# Dismantling the information flow in complex interconnected systems

Arsham Ghavasieh<EMAIL_ADDRESS>Fondazione Bruno Kessler, Via Sommarive
18, 38123 Povo (TN), Italy  Department of Physics, University of Trento, Via
Sommarive 14, 38123 Povo (TN), Italy  Giulia Bertagnolli  Fondazione Bruno
Kessler, Via Sommarive 18, 38123 Povo (TN), Italy  Department of Mathematics,
University of Trento, Via Sommarive 14, 38123 Povo (TN), Italy  Manlio De
Domenico<EMAIL_ADDRESS>Fondazione Bruno Kessler, Via Sommarive
18, 38123 Povo (TN), Italy  Department of Physics and Astronomy “Galileo
Galilei”, University of Padova, Padova, Italy

###### Abstract

Microscopic structural damage, such as lesions in neural systems or
disruptions in urban transportation networks, can impair the dynamics crucial
for systems’ functionality, such as electrochemical signals or human flows, or
any other type of information exchange, respectively, at larger topological
scales. Damage is usually modeled by progressive removal of components or
connections and, consequently, systems’ robustness is assessed in terms of how
fast their structure fragments into disconnected sub-systems. Yet, this
approach fails to capture how damage hinders the propagation of information
across scales, since system function can be degraded even in absence of
fragmentation —e.g., pathological yet structurally integrated human brain.
Here, we probe the response to damage of dynamical processes on the top of
complex networks, to study how such an information flow is affected. We find
that removal of nodes central for network connectivity might have
insignificant effects, challenging the traditional assumption that structural
metrics alone are sufficient to gain insights about how complex systems
operate. Using a damaging protocol explicitly accounting for flow dynamics, we
analyze synthetic and empirical systems, from biological to infrastructural
ones, and show that it is possible to drive the system towards functional
fragmentation before full structural disintegration.

††preprint: APS/123-QED

##  I Introduction

Despite the diversity of physical attributes, all complex systems can be
viewed as large collections of units exchanging information to function
properly —e.g., spreading of diseases through social systems, human flows
between urban areas through transportation networks, financial transactions
between financial agents in stock markets and electrochemical signals
exchanged among neurons in the human brain. Of course, regulation and
maintenance of such pairwise communications necessitates the presence of an
underlying structure, a network, exhibiting high resistance to disintegration
Albert  _et al._ (2000); Callaway  _et al._ (2000); Holme  _et al._ (2002);
Centola (2008); Trajanovski  _et al._ (2013); Allard and Hébert-Dufresne
(2019); D'Souza  _et al._ (2019); Radicchi and Bianconi (2017) even under
sever damage—e.g., genetic mutations in gene-gene interaction networks Manke
_et al._ (2006), extinction of species in ecosystems Dunne  _et al._ (2002),
failure of Internet routers Doyle  _et al._ (2005) or unavailability of
transportation means De Domenico  _et al._ (2014). During the last decade,
network integrity—i.e., the existence of  pathways between every pair of units
guaranteeing  that they can exchange information—has been widely used as a
proxy of network robustness, under progressive removal of units or connections
Iyer  _et al._ (2013); De Domenico  _et al._ (2014), indicating that in many
empirical systems the full disintegration happens only if a large fraction of
the system is damaged. While structural integrity is a necessary condition for
information exchange, it is barely sufficient to quantify the effect of damage
on the flow of information within the systems (See Fig. 1) , a property
crucial for system’s functionality. For instance, it has been shown that by
analyzing the perturbation of information flow, one can distinguish the
healthy brain from the pathological one, even in absence of significant
structural differences among the two types of connectomes Benigni  _et al._
(2021). Accordingly, a number of methods have been developed to capture unit-
unit communications, considering the coupling between the structure and
dynamical processes governing the flow of information Masuda  _et al._ (2017);
Arenas  _et al._ (2008); De Domenico  _et al._ (2016) and to account for the
heterogeneity and intervening of temporal and spatial information propagation
scales Arenas  _et al._ (2006); Lambiotte  _et al._ (2014).  Yet, the
robustness literature is still predominantly limited to structural analyses,
missing a functional perspective.

Of course, in special cases, like for very sparse networks, or when only
short-range interactions are under investigation, the role played by the
structure is expected to be dominant. In those cases, structural indicators
provide a reliable description of the flow dynamics and shortest paths,
sequences of minimum number of links connecting pairs of nodes, might reliably
describe the flow pathways in the system. However, in most scenarios,  damage
can locally or globally perturb the information dynamics in empirical systems—
e.g., lesions impairing the flow of electrochemical signals within the larger
or smaller sections of the brain, to failures of transportation systems
hindering human flows at the level of districts, cities or regions—, and
ignoring the dynamics of how information spreads and the multiscale nature of
interactions can lead to poor results. To this end, a novel framework is
needed to go beyond the traditional structural paradigm and investigate
functional robustness of interconnected systems, characterized in terms of the
system’s ability to maintain the flow exchange under random or targeted
disruptions.

Here, we assess functional robustness in terms of the effect of unit removal
on the information dynamics: a decrease in the average received information
(ARI) per unit, modeled using diffusion processes coupled with the network,
and an increase in the dispersion of units in the  diffusion manifold  induced
by the information dynamics, quantified as their average squared diffusion
distance (ASDD). We use statistical physics of complex information dynamics
Ghavasieh  _et al._ (2020) to identify and target the critical components of
the system based on the impact of their removal on the Von Neumann entropy
Ghavasieh  _et al._ (2021). We demonstrate that this method directly reduces
the overlap between flows originating from different parts of the system and,
consequently, leads to efficient dismantling of the information flow —i.e.,
functional dismantling.

Our results show that while attack strategies based on topological indicators
such as degree, closeness and betweenness are effective in a limited number of
scenarios, they are not distinguishable from random failures when the network
is not sufficiently sparse or when the mid- or long-range communications are
the target.  Instead, our approach can identify the most central components of
the human connectome, the neural network of the C. Elegans, European airline
network and Chilean power grid, at multiple propagation time scales, whose
removal effectively disturbs the flow of information in those systems.

Figure 1: Impairment of information flow and functional degradation. (a) The
structure of a complex system is shown, where two components A and B are
highlighted in blue. To perform a system specific function, A and B must
communicate—e.g., Broca and Wernicke areas in human brain for speech
production, London and Berlin airports for flow of business people, industry
section and academia exchanging ideas and knowledge for technological
development— and the flow of information between them can be modeled using
differential equations coupling the structure, noted as $G$, and dynamical
processes, governed by the control operator $\hat{H}$. (b) The dependence of
functionality on the scale of information propagation in the system, from
short- to long-range, and damage destroying part of the system. A and B can
not perform the system specific function, if the communication range is short
or in cases where the structure is damaged while the communication scale is
not large enough. Note that the damage depicted here is not severe enough to
dismantle the structure, yet it has a considerable effect on the flow and,
thus, functionality.

##  II From structural to functional robustness

The size of the largest connected component (LCC)—i.e., the number of nodes in
the largest group of nodes that are connected together via sequences of
links—has been extensively used as a proxy for structural robustness of
networks Albert  _et al._ (2000); Holme  _et al._ (2002); Iyer  _et al._
(2013). In this case, the network is considered to be structurally dismantled
when the size of LCC drops, consequent to removal of a fraction of links or
nodes representing the damage imposed on the system, either in the form of
random failures or targeted attacks. While the former can be easily simulated,
using an algorithm that randomly removes nodes from the network and keeps a
record of the size of LCC, there is no unique way to model the latter. To
compensate, a variety of centrality measures have been introduced to rank the
nodes according to different criteria of importance, and remove them, one by
one, aiming to pose the maximum possible damage to the structure Wandelt  _et
al._ (2018). However, we still lack a universal centrality measure proven to
be the most effective in quick structural dismantling of all network types.

In general, there are two types of attack strategies: i) static, where the
ranking of the nodes is computed only once at the beginning of the algorithm
and ii) iterative, where the centrality of nodes are updated, after every node
removal. The first category has significantly lower computational cost, while
the second one is more effective in leading networks to quick structural
impairment. On the one hand, iterative betweenness —i.e., a centrality that
ranks the nodes according to the number of shortest paths crossing them—,
shows impressive performance Wandelt  _et al._ (2018) with reasonably low
computational complexity. On the other hand, new measures of centrality have
been introduced to combine topological metrics with meta data analysis Artime
and Domenico (2021) or based on machine learning Grassia  _et al._ (2021),
geometry Bertagnolli  _et al._ (2019) and, very recently, statistical physics
Ghavasieh  _et al._ (2021).

In this work, we aim to study functional robustness, in contrast with the
structural. Our goal is to test the performance of the widely used, yet purely
topological, metrics including iterative degree, betweenness, eigenvector and
closeness centrality, in dismantling the flow of information, instead of LCC.
Additionally, we compare the structural metrics with a recent centrality
measure named entanglement, that ranks the nodes according to the impact of
their removal on the diversity of flow pathways in the system, providing a
suitable candidate for effective functional dismantling. It is worth
mentioning that entanglement centrality has also been previously studied
against a large number of widely used structural and dynamical centrality
measures and shown to outperform them, or perform as well as the best of them,
in dismantling a wide range of networks structurally Ghavasieh  _et al._
(2021). For this reason, here, we study the effectiveness of the best
performing structural measures in comparison with entanglement, excluding
other dynamical metrics.

