# Beyond 5G Domainless Network Operation enabled by Multiband: Toward Optical
Continuum Architectures

Óscar González de Dios
Telefonica I+D, Spain
<EMAIL_ADDRESS>
Ramon Casellas
CTTC/CERCA, Spain
<EMAIL_ADDRESS>
Filippo Cugini
CNIT, Italy
<EMAIL_ADDRESS>
José Alberto Hernández
Universidad Carlos III de Madrid, Spain
<EMAIL_ADDRESS>

###### Abstract

Both public and private innovation projects are targeting the design,
prototyping and demonstration of a novel end-to-end integrated packet-optical
transport architecture based on Multi-Band (MB) optical transmission and
switching networks. Essentially, MB is expected to be the next technological
evolution to deal with the traffic demand and service requirements of 5G
mobile networks, and beyond, in the most cost-effective manner. Thanks to MB
transmission, classical telco architectures segmented into hierarchical levels
and domains can move forward toward an optical network continuum, where edge
access nodes are all-optically interconnected with top-hierarchical nodes,
interfacing Content Delivery Networks (CDN) and Internet Exchange Points
(IXP). This article overviews the technological challenges and innovation
requirements to enable such an architectural shift of telco networks both from
a data and control & management planes.

_K_ eywords 5G $\cdot$ Optical Networks $\cdot$ Multiband $\cdot$ Optical
Continuum

##  1 Introduction

While mobile traffic currently accounts for approximately 9% of the total IP
traffic generated by the Internet, it is estimated that the upcoming massive
deployment of 5G/IoT devices and services will double this value by the end of
2023, reaching 20% of total IP traffic [1]. Such a tremendous increase of
connected devices and traffic is expected to push the underlying optical
network infrastructure beyond its current limits. In fact, during the COVID-19
pandemic, many telecommunication operators (aka telcos) have reported extra
traffic increase rates of 50% [3].

Besides, today’s 5G transport is IP-centric supported by several optical
networks spanning different domains (backbone, regional, metro, aggregation,
access, etc). The photonic layer, often built with (Reconfigurable) Optical
Add/Drop Multiplexers (ROADM) and Optical Line Terminals (OLT), is often used
to interconnect IP routers among different domains in a hierarchical
architecture of optically isolated networks, namely access, aggregation, metro
and core. Indeed, electronic intermediate termination is currently present at
the boundaries between network domains.

Also, the traffic patterns are changing, and the network needs to migrate
toward a data-center cloud ready infrastructure since the large majority of
the traffic is no longer directed towards the IXP and Internet, but instead
toward the caches in CDN scenarios.

Innovation in optical networking solutions are required to provide sufficient
and sustainable capacity capable of dealing not only with the expected
increase in traffic and users/things, but also with the traffic pattern shifts
expected in the next decade.

One promising approach to increase bandwidth capacity without changing the
optical infrastructure relies on Multi-Band (MB) transmission. Indeed, a
number of research efforts, both from public and private funding, have
realized the need to invest on MB networks as a way to squeeze the capacity of
the already existing optical infrastructure, before deploying new fiber, as a
cost-effective solution.

MB expands the available capacity of optical fibres, by enabling transmission
within S, E, and O bands, in addition to commercially available C and/or C+L
bands. The total bandwidth available in MB is around 53 THz, which translates
into more than 10x transmission capacity increase with respect to the C-band
only.

To realize MB networks, technology advances are required, both in the data and
the control and management planes. The devices needed for MB data plane
include new optical amplifiers, filter-less sub-systems, add/drop
multiplexers, ROADMs, and transmission systems. These technology advances
complement novel packet-optical white boxes using flexible MB sliceable
bandwidth/bitrate variable transceivers (BVTs) and novel pluggable optics
(e.g. of type Point-to-Multi-Point - PtMP such as XR, or 400 ZR/ZR+). The
availability of MB transmission will also lead to a complete redesign of the
end-to-end architecture, removing boundaries between network domains, reducing
electronic intermediate terminations, and providing a network continuum
between X-haul/access, aggregation, metro, and core segments.

The control plane needs to be significantly extended to support MB elements
and a ‘domain-less’ network architecture. The new control plane will
extensively rely on physical layer abstraction, new MB impairment modelling,
and pervasive telemetry data collection to feed AI/ML algorithms that will
thus lead to a Zero-Touch Networking (ZTN) paradigm for MB networks.
Ultimately this will result into a new, open network operating system deployed
at the packet-optical white box.

This article attempts to shed light into the question on whether or not a new
MB network infrastructure may allow telco architectures move toward an optical
network continuum, where domain boundaries are removed and access nodes
(central offices and mobile base stations) are all-optically interconnected to
core and cache nodes bypassing classical electronic boundaries and aggregation
stages. In addition, it shows what challenges and requirements are foreseen to
fully realize such a radical architecture shift.

The remainder of this work is then organised as follows: Section 2 is devoted
to explain the telco architecture shift toward a domainless architecture.
Section 3 overviews the main challenges in the data plane while section 4 does
the same for the control and management planes. Finally, sections 5 and 6 show
some preliminary assessment justifying the potential benefits of such a
transition and the need to further investigate the challenges highlighted in
this paper. Section 7 concludes this work with its main findings and
conclusions.

##  2 Telco hierarchical architecture

Today’s network architecture is built as a combination of IP/MPLS and optical
WDM transport technologies, featuring a hierarchical structure spanning
multiple domains, namely access, aggregation, metro and core domains (see Fig.
1). These network domains span country-wide connectivity, including several
metro/regional networks interconnected at city level and rural networks to
reach small villages, both for fixed and mobile traffic.

Figure 1: Hierarchical and domain-segment architecture of telco operators

In a classical telco architecture, several Hierarchical Levels (HL) are often
encountered with the following characteristics [12]:

  * •

HL5 typically comprise radio access points and mobile base stations, which are
connected to cell site gateways, where traffic is encapsulated in L3VPNs to
reach the packet core.

  * •

HL4 nodes (SWS nodes in Fig. 1) mostly comprise residential access, typically
DSLAMs aggregating xDSL traffic and OLTs aggregating FTTx households.

  * •

HL3 (SWD nodes in Fig. 1) comprise a first metro node which collects the
traffic from HL4 and HL5 nodes and performs IP grooming toward the next level.

  * •

HL2 nodes (SWC nodes in Fig. 1) often host MAN services for content
distribution, such as CDN caching and IPTV services.

  * •

HL1 nodes (PE nodes in Fig. 1) comprise interconnection to the WAN.

The role and network size of each node is also summarised in Fig. 2, where a
simplified hierarchical architecture is shown.

Figure 2: Role and size of hierarchical levels

Such high number of domains makes network operations complex and requires
multiple O/E and E/O conversions and the traversal of intermediate IP routers.
In addition to this, telco networks are evolving toward a data-center centric
network, where most of the end-user traffic is terminated at the CDNs and OTT
caches (Google, Facebook, Netflix, etc) [12]. The rise of datacenters spread
across the telcos are growing exponentially, and the edge computing paradigm
advocates for introducing many more small micro-data centers closer to the
users, on attempts to provide ultra-low latency and improved user experience
[9].

