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  <front>
    <journal-meta />
    <article-meta>
      <title-group>
        <article-title>Network routing optimization using Digital Twins</article-title>
      </title-group>
      <contrib-group>
        <contrib contrib-type="author">
          <string-name>Mohamed Zalat</string-name>
          <xref ref-type="aff" rid="aff0">0</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Chris Barber</string-name>
          <xref ref-type="aff" rid="aff1">1</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>David Krauss</string-name>
          <xref ref-type="aff" rid="aff1">1</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Babak Esfandiari</string-name>
          <xref ref-type="aff" rid="aff0">0</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Thomas Kunz</string-name>
          <xref ref-type="aff" rid="aff0">0</xref>
        </contrib>
        <aff id="aff0">
          <label>0</label>
          <institution>Carleton University</institution>
          ,
          <addr-line>Ottawa, Ontario</addr-line>
          ,
          <country country="CA">Canada</country>
        </aff>
        <aff id="aff1">
          <label>1</label>
          <institution>Ciena</institution>
          ,
          <addr-line>Hanover, Maryland</addr-line>
          ,
          <country country="US">USA</country>
        </aff>
      </contrib-group>
      <abstract>
        <p>In the context of network Trafic Engineering (TE), the Open Shortest Path First (OSPF) configuration of a communication network is optimized based on the maximum link utilization or the average link utilization of the network. In this paper, we introduce an OSPF Interior Gateway Protocol (IGP) weight optimization technique using a Digital Twin (DT) of the network that optimizes the weights based on Quality of Service (QoS) metrics, specifically the average End-to-End (E2E) trafic delay. Our results show that we can significantly minimize the average trafic delay of a network by optimizing the OSPF configuration using a DT.</p>
      </abstract>
      <kwd-group>
        <kwd>eol&gt;Trafic Engineering</kwd>
        <kwd>OSPF</kwd>
        <kwd>IGP</kwd>
        <kwd>Optimization</kwd>
        <kwd>Network Digital Twins</kwd>
        <kwd>Digital Twins</kwd>
      </kwd-group>
    </article-meta>
  </front>
  <body>
    <sec id="sec-1">
      <title>1. Introduction</title>
      <sec id="sec-1-1">
        <title>1.1. Motivation</title>
        <p>We take inspiration from their work to introduce an OSPF optimization technique that considers
QoS metrics. Using a Network Digital Twin (NDT) model, such as TwinNet, one can produce
architectures capable of optimizing the OSPF configuration of a network for the QoS metrics of
their choice.</p>
      </sec>
      <sec id="sec-1-2">
        <title>1.2. Contributions</title>
        <p>To optimize the OSPF configuration of a network based on QoS metrics, we propose using a
NDT: a Digital Twin (DT) of a communication network that is capable of optimizing the network
and reflecting the behavior of the network [ 9]. Digital Twins (DT) are recently emerging in the
ifeld of communication networks and there is a work-in-progress Internet Research Task Force
(IRTF) draft on a reference architecture of a NDT [10]. The NDT takes information about the
topology (including the OSPF configuration) and trafic flows going through its physical twin
(the physical network we are optimizing) and updates the OSPF configuration of its physical
twin based on the predicted optimal configuration.</p>
        <p>We implement our NDT by combining RouteNet-F [11], a GNN model that predicts QoS
metrics for a given network and its trafic flows, with a genetic algorithm designed to minimize
trafic delays of an OMNeT++ simulated network (the physical twin) [ 12]. We show how our
method improves the trafic delays of randomly generated network topologies by optimizing
their OSPF configuration.</p>
      </sec>
      <sec id="sec-1-3">
        <title>1.3. Structure</title>
        <p>In Section 2, we summarize the state-of-the-art TE methods for OSPF optimization. In Section 3,
we introduce some background which we base our NDT implementation on. In Section 4, we
describe the method we propose to perform OSPF optimization using a NDT. In Section 5,