In the following, we firstly introduce ARI and ASDD, as proxies for functional
robustness. Then, we review the definition of entanglement centrality derived
from statistical physics of complex information dynamics. Finally, we show
that topological metrics fail to identify the nodes central for the flow
dynamics, under a number of scenarios. We demonstrate that, surprisingly, the
effectiveness of attacks guided by topological measures on the functional
robustness is not distinguishable from random damage. This result highlights
that topological information, including the distribution of degree and
shortest paths, is not sufficient to capture the information flow in synthetic
and empirical networks and, therefore, shortest paths are not determinant of
the node-node interactions.

##  III Quantifying functional robustness

Figure 2: Multiscale information propagation. Top row shows the propagation
from an arbitrary node (having high field concentration at $\tau=0$). At small
temporal scales, the field propagation is limited to the locality
($\tau=0.1,0.5$), while the global interactions are allowed at larger temporal
scales ($\tau=5$). The equilibrium of the dynamics happens when the field
reaches the final distribution, independent of its initial conditions. Below
each plot in the top row, the propagation geometry constructed from (3),
embedded in 2 dimensions using multidimensional scaling, is shown. Each dot
corresponds to a node and node colors indicate the average diffusion distance
between all vertices in the network. The nodes become closer in the diffusion
manifold as one tunes $\tau$ to larger values, from separate units to
functional modules and, eventually, reaching full integration. Similar
coloring can be obtained using average squared diffusion distance instead of
average diffusion distance.

Different levels of abstraction can be implemented in modeling a complex
phenomenon, from agent based modeling Abar  _et al._ (2017), where one tries
to insert every available detail into the equations, to the mean-field
approaches, where even the agents might be coarse grained Noid  _et al._
(2008); Aguilera  _et al._ (2021) into fields with no regard for the
connectivity patterns, heterogeneity, etc. Comparing the two approaches, one
immediately observes a trade-off. High resolution modeling is often preferred
when exact calculations are necessary for decision or policy making. Yet, low
resolution modeling is also powerful, as it captures the most important
factors and their associations, providing understanding at the expense of
precision. For instance, most network science studies during the last two
decades ground themselves in structural analysis of real-world systems,
counting the number of connections, finding the geodesic distances as
plausible transportation routes and studying the modularity and hierarchy of
the structure, all based on distribution of links among the nodes. One
abstraction level above the structural analysis is to couple the network with
dynamical processes, such as classes of diffusion, that are general enough to
describe a wide range of transport phenomena Ghavasieh and De Domenico (2020),
at least to the first order approximation, and suitable to model
communications between the nodes.

To mathematically describe the coupling, we identify the nodes of a network as
canonical vectors $|x_{i}\rangle,\,i=1,2,\dots,N$ and encode their connections
in the operator $\hat{W}$ which in the space of nodes– shaped by the canonical
vectors $|x_{i}\rangle$– represents the adjacency matrix, where $\langle
x_{j}|\hat{W}|x_{i}\rangle=W_{ij}$ is the weight of the link from $i$-th to
$j$-th node,  in accordance with notation of a fundamental reference of this
work Ghavasieh  _et al._ (2020). The field $|\phi(\tau)\rangle$ is assumed on
top of the network, and its amount at node $i$ at time $\tau$ is shown as
$\langle x_{i}|{\phi}(\tau)\rangle$. The flow of the field from one node to
another will be used as a proxy for information exchange between the two. The
evolution is governed by the linear or, in case of non-linear dynamics,
linearized equation

$\displaystyle\partial_{\tau}|\phi(\tau)\rangle=-\hat{H}(\hat{W})|\phi(\tau)\rangle,$
(1)

where $\hat{H}(\hat{W})$ is a control operator. Note that the equation is
exact for a range of dynamical processes including random walks, consensus
dynamics, synchronization near the meta-stable manifold and continuous
diffusion. As explained before, in this work we focus on diffusion dynamics
where the control operator becomes the Laplacian matrix
$\hat{H}(\hat{W})=\hat{K}(\hat{W})-\hat{W}$, with $\hat{K}(\hat{W})$ being the
diagonal degree matrix where $\langle
x_{j}|\hat{K}(\hat{W})|x_{i}\rangle=\delta_{ij}k_{i}$ with the Kronecker delta
function $\delta_{ij}$ that is equal to $1$ when $i=j$ and $0$ otherwise, and
$k_{i}=\sum\limits_{j=1}^{N}W_{ij}$ being the degree of node $i$.

Solving the linear equation, one obtains the propagator
$\hat{U}(\tau,\hat{W})=e^{-\tau\hat{H}(\hat{W})}$. Consequently, information
flow from node $i$ to node $j$ can be described using a propagator method
similar to the Euclidean path integrals $\langle
x_{j}|\hat{U}(\tau,\hat{W})|x_{i}\rangle$, and by changing $\tau$ from low to
high values, the propagator can encode short-, middle- and long-range
communications Ghavasieh  _et al._ (2020).

In this case, the average received flow per node reads

$\displaystyle\mathcal{A}(\tau,\hat{W})=\frac{1}{N}\sum\limits_{i\neq
j}\langle x_{j}|e^{-\tau\hat{H}(\hat{W})}|x_{i}\rangle,$ (2)

where $N$ is the number of nodes in the network. Significant damage is
expected to lower the average received information per node. In the following
sections, we apply random and targeted removal of the nodes to a range of
synthetic and empirical networks and study the alterations of ARI along
dismantling trajectories, keeping a record of the effect of progressive damage
on the communications. Note that the nodes removed in the process do not
exchange information with the network, having no impact on the summation in
(2), they can be assumed excluded from the analysis. Therefore, after removal
of $m$ nodes, the denominator of (2) would be equal to $N-m$.

Based on the same propagator, one can consider the Euclidean distance between
the propagation of the field at time $\tau$ from nodes $i$ and $j$

$\displaystyle D_{\tau}(i,j)$ $\displaystyle=\left\lVert
e^{-\tau\hat{H}(\hat{W})}|x_{i}\rangle-e^{-\tau\hat{H}(\hat{W})}|x_{j}\rangle\right\rVert_{2}$
(3) $\displaystyle=\sqrt{\sum_{k}\left(\langle
x_{k}|e^{-\tau\hat{H}(\hat{W})}|x_{i}\rangle-\langle
x_{k}|e^{-\tau\hat{H}(\hat{W})}|x_{j}\rangle\right)^{2}}$

which induces a metric, called diffusion distance De Domenico (2017), on the
network  and, consequently a (geo)metric structure on the set of nodes, called
diffusion geometry De Domenico (2017). According to $D_{\tau}$ nodes $i$ and
$j$ are close if the flows emanating from them are similar or, in
probabilistic terms, if the probability that two independent random walkers
starting from nodes $i$ and $j$ respectively meet in any node $k$ at time
$\tau$ is high.  Observe also that the information dynamics maps each node
$i\in V$ to a point $e^{-\tau\hat{H}(\hat{W})}|x_{i}\rangle$ on a hyperspace
of $\mathbb{R}^{N}$ Bertagnolli and De Domenico (2021), called diffusion
space, where nodes connected by many short walks lie close to each other.
Since the diffusion distance integrates the information on the connectivity of
the network at different scales, it is also sensitive to those structural
changes forming bottlenecks in the information flow.

We summarize the geometric information provided by the diffusion distance
through the average squared diffusion distance (ASDD)

$\mathcal{D}^{2}(\tau,\hat{W})=\frac{1}{2N^{2}}\sum_{i,j}D_{\tau}^{2}(i,j)$
(4)

which can be shown to be a measure of dispersion in the diffusion space,
exploiting the fact that the propagator $e^{-\tau\hat{H}(\hat{W})}$ is a
stochastic matrix—the Laplacian $\hat{H}(\hat{W})$ is, indeed, a $Q-$matrix
Norris (1997); Bertagnolli and De Domenico (2021) and, under particular
assumptions on $\phi$ (1) describes an edge-centric random walk Masuda  _et
al._ (2017), see the Appendix A. Consequently, we expect that the network
dismantling increases the dispersion of the nodes (points) in the diffusion
space.

For a visual illustration of information propagation and the functional
geometry of networks, see Fig. 2. Obviously, when $\tau$ is sufficiently
small, information exchange is limited to the locality of the nodes, among
first neighbors, and through the shortest paths connecting them to the
spatially distant ones. Also, at this temporal scale, the diffusion distance
between the nodes is expected to be strongly determined by the shortest paths
connecting them—i.e., the local geometry of the network. While, as one
increases $\tau$ in a network that is not extremely sparse, the longer paths
(and their number) become important, modulating the amount of exchanged field
among the nodes and the mutual diffusion distance between them. In this
regime, we expect that both ARI and ASDD can not be understood in terms of
purely structural metrics such as betweenness and degree.

##  IV Entanglement centrality

Originally, the notion of density matrices has been introduced in quantum
mechanics to capture the pairwise coherence between quantum states in a
physical system Fano (1957). Similarly, complex networks, as collections of
pairwise connections between objects, cannot be fully described by vectors or
distribution functions, without information loss. Therefore, following this
analogy and inspired by the framework of quantum statistical physics, the
state of complex networks has been derived in terms of density matrices
Ghavasieh  _et al._ (2020), describing statistical ensembles of stream
operators given by the outer product of eigenvectors of a control operator
$\hat{H}$ that guides the flow dynamics— note that such density matrix has
been only used to analyze classical complex networks and its applicability in
case of quantum complex networks is still open to be explored. The framework
has been successfully applied to analyze and improve the transportation
properties of social and infrastructural multiplex networks Ghavasieh and De
Domenico (2020), cluster the human microbiome De Domenico and Biamonte (2016)
extract the mesoscale organization and functional diversity of the human brain
Nicolini  _et al._ (2020); Benigni  _et al._ (2021), and characterize
functional modules in fungal networks Ghavasieh and Domenico (2021).