In spite of the introduction of ever-faster high-capacity channels (400 Gb/s
and upcoming 800 Gb/s), the network infrastructure remains static at the
switching level, where packets need to traverse multiple domains (IP and
optical) and still perform several IP routing hops (typically between 4 and 6
hops, according to Telefonica’s estimates).

In this sense, multi-band networks provide a good opportunity to move from
this rigid domain-based architecture into a flexible domainless architecture
as shown in Fig. 3 where HL4 nodes directly interconnect with HL2 or HL1 nodes
bypassing intermediate HL3 nodes. In order to create a sustainable network
continuum from access to core and DC, current rigid boundaries need to be
broken. This optical continuum is expected to drastically reduce O/E/O needs
and will allow to provide differentiated transparent connectivity services
across network segments.

To provide such comprehensive and ambitious approach, a number of research
innovations in both the data and control & management planes are required,
namely:

  * •

A MB optical transport network infrastructure, based on specifically designed
MB node architectures, MB switching, MB amplification, and MB transmission
solutions, cost-effectively addressing the 10x bandwidth increase, providing
open and fast programmability, and enabling an end-to-end continuum that
drastically reduces electronic regeneration across network segments.

  * •

A MB low-latency optical X-haul infrastructure exploiting pluggable-based PtMP
heterogeneous solutions to enable massive Beyond-5G (B5G) and small cell
deployments and their integration within the transport infrastructure and the
support of QoS-guaranteed services (e.g., in case of mobility).

  * •

A modular zero-touch control plane, able to orchestrate services across the
end-to-end underlying infrastructure, jointly optimizing the packet and MB
optical layers, while ensuring scalability and reliability, targeting zero-
touch operations and relying on cluster-based distributed systems that
synchronize state and exchange Machine Learning (ML) models.

The next sections further elaborate on such required innovations in both the
data and control & management planes toward an all-optical MB-based network
continuum.

##  3 Multi-band data plane: challenges from an Operator’s perspective

Fig. 3 shows the proposed innovative data plane infrastructure based on MB
technology to provide a network continuum from 5G access to data centers, with
differentiated per-band optical transparency and services. Connectivity
services over short distances will exploit the shorter wavelength range, where
higher fibre attenuation is experienced and amplification systems may not be
cost-effective [4], adopting a pay-as-you-grow strategy. As shown in [15] a
differentiated use of bands provides remarkable benefits to increase capacity
while mitigating the complexity of additional hardware subsystems. For
example, channels in the S band (yellow lines in Fig. 3) may be present on
some links only, transparently crossing some intermediate nodes and domain
boundaries, and being able to be added/dropped only in a few selected nodes.
This way, seamless capacity scaling of the end-to-end network might be
achieved, addressing the scaling disparities in the various optical network
segments (access, aggregation, metro) while reducing the need for O/E/O
regeneration at domain borders.

MB will leverage on 365 nm spectrum resources, from 1260 to 1625 nm, i.e. 53.4
THz, more than 12-fold the C-band spectral range. The orange area in Fig. 3
(right) depicts the range of the currently used C-band.

Figure 3: MB network architecture with per-band differentiated reach and
services (left); MB switching/ROADMs (bottom); MB considered spectrum with
measured loss of G.652D fibre (right)

To fully take advantage of MB resources in Telco networks and efficiently
implement the aforementioned data plane infrastructure with differentiated
per-band optical transparency, a number of challenges need to be specifically
addressed at the data plane level.

First, new cost-effective devices need to become commercially available. These
include amplifiers, transmission modules, and optical switching elements.

MB amplification has been demonstrated for transmission bandwidths up to
around 100 nm. In most cases, hybrid amplification leveraging on EDFA, Raman,
and Semiconductor Optical Amplifiers (SOA) were used. In [5], the maximum
experimented Raman amplification bandwidth was 83 nm. Such demonstration was
done without exploiting mature control techniques (e.g., no gain control).
Novel S-band optical amplifiers (Thulium-Doped Fiber Amplifier - TDFA) are
nowadays commercially available at a cost significantly higher than EDFAs and
with lower performance (e.g., non-flat noise figure of around 7 to 8 dB).

Different transceiver solutions have been proposed in the literature towards
supporting the fast evolution of current optical networks. In this context,
point-to-multipoint coherent transceivers, also called sliceable bandwidth
variable transceivers (S-BVTs), have assumed an especially relevant role in
the research community during the last years, following the roadmap of optical
networks. Current S-BVTs consist of a set of (white box) BVTs that transmit
multiple slices/signals within the C band. The S-BVT enables disaggregation of
the different optical components (BVTs) and also of hardware (HW) from
software (SW) [14]. Several transceiver designs and configurations have been
presented in the literature targeting different network segments. Many digital
signal processing (DSP) and front-end transceiver options have been covered,
also including approaches relaying on photonic technologies, according to the
specific target application [13]. Transmitter/receiver modules on different
bands have been so far designed mainly for low cost and short reach access
solutions. That is, significant improvements are needed to make them cost-
effective for medium/long term and dense multi-band WDM.

In terms of MB node solution, only prototypes at the research level have been
demonstrated including some switching capabilities or wavelength blockers
operating across multiple bands (e.g. [11]). Significant work is needed to
efficiently implement node solutions enabling differentiated per-band optical
switching and bypass or blocking.

Besides data plane components and system solutions, advanced MB modeling needs
to be efficiently implemented for the design of production-ready MB networks
as well as to be encompassed within enhanced spectrum assignment algorithms,
accounting for target reach and detailed in/cross-band impairments. Concerning
transmission impairments like amplified spontaneous emission (ASE) and four-
wave mixing (FWM), the use of MB networks notably changes the non-linearities,
such that further extensions are required. The impact of these impairments in
network capacity is assessed by means of Routing, Modulation and Spectrum
Assignment (RMSA) algorithms [19], albeit the studies are addressing C-band
only. Moreover, ML techniques have been proposed to improve the accuracy of
these estimations [17] as an independent approach.

##  4 Zero-touch Control and Service Orchestration

The control plane represents the intelligence of a network; in this sense,
next-gen control planes aim at the design and development of zero-touch
control and self-management of the underlying end-to-end data plane
infrastructure and for the dynamic provisioning, management and orchestration
of services, relying on dynamic planning covering the joint orchestration of
IT and Network equipment. At present, network operators deploy optical nodes
provided as an end-to-end closed solution, with proprietary device
configuration interfaces. However, there is a trend to change this status-quo
toward open optical networks, which require: (1) interoperability through
standard application programming interfaces and (2) SDN Control Plane
capabilities, based on standard models, to glue multivendor solutions [6].

###  4.1 Innovation on control of MB transport infrastructure

The control plane needs to be significantly extended to support MB elements
and a ‘domain-less’ network architecture. New open and standard YANG models
and interfaces for the forthcoming MB devices (i.e. ROADMs, amplifiers,
pluggables and S-BVTs) need to be designed and validated. The optical MB
infrastructure needs to be seamlessly integrated with the packet layer, thanks
to the removal of standalone transponders and the extensive use of packet-
optical white boxes, equipped with pluggable modules, P4-based packet
forwarding programmability [16], and embedded monitoring and self-
diagnosis/adaptation capabilities based on local control loops. To this
extent, new open packet-opto operating systems are necessary [2].