we show how our OSPF optimization method using NDT was able to optimize the simulated
network topologies. Lastly, we provide a concluding paragraph with the future work in Section 6.</p>
      </sec>
    </sec>
    <sec id="sec-2">
      <title>2. OSPF weight optimization state of the art</title>
      <p>OSPF weight optimization is the process of setting the Interior Gateway Protocol (IGP) metric of
each link in the network to optimize for some metric of the network, such as maximum link
utilization, average End-to-End (E2E) loss, etc. The IGP metric of a link in the OSPF protocol,
also referred to as the weight of the link, is the cost of taking this link. Trafic flowing to a
destination node from a source node is routed through the minimal-cost path.</p>
      <p>
        Current methods in the literature only optimize for bottleneck link utilization or average link
utilization [
        <xref ref-type="bibr" rid="ref2 ref5 ref6">6, 5, 2</xref>
        ], or for energy eficiency [ 13]. One method includes Service-Layer Agreement
(SLA) requirements and performance under link failures in addition to link utilization [14]. The
problem of finding an OSPF configuration that minimizes the bottleneck link utilization for a
given topology and a set of trafic flows/demands is a known NP-hard problem [15]. Current
methods either use a neighborhood search technique using a set of heuristics, or a Machine
Learning (ML) approach. In this section, we cover the state-of-the-art of OSPF optimization for
link utilization starting with the first method, IGP Weight Optimization [
        <xref ref-type="bibr" rid="ref6">6</xref>
        ], a technique that
uses local search heuristics to find the optimal OSPF configuration.
      </p>
      <sec id="sec-2-1">
        <title>2.1. Local Search approach</title>
        <sec id="sec-2-1-1">
          <title>2.1.1. IGP Weight Optimization</title>
          <p>
            IGP Weight Optimization was proposed by Fortz and Thorup [
            <xref ref-type="bibr" rid="ref6">6</xref>
            ]. It takes in the topology of the
network, the associated weight and capacity of each link in the network, and the bandwidth
demand between each source-destination node pair. It then uses a local search algorithm to find
a weight configuration that minimizes the total utilization cost of all links in the network. They
define the utilization cost of a link as a function that scales exponentially with the utilization
percentage of the link.
          </p>
          <p>The neighbors of the current candidate OSPF configuration of a given network topology are
defined as the following:
• Single link weight change.
• Splitting trafic to a destination node evenly across a random number of neighbors at a
transit node (refer to Figure 1).</p>
          <p>
            To prevent looping through visited OSPF configurations, they maintain a hash table of
all visited OSPF configurations. This algorithm can be terminated at any point during the
search to find a better OSPF configuration; however, it is not guaranteed that it will find the
global optimum configuration. Rusek et al. [
            <xref ref-type="bibr" rid="ref2">2</xref>
            ] recently reproduced the results of Fortz and
Thorup [
            <xref ref-type="bibr" rid="ref6">6</xref>
            ] to compare IGP weight optimization to their ML approach. They show that it can
ifnd significantly better OSPF configurations that minimize the maximum link utilization in
a short period of time on 4 real-world topologies (Janos-US, GEANT, Nobel-Germany, and
COST266).
          </p>
        </sec>
        <sec id="sec-2-1-2">
          <title>2.1.2. Dynamic IGP Weight Optimization</title>
          <p>
            Dynamic IGP Weight Optimization was proposed by Brun and Garcia [
            <xref ref-type="bibr" rid="ref5">5</xref>
            ]. It uses a very similar
local search algorithm as IGP Weight Optimization with some slight modifications to the
neighborhood space to find an OSPF configuration that minimizes the maximally utilized link.
The neighbors of a candidate OSPF configuration are defined as the following:
• A single link weight change that deviates a trafic flow from the link if the trafic flow has
multiple minimum cost paths.
• A single link weight change that introduces an alternative path for a trafic flow (introduces
a new minimum cost path).