Interestingly, the Von Neumann entropy has been shown to be a measure of
diversity of flow pathways in the systems Ghavasieh  _et al._ (2020).
Consequently, the effect of node removal on the diversity of flow pathways,
reflected in the Von Neumann entropy, has been used as a measure of
centrality, named _network entanglement_. The effectiveness of network
entanglement in structural dismantling has been compared against a range of
mostly used structural and dynamical centrality measures Ghavasieh  _et al._
(2021). It has been shown that attacks guided by entanglement dismantled
synthetic and empirical networks faster than other measures, or as fast as the
best of them. Yet, the computational complexity of entanglement has been shown
to be higher than many of these measures, being around $\mathcal{O}(N^{3})$
after mathematical approximations and treatment. This makes other structural
and dynamical metrics, even though with lower performance in guiding
structural or functional attacks, more scalable. Nevertheless, entanglement
allows for interpretation and better understanding of the multiscale nature of
node importance in terms of the effect of its removal on the diversity of flow
pathways and, therefore, assessing functional robustness, as shown in the
following. For a more comprehensive review of network entanglement and the
slightly different version of it used here, please see appendinx. B.

In the following, we compare the effectiveness of attack strategies guided by
entanglement with those based on structural metrics including the ones with
highest performance in structural dismantling, like iterative betweennes.

##  V Synthetic networks

Figure 3: Functional dismantling of synthetic networks. 10 independent
realization of 4 classes of network ($N=256$ and average degree $\approx 12$)
have been considered: Barabasi-Albert (BA), Erods-Renyi (ER), Stochastic Block
Model with four communities (SBM4) and Watts-Strogatz (WS). ARI and ASDD at
different temporal scales ($\tau=1,3,5$) have been studied under random
failures and targeted attacks based on structural centrality measures
including iterative degree, betweenness, closeness and eigenvector and
entanglement, as a functional metric. The gray area in each plot shows the
region in which the effectiveness of structural measures is comparable to
random selection of nodes. Under all scenarios considered here, entanglement
centrality is able to impose larger impact on ARI and ASDD.

We study the functional robustness of four different network classes,
including Barabasi-Albert Barabási and Albert (1999), Erods-Renyi Erdös and
Rényi (2011), stochastic block model with four communities and Watts-Strogatz
Watts and Strogatz (1998) models, as useful models of real-world systems. For
each class, 10 independent realizations with $N=256$ nodes and average degree
approximately equal to $12$ are evaluated under random failures and targeted
attacks guided by structural measures including iterative degree, betweenness,
closeness and eigenvector centrality.

In most cases and as long as the size of damage—i.e., the percentage of
removed nodes—is not considerably large, the effectiveness of these structural
metrics on ARI and ASDD is not distinguishable from that of random failures
(See Fig.3). It is worth highlighting that even the performance of iterative
betweenness, which is known to be highly efficient in dismantling the networks
structurally, is comparable to random disruptions unless the size of damage is
considerably large. The only counter example among the considered cases is for
Barabasi-Albert networks at small temporal scales, $\tau=1$, due to the
characteristic heterogeneity of its degree distribution. Generally, it is
evident that the ability of structural measures in capturing centrality is
higher in case of smaller propagation times. This means that if the flow
dynamics on a real network is local—e.g., the spreading of a niche content on
an online social network like Twitter cannot happen forever and reach everyone
since after some time $T>0$ the tweet is scrolled down the timeline—then
attacking topologically central nodes is efficient also for dismantling the
information flow. Of course, this is expected as small $\tau$ characterizes
local interactions between the nodes and the local interactions can be
directly described by the adjacency matrix, regardless of the coupling between
network and diffusion dynamics.

According to our results, entanglement as a functional metric outperforms the
structural ones, under almost all considered circumstances. This is even more
interesting in the light of the previous works, showing that this measure has
no particular correlation with and cannot be captured by any of a wide range
of structural and dynamical centrality measures Ghavasieh  _et al._ (2021).

##  VI Empirical networks

Figure 4: Functional dismantling of the Human Connectome. Attacks based on
entanglement centrality provide the upper bound for functional damage on the
human connectome, among the considered centrality measures. The overall
effectiveness of structural measures are indistinguishable from each other and
comparable with random failures at large temporal scales. The spatial
locations of nodes match the positions of brain regions given by the dataset.
The size of the nodes is proportional to their degree. Dark gray indicates the
top 10% of the nodes according to entanglement centrality, at specific $\tau$.
As an example, the effect of removal of 20% of nodes guided by betweenness and
entanglement centrality, on the structure and _emergent functional state_
Ghavasieh and Domenico (2021)— i.e., a network in which the edge weights are
given by the amount of flow exchange between the nodes— has been illustrated,
at $\tau=30$.  Figure 5: Functional dismantling of C. Elegans neural system.
The neural network of the nematode worm C. Elegans ($N=297$) is considered.
The effect of random failures and targeted attacks guided by iterative
betweenness, degree, eigenvector, closeness and entanglement centrality
measure on ARI and ASDD is plotted at multiple propagation time scales
$\tau=1,3,10,30$. Attacks based on entanglement centrality provide the upper
bound for damage among the considered centrality measures. The spatial
locations of nodes match the positions of brain regions given by the dataset.
The size of the nodes is proportional to their degree. Dark gray indicates the
top 10% of the nodes according to entanglement centrality, at specific $\tau$.
Figure 6: Functional dismantling of Chilean Power Grid. The Chillean power
grid network ($N=218$) indicating power plants and substations is considered.
The effect of random failures and targeted attacks guided by iterative
betweenness, degree, eigenvector, closeness and entanglement centrality
measure on ARI and ASDD is plotted at multiple propagation time scales
$\tau=1,3,10,30$. Attacks based on entanglement centrality provide the upper
bound for damage among the considered centrality measures. The overall
effectiveness of structural measures are almost indistinguishable from each
other. The spatial locations of nodes match the positions of brain regions
given by the dataset. The size of the nodes is proportional to their degree.
Dark gray indicates the top 10% of the nodes according to entanglement
centrality, at specific $\tau$.  Figure 7: Functional dismantling of EU
Airlines. the European airlines network ($N=450$) is considered. The effect of
random failures and targeted attacks guided by iterative betweenness, degree,
eigenvector, closeness and entanglement centrality measure on ARI and ASDD is
plotted at multiple propagation time scales $\tau=1,3,10,30$. Attacks based on
entanglement centrality provide the upper bound for damage among the
considered centrality measures. The overall effectiveness of structural
measures are almost indistinguishable from each other. The spatial locations
of nodes match the positions of brain regions given by the dataset. The size
of the nodes is proportional to their degree. Dark gray indicates the top 10%
of the nodes according to entanglement centrality, at specific $\tau$.

In this section, we analyze the functional robustness of four empirical
interconnected systems, including two natural and two man-made: an averaged
human connectome ($N=188$) provided by the NKI-Rockland sample including brain
regions and their structural connections constructed using Diffusion Tensor
Imaging (DTI) Nooner  _et al._ (2012), the neural network of the nematode worm
C. Elegans ($N=297$) representing neurons linked by their neural junctions
Watts and Strogatz (1998), the reduced Chillean power grid network Kim  _et
al._ (2018); Son  _et al._ (2018) ($N=218$) representing power plants and
substations, and the European airlines network ($N=450$) Cardillo  _et al._
(2013), originally a multilayer with thirty-seven layers that we aggregate
into a single layer network.

While most well-known structural centrality measures have been generalized to
be able to cope with weighted networks, their success is mostly shown for
binary adjacency matrices in the literature. For this reason, these measures
are expected to be less competent for the analysis weighted network against a
functional measure like entanglement. Consequently, for the weighted real-
world networks studied here, such comparison can be unfair and, more
importantly, the result of such juxtapositions would not support the core
message of this study, which is to show that structural information can not
provide a multiscale proxy for node-node interactions. Therefore, we binarize
the weighted adjacency matrices, keeping only the elements above the threshold
of one sigma—i.e., one standard deviation above the average of elements—and
use the binarized networks to perform the analysis of functional robustness.

According to our results, attacks guided by entanglement exhibit an upper
bound for the damage reflected in ARI and ASDD, especially when long range
interactions are under investigation, in almost all scenarios (See Figures 4,
5, 7 and 6). In the human connectome and C. Elegans network the gap between
structural metrics and entanglement is more evident, compared with the
technological networks. Brain networks are characterized by different
interlinked regions, which are able to i) carry on specific tasks processing
information inside each module and ii) integrate information functioning as a
whole and enabling cognition, thanks to long-range links, a multiscale modular
organization Betzel and Bassett (2017), rich-clubs and cores Van Den Heuvel
and Sporns (2011) etc. Not only the network topology shapes flows in the
brain, but also different communications dynamics do Avena-Koenigsberger  _et
al._ (2018), since the creation of each synaptic connection involves a trade-
off between costs (in terms of energy) and benefits (e.g., increasing the
routing or the diffusion efficiency of the network, or the network
resilience). Hence, removing highly entangled nodes, may not disrupt the
network topologically but may have a non-trivial impact the brain functions.
The multiscale property of entanglement allows for the identification of
different groups of nodes that are important for short, middle and long range
signaling. For example, in the human connectome at $\tau=1$ and $\tau=3$ the
important regions are distributed almost equally across the right and left
hemispheres with a tilt from the occipital at $\tau=1$ to the temporal areas
at $\tau=3$. However, at middle propagation scales, $\tau=10$, most central
regions populate in the right hemisphere and at the large scales, $\tau=30$,
this asymmetry shifts to the left hemisphere.