Specific procedures that enable self-management, self-diagnosis, and self-
optimization through adaptation procedures of such devices are required. These
include automatic inventory management, MB impairment-aware path computation,
MB topology abstractions, and automatic identification of candidate pre-
validated configurations. All these procedures need to be effectively
supported by AI/ML and telemetry-based monitoring (with scenario dependent
volume and rate of data) providing predictive capabilities for innovative SDN-
based autonomous and ZTN.

###  4.2 Innovation in Open Packet-Opto Operating System

Fig. 4 shows the architecture of a possible example of white box open
operating system, relying on emerging initiatives from open hardware. A
disaggregated bare metal switch with coherent pluggable interfaces and FPGA-
controlled novel MB transmission prototypes can be abstracted via a base OS
(e.g. Open Networking Linux - ONL).

Then, specifically designed Transceiver Abstraction Interfaces (TAI) have to
be developed to support novel pluggable modules for MB, PtMP digital
subcarrier multiplexed (DSCM), and OLT modules. Self-adaptation of transceiver
parameters (e.g. auto-tunability, operational mode adaptation, etc.) will have
to be efficiently supported, also relying on finite state machines for
proactive enforcement. At the upper layer, extensive monitoring will be
needed, including optical parameters collected via TAI, metadata from the P4
pipeline, and in-band telemetry. Furthermore, AI-based local elaborations with
feature extraction at the data plane will feed a novel module providing local
control loops and autonomous adaptations of the whole node functionalities
(e.g. fast recovery/rescheduling of low latency flows toward different
bands/pluggables). The traditional local SDN agent will have to be extended to
provide both optical and packet continuous interaction with the new SDN
domain-less control plane. The OS should rely on existing modules that are
part e.g. of the SONiC OS, a reliable platform supporting, besides legacy
protocols (e.g. Border Gateway Protocol - BGP), effective integration with the
Kubernetes platform. The integrated solution will create a powerful open
packet-optical operating system with large potential impact, specifically
tailored to meet network operators’ needs.

Figure 4: Open packet-opto operating system (in orange the novel SW
components).

###  4.3 Innovations in SDN controller architecture

SDN controllers tend to be monolithic and cover mainly a single optical
domain. However, there is a clear need to design and develop new controller
architectures (e.g. using a micro-services approach combining multiple
functional elements); to consider new control plane deployment models (where
such functional elements are deployed as virtualized functions or containers);
to tightly integrate configuration, control, and monitoring and to address
scalability issues by deploying multiple controllers, whose arrangement is
dynamic, involving high volume communications and new interfaces between
controllers. Gigabit/s data links supporting control plane communications are
becoming available and this allows improved distributed systems, fast state
synchronization between controllers, huge volume monitoring, and real time
telemetry and closed control loops based on AI/ML. This is realized via
collaborative knowledge-management based on data and model exchange among node
agents, as well as between the network controller and node agents encompassing
centralized vs distributed vs coordinated autonomous decision making. Modern
SDN architectures shall also address infrastructure sharing and network
virtualization/slicing. Indeed, MB is an opportunity for efficient network
sharing at the optical layer, including the virtualization of access networks
(vPON, vOLT) towards the B5G Radio Access Network (RAN), along with new
business models related to multi-tenancy.

###  4.4 Innovations regarding end-to-end orchestration

As stated, there is a need to address multi-domain SDN to support end-to-end
orchestration of highly dynamic services, defining architectures that span
dynamic network segments that today correspond to the access, metro and part
of the core and allow seamless integration with the RAN, the integration with
computing domains and the "federation" of operations and vendor islands, which
are heterogeneous in nature.

Dedicated Network Planning subsystems and framework will assist the e2e
orchestration and the SDN controllers in the decisions regarding placement and
networking resource allocation divided in two blocks: In-Operation
Provisioning assists the major components in the Infrastructure Control and
Service Orchestration platform in the form of decision-making for resource
allocation both in terms of IT and networking infrastructure

##  5 Potential impact and benefits for Telco networks

This section aims to provide a preliminary comparison in terms of potential
benefits for telco architectures based on MB networks with respect to existing
solutions.

As a baseline for comparison, we assume a classical telco architecture using N
parallel fiber systems exploiting the C-band only, allowing 40 channels at 400
Gb/s per channel in a hierarchical architecture, where traffic is originated
at the HL4s and directed toward the HL1/2 nodes, using HL3 nodes as
intermediate nodes where IP grooming is performed (see section 2) with $\eta$
oversusbscription ratio (typically $\eta$ is 1/2 or 1/3 at the metro/core, and
1/10 in the access) [18, 10].

In this benchmark scenario, let us consider a hierarchical architecture where
the lower, intermediate and upper hierarchical layers comprise $H_{4}$,
$H_{3}$ and $H_{1/2}$ nodes respectively. In this setting, traffic is sourced
from layers HL4 and HL3 and directed toward the closest HL1/2 node; in
addition, intermediate HL3 nodes collect and aggregate traffic from a number
of source HL4 nodes and perform grooming at the IP layer and forward this
traffic to the closest HL1/2 node, using some over-subscription parameter
$\eta$ at the grooming stage. In this setting, the number of 400G transponders
needed at the HL4, HL3 and HL1/2 nodes to satisfy the traffic sourced at the
HL4 nodes are:

$\displaystyle HL4\quad N_{tx,HL4}$ $\displaystyle=$
$\displaystyle\Big{\lceil}\frac{A_{4}}{400G}\Big{\rceil}H_{4}$ (1)
$\displaystyle HL3\quad N_{tx,HL3}$ $\displaystyle=$
$\displaystyle\Big{\lceil}\frac{A_{4}}{400G}\Big{\rceil}H_{4}+\Big{\lceil}\frac{H_{4}}{H_{3}}\frac{\eta
A_{4}}{400G}\Big{\rceil}H_{3}$ (2) $\displaystyle HL12\quad N_{tx,HL12}$
$\displaystyle=$ $\displaystyle\Big{\lceil}\frac{H_{4}}{H_{3}}\frac{\eta
A_{4}}{400G}\Big{\rceil}H_{3}$ (3)

where $A_{4}$ refers to the average source traffic originated per HL4 node.
Also, the intermediate HL3 layer assumes $1/2$ over-subscription ratio. In
total, the number of 400G transponders needed in this scenario can be
approximated as:

$N_{tx,IPoWDMgrooming}\approx(1+2\eta)\cdot\frac{A_{4}}{400G}H_{4}$ (4)

In an optical-continuum scenario where intermediate HL3 nodes are avoided, the
number of 400G transponders required to interconnect HL4s directly to HL1/2s
are:

$\displaystyle HL4\quad N_{tx,HL4}$ $\displaystyle=$
$\displaystyle\Big{\lceil}\frac{A_{4}}{400G}\Big{\rceil}H_{4}$ (5)
$\displaystyle HL12\quad N_{tx,HL12}$ $\displaystyle=$
$\displaystyle\Big{\lceil}\frac{A_{4}}{400G}\Big{\rceil}H_{4}$ (6)

for the same $A_{4}$ offered traffic, thus yielding a total of approximately:

$N_{tx,IPoWDMcont}\approx 2\cdot\frac{A_{4}}{400G}H_{4}$ (7)

As an example, consider a large MAN with 200 HL4 nodes, 40 intermediate HL3
nodes and 5 HL1/2 nodes providing connectivity with CDN contents and IXP, and
further assume a traffic demand below 400G per HL4 node, say 300G peak offered
traffic. This amount corresponds to an HL4 node serving 10,000
customers/things (both fixed and mobile) offering approximately 30 Mb/s
offered traffic per customer/thing, in line with the estimates of [8].