          </p>
          <p>In addition to those diferences, Dynamic IGP Weight Optimization estimates the
sourcedestination demand matrix from a time series of link demands rather than having prior
knowledge of those demands. The resulting OSPF configuration recommendation is then based on the
worst-case demand scenario out of the set of possible demands inferred from the link demands.
However, this makes it much more computationally expensive than IGP Weight Optimization.</p>
        </sec>
      </sec>
      <sec id="sec-2-2">
        <title>2.2. Machine Learning approach</title>
        <sec id="sec-2-2-1">
          <title>2.2.1. Routing by Back Propagation (RBB)</title>
          <p>
            Routing by Back-propagation (RBB) is the latest OSPF optimization technique proposed by
Rusek et al. [
            <xref ref-type="bibr" rid="ref2">2</xref>
            ] It makes use of a GNN model [
            <xref ref-type="bibr" rid="ref7">7</xref>
            ] that is trained to approximate the minimum
cost path between each source-destination node pair (i.e. Dijkstra’s algorithm [16]) for a given
topology with a set of link weights. This pre-trained GNN is then used along with the capacity
of each link and the trafic demands between each source-destination pair to backpropagate
the input IGP weights of each link to minimize the maximally utilized link. This method is
much faster than the aforementioned techniques, with the authors reporting an average 25%
reduction in the maximally utilized link utilization from the default OSPF configuration after 3
backpropagation steps.
          </p>
          <p>While the methods we listed in this section can find much better OSPF configurations in a
short period, none of them consider any variable beyond link utilization for OSPF configuration.
Before introducing our NDT for OSPF optimization, we introduce some of the related work
used in our NDT implementation in the next section.</p>
        </sec>
      </sec>
    </sec>
    <sec id="sec-3">
      <title>3. Background</title>
      <sec id="sec-3-1">
        <title>3.1. RouteNet-Fermi</title>
        <p>RouteNet-Fermi, also known as RouteNet-F, is a GNN model of a communication network
proposed by the same authors of TwinNet, Ferriol-Galmés et al. [11] It takes the topology of
the network (which includes link bandwidths, queuing policy of each interface, the number
of bufers at each interface and their respective sizes), the trafic flows between each
sourcedestination node (including the average bandwidth used, the packet size distribution and time
distribution), and the route each trafic flow takes in the topology as input. The resulting output
of RouteNet-F is the predicted delay, jitter, and loss of each trafic flow. Their results show that
RouteNet-F can generalize and accurately predict QoS metrics on unseen topologies and trafic
lfows. It serves as a ML based network model that is capable of predicting QoS metrics much
faster than tools such as OMNeT++ [12]. For those reasons, we chose to use RouteNet-F as our
digital network model in our NDT implementation in Section 4.</p>
        <p>In the following section, we introduce genetic algorithms, the optimization technique that is
coupled with RouteNet-F for our NDT implementation of QoS based OSPF optimization.</p>
      </sec>
      <sec id="sec-3-2">
        <title>3.2. Genetic algorithms</title>
        <p>Genetic algorithms are search and optimization algorithms proposed by Holland [17]. They
are inspired by the process of natural selection and evolution. They are characterized by the
following seven steps:
• Initialization where a population of potential solutions for an optimization problem is
randomly generated.
• Evaluation where each solution is evaluated using a fitness function.
• Selection where solutions with a higher score are selected for “reproduction”.
• Recombination where a pair or pairs of the highest-scoring solutions are randomly
combined to create the new “population”.
• Mutation where solutions in the new population are randomly modified to prevent the
algorithm from converging early.