As mentioned earlier, for very sparse networks or when one is interested in
exploring only short range interactions limited to the locality, the structure
is expected to plays a dominant role and, therefore, structural centrality
measures become reliable for dismantling the flow. Our results clearly confirm
the former expectation, as the structural centrality measures exhibit a better
performance at small temporal scales, where nodes interact inside their
neighborhood and the adjacency matrix provides a good proxy for the flow, in
all considered cases. It can also be mathematically demonstrated in terms of a
Taylor expansion of the propagator at $\tau\ll 1$, as
$\hat{U}(\tau,\hat{W})\approx\hat{I}-\tau\hat{H}(\hat{W})=\hat{I}-\tau\hat{K}(\hat{W})+\tau\hat{W}$.
The off-diagonal elements are directly given by the adjacency operator
$\hat{W}$, with $\hat{I}$ being the identity and $\hat{K}(\hat{W})$ the
diagonal degree matrix. The latter expectation is also supported by our
empirical network analysis. As shown for European Airlines and Chilean Power
Grid, the sparseness of technological networks make them more vulnerable to
functional impairment under attacks based on structural metrics. For example,
the Chilean power grid network with $N=218$ has only $527$ links.
Nevertheless, even in such cases, entanglement centrality outperforms the
structural measures, provides insights into the multiscale nature of node
importance for information flow and of the process-driven network geometry,
and identifies groups of nodes that are specifically important at every scale
and whose removal has also the strongest impact on the diffusion geometry of
the network.

##  VII discussion

Instead of the traditional approach to the problem of network robustness that
is based on quantifying the resistance of networks to structural impairment,
we explored the functional effect of progressive damages. To this aim,
exploiting diffusion dynamics coupled with the structure, we introduced two
descriptors: the average received information per node and the average squared
diffusion distance between the nodes. Thus, we assessed the effect of random
and targeted attacks guided by a number of widely used centrality measures on
these metrics, in a broad range of scenarios including synthetic and empirical
networks.

Our results indicate that removing the nodes with high topological centrality,
for example the ones having high degree or betweenness,  might have a
surprisingly insignificant effect on the flow dynamics. More specifically, the
effectiveness of attack strategies guided by such structural metrics  can be
indistinguishable from that of random removal of nodes, most evident when mid-
to long-range propagation time-scales are considered and the network is not
extremely sparse. This finding stands as further evidence supporting the
paradigm shift happening in today’s network science, based on the fact that
structural information, on their own, can be unreliable. For instance, the
limitation of iterative betweenness in identifying the important nodes
suggests that the transport phenomena can not be captured only in terms of
shortest paths, an approximation taken for granted in the literature.
Alternatively, we have shown that using statistical physics of complex
information dynamics, one can identify nodes whose removal directly impacts
the diversity of flow pathways in the system as a whole and the system’s
geometry induced by the information diffusion, leading to a quick impairment
of the flow dynamics  in a range of synthetic and a multitude of empirical
systems across scales, from the neural network of C. Elegans and Human
Connectome to Chilean Power Grid and European Airlines.

Overall, apart from the practical aspects such as flow dismantling and
multiscale identification of central units, our work provides insights into
both applicability and limitation of structural metrics in capturing complex
collective phenomena, such as information dynamics. In other words, the
functional robustness framework has been able to differentiate two regimes:
i) where the role played by the network is dominant and structural metrics can
be reliably used as fast and effective tools to proxy interactions and ii)
where the structural measures are not sufficiently sophisticated to capture
the complexity arising from the coupling between structure and dynamics. In i)
the system’s units are far apart in the diffusion space, because at very small
time scales the flows are localized around their source and the dynamical
proximity of the nodes in the network, captured by the diffusion distance,
depends mostly on the local connectivity. While in regime ii) the intermediate
time scales of the dynamics allow the integration of local and global
connectivity information, so that the geometry is able to reveal not only the
fine details of the topology, but also the interplay between the global
structure and the flow pathways between pairs of nodes. The link between
geometry, flow dynamics and functional robustness is the spectrum of the
Laplacian matrix: Its eigenvalues quantify the diversity of flow pathways in
the system, but also the characteristic times of a random walk among its units
and the shape of the system in its diffusion space.

Our work introduces a novel functional perspective into the robustness
analysis of interconnected systems, indicates the failure of structural
metrics in identifying the nodes central for information dynamics and
highlights the power of methods grounded in statistical physics and geometry
in unraveling the complex interplay between the structure and dynamics.

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##  Appendix A Diffusion geometry

### On the diffusion distance

We start by showing that the Laplacian matrix
$\hat{H}(\hat{W})=\hat{K}(\hat{W})-\hat{W}$ is a $Q-$matrix on the set of
nodes $V$ of the network $G=(V,E)$ or, more precisely, that
$-\hat{H}(\hat{W})$ does. In the jargon of probability theory, a $Q-$matrix
Norris (1997) on $V$ is a matrix $Q=\left(q_{ij}:i,j\in V\right)$ such that

  * (i)

$0\leq-q_{ii}<+\infty$ for all $i$

  * (ii)

$q_{ij}\geq 0$ for all $i\neq j$

  * (iii)

$\sum\limits_{j\in V}q_{ij}=0$ for all $i$.

The check is immediate. This is equivalent (Norris, 1997, Thm. 2.1.2) to
proving that $e^{-\tau\hat{H}(\hat{W})}$ is a stochastic matrix for all
$\tau\geq 0$, i.e., it has non-negative elements and its rows sum up to 1.
From a $Q-$matrix $Q$ it is always possible to obtain its jump matrix
$\Pi=\left(\pi_{ij}:i,j\in V\right)$ setting

$\displaystyle\pi_{ij}$
$\displaystyle=\begin{cases}\frac{q_{ij}}{-q_{ii}}&j\neq i\And q_{ii}\neq 0\\\
0&j\neq i\And q_{ii}=0\\\ \end{cases}$ $\displaystyle\pi_{ii}$
$\displaystyle=\begin{cases}0&q_{ii}\neq 0\\\ 1&q_{ii}=0.\end{cases}$

Observe that the jump matrix corresponding to our $\hat{H}(\hat{W})$ is
$\hat{K}(\hat{W})^{-1}\hat{W}$—$q_{ij}=W_{ij}$ and $-q_{ii}=k_{1}$—with ones
on the diagonal for isolated nodes, if any. Finally, by (Norris, 1997, Thm.
2.8.2) we have that the master equation (1) defines a continuous-time Markov
chain with generator $-\hat{H}(\hat{W})$ on $V$, i.e., a right-continuous
process with (independent) exponential holding times of rates $k_{i}$. In
network terms this is an edge-centric continuous-time random walk with rate
$k_{i}$ of leaving the $i-$th node Masuda  _et al._ (2017) and $\langle
x_{j}|e^{-\tau\hat{H}(\hat{W})}|x_{i}\rangle=:p_{\tau}(i,j)$ represents the
transition probability from $i$ to $j$ in time $\tau$ or equivalently, for
fixed initial node $i$,
$e^{-\tau\hat{H}(\hat{W})}|x_{i}\rangle=p_{\tau}(i,\cdot)=:p_{\tau}(i)$ is a
bump function on $i$ with increasing support as $\tau$ grows Coifman and Lafon
(2006). The diffusion distance (3) can then be seen as a distance between bump
functions and, in this case, it is useful to write its eigenmode expansion
using the (inverse) graph Fourier transform Sandryhaila and Moura (2013);
Masuda  _et al._ (2017). The Laplacian matrix $\hat{H}(\hat{W})$ is real and
symmetric, for undirected networks, it can then be diagonalized
$\hat{H}(\hat{W})=\hat{Q}\Lambda\hat{Q}^{*}$, where $\hat{Q}=\hat{Q}(\hat{W})$
is an $N\times N$ matrix of the orthonormalized eigenvectors
$|\varphi_{\ell}\rangle,\ell=0,1,...N-1$ of $\hat{H}(\hat{W})$ such that
$\langle\varphi_{\ell}|\varphi_{\ell^{\prime}}\rangle=\delta_{\ell\ell^{\prime}}$,
i.e., $\hat{Q}\hat{Q}^{*}=\hat{Q}^{*}\hat{Q}=\hat{I}$, and
$\hat{\Lambda}=\hat{\Lambda}(\hat{W})$ is the diagonal matrix of its
eigenvalues $\lambda_{\ell},\ell=0,1,...N-1$. Consequently,
$e^{-\tau\hat{H}(\hat{W})}=\hat{Q}e^{-\tau\hat{\Lambda}}\hat{Q}^{*}$ and

$\displaystyle p_{\tau}(i,k)$
$\displaystyle=\sum_{\ell=0}^{N-1}a_{\ell}(i;\tau)\varphi_{\ell}(k)$ with
$\displaystyle\begin{cases}&a_{\ell}(i;\tau)=e^{-\tau\lambda_{\ell}}a_{\ell}(i;0)\\\
&a_{\ell}(i;0)=\langle
x_{i}|\varphi_{\ell}\rangle=:\varphi_{\ell}(i)\end{cases}$

Eq. (3) can then be re-written as

$\displaystyle D^{2}_{\tau}(i,j)$ $\displaystyle=\lVert
p_{\tau}(i)-p_{\tau}(j)\rVert^{2}=\sum_{\ell}\left(a_{\ell}(i;\tau)-a_{\ell}(j;\tau)\right)^{2}$
$\displaystyle=\sum_{\ell}e^{-2\tau\lambda_{\ell}}\left(\varphi_{\ell}(i)-\varphi_{\ell}(j)\right)^{2}$
(5)

Finally, observe that if network $G$ is undirected and fully connected then
the eigenvalues of $\hat{H}(\hat{W})$ are $\lambda_{0}=0$ and
$\lambda_{\ell}=N$ for all $\ell=1,\dots,N-1$. $\varphi_{0}$ is constant and
can be removed from the summation (5) hence
$D^{2}_{\tau}(i,j)=e^{-2N\tau}\sum_{\ell\geq
1}\left(\varphi_{\ell}(i)-\varphi_{\ell}(j)\right)^{2}$, so that distances are
determined uniquely by the eigenspace of $\lambda=N$. Equivalently, this can
be seen using the fact that in this case $\hat{H}(\hat{W})=(N-1)I-\hat{W}$
and, since $I\hat{W}=\hat{W}I$,
$e^{-\tau\hat{H}(\hat{W})}=e^{-\tau(N-1)}e^{\tau\hat{W}}$, where
$e^{\tau\hat{W}}$ contains the information about the number of paths of
increasing length between pairs of nodes, which does not depend on its
extremes for fully connected networks Estrada and Hatano (2008).