In this scenario, each HL4 node has got one 400G transponder to its
intermediate HL3; the intermediate HL3 nodes have got another transponder to
each HL4. In addtion, there is one HL3 per 5 HL4 nodes, thus aggregating the
traffic of $5\times 300G=1,500G$ but, thanks to the $\eta=0.5$ over-
subscription ratio, the HL3 needs to provision capacity to only $750G$ toward
the upper HL1/2 node, i.e. two 400G transponders, making a total of seven 400G
transponders per HL3 node (5 in the downstream direction toward 5 HL4 nodes
plus another 2 toward its HL1/2 node). Finally, the HL1/2 nodes need a 1-to-1
mapping of transponders to the HL3, thus, the total number of transponders per
hierarchical layer is:

  * •

At HL4, 1 transponder per node times 200 nodes = 200 transponders

  * •

At HL3, 5x40=200 transponders interfacing the HL4s plus another 2x40 = 80
transponders interfacing the HL12, for a total of 280 transponders.

  * •

At HL12, 5x16=80 transponders interfacing the HL3s

As shown, the total number of 400G transponders required in this scenario is
560 transponders. In the optical continuum scenario avoiding intermediate HL3
nodes, the number of transponders needed are 200 in the HL4 layer plus another
200 in the HL1/2 layer, accounting for a total of 400 transponders. This
implies a total saving of 28.5% transponders with respect to the classical
layered architecture approach.

In addition to providing a faster direct all-optical connectivity, another
advantage of this architecture is the removal of expensive high-capacity IP
routers at intermediate HL3 nodes. Following the cost-estimates of [7] and
updated in [9], where the cost of a 400G transponder is 12 normalised Cost
Units (CU) and a Large-size IP router costs 64 Cost Units, then the total
savings including transponders and intermediate IP Routers could rise up to
35-40% approximately, since the layered architecture accounts for $580\times
12+20\times 60=7728~{}CU$ while the optical continuum is only $400\times
12=4800~{}CU$.

Although more detailed and extensive techno-economic analysis needs to be
conducted, the above example shows the potential cost-savings of employing IP
offloading of intermediate HL3 nodes, by interconnecting directly HL4 nodes
with top-level HL1/2. Obviously, this approach requires more wavelengths
available per fiber, which is only possible in MB networks (900 wavelengths MB
vs only 80 wavelengths in C-band only [4]).

##  6 MB pluggable S-BVT

This section provides a preliminary assessment of the potential benefits
achievable by the introduction of PtMP coherent pluggable modules supporting
MB transmission. Fig. 5 shows three reference solutions to enable any-to-any
connectivity at the packet level.

Figure 5: Connectivity solutions based on: direct interconnection in full mesh
(left); hierarchical electronic grooming (center); proposed direct
interconnection through point-to-multipoint pluggable transceivers over MB in
packet-optical nodes.

Fig. 5 (left) shows a traditional approach based on direct connectivity among
the considered $N$ network nodes. In the figure, node $A$ has a dedicated
interface and a dedicated direct link (i.e. optical channel) toward any other
node of the considered network. This approach suffers from two main
scalability problems already for relatively limited values of $N$. First, it
leads to a number N-1 of interfaces per router which is impractical to
support. Second, the number of required channels for a full mesh optical
interconnection rapidly exhausts the traditionally available spectrum in the C
band.

For these two reasons, the traditional approach to enable any-to-any
connectivity at the IP level leverages on a hierarchical infrastructure, shown
in Fig. 5 (center). In this exemplary case, node $A$ has a single interface
and a single optical channel towards a transit router where traffic grooming
is performed (yellow node in the figure). The transit router belongs to a
hierarchical backbone which assures, through possible multi-hop
interconnection, the required full mesh logical connectivity. This approach
can be implemented guaranteeing limited use of interfaces per node and limited
number of channels in the network, being able to cope with both node and
C-band available resource. However, this approach is not free from
limitations. The backbone routers are expensive and power-hungry electronic
nodes. Furthermore, the end-to-end latency is increased due to the presence of
intermediate hops and queuing stages, potentially subject to congestion
issues.

Fig. 5 (right) shows the proposed approach leveraging on novel PtMP coherent
pluggable modules (i.e. pluggable S-BVT) operating in MB range and equipped
within routers as packet-optical nodes. Such PtMP interfaces enable the
implementation of direct any-to-any connectivity without requiring N-1
interfaces in the node. For example, a 400 Gb/s pluggable module could be
implemented as a sliceable 4x100Gb/s solution towards 4 different nodes as in
the figure. The second potential limitation in terms of number of required
channels is solved by the availability of spectrum resources enabled by
multiple bands. The presence of intermediate electronic grooming would be
removed or strongly reduced leveraging on transparent optical
switching/splitting solutions (gray nodes in the Fig. 5 right).

It is important to highlight that PtMP coherent pluggable modules are already
commercially available (e.g., XR Optics) although operating in the traditional
bands only and mainly targeting x-haul scenario. In this work, we envision the
evolution to operate over MB and across metro-regional distances.

A possible implementation of MB PtMP pluggable modules in the previously
considered benchmark scenario could consider to bypass HL3 nodes and
interconnect HL4 directly to HL1/2 nodes. The number of PtMP pluggable
following a 1:m fanout (i.e. 1:4 implies 4 x 100G) per layer follows:

$\displaystyle HL4\quad N_{tx,HL4}$ $\displaystyle=$
$\displaystyle\Big{\lceil}\frac{A_{4}}{400G/m}\Big{\rceil}H_{4}$ (8)
$\displaystyle HL12\quad N_{tx,HL12}$ $\displaystyle=$
$\displaystyle\Big{\lceil}\frac{A_{4}}{400G/m}\Big{\rceil}H_{4}\cdot\frac{1}{m}$
(9)

which, in total, the number of transceivers approximately follows:

$N_{tx,PtMP}\approx\frac{A_{4}}{400G/m}H_{4}\cdot\left(1+\frac{1}{m}\right)$
(10)

Following the example in the previous section, where each HL4 offers 300G
toward the closest HL1/2, we have that every HL1/2 concentrates the traffic of
200/5 = 40 HL4 nodes, i.e. 12,000G or 12T. In that case, the total number of
needed PtMP pluggables is 200 modules for the HL4 nodes plus another 12T/400G
= 30 modules per HL1/2, leading to a total of 200 + 5x30 = 350 PtMP modules.
This number is significantly smaller than the 580 needed in the IPoWDM with IP
grooming at the HL3 nodes, and also smaller than the 400 case in the optical
continuum case with fixed 400G transponders.

Also in this case, more detailed and extensive techno-economic analysis needs
to be conducted to accurately assess the potential of novel PtMP pluggable
modules over multi-bands.