• Replacement where new solutions replace the old solutions in the population when
creating the next generation.</p>
        <p>• Termination where the algorithm terminates at a specified stopping criterion.</p>
        <p>Genetic algorithms are very simple to implement and are applicable to any fitness function
rather than being specialized such as heuristic local search techniques. They are also less likely
to be stuck in a local optimum due to the mutation step. However, they come at a computational
expense. When using a ML digital model such as RouteNet-F to provide the metrics for our
iftness function, we can aford to use genetic algorithms and find a much better solution than
the current solution within a short period. We decided to use genetic algorithms for their
simplicity and generalization as our goal is to show how we can use a NDT for optimizing OSPF
configuration based on QoS metrics. While using a local search technique with heuristics such
as those mentioned in Section 2 can improve the time it takes to find a better solution, using a
genetic algorithm gives us the flexibility to choose our fitness function. In the following section,
we describe the architecture of our NDT for OSPF optimization and the genetic algorithm we
use.</p>
      </sec>
    </sec>
    <sec id="sec-4">
      <title>4. Methodology</title>
      <p>Our proposed NDT architecture for OSPF optimization is shown in Figure 3. We use RouteNet-F
as the model to predict the QoS metrics of the current trafic flows going through the network
and couple it with a genetic algorithm to find an OSPF configuration that minimizes the average
trafic delay in the network. Our genetic algorithm is initialized by mutating the current OSPF
configuration to create a population of 2000 solutions in our experiments. Mutations have
a 20% chance of occurring on each link in all generations. The solution with the minimum
average trafic delay is then recombined with itself to create a new population. If no solution
from the new generation has a lower average trafic delay than the previous generation for 4
generations our algorithm terminates. The population size and mutation probability as well
as our termination criteria are hyper-parameters of our model and were selected based on our
intuition. Algorithm 1 summarizes the implementation of our NDT for OSPF optimization.</p>
      <p>To evaluate our implementation, we randomly generated three topologies, a 6-node topology,
a 10-node topology, and a 20-node topology with randomly generated Poisson trafic flows
ranging from 200 bits per second to 2000 bits per second for each source-destination pair. The
link capacity of each link was set to 100kbps and all queuing policies were set to FIFO with a
single 32kbits queue. We used the RouteNet-F model trained on the fat-tree  = 128 dataset
provided in the RouteNet-F paper [11]. For our networks (i.e. the “physical” twin) we used the
same OMNeT++ simulation used to train RouteNet-F. The OSPF weights were also randomly
initialized to either have a weight of 5 or 1 in our simulated networks. The genetic algorithm
was also constrained to this set of weights to minimize its search space. In the following section,
we report and discuss the results of our experiments.</p>
    </sec>
    <sec id="sec-5">
      <title>5. Results and discussion</title>
      <p>In Figure 4, we plot the Cumulative Distribution Function (CDF) of the trafic flow delays going
through each of our simulated topologies before and after optimization. We obtained the delays
Algorithm 1: Digital Twin for Trafic Engineering</p>
      <p>Input:  ,  (where  is the real network, and  is the digital model)
Parameter :  generations without improvement before termination,  size of generation
 ← retrieve topology and features of 
  ← retrieve estimated distribution of each flow in 
  ← 
 ← predicted average flow delay of  using 
 ℎ ← 0
while True do
  ← generate  weight configurations from  using
genetic algorithm and recompute routes
 ← predict the average flow delay of   using 
 ℎ ←  ℎ + 1
if min(predictedDelays) &lt; bestDelay then
 ← ()
  ←  [.()]</p>
      <p>ℎ ← 0
end
if generationsWithoutImprovement ≥  then</p>
      <p>break
end
end
 ←</p>
      <p>Apply   to  and retrieve average flow delays from 
from the OMNeT++ simulated topology (i.e. our “network physical twin”) in our experiments.
Our results show that our technique successfully reduced the average trafic delay in all our
randomly generated networks. Our method shifts the distribution of trafic delays closer to
lower delays and consistently reduced the maximum trafic delay in the network in all our
topologies. All our NDTs found a solution within 5 minutes on a GTX1080Ti GPU and an Intel
i7 7700k CPU. Our promising results indicate that our NDT for OSPF optimization based on
QoS metrics is viable and more importantly, allows us to optimize for other QoS metrics beyond
delay with ease. We can optimize for other metrics by simply changing the fitness function.