### Average square diffusion distance (ASDD)

Eq. (1) maps nodes in a network to a cloud of points in the diffusion space,
whose dispersion can be quantified using a generalized scalar measure of
variance as the trace of their (sample) covariance matrix. We here show that
this is equivalent to computing the ASDD.

Let us fix $\tau>0$, so we can drop it from the notation, and call
$p_{ij}=\langle x_{j}|e^{-\tau\hat{H}(\hat{W})}|x_{i}\rangle$. Eq. (4) becomes

$\displaystyle\mathcal{D}^{2}(\tau,\hat{W})$
$\displaystyle=\frac{1}{2N^{2}}\sum_{i,j=1}^{N}\sum_{k=1}^{N}\left(p_{ik}-p_{jk}\right)^{2}$

while the trace of the sample covariance matrix $C$ corresponding to the $N$
vectors
$\left\\{e^{-\tau\hat{H}(\hat{W})}|x_{i}\rangle,~{}i=1,\dots,N\right\\}$ is

$\displaystyle\text{Tr}\left(C\right)=\frac{1}{N}\sum_{i=1}^{N}\sum_{k=1}^{N}\left(p_{ik}-\bar{p}_{i}\right)^{2}.$

The equivalence can be proved using a known property of the variance—also
known as variance deformation formula Zhang  _et al._ (2012)—, but it is also
easily proved “by hand” in case particular case of undirected networks, where
the Laplacian matrix $\hat{H}(\hat{W})$ is symmetric. Firstly, observe that
for all $i$ $\bar{p}_{i}=\frac{1}{N}\sum_{k=1}^{N}p_{ik}=\frac{1}{N}=:\bar{p}$
Then,

$\displaystyle\text{Tr}\left(C\right)$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\left(\sum_{k=1}^{N}p_{ik}^{2}-2\bar{p}\sum_{k=1}^{N}p_{ik}+\sum_{k=1}^{N}\bar{p}^{2}\right)^{2}$
$\displaystyle=\frac{1}{N}\left(\sum_{i=1}^{N}m^{2}_{i}-1\right)$

where we called $m^{2}_{i}=\sum_{k=1}^{N}p_{ik}^{2}$ the raw second moment of
the $i-$th random vector. Similarly,

$\displaystyle\mathcal{D}^{2}(\tau,\hat{W})$
$\displaystyle=\frac{1}{2N^{2}}\sum_{i,j=1}^{N}\sum_{k=1}^{N}p_{ik}^{2}-2\sum_{k=1}^{N}p_{ik}p_{jk}+\sum_{k=1}^{N}p_{jk}^{2}$
$\displaystyle=\frac{1}{2N^{2}}\sum_{i,j=1}^{N}\left(m_{i}^{2}-2(P^{2})_{ij}+m_{j}^{2}\right)$
$\displaystyle=\frac{1}{2N^{2}}\left(2N\sum_{i=1}^{N}m_{i}^{2}-2\sum_{i=1}^{N}\sum_{j=1}^{N}(P^{2})_{ij}\right)$
$\displaystyle=\frac{1}{N}\left(\sum_{i=1}^{N}m_{i}^{2}-1\right)$

where we used the undirected network assumption and the semigroup property of
$e^{-\tau\hat{H}}$ to write $p_{jk}=p_{kj}$ yielding
$\sum_{k=1}^{N}p_{ik}p_{jk}=(P^{2})_{ij}$ and then $\sum_{j}(P^{2})_{ij}=1$.

Finally, two minor observations: firstly, here we use the biased sample
covariance, but its unbiased version, with factor $\frac{1}{N-1}$ instead of
$\frac{1}{N}$, can also be used provided that the sum of squared distances is
also divided by $\frac{1}{N(N-1)}$ instead of $\frac{1}{N^{2}}$, i.e., the
zeros on the diagonal are not counted in in the sum. Secondly, the factor
$\frac{1}{2}$ in the ASDD definition can also be seen as a re-scaling into
$[0,1]$ of the diffusion distances, which are indeed bounded in $[0,\sqrt{2}]$
Bertagnolli and De Domenico (2021).

##  Appendix B Network entanglement centrality

It is possible to describe the macroscopic state of complex interconnected
systems, in terms of physical quantities such as entropy De Domenico and
Biamonte (2016) and free energy, in a multi-resolution approach determined by
the propagation time-scale $\tau$ of signals, introduced in the text. The
density matrix derived from the formalism Ghavasieh  _et al._ (2020) reads

$\displaystyle\hat{\rho}(\tau,\hat{W})=\frac{\hat{U}(\tau,\hat{W})}{\text{Tr}\left(\hat{U}(\tau,\hat{W})\right)},$
(6)

and the corresponding Von Neumann entropy is given by

$\displaystyle\mathcal{S}(\tau,\hat{W})=-\text{Tr}\left(\hat{\rho}(\tau,\hat{W})\log_{2}{\hat{\rho}(\tau,\hat{W})}\right).$
(7)

Interestingly, the Von Neumann entropy measures the mixedness of the ensemble
of stream operators directing the information propagation. In other words, it
provides a proxy for diversity of flow pathways in the system, characterizing
the functional diversity of nodes in sending or receiving information
Ghavasieh  _et al._ (2020). More specifically, entropy is shown to be
inversely proportional to the average overlap of the flow emanating from every
pair of nodes $i$ and $j$, given by $\hat{U}(\tau,\hat{W})|x_{i}\rangle$ and
$\hat{U}(\tau,\hat{W})|x_{j}\rangle$ (See Fig. 8). For instance, in a totally
dismantled network of $N$ isolated nodes and no connections, the flow vectors
initiated by nodes have zero overlap, leading to highest possible entropy
$\log_{2}{N}$ De Domenico and Biamonte (2016) and largest possible pairwise
diffusion distances $D_{\tau}(i,j)=\sqrt{2}$ for all $i,j$ and $\tau$ and,
consequently, to the maximum spreading of the nodes in the diffusion space,
i.e., $\mathcal{D}^{2}(\tau,\hat{(}W))=1$. Conversely, the propagation vectors
have the highest possible overlap in a fully connected network. Also in this
case the diffusion distance is constant for all pairs of nodes and is uniquely
determined by $\tau$ and by the number of nodes (see the Appendix A).

Figure 8: Flow overlap. Propagation of flow from two nodes, represented in red
and yellow, as arrows emanating from them. The overlap region is colored in
gray—i.e., the nodes within the gray zone receive information from both
initiators. The overlap between the flow emanating from any pair of nodes, $i$
and $j$, can be calculated using cosine similarity between the corresponding
propagation vectors $\hat{U}(\tau,\hat{W})|x_{i}\rangle$ and
$\hat{U}(\tau,\hat{W})|x_{j}\rangle$. Conversely, average cosine dissimilarity
of propagation vectors indicates the diversity of flow pathways in the system,
which is proportional to the system’s Von Neumann entropy Ghavasieh  _et al._
(2020).

Assume the initial entropy a network, before damage, is
$\mathcal{S}^{(0)}(\tau,\hat{W})$. At the final stage, where all the nodes are
detached—i.e., all the links connecting them are removed—and the full
structural and functional dismantling is achieved, it follows
$\mathcal{S}^{(f)}(\tau,\hat{W})=\log_{2}{N}$. If we want the fastest
transition from $\mathcal{S}^{(0)}(\tau,\hat{W})$ to
$\mathcal{S}^{(f)}(\tau,\hat{W})$, in what order should we rank and,
respectively, disrupt the units? The simplest answer to this question is that
at each step, we remove the node whose removal has the maximum increase in the
Von Neumann entropy. For instance, let $\mathcal{S}^{(m-1)}(\tau,\hat{W})$ be
the entropy of the network before the removal of $m$-th node and
$\mathcal{S}^{(m)}_{i}(\tau,\hat{W})$ be the entropy of the network if one
removes the $i$-th node at the $m$-th step (See Fig. 9). The entanglement
Ghavasieh  _et al._ (2021) between the $i$-th node and the network at that
step is defined as

$\epsilon_{i}^{(m-1)}(\tau,\hat{W})=\mathcal{S}^{(m)}_{i}(\tau,\hat{W})-\mathcal{S}^{(m-1)}(\tau,\hat{W}),$
(8)

that can be used as a measure quantifying the importance of that node for
diversity of flow pathways. At each step, we identify the node with maximum
entanglement by

$\displaystyle\max_{i}[\epsilon_{i}^{(m-1)}(\tau,\hat{W})]=\max_{i}[\mathcal{S}^{(m)}_{i}(\tau,\hat{W})].$
(9)

Note that the second term in (8) is independent of the node to be removed and,
therefore, it vanishes from the maximization. Assuming that the $i$-th node
has the highest entanglement at the $m$-th step, the entropy at $m$-th step
reads $\mathcal{S}^{(m)}(\tau,\hat{W})=\mathcal{S}^{(m)}_{i}(\tau,\hat{W})$.
Note that the algorithm must exclude the nodes that are already isolated,
before performing the maximization, since the detachment process does not make
sense for them.