##  7 Summary

This article overviews the concept of optical continuum architectures enabled
by multiband network technology. Indeed, emerging multiband techniques may
allow up to 900 wavelengths per optical fibre using the S, O, E, C and L
bands, thus providing a potential 10x increase in network capacity which can
be exploited to providing direct lightpaths from 5G access/x-haul nodes toward
the core and CDN nodes. This approach allows removing/reducing boundaries
between domains (access, aggregation, metro) and further limiting the number
of E/O and O/E terminations and IP routers at intermediate layers, with
subsequent cost savings and simplified network operations.

However, for this approach to become a reality, a number of technical
challenges in both the data and control and management planes need to be
addressed. Not only new MB devices need to be developed and matured, but also
the necessary open interfaces and APIs need to evolve to allow SDN-based zero-
touch open control. Doubtlessly, the next few years will witness a large
number of research articles addressing some or many of these aspects toward
multiband networks.

## Acknowledgements and Funding

The authors would like to thank all B5G-OPEN Partners for the fruitful
discussions and valuable contributions.

This work is partially funded through the European Commission B5G-OPEN project
(grant no. 101016663), see https://www.b5g-open.eu/, last access February
2023, for further details.

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# Scaling Features of Price-Volume Cross Correlation

Jamshid Ardalankia  Mohammad Osoolian<EMAIL_ADDRESS>Emmanuel
Haven<EMAIL_ADDRESS>G.Reza Jafari<EMAIL_ADDRESS>Department of Financial
Management, Shahid Beheshti University, G.C., Evin, Tehran, 19839, Iran
Center for Complex Networks and Social Datascience, Department of Physics,
Shahid Beheshti University, G.C., Evin, Tehran, 19839, Iran  Faculty of
Business Administration, Memorial University, St. John’s, Canada and IQSCS, UK
Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran, 19839,
Iran  Department of Network and Data Science, Central European University,
1051 Budapest, Hungary

###### Abstract

Price without transaction makes no sense. Trading volume authenticates its
corresponding price, so there exists mutual information and correlation
between price and trading volume. We are curious about fractal features of
this correlation and need to know how structures in different scales translate
information. To explore the influence of investment size (trading volume),
price-wise (gain/loss), and time-scale effects, we analyzed the price and
trading volume and their coupling by applying the MF-DXA method. Our results
imply that price, trading volume and price-volume coupling exhibit a power law
and are also multifractal. Meanwhile, considering developed markets, the
price-volume couplings are significantly negatively correlated. However, in
emerging markets, price has less of a contribution in price-volume coupling.
In emerging markets in comparison with the developed markets, trading volume
and price are more independent.

###### keywords:

Price-Volume Cross Correlation , Stock Market , Multifractal Behavior , MF-DFA

††journal: Physica A

##  1 Introduction

The beginning of a comprehensive analysis of markets requires studying the
markets’ features with the consideration of some sort of scale-wise and
magnitude-wise phenomena. This implies that the researcher needs to consider
scaling behaviors of financial times series during the analysis. This was
firstly done by Lux and Ausloos [1]. Also, some scholars investigated the
multifractal features [2, 3, 4, 5, 6, 7, 8] of stock markets [1, 9, 10, 11,
12, 13], foreign exchange markets [5, 14], commodity markets [15],
cryptocurrency markets [16], macro-economics time series [17], and financial
multifractal network analyses [18]. Regarding to other scientific fields, some
scholars have carried out fractal analysis on some phenomena in meteorology
[19, 20] , text structure analysis [21], hydrogeology [22], and astronomy
[23].

Among different variables in a market, the phenomenon which leads to price
discovery in an auction, is translation of trading volume. In this regard,
some scholars [24, 11, 25, 26, 27, 28, 29, 30, 31, 32, 33] investigated on the
existence of information translation among price and trading volume.
Nevertheless, high trading volume may help one to distinguish how much
tendency there exists to a certain price. Hence, if a certain price
corresponds to a relatively high trading volume, it can be an indication of
high price reliability [25], at that time, in the market. It may lead to small
spreads and also small liquidation volatilities [34] within that time-scale.
To clarify the importance of investigating price-volume information together,
Ausloos and Ivanova [24] combined classical technical analysis with
thermodynamic proxies. Considering that from a physics point of view, it is
doubtful to investigate price movements without price-volume dependencies,
they obtained some fruitful measures which can model market behavior in terms
of trends and amplitude of volatilities by price and trading volume time
series. Since trading volume is a touchstone of how traders work in the market
and translate information via their transactions, this phenomenon can shape
movements in a market.

To clarify the multifractal features of price-volume coupling, Podobnik et al.
[11] obtained a power-law in price-volume cross correlations. They
investigated logarithmic changes of price and its corresponding volume and
showed that just the cross correlation of absolute values of volatilities are
statistically significant. It is noteworthy to state that in the developed
markets, the entanglement of price return fluctuations and trading volume
[25], leads to a situation whereby an increase in volume shall –to some extent
as a turning point– increase price fluctuations. But then, a further increase
in trading volume leads to lowering the price fluctuation. Meanwhile, Guo et
al. [26] investigated the price-volume cross-correlation of commodities in
futures markets and compared their multifractal behaviors.

In reference to the theme of this current paper, Podobnik et al. [11]; Nasiri
et al. [25] and Guo et al. [26], found that price-volume cross volatilities,
may have scaling behavior at the levels of time and magnitude-scales. Since it
makes no sense to study price volatilities without considering the reliability
which a certain price owes to trading volume, we investigate the scaling
behavior of the cross correlation of price-volume volatilities. Different
scaling behavior toward events of different magnitudes, causes multifractal
behavior [5].

In order to reveal scaling behavior, one needs to differentiate large-scale
and small-scale patterns in a stock market as complex system [6, 3, 35, 36].
Since large-scale patterns (main trends) are somehow evident they do not
present much latent information [5]. Hence, initially it is necessary to apply
“detrended fluctuation analysis” (DFA) for non-stationary signals. See Peng et
al. [2] and also [6, 37, 38, 39, 40].

Because of psychological biases and irrational decisions of investors [41, 42,
43, 44, 45, 46]; out-of-market effects [41, 47]; uncertainty about the
transparency of fundamental analysis [42, 48]; the coexistence of collective
effects and noise [41, 49]; and the lagging diffusion of internal and external
information between several dynamics, a situation occurs where some
assumptions of the EMH (Efficient Market Hypothesis) become doubtful. In spite
of the complexity aspects in financial markets such as self-organizing (in
order to increase adaptability) and scaling patterns of nonlinear dynamics, we
observe power-law (scale invariance) behavior in some scales [41, 50].
Industrial, economic and political cycles which are dynamic and ever changing,
may cause a variability of statistical properties of time series in different
scales. This is what we call ‘non-stationarity’. These behaviors cause
multifractal correlations between time series. Systems which contain
persistent nonlinear interactions as inputs and outputs of the constituents,
may cause some simultaneous and some lagging effects to emerge. In the case of
persistent information, these effects cause long-range auto-correlation (for
one constituent) and long-range cross-correlation (between several
constituents) [35, 38, 51, 52].