For instance, to optimize trafic loss or link utilization, we can change the fitness function to
be negatively proportional to the maximum trafic loss or the maximum link utilization in the
network without modifying the architecture.</p>
      <p>Coupling our NDT architecture for optimizing a network with a smarter optimization
algorithm and a desired optimization criteria opens the door for new methods of automating TE
and network optimization.</p>
      <p>(a) 6 Node topology.
(b) 10 Node topology.</p>
      <p>(c) 20 Node topology.</p>
    </sec>
    <sec id="sec-6">
      <title>6. Conclusions and future work</title>
      <p>In this paper, we showed how one can use our NDT architecture to perform OSPF optimization
based on QoS metrics. Our NDT was able to find OSPF configurations that significantly reduced
the trafic delays in the twinned networks. However, our genetic algorithm was limited to a
set of two weights and may take longer to terminate when the set of weights is large as it
increases the search space. A better termination criterion in this case would be the percentage
improvement in the last  generations.</p>
      <p>Using a smarter optimization algorithm such as a local search heuristic can also improve
performance and the speed of finding a solution. However, the genetic algorithm provides a
baseline implementation that can be extended further.</p>
      <p>Our future work would include comparing our NDT for OSPF optimization to the
state-of-theart OSPF optimization techniques using the average trafic delay, maximum link utilization and
the time to arrive at a solution. Moreover, applying this architecture to a real network rather
than a simulated network would show how well the NDT for OSPF optimization performs in
practice and if changes are required to adjust for a real network.</p>
    </sec>
    <sec id="sec-7">
      <title>Acknowledgments</title>
      <p>We acknowledge the support of Ciena and the Natural Sciences and Engineering Research
Council of Canada (NSERC).
[9] P. Almasan, M. Ferriol-Galmés, J. Paillisse, J. Suárez-Varela, D. Perino, D. López, A. A. P.</p>
      <p>Perales, P. Harvey, L. Ciavaglia, L. Wong, et al., Digital twin network: Opportunities and
challenges, arXiv preprint arXiv:2201.01144 (2022).
[10] C. Zhou, H. Yang, X. Duan, D. Lopez, A. Pastor, Q. Wu, M. Boucadair, C. Jacquenet,
Digital Twin Network: Concepts and Reference Architecture, Internet-Draft
draft-irtfnmrg-network-digital-twin-arch-03, Internet Engineering Task Force, 2023. URL: https://
datatracker.ietf.org/doc/draft-irtf-nmrg-network-digital-twin-arch/03/, work in Progress.
[11] M. Ferriol-Galmés, J. Paillisse, J. Suárez-Varela, K. Rusek, S. Xiao, X. Shi, X. Cheng, P.
BarletRos, A. Cabellos-Aparicio, Routenet-fermi: Network modeling with graph neural networks,
IEEE/ACM Transactions on Networking (2023).
[12] A. Varga, Omnet++, Modeling and tools for network simulation (2010) 35–59.
[13] F. Francois, N. Wang, K. Moessner, S. Georgoulas, K. Xu, On igp link weight optimization
for joint energy eficiency and load balancing improvement, Computer Communications
50 (2014) 130–141.
[14] A. Nucci, S. Bhattacharyya, N. Taft, C. Diot, Igp link weight assignment for operational
tier-1 backbones, IEEE/ACM Transactions on Networking 15 (2007) 789–802.
[15] F. Giroire, S. Pérennes, I. Tahiri, On the hardness of equal shortest path routing, Electronic</p>
      <p>Notes in Discrete Mathematics 41 (2013) 439–446.
[16] E. W. Dijkstra, A note on two problems in connexion with graphs, Numerische mathematik
1 (1959) 269–271.