It is worth mentioning that the detachment process considered here (See Fig.
9) is slightly different from the original definition Ghavasieh  _et al._
(2021), where the node was removed with its incident edges shaping an
independent star network. In fact, the previous definition has a nice
property, used to prove that the behavior of entanglement of a node at very
small $\tau$ is determined by its degree. However, it seems also natural to
think that the detached node becomes isolated from the rest of the network, as
considered here.

Figure 9: Detachment process. Node-network entanglement is defined in terms of
the entropy change due to detachment of one node. Network Von Neumann entropy
provides a proxy for the diversity of flow pathways Ghavasieh  _et al._
(2020): Therefore, the node having highest entanglement with the network is
important, as its removal severely affects information exchange. This makes
entanglement a powerful measure to identify the nodes that are functionally
critical for information dynamics and to find attack strategies to
functionally dismantle networks.

##  Appendix C Definitions

In this appendix we collect some common expressions used in the literature and
in this work, with the relative (tentative) definitions and examples from
different fields.

Expression |  Definition and examples
---|---
Structure |  The arrangement of and relations between the components of complex systems, often modeled in terms of networks where components are nodes and their connections are links. Sometimes topological structure, or simply ”the network topology”, is used to indicate the arrangement of dyadic interactions between units, in contrast with ”weighted structure”, which also includes the intensity of the interactions.
• In physics: The connections between states of physical systems determining
the rates of transitions between them. • In biology: The network encoding
connections between cells, organs or species. • In chemistry: The the
interrelations between chemical compounds in a chemical reaction network. • In
social sciences: The relationships between individuals in a social network. •
In transportation systems: The network of transportation routes connecting
districts, urban areas, regions, countries or continents.
Dynamical process |  Quantities or fields change with respect to time, according to rules imposed by differential equations each known as a dynamical process.
• In physics: The thermalization protocol describing the transitions between
physical states of a system. • In biology: The biochemical equation governing
the spreading of chemicals or electrical signals among cells, organs or
species. • In chemistry: The reaction-diffusion equations describing the
behavior of the population corresponding to a chemical reaction network. • In
social sciences: The consensus dynamics between individuals in a social
network, or the dynamical equations describing the spread of pathogens or news
between them. • In transportation systems: The equation describing the flow of
people or goods through the network of transportation routes connecting
districts, urban areas, regions, countries or continents.
Information exchange |  A general term to describe the effect of components of complex systems on each other, often modeled in terms of the flow of a physical quantity between pairs of nodes, governed by dynamical processes. Communication is here used as a synonym.
• In physics: The exchange of particles between two physical states, induced
by a themalization protocol or external forces. • In biology: The exchange of
electrochemical signals between two cells, organs or species. • In chemistry:
The influence of two chemical compounds on each other, leading to a change in
their populations, in a chemical reaction network. • In social sciences: The
consensus dynamics between individuals in a social network, or the dynamical
equations how one individual infects the other with a pathogen or informs the
other about a news. • In transportation systems: The exchange of people or
goods between two nodes, through the network of transportation routes
connecting districts, urban areas, regions, countries or continents.
Information flow |  Emanation of a quantity or field, whose exchange between the components proxies their communications, from a source, often considered to be one of the components, into the system, through the links.
• In physics: The flow of particles from one physical state into others,
induced by a themalization protocol or external forces. • In biology: The flow
of electrochemical signals from cells, organs or species. • In chemistry: The
impact of a chemical compound on the populations of others, in a chemical
reaction network. • In social sciences: The flow of pathogen or news from an
individual to the rest. • In transportation systems: The flow of people or
goods from one of the nodes, through the network of transportation routes
connecting districts, urban areas, regions, countries or continents.
Impairment of information flow |  When damage significantly lowers the exchange of the field, that proxies communications, between the components across the system.
• In physics: When external perturbations lower the flow of particles from one
physical state into others, induced by a themalization protocol or external
forces. • In biology: When damage lowers the flow of electrochemical signals
from cells, organs or species. • In chemistry: When external perturbations
lowers the impact of a chemical compound on the populations of others, in a
chemical reaction network. • In social sciences: When damage lowers the flow
of pathogen or news from an individual to the rest. • In transportation
systems: When damage lowers the flow of people or goods from one of the nodes,
through the network of transportation routes connecting districts, urban
areas, regions, countries or continents.
Function |  System specific tasks expected to be performed, that can involve a single component or a group of components that exchange information with each other.
• In physics: The activity of a classical or quantum heat engine with certain
properties like power and efficiency. • In biology: The physiological activity
of a cell, organ, system or body. • In chemistry: The characteristic behavior
of a chemical compound or groups of chemicals linked in a chemical reaction
network. • In social sciences: The activity or behavior of an individual or a
group of them in a society. • In transportation systems: Financial activities
that depend on the flow of people or goods from one of the nodes, through the
network of transportation routes connecting districts, urban areas, regions,
countries or continents.
Operation |  While function and operation are used interchangeably in the literature, the more precise definition of operation is the method and mechanism by which a system or a part of it performs its function.
Degradation of function |  When damage disturbs the components performing a function or impairs the flow of information between groups of component, preventing them from performing a function.
• In physics: Perturbed activity of a classical or quantum heat engine that
reflects in certain properties like power and efficiency. • In biology:
Significant perturbation of physiological activity of a cell, organ or body. •
In chemistry: Significant perturbation of the dynamics of a groups of
chemicals linked in a chemical reaction network. • In social sciences:
Significant disturbance in activity of an individual or a group of them in a
society. • In transportation systems: Hindering of the flow of people or goods
from one of the nodes, through the network of transportation routes connecting
districts, urban areas, regions, countries or continents.


# A Radio Counterpart to a Jupiter-Mass Binary Object in Orion

Luis F. Rodríguez Instituto de Radioastronomía y Astrofísica, Universidad
Nacional Autónoma de México
Apartado Postal 3–72, 58090 Morelia, Michoacán, México  Mesoamerican Center
for Theoretical Physics
Universidad Autónoma de Chiapas
Carretera Emiliano Zapata Km 4, 29050 Tuxtla Gutiérrez, Chiapas, México
Laurent Loinard Instituto de Radioastronomía y Astrofísica, Universidad
Nacional Autónoma de México
Apartado Postal 3–72, 58090 Morelia, Michoacán, México  Luis A. Zapata
Instituto de Radioastronomía y Astrofísica, Universidad Nacional Autónoma de
México
Apartado Postal 3–72, 58090 Morelia, Michoacán, México

###### Abstract

Using James Webb Space Telescope near-infrared data of the inner Orion Nebula,
Pearson & McCaughrean (2023) detected 40 Jupiter-Mass Binary Objects (JuMBOS).
These systems are not associated with stars and their components have masses
of giant Jupiter-like planets and separations in the plane of the sky of order
$\sim$100 au. The existence of these wide free-floating planetary mass
binaries was unexpected in our current theories of star and planet formation.
Here we report the radio continuum (6.1 and 10.0 GHz) Karl G. Jansky Very
Large Array detection of a counterpart to JuMBO 24. The radio emission appears
to be steady at a level of $\sim$50 $\mu$Jy over timescales of days and years.
We set an upper limit of $\simeq 15$ km s-1 to the velocity of the radio
source in the plane of the sky. As in the near-infrared, the radio emission
seems to be coming from both components of the binary.

Exoplanets (498) – Free floating planets (549) – Radio astrometry (1337) –
Radio continuum emission (1340)

††facilities: NASA/ESA/CSA(JWST), NRAO(VLA)††software: AIPS (Greisen, 2003),
CASA (McMullin et al., 2007)

##  1 Introduction

In a new James Webb Space Telescope (JWST) near-infrared survey of the inner
Orion Nebula and Trapezium Cluster, Pearson & McCaughrean (2023) detected 40
Jupiter- Mass Binary Objects (JuMBOS). Additionally, two triple objects were
reported. These multiple systems are not associated to stars and their
components have masses in the range of 0.6 to 14 Jupiter masses and
separations in the plane of the sky between 28 and 384 au (0.07 and 0.99
arcsec at a distance of 388 pc; Kounkel et al. 2017). The existence of these
wide planetary-mass binaries is surprising since our current knowledge of star
and planet formation cannot account for them. A possible theoretical
explanation has been offered by Wang et al. (2023) in terms of an ejection
after a close stellar flyby between two solar-like stars. Rogue planets
(single planets not gravitationally bound to any star or brown dwarf, also
known as free-floating or orphan planets) have been detected in the optical
and near-infrared (Lucas & Roche, 2000; Zapatero Osorio et al., 2000). This
latter class of single sources, however, is compatible with the known physical
conditions of forming stellar and planetary systems (Hurley & Shara, 2002;
Kroupa & Bouvier, 2003; Whitworth & Zinnecker, 2004; Miret-Roig, 2023). We
have searched for radio counterparts to the 42 JuMBOS in Orion using sensitive
observations from the archive of the Karl G. Jansky Very Large Array (VLA) of
the National Radio Astronomy Observatory. Here we report the radio detection
of JuMBO 24 in three epochs distributed across a decade, in 2012, 2018, and
2022. This paper is organized as follows. In Section 2 we present the
observations, while in Section 3 we discuss the implications of this
detection. Finally, in Section 4 we summarize our conclusions.