Since real-world financial time-series may be non-stationary –possess some
trends– and may be limited in length, we need a method which considers
limited-length effects and non-stationarity effects [23]. The mentioned
methodology has been applied successfully in finance [1, 11, 53, 54, 55, 56,
57, 58, 17, 52] and has been developed to Fourier-DFA and MF-DFA [7].

Besides these references, Podobnik and Stanley [10] introduced detrended cross
correlation analysis (DXA) for studying two power-law non-stationary time
series. Then, Zhou [59] developed the DXA method to multifractal DXA (MF-DXA)
to investigate multifractal behaviors of two power-law non-stationary time
series. Moreover, to investigate coupling behavior among more than two non-
stationary power-law time series, some researchers introduced Coupling-DXA
method [5]. They believed that studying more than two signals in complex
systems leads researchers to better understanding the FOREX market structure,
and they showed that several (more than two) constituents have a scaling
coupling behavior. In the case of combining the MF-DXA within magnitude-wise
scales, several researchers [4, 60, 11, 61, 62] have done some efforts on
detrended covariance.

Also, some scholars specifically cast light onto the detrending methods in
financial markets. As a heuristic method, Caraiani and Haven [14] applied EMD
(Empirical Mode Decomposition) method on the detrending process of MF-DFA.
They investigated multifractality in the currencies and non-linearity of the
market. In this paper, we apply multifractal detrended cross correlation
analysis (MF-DXA) to investigate the scaling behavior of volatilities of
price-volume coupling in markets such as the DJIA, S&P500, TOPIX, TSE and
SSEC. It is shown that, the price and trading volume and their cross
correlations, contain multifractal behaviors. Moreover, the correlation
coefficients of their volatilities decrease with an increase of the time-
scale. In section 2, a review on methodology is presented. Then in section 3,
we analyze and describe the input data used in the methodology. In section 4,
the empirical results are discussed and then in section 5, we provide for the
conclusion.

##  2 Methodology

### MF-DXA

The general steps of MF-DXA methodology [1, 6, 2, 37, 38, 39, 40] are as
follows. Initially we convert the time series to standardized logarithmic
changes. After demeaning each data-point, we execute a cumulative summation
which is called profile series (Eq. 1).

$\displaystyle X_{(i)}$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{i}[x_{k}-<x>]$ $\displaystyle Y_{(i)}$
$\displaystyle=$ $\displaystyle\sum_{k=1}^{i}[y_{k}-<y>]$ $\displaystyle i$
$\displaystyle=$ $\displaystyle 1,2,....,N.$ (1)

Where $<...>$ is the average value of the time series; $N$ is the total
length; and $X_{(i)}$ and $Y_{(i)}$ are profile series. To investigate time-
scale effects on series, we consider several time-scale windows with the
length of $s$ which are iterated on the time series. Variable $N_{s}$ is
obtained by $int(\frac{N}{4})$ which provides the maximum number of segments
(with length of $s$) in each time series. Since the length $N$ may not be an
integer multiple of scale length $s$, we repeat the same iteration from ending
data point to starting one. It means that each non-overlapping scale window,
is allocated to sequential time locations without ignoring any data points
(Eq. 3). Consequently, the total number of time-wise scales would be $2N_{s}$.

In order to convert profile series to non-stationary type series, we should
eliminate the main trends (large-scale behaviors) from the time series. Among
lots of detrending methods (such as polynomial detrending, Fourier
detrending), it is better to start from the simplest one to avoid over-fitting
and also to avoid eliminating excessive amount of information. So in this
study, one-degree linear detrending is applied. As follows, detrending
covariance for each time-wise scale ($s$) and windows location ($\nu$) is
obtained by subtracting the local trend of each window. After detrending,
local deviation in each non-overlapping window is obtained.

$N_{s}=int(\frac{N}{s})$ (2)

$\displaystyle F_{(s,\nu)}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{s}\sum_{j=1}^{s}|X_{(\nu-1)s+j}^{(j)}-\tilde{X}_{\nu}^{(j)}||Y_{(\nu-1)s+j}^{(j)}-\tilde{Y}_{\nu}^{(j)}|$
$\displaystyle\nu$ $\displaystyle=$ $\displaystyle 1,2,..,N_{s}$
$\displaystyle F_{(s,\nu)}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{s}\sum_{j=1}^{s}|X_{N-(\nu-
N_{s})s+j}^{(j)}-\tilde{X}_{\nu}^{(j)}||Y_{N-(\nu-
N_{s})s+j}^{(j)}-\tilde{Y}_{\nu}^{(j)}|$ $\displaystyle\nu$ $\displaystyle=$
$\displaystyle N_{s}+1,N_{s}+2,...,2N_{s};$ (3)

where $\tilde{X}_{\nu}$ and $\tilde{Y}_{\nu}$ are local trends which are
fitted polynomials in each window $\nu$ with the local length of $s$.
For small (large) scale windows, the detrended covariance will be affected by
small and rapid (large and slow) fluctuations. Because of the relatively short
(long) scales, on average, there will be an increase (decrease) of local
effects in the detrended covariance function. Accordingly, small-scale (large-
scale) behaviors are observed significantly in small (large) windows. We can
further explain this by saying that monofractals are normally distributed [63]
and the volatilities can be explained just by the second statistical moment,
i.e. the ‘variance’. On the contrary, when it comes to multifractals, in large
(small) scales, local variations are excessively large (small). As a result,
to consider different behaviors of fluctuations, we magnify our concentration
from small and frequent, to large and rare fluctuations by weighting them.
Hence, for considering effects of events on magnitude scales, the parameter
$q$ is applied. When a big $q$ is considered, a high weight is applied to the
tails (rare and enormous events) of the logarithmic changes histogram.To be
unbiased toward small or large variations, $q=0$ is applied.

$F_{q}(s)=[\frac{1}{2N_{s}}\sum_{\nu=1}^{2N_{S}}{F^{2}}(s,\nu)^{q/2}]^{\frac{1}{q}}\qquad
q\neq 0.$
$F_{q}(s)=exp{\\{\frac{1}{4N_{s}}\sum_{\nu=1}^{2N_{s}}\ln[F^{2}(s,\nu)]\\}}\qquad
q=0.$

In the case of the second moment, $F_{q}(s)$ leads to
$\sigma=\sqrt{\sigma_{1}\sigma_{2}}$ [5, 10].
If the time series are scale-invariant and have long-range correlation, the
MF-DXA approach would contain a scaling behavior as follows:

$F_{q}(s)\sim s^{h_{xy}(q)},$

where $F_{q}(s)$ is a fluctuation function in order of $q$ and scale length of
$s$.
The slopes of $\log(F)-\log(s)$ are called the generalized Hurst exponent
$h(q)$. For each moment $q$, there is an $F(s)$. Whilst within a range of
time-scales, the slope of $log(F)-log(s)$ for various moment $q$ is constant,
the time series is scale-invariant. On the contrary, if there is a change in
the value of slope $h(q)$ for a single $q$, it is called a ‘cross-over’ and
the time series is now scale variant. For a range of $q$, a spectrum of $h(q)$
is obtained. The degree of multifractality [64] as a risk measure [15] is as
follows:

$\Delta h=h_{max}(q)-h_{min}(q).$

If $\Delta h=0$, the system is monofractal so the time series does not have
segments with extreme small and extreme large fluctuations. Hence, its
detrended covariance within a same time-scale window length, when powered by
$q$th-order in different windows location $\nu$, will yield no peak. On the
contrary, for $\Delta h\neq 0$ , the system is multifractal [65]. In this
case, the average value for residuals of local trends for different $s$ and
$\nu$, are not similar. So, large variations dominate the results for a
relatively large $q$, and small variations dominate the results for a
relatively small $q$.
It is worthy to state, that if the Hurst exponent $h^{Price}_{(q=2)}=0.5$, the
time series is a random walk. For a Hurst exponent $h^{Price}_{(q=2)}<0.5$
$(h^{Price}_{(q=2)}>0.5)$, the time series tends to be anti-persistent
(persistent) and is negatively-correlated (positively-correlated). Also, the
Hurst exponent values are as following: for the developed markets
$0<H_{Price}<0.5$, and for the emerging markets $0.5<H_{Price}<1$. Rényi’s
exponent (scaling exponent function) is as follows [4]:

$\tau_{xy}(q)=qh_{xy}(q)-1.$

If $\tau_{xy}(q)$ (which is a derivative of $h_{xy}(q)$) is a linear function
of $q$, the time series is monofractal. To examine the singularity content of
time series, the singularity spectrum is as follows:

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle
h_{xy}(q)+qh_{xy}^{\prime}(q)$ $\displaystyle f_{xy}(\alpha)$ $\displaystyle=$
$\displaystyle q(\alpha-h_{xy}(q))+1;$ (4)

where $\alpha$ is the singularity of a time series and $f_{xy}(\alpha)$ is the
multifractality spectrum. In addition, the multifractality strength is
observed by the singularity width $\Delta\alpha$, as below:

$\Delta\alpha=\alpha_{max}-\alpha_{min};$ (5)

where $\alpha_{max}$ relates to $q_{min}$ and $\alpha_{min}$ relates to
$q_{max}$ . If $\Delta\alpha=0$, the time series is monofractal and the
response of the cross correlation toward different events of $q$, in large and
small lengths of the time-scale is identical, and the multifractality spectrum
is just a point.
Until this section, the fluctuation function $F(s)$ for each $q$, is obtained
for the coupling of time series.

### MF-DFA

When applying the detrended covariance function for just a single time series,
the detrended variance function is presented. The rest of the methodology is
similar to the MF-DXA. This process needs to be applied for price and trading
volume separately.

### Correlation

The absence of any correlation leads to $\rho_{DXA}=0$. Based on Podobnik et
al. [51] this claim is just valid for an unlimited-length of time series. In
the case of limited length, the situation of $\rho_{DXA}\neq 0$, but no
correlation is probable. If the investigated time series have power law -as we
will show in the rest of this research- with the help of cross correlation
statistic [51, 57, 15], by the equation below we will show the scales which
the cross correlation coefficient is statistically significant:

$Q_{cc}(m)=N^{2}\sum_{i=1}^{m}\frac{X_{i}^{2}}{N-i},$ (6)

where $X_{i}$ is cross correlation function as follows:

$X_{i}=\frac{\sum_{k=i+1}^{N}x_{k}y_{k-i}}{\sqrt{\sum_{k=1}^{N}x_{k}^{2}}\sqrt{\sum_{k=1}^{N}y_{k}^{2}}};$
(7)

where $x_{k}$ and $y_{k}$ are detrended logarithmic changes of time series.
Since the cross correlation statistic $Q_{cc}(m)$ is somehow similar to the
$\chi^{2}(m)$ (Chi-squared) distribution, the critical value is measured by
$\chi^{2}(m)$. So, when $Q_{cc}(m)$ is less than the critical value (null
hypothesis) it means there is no significant cross correlation. On the
contrary, if $Q_{cc}(m)$ is more than the critical value (alternative
hypothesis), then there exists no reason to reject the existence of cross
correlation.
In order to reveal cross-correlation scaling behavior, we calculate the
correlation by DXA fluctuation as follows:

$\rho_{DXA}=\frac{F_{DXA_{price-
volume}}^{2}(s)}{F_{DFA_{price}}(s)F_{DFA_{volume}}(s)};$ (8)

where $\rho_{DXA}$ stands for the cross-correlation coefficient of price
fluctuation function and the trading volume fluctuation function and $s$ is
the time-scale length. $F_{DFA}$ is the fluctuation function of each time
series individually (price or trading volume) and $F_{DXA}$ is the fluctuation
function of cross-correlation of price-volume.

##  3 Empirical Data

In this study, we gathered the price index and trading volume of DJIA, S&P500,
TOPIX, TSE and SSEC markets during March $21^{st}$, 2013 until March $20^{th}$
,2018. This includes around 1300 trading days from which we can investigate
the multifractal behavior of time series volatilities and their related cross
correlations.

Figure 1: Descriptive statistics of investigated markets are demonstrated.

In Fig.1 top panel, the descriptive statistics for one of the markets is
shown. Also, as illustrated in Fig.1 bottom panel, none of the time series are
normally distributed. Hence, they are not monofractal [63]. Initially, after
calculating the logarithmic changes of time series, they were standardized.
Then

$\displaystyle r_{price}$ $\displaystyle=$
$\displaystyle\ln{P_{t}}-\ln{P_{t-1}}$ $\displaystyle r_{volume}$
$\displaystyle=$ $\displaystyle\ln{V_{t}}-\ln{V_{t-1}},$ (9)

where $t$ is daily data point, and also, $P$ and $V$ refer to price and
trading volume, respectively. The above mentioned results are used as inputs
to the methodology.

##  4 Empirical Results

### MF-DXA

After detrending in each non-overlapping segments, $\nu$, with length of $s$
for price, volume, and price-volume cross series, we shall in order to
estimate the power-law relation, compute $\log(F)-\log(s)$. This is
illustrated in Fig. 2 for the markets we investigated. In Fig. 3 and Fig. 4,
the multifractality features of price-volume coupling for the investigated
markets are presented.

Figure 2: $log(F)-log(s)$ for price index (left) and trading volume (middle)
and their cross correlation (right) are demonstrated. From top to bottom,
$log(F)-log(s)$ for TSE, SSEC, TOPIX, S&P500, DJIA, are shown.

Figure 3: Multifractality features such as Hurst exponent spectrum for price
index (left) and trading volume (middle) and their price-volume correlation
(right) are demonstrated. From top to bottom, multifractality features of TSE,
SSEC, TOPIX, S&P500, DJIA, are shown.

Figure 4: Top: Multifractality features of price-volume coupling: Hurst
exponent spectrum (left), singularity spectrum (middle) and scaling exponent
(right) are shown. Bottom: Multifractality features of price time series,
volume time series, and price-volume coupling are presented.