[17] J. H. Holland, Genetic algorithms, Scientific American (1992).</p>
    </sec>
  </body>
  <back>
    <ref-list>
      <ref id="ref1">
        <mixed-citation>
          [1]
          <string-name>
            <given-names>Z.</given-names>
            <surname>Wang</surname>
          </string-name>
          ,
          <string-name>
            <given-names>Z.</given-names>
            <surname>Li</surname>
          </string-name>
          , G. Liu,
          <string-name>
            <given-names>Y.</given-names>
            <surname>Chen</surname>
          </string-name>
          ,
          <string-name>
            <given-names>Q.</given-names>
            <surname>Wu</surname>
          </string-name>
          , G. Cheng,
          <article-title>Examination of wan trafic characteristics in a large-scale data center network</article-title>
          ,
          <source>in: Proceedings of the 21st ACM Internet Measurement Conference</source>
          ,
          <year>2021</year>
          , pp.
          <fpage>1</fpage>
          -
          <lpage>14</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref2">
        <mixed-citation>
          [2]
          <string-name>
            <given-names>K.</given-names>
            <surname>Rusek</surname>
          </string-name>
          ,
          <string-name>
            <given-names>P.</given-names>
            <surname>Almasan</surname>
          </string-name>
          ,
          <string-name>
            <given-names>J.</given-names>
            <surname>Suárez-Varela</surname>
          </string-name>
          ,
          <string-name>
            <given-names>P.</given-names>
            <surname>Chołda</surname>
          </string-name>
          ,
          <string-name>
            <given-names>P.</given-names>
            <surname>Barlet-Ros</surname>
          </string-name>
          ,
          <string-name>
            <given-names>A.</given-names>
            <surname>Cabellos-Aparicio</surname>
          </string-name>
          ,
          <article-title>Fast trafic engineering by gradient descent with learned diferentiable routing</article-title>
          ,
          <source>in: 2022 18th International Conference on Network and Service Management (CNSM)</source>
          , IEEE,
          <year>2022</year>
          , pp.
          <fpage>359</fpage>
          -
          <lpage>363</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref3">
        <mixed-citation>
          [3]
          <string-name>
            <given-names>S.</given-names>
            <surname>Gay</surname>
          </string-name>
          ,
          <string-name>
            <given-names>P.</given-names>
            <surname>Schaus</surname>
          </string-name>
          ,
          <string-name>
            <given-names>S.</given-names>
            <surname>Vissicchio</surname>
          </string-name>
          , Repetita:
          <article-title>Repeatable experiments for performance evaluation of trafic-engineering algorithms</article-title>
          ,
          <source>arXiv preprint arXiv:1710.08665</source>
          (
          <year>2017</year>
          ).
        </mixed-citation>
      </ref>
      <ref id="ref4">
        <mixed-citation>
          [4]
          <string-name>
            <given-names>R.</given-names>
            <surname>Hartert</surname>
          </string-name>
          ,
          <string-name>
            <given-names>S.</given-names>
            <surname>Vissicchio</surname>
          </string-name>
          ,
          <string-name>
            <given-names>P.</given-names>
            <surname>Schaus</surname>
          </string-name>
          ,
          <string-name>
            <given-names>O.</given-names>
            <surname>Bonaventure</surname>
          </string-name>
          ,
          <string-name>
            <given-names>C.</given-names>
            <surname>Filsfils</surname>
          </string-name>
          ,
          <string-name>
            <given-names>T.</given-names>
            <surname>Telkamp</surname>
          </string-name>
          ,
          <string-name>
            <given-names>P.</given-names>
            <surname>Francois</surname>
          </string-name>
          ,
          <article-title>A declarative and expressive approach to control forwarding paths in carrier-grade networks</article-title>
          ,
          <source>ACM SIGCOMM computer communication review 45</source>
          (
          <year>2015</year>
          )
          <fpage>15</fpage>
          -
          <lpage>28</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref5">
        <mixed-citation>
          [5]
          <string-name>
            <given-names>O.</given-names>
            <surname>Brun</surname>
          </string-name>
          ,
          <string-name>
            <surname>J.-M. Garcia</surname>
          </string-name>
          ,
          <article-title>Dynamic igp weight optimization in ip networks</article-title>
          ,
          <source>in: 2011 First International Symposium on Network Cloud Computing and Applications</source>
          ,
          <year>2011</year>
          , pp.