##  2 Observations

###  2.1 VLA Observations

We analyzed the VLA data of the three projects listed in Table 1. The data
were calibrated in the standard manner using the CASA (Common Astronomy
Software Applications; McMullin et al. 2007) package of NRAO and the pipeline
provided for VLA111https://science.nrao.edu/facilities/vla/data-
processing/pipeline observations. A robust weighting of 0 (Briggs, 1995) was
used in the imaging to optimize the compromise between sensitivity and angular
resolution. The images were made with a $(u,v)$ range set to
$>100~{}k\lambda$, to suppress structures larger than 2′′ that limit the
signal-to-noise ratio in the Orion field. The project SD0630 consists of five
sessions made on 2012 September 30 and October 2, 3, 4, and 5. The projects
17A-069 and 22A-285 were made on single sessions on 2018 May 11 and 2022 July
2, respectively. In Table 1 we also give the mean epoch of the observations,
the configurations used, the central frequency and the bandwidth of the
observations, the parameters of the synthesized beam, the radio flux density
and position of JuMBO 24, and the 4-$\sigma$ upper limit to the absolute
circular polarization. The gain calibrator, J0541$-$0541, was the same for all
observations so the astrometry can be directly compared more easily. The
absolute flux calibrator was J0137$+$331 (3C 48) for projects SD0630 and
22A-285, and J0521$+$1638 (3C 138) for project 17A-069.

We find that the near-infrared source JuMBO 24 coincides within 0.05 arcsec
with a radio continuum source present in all three VLA images. The position of
JuMBO 24 given by Pearson & McCaughrean (2023) is RA(J2000) =
$05^{h}35^{m}19\hbox to0.0pt{.\hss}^{s}50616$, DEC(J2000) =
$-05^{\circ}23^{\prime}39\hbox to0.0pt{.\hss}{{}^{\prime\prime}}7303$. The
positions of the radio counterpart source for the three epochs are given in
Table 1. This radio source first appeared in the list of Forbrich et al.
(2016, their source 441). JuMBO 24 was originally detected in near-infrared
Hubble Space Telescope observations in the F160W (1.4-1.8 $\mu$m) and F110W
(0.8-1.4 $\mu$m) filters, with magnitudes of 17.11 and 18.33, respectively
(Luhman et al., 2000, their source 274). Slesnick et al. (2004) reported a
K-band magnitude of 19.51 (their source 728). These authors used near-infrared
photometry and spectroscopy as well as evolutionary stellar models (D’Antona &
Mazzitelli, 1994; Baraffe et al., 1998) to determine an infrared spectral type
of M5.5$\pm$1.5 and a mass of $\sim 52$ MJ, under the assumption of a single
star. The JWST data and the interpretation of Pearson & McCaughrean (2023)
indicate that JuMBO 24 is a binary with both components having the same giant-
planet mass of $11.5~{}M_{J}$ and a separation of 28 au in the plane of the
sky. JuMBO 24 stands out in the list of these objects because it has the
largest total mass ($23~{}M_{J}$) and the closest separation in the plane of
the sky (28 au). Also, the primary of JuMBO 24 has the second smallest
extinction, $A_{V}$ = 3.6, behind only JuMBO 27, that has $A_{V}$ = 2.4.

Table 1: Parameters of the VLA Observations

| Mean |  | $\nu$ | $\Delta\nu$ | Synthesized Beam | Observational results for JuMBO 24
---|---|---|---|---|---|---
Project | Epoch | Configuration | (GHz) | (GHz) | ($\theta_{max}\times\theta_{min}$; PA) |  $F_{\nu}~{}(\mu$Jy) | RA(J2000) | DEC(J2000) | $|V|$
SD0630 | 2012.762 | A, BnA$\rightarrow$A | 6.1 | 2.0 | $0\hbox to0.0pt{.\hss}{{}^{\prime\prime}}25\times 0\hbox to0.0pt{.\hss}{{}^{\prime\prime}}21;+2^{\circ}$ | 45.4$\pm$4.5 | $19\hbox to0.0pt{.\hss}^{s}5061\pm 0\hbox to0.0pt{.\hss}^{s}0006$ | $39\hbox to0.0pt{.\hss}{{}^{\prime\prime}}730\pm 0\hbox to0.0pt{.\hss}{{}^{\prime\prime}}012$ | $\leq 15\%$
17A-069 | 2018.359 | A | 10.0 | 4.0 | $0\hbox to0.0pt{.\hss}{{}^{\prime\prime}}21\times 0\hbox to0.0pt{.\hss}{{}^{\prime\prime}}17;-17^{\circ}$ | 60.0$\pm$13.0 | $19\hbox to0.0pt{.\hss}^{s}5065\pm 0\hbox to0.0pt{.\hss}^{s}0017$ | $39\hbox to0.0pt{.\hss}{{}^{\prime\prime}}727\pm 0\hbox to0.0pt{.\hss}{{}^{\prime\prime}}024$ | $\leq 40\%$
22A-285 | 2022.501 | A | 6.1 | 4.0 | $0\hbox to0.0pt{.\hss}{{}^{\prime\prime}}27\times 0\hbox to0.0pt{.\hss}{{}^{\prime\prime}}21;+10^{\circ}$ | 50.0$\pm$7.2 | $19\hbox to0.0pt{.\hss}^{s}5059\pm 0\hbox to0.0pt{.\hss}^{s}0016$ | $39\hbox to0.0pt{.\hss}{{}^{\prime\prime}}698\pm 0\hbox to0.0pt{.\hss}{{}^{\prime\prime}}017$ | $\leq 20\%$

Note. — Columns 4 and 5 are the central frequency and bandwidth of the
observations. The position given in columns 8 and 9 show only the seconds of
the RA and the arcseconds of the DEC, measured from RA(J2000) = $05^{h}35^{m}$
and DEC(J2000) = $-05^{\circ}23^{\prime}$, respectively. Column 10 gives the
4-$\sigma$ upper limit to the absolute circular polarization.

###  2.2 ALMA Observations

We searched for JuMBO counterparts in the Atacama Large
Millimeter/Submillimeter Array archive. We analyzed several data sets:
2013.1.00546.S, 2015.1.00534.S, 2015.1.00262.S, 2015.1.00669.S,
2017.1.01353.S, 2017.1.01639.S, and 2018.1.01107.S. However, we do not find
any evident millimeter or even submillimeter continuum counterpart for any of
the 42 JuMBOs. In particular, the data set 2015.1.00669.S (PI: Hacar, Alvaro)
is a large mosaic made in Band 3 that covers a large area (OMC2, OMC1-N, OMC1,
OMC1-S). The rms noise for this large mosaic is 150 $\mu$Jy Beam-1, so we do
not detect any counterpart at a 5-$\sigma$ level of 750 $\mu$Jy Beam-1. Future
much deeper ALMA observations of the Orion Nebula could reveal thermal
emission related with the JuMBOS.

##  3 Discussion

The first centimeter detection of a single planetary mass object, SIMP
J01365662+0933473, was achieved by Kao et al. (2018). The first identification
of a centimeter continuum source with a planetary-mass binary object presented
here is also of great relevance by the information the radio data can provide.
We exemplify these possibilities in what follows. Further work, particularly
in relation to the origin of the radio emission is deferred to forthcoming
papers.

###  3.1 The Probability of a Random Association

In this section we estimate how likely it is that the positional agreement of
JuMBO 24 with one of the radio sources in the region is fortuitous and not
real. In their study of Orion, Forbrich et al. (2016) detected a total of 556
radio sources in a region of $13\hbox to0.0pt{.\hss}’6\times 13\hbox
to0.0pt{.\hss}’6$. Most of the sources, however, are enclosed in a region with
radius of $\sim 3’$ (see Figure 5 of Forbrich et al. (2016)). Of the 556
sources, 358 have flux densities equal or larger than 50 $\mu$Jy, the
characteristic value for the radio source proposed to be associated with JuMBO
24. These results give a solid angle density of $3.5\times 10^{-3}$ radio
sources per arcsec2.

The most accurate position for the radio source is that obtained from the
multiepoch project SD0630. Convolving the statistical errors of the position
of the radio source (Table 1) with the systematic positional error of $0\hbox
to0.0pt{.\hss}{{}^{\prime\prime}}01$, as estimated by Dzib et al. (2017), we
obtain a $\pm 1-\sigma$ error box of $0\hbox
to0.0pt{.\hss}{{}^{\prime\prime}}027\times 0\hbox
to0.0pt{.\hss}{{}^{\prime\prime}}031$. The a priori probability of a source
randomly falling in this error box is then $2.9\times 10^{-6}$. However, the
coincidence of the radio source with any of the 42 JuMBOs in Orion would have
been considered significant, so assuming that these possible coincidences are
independent events we have to multiply the probability of a given JuMBO source
being spatially associated times 42. We finally get an a priori probability of
one JuMBO randomly coinciding in position with one radio source to be only
$1.2\times 10^{-4}$, supporting a real association.