Altering a power law relation in different time-scales, means there exists
scale variance and it is an indication different power-laws are needed to
describe the corresponding time-series in different time-scales. On the other
hand, a system is scale invariant when it generates its structure in different
intervals. To obtain valid power-laws, proper scales are extracted which are
shown in Fig. 4. It is worthy to say that for window lengths near $N$, the
local trend of the segments become more similar to the whole time series
rather than the too small segments, and in this process $F(s)$ becomes
relatively independent of iterations on $\nu$.
Another reason for this phenomenon is the effect size of variations. Large
segments (small segments) contain large-scale (small-scale) behaviors of the
time series. Hence, applying the $q$th-order effect on the fluctuation
function, leads to a divergence of $\log(F)-\log(s)$ for small scales.
Nevertheless, for $s\ll N$ (local deviation of detrended segments are small on
relatively small scales), the fitting polynomial –a linear local trend in this
study– is well fitted on the segments and results in a magnification of the
divergence of $F(s)$ for different $q$ values.
The generalized Hurst exponent presents multifractality of correlations. Based
on Podobnik and Stanley [10], the Binomial measure from the $p$-model for the
cross-correlation exponent of the fractionally auto-regressive integrated
moving average (FARIMA) with identical stochastic noises, is equal to their
arithmetic mean at $q=2$. However, Zhou [59] illustrated that the same
relation for $q$ values other than 2, is valid [15] which is presented in red
color in the left panels in 3.

In Fig.3, price multifractalities of all investigated markets are larger than
volume multifractalities of the corresponding markets. If the time series is
multifractal and $q$ is extremely large for the system, it may yield to
$f_{(\alpha)}<0$ [66]. On the other hand, $f_{(\alpha)}<0$, may occur on
extremely large or small singularity spectra [9]. Since a negative dimension
is an unreal solution, this may warn us that the timeseries histogram is
spars– which is a characteristic of real-world financial timeseries due to the
fact that they may not be Guassian. Hence, those events near the tails of the
histogram increase and eliminate limited-length effects. In addition, the
right panels of Fig. 3, which relate to the nonlinearity of scaling exponents,
prove that price multifractalities occur more often than volume
mulifractalities of the markets we investigated. As a result, the
multifractality of price-volume coupling can be found to be between the
corresponding price and trading volume multifractalities.
As shown in Fig. 4, among the investigated markets, TSE has the least
multifractality degree of price-volume coupling. Conversely, for the developed
markets, the multifractality degrees of price-volume coupling are large than
that of emerging markets. However, considering the singularity spectrum, low
efficiency of the markets contributes to high singularity strength in price-
volume couplings. Furthermore, low efficiency in the markets coincides with a
scaling exponent of price-volume coupling with low curvature. It is notable
that, the price-volume coupling of the developed markets such as DJIA, S&P500
and TOPIX (with $H^{price}<0.5$), are highly affected by trading volume.
Hence, their price-volume couplings are not a random walk (on the contrary to
the TSE). What is noteworthy, is that the singularity spectrum for an emerging
market such as TSE (with $H^{price}>0.5$) is more left-hooked than that of the
developed ones. The price-volume cross correlation of an emerging market is
more led by some attributes other than volume of transactions (such as the
manipulation of the supply and demand at certain prices which causes markets
to be led more by price rather than trading volume!). This is proven by the
left panels of Fig.3.

### Cross Correlation Coefficient

Firstly, by applying Eq. 8 the scales with significant cross correlation
coefficients are distinguished, as shown in Fig. 5. After a statistical
confirmation of correlation significance, by applying Eq. 9, we evaluate the
scaling behavior of correlation coefficients of the investigated markets
throughout different time-scales, Fig. 5, bottom.

Figure 5: Top: Cross correlation statistics of the investigated market versus
critical value are shown. Bottom: Scaling behavior of correlation coefficients
of the investigated market are demonstrated.

We filtered the area with significant cross correlations in order to
investigate the cross correlation coefficients in the next step. Fig. 5
contains cross correlation statistics versus critical value, and Fig. 5,
bottom, shows the corresponding correlation coefficients in a multiscale
pattern–all for $h_{(q=2)}$– of price-volume coupling for investigated
markets. As shown, the cross correlation statistics are significant for the
studied range of time-scales. As illustrated in Fig. 5, bottom, the large
temporal segments size contribute to the small cross-correlation coefficients
and large fluctuation functions (such as ${DFA}_{price}$, ${DFA}_{volume}$,
${DXA}_{price-volume}$). When it comes to the large scales, the multifractal
behavior of fluctuation functions corresponding to price and volume, does not
present an extreme correlation and the scaling behavior of price and trading
volumes are totally different in relatively large scales. Hence, in large
scales, it makes no sense to describe price volatilities by trading volume
volatilities. It is noteworthy that, in Fig. 5, bottom, the price-volume cross
correlations of domains maintaining a power law in both price time series and
volume time series in each market, will show that the decrease of the
correlation coefficient co-occurs faster by an increase of time-scales in the
emerging markets such as TSE and SSEC rather than that of the developed ones.
Consequently, as shown in Fig. 5 and Fig. 5, bottom, by increasing time-scale,
the behavior of emerging markets deviate from the behavior of developed
markets. In the period of our study, it is obvious that among all investigated
markets, the scaling correlation coefficients of the developed markets
maintain to 0.4 to around 0.6. On the other hand, the scaling correlation
coefficients of the emerging markets maintain around less than 1 and
ultimately tend to somehow 0.2 to 0.35 in large time-scales. It means that the
decrease in scaling correlation coefficients of emerging markets is more
sensitive rather than that of the developed markets toward time-scale size.

##  5 Conclusion

In multifractal time series, we need to consider several statistical moments
to describe the behavior of the system. Therefore, investors need to execute a
multifractal analysis to be more familiar with the system’s behavior.
Nevertheless, in the aforementioned markets, the generalized Hurst exponent,
$h_{(q)}$, is a function of q. By illustrating the singularity spectrum, this
study shows nonlinear behavior of price, volume, and price-volume structure.
As a consequence, studying a single variable without considering simultaneous
collective effects and their cross effects, may be biased.

Since the Hurst exponent of price is larger than 0.5 ($h_{(q=2)}^{price}>0.5$)
for TSE, it is an emerging market with persistent behavior of price
volatilities. Conversely, because of $h_{(q=2)}^{price}<0.5$, DJIA, S&P500 and
TOPIX, are classified as developed markets with short timescale and anti-
persistent behavior. The investigated markets are affected by their memory.
Hence, the models which analyze these markets based on the Efficient Market
Hypothesis may no longer estimate the market behavior accurately. In an
inefficient market, the short timescale and large timescale phenomena exist in
price.
Long-range timescales leads to more predictability for price. Long-range
behaviors cause more persistent information effects and lessen the speed of
fading-out information from past dynamics. Furthermore, the trading volume
volatilities present negatively correlated behavior. Furthermore, the
multifractal behavior of volume DFA is less than that of price DFA.
Moreover, the cross correlation coefficients show scaling behaviors which are
totally significant in the investigated time-scales and decreases by
increasing time-scales.

Since for the price-volume coupling we have
$0.5>H^{TSE}>H^{SSEC}>H^{developed}$, it can be concluded that the price-
volume couplings for the investigated developed markets are significantly
negatively correlated and they possess significant valid information. As a
whole, in emerging markets, market behavior is guided more by a phenomenon
other than trading volume rather than just trading volume.

Multifractal volatilities in financial markets have become highly important in
risk management. Efficient risk management requires understanding of
information translation between couplings of stock markets structures and
internal and external dynamics. One of the most practical measures of this
study is applying the cross correlation matrix based on Laloux et al. [67] and
Plerou et al. [68] for several fluctuation functions to measure market risk.

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