          <fpage>36</fpage>
          -
          <lpage>43</lpage>
          . doi:
          <volume>10</volume>
          .1109/NCCA.
          <year>2011</year>
          .
          <volume>13</volume>
          .
        </mixed-citation>
      </ref>
      <ref id="ref6">
        <mixed-citation>
          [6]
          <string-name>
            <given-names>B.</given-names>
            <surname>Fortz</surname>
          </string-name>
          ,
          <string-name>
            <given-names>M.</given-names>
            <surname>Thorup</surname>
          </string-name>
          ,
          <article-title>Internet trafic engineering by optimizing ospf weights</article-title>
          ,
          <source>in: Proceedings IEEE INFOCOM 2000. conference on computer communications. Nineteenth annual joint conference of the IEEE computer and communications societies (Cat. No. 00CH37064)</source>
          , volume
          <volume>2</volume>
          , IEEE,
          <year>2000</year>
          , pp.
          <fpage>519</fpage>
          -
          <lpage>528</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref7">
        <mixed-citation>
          [7]
          <string-name>
            <given-names>P. W.</given-names>
            <surname>Battaglia</surname>
          </string-name>
          ,
          <string-name>
            <given-names>J. B.</given-names>
            <surname>Hamrick</surname>
          </string-name>
          ,
          <string-name>
            <given-names>V.</given-names>
            <surname>Bapst</surname>
          </string-name>
          ,
          <string-name>
            <given-names>A.</given-names>
            <surname>Sanchez-Gonzalez</surname>
          </string-name>
          ,
          <string-name>
            <given-names>V.</given-names>
            <surname>Zambaldi</surname>
          </string-name>
          ,
          <string-name>
            <given-names>M.</given-names>
            <surname>Malinowski</surname>
          </string-name>
          ,
          <string-name>
            <given-names>A.</given-names>
            <surname>Tacchetti</surname>
          </string-name>
          ,
          <string-name>
            <given-names>D.</given-names>
            <surname>Raposo</surname>
          </string-name>
          ,
          <string-name>
            <given-names>A.</given-names>
            <surname>Santoro</surname>
          </string-name>
          ,
          <string-name>
            <given-names>R.</given-names>
            <surname>Faulkner</surname>
          </string-name>
          , et al.,
          <article-title>Relational inductive biases, deep learning, and graph networks</article-title>
          , arXiv preprint arXiv:
          <year>1806</year>
          .
          <volume>01261</volume>
          (
          <year>2018</year>
          ).
        </mixed-citation>
      </ref>
      <ref id="ref8">
        <mixed-citation>
          [8]
          <string-name>
            <given-names>M.</given-names>
            <surname>Ferriol-Galmés</surname>
          </string-name>
          ,
          <string-name>
            <given-names>J.</given-names>
            <surname>Suárez-Varela</surname>
          </string-name>
          ,
          <string-name>
            <given-names>J.</given-names>
            <surname>Paillissé</surname>
          </string-name>
          ,
          <string-name>
            <given-names>X.</given-names>
            <surname>Shi</surname>
          </string-name>
          ,
          <string-name>
            <given-names>S.</given-names>
            <surname>Xiao</surname>
          </string-name>
          , X. Cheng, P.
          <string-name>
            <surname>Barlet-Ros</surname>
            ,
            <given-names>A.</given-names>
          </string-name>
          <string-name>
            <surname>Cabellos-Aparicio</surname>
          </string-name>
          ,
          <article-title>Building a digital twin for network optimization using graph neural networks</article-title>
          ,
          <source>Computer Networks</source>
          <volume>217</volume>
          (
          <year>2022</year>
          )
          <fpage>109329</fpage>
          .
        </mixed-citation>
      </ref>
    </ref-list>
  </back>
</article>