs. 2 and 3). Vaccination transitions can also be added, as in
[22]. More generally, this pair-based proxy method will be applicable to
predict extinction in any network system that is well described by a pair-
based approximation for dynamics of nodes and links, including games on
networks (e.g., [34, 35]). Further, we expect that our method of continuously
varying a parameter while tracking the path to extinction will be useful in
other contexts where finding the path a priori is difficult due to high
dimensionality, as in finite mode projection of partial differential equations
[36], and pattern switching along paths in continuous systems [37].

BL was a National Research Council post doctoral fellow. IBS was supported by
NRL base funding (N0001414WX00023) and Office of Naval Research
(N0001414WX20610). LBS was supported by the Army Research Office, Air Force
Office of Scientific Research, and by Award Number R01GM090204 from the
National Institute of General Medical Sciences.

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