###  3.2 Upper Limits to the Proper Motion

Using the radio positions determined at the three epochs we can search for the
proper motions of the source. As noted before, the fact that the same nearby
($\sim 1\hbox to0.0pt{.\hss}^{\circ}5$) gain calibrator (J0541$-$0541) was
used in all observations implies a sound astrometry. A least-squares fit to
the positions given in Table 1, after adding in quadrature a systematic error
of $0\hbox to0.0pt{.\hss}{{}^{\prime\prime}}01$, yields the following values:

$\mu_{\alpha}\cos\delta=+1.8\pm
2.8\text{~{}mas~{}yr}^{-1}~{}~{};~{}~{}\mu_{\delta}=-3.0\pm
2.5\text{~{}mas~{}yr}^{-1}.$

This result implies that there are no significant proper motions at a
3-$\sigma$ level of $\sim$8 mas yr-1, that corresponds to a transverse
velocity of $\sim 15$ km s-1 at a distance of 388 pc (Kounkel et al., 2017).
These upper limits rule out large velocities in the plane of the sky, although
it should be pointed out that significant velocities could be present in the
radial direction. Our measurements can be registered to an approximate rest
frame for the Orion Cluster by subtracting the average proper motion of the
radio stars in the core of Orion (Dzib et al., 2017),
$\mu_{\alpha}\cos\delta=+1.07\pm
0.09\text{~{}mas~{}yr}^{-1};\mu_{\delta}=-0.84\pm
0.16\text{~{}mas~{}yr}^{-1}$. We obtain:

$\mu_{\alpha}\cos\delta=+0.7\pm
2.8\text{~{}mas~{}yr}^{-1}~{}~{};~{}~{}\mu_{\delta}=-2.2\pm
2.5\text{~{}mas~{}yr}^{-1}.$

We conclude that JuMBO 24 is not moving at large velocities compared to the
radio stars in Orion. van Elteren et al. (2019) have made numerical
simulations for the evolution of young and dense stellar clusters. They find
that single free-floating planets ejected from their original system after a
strong encounter with another star will have velocities typically three times
those of the stars in the cluster. Dzib et al. (2017) find a typical velocity
dispersion of $\sim 2-3$ km s-1 along a plane-of-the-sky coordinate for the
radio-emitting stars in the core of the Orion Nebula Cluster. Assuming that
the results of van Elteren et al. (2019) can be extended to free-floating
binary planets, we expect velocity dispersions of $\sim 6-9$ km s-1 for the
JuMBOs as well as for the $\sim$500 single planetary-mass objects reported in
Orion by Pearson & McCaughrean (2023). Future highly accurate determinations
of the kinematics of the Orion JuMBOs will help understand the origin of these
unexpected sources.

###  3.3 Time Monitoring

The two measurements at 6.1 GHz given in Table 1 are separated by a decade
and, within the noise, they are consistent with being equal. Another evidence
that weighs in favor of a steady flux is the analysis of the five different
epochs observed in project SD0630. In Table 2, we give the epochs and flux
densities for each individual observing session, where we can see that all
flux densities are consistent within noise with a value of $\simeq
45~{}\mu$Jy. While more intensive observations are needed to establish firmly
the time behavior of the radio emission from JuMBO 24, all existing data are
consistent with a steady flux density over timescales of days to years.

Table 2: Flux Densities of JuMBO 24 at 6.1 GHz from Project SD0630 | Mean UTC | Flux Density
---|---|---
Date | (hours) | ($\mu$Jy)
2012 Sep 30 | 11.3 | 52.9$\pm$6.5
2012 Oct 02 | 11.2 | 39.7$\pm$5.5
2012 Oct 03 | 14.8 | 55.0$\pm$11.0
2012 Oct 04 | 13.4 | 44.9$\pm$7.1
2012 Oct 05 | 12.3 | 40.5$\pm$6.6

###  3.4 Nature of the Radio Emission

The objects with known radio emission that are more similar to the components
of the JuMBOs are the ultracool dwarfs (Kao & Sebastian Pineda, 2022). The
radio emission from the latter objects has two main components: a periodically
bursting emission arising in aurorae and a slowly varying, quiescent emission
with a low degree of circular polarization arising from a structure similar in
morphology to the Jovian radiation belts (Climent et al., 2023; Kao et al.,
2023). The bursting component is circularly polarized and it is believed to be
produced by the electron cyclotron maser instability (Hallinan et al., 2006),
the same coherent emission process that creates the auroral radio emission
from the planets of the Solar System (Pineda et al., 2017). On the other hand,
the quiescent component is usually interpreted as being caused by radio
emission from relativistic electrons giving rise to gyrosynchrotron or
synchrotron mechanisms, according to how relativistic the electrons are. Now
it is known that this quiescent emission is produced in radiation belts
(Climent et al., 2023; Kao et al., 2023). Given that the radio emission from
JuMBO 24 is steady in time, one would expect it to be produced by relativistic
electrons spiraling in the magnetic fields of a radiation belt.

The spectral index of the radio emission is an indicator of the nature of the
radiation. The observations of SD0630 were made using two bandwidths of 1 GHz
each, centered at 4.8 and 7.4 GHz, respectively. We obtain that the flux
densities at these two frequencies are 53.8$\pm$6.0 and 43.0$\pm$5.8 $\mu$Jy,
respectively. Assuming that the continuum spectrum is described by a power
law, $S_{\nu}\propto\nu^{\alpha}$, we derive a spectral index of
$\alpha=-0.52\pm 0.40$. We can also derive a spectral index assuming that the
source has a steady flux density and combining the values given in Table 1.
From these data we derive $\alpha=+0.50\pm 0.43$. The weighted mean and rms of
these two determinations is $\alpha=-0.05\pm 0.51$. Given the large
uncertainty, it is difficult to discriminate between possible emission
mechanisms, but we can rule out an optically-thick blackbody in the Rayleigh-
Jeans regime ($\alpha$ = 2) and the negative steep indices characteristic of
pulsars ($\alpha\leq$ –2).

We did not detect circular polarization at any epoch. However, the upper
limits are not very stringent. The presence of circular polarization is a
strong indicator of either electron cyclotron maser instability, cyclotron or
gyrosynchrotron radiation but, of course, its absence does not rule out these
mechanisms.

As noted above, the ultracool dwarfs are the closest thing to giant planets
that have been detected in the radio. The radio luminosities of nine quiescent
binary ultracool dwarf systems discussed by Kao & Sebastian Pineda (2022) are
in the range of $10^{11.9-14.6}$ erg s-1 Hz-1. JuMBO 24 has a radio luminosity
of $10^{16.0}$ erg s-1 Hz-1. We then conclude that the radio luminosity of
JuMBO 24 is orders of magnitude larger than naively expected.

###  3.5 Position and Morphology of the Radio and Infrared Sources

In Figure 1 we show an image of the radio continuum emission obtained from
concatenating all the data of project SD0630. We can see that the radio source
coincides closely with the near-infrared position given by Pearson &
McCaughrean (2023). A Gaussian ellipsoid fit to the radio image using the task
IMFIT of CASA gives a deconvolved angular size of $0\hbox
to0.0pt{.\hss}{{}^{\prime\prime}}179\pm 0\hbox
to0.0pt{.\hss}{{}^{\prime\prime}}045\times 0\hbox
to0.0pt{.\hss}{{}^{\prime\prime}}143\pm 0\hbox
to0.0pt{.\hss}{{}^{\prime\prime}}074;60^{\circ}\pm 89^{\circ}$. Although the
radio source is only marginally resolved, these angular dimensions and the
orientation of the major axis are consistent with the value obtained from
Gaussian ellipsoid fits to the JWST images. For instance a fit to the image at
2.77 $\mu$m results in $0\hbox to0.0pt{.\hss}{{}^{\prime\prime}}170\pm 0\hbox
to0.0pt{.\hss}{{}^{\prime\prime}}002\times 0\hbox
to0.0pt{.\hss}{{}^{\prime\prime}}132\pm 0\hbox
to0.0pt{.\hss}{{}^{\prime\prime}}002;97^{\circ}\pm 2^{\circ}$. The shorter
wavelength JWST observations show that the cause of the extension along the
east-west direction is the binarity of the source, which is comprised of two
objects separated by about 100 mas precisely along the east-west direction.
This result suggests that, as in the near-infrared, there is significant radio
emission from both components of the binary. This result is similar to that
obtained for the compact ultra-cool dwarf binary VHS 1256$-$1257AB, where the
optical and radio emissions show compatible dimensions and orientation
(Rodriguez et al., 2023).

Figure 1: VLA image of the 6.1 GHz emission from JuMBO 24, made concatenating
all the data of project SD0630. The contours start at, and increase by steps
of, 9 $\mu$Jy beam-1. The color scale indicates intensity as shown by the
color bar to the right. The white rectangle indicates the position of JuMBO 24
as reported by Pearson & McCaughrean (2023): RA(J2000) = $05^{h}35^{m}19\hbox
to0.0pt{.\hss}^{s}50616$, DEC(J2000) = $-05^{\circ}23^{\prime}39\hbox
to0.0pt{.\hss}{{}^{\prime\prime}}7303$. The red ellipse shows the angular size
and orientation of the target in the 2.77 $\mu$m JWST images ($0\hbox
to0.0pt{.\hss}{{}^{\prime\prime}}170\times 0\hbox
to0.0pt{.\hss}{{}^{\prime\prime}}132;97^{\circ}$). The synthesized beam of the
radio map is shown at the bottom left of the image.

##  4 Conclusions

We report the detection of radio continuum emission at 6 to 10 GHz associated
with the recently identified Jupiter-Mass Binary Object JuMBO 24 in the Orion
Nebula Cluster. Within the errors, the radio emission appears steady on
timescales from days to years and exhibits no circular polarization. No
significant proper motion is detected between the three observing epochs,
ruling out large velocities ($\geq 15~{}km~{}s^{-1}$) in the plane of the sky.
The radio emission is marginally resolved in the same direction as the
infrared source detected by the JWST, suggesting that the radio emission comes
from a combination of the two planetary mass objects. Additional radio
observations are necessary to pin down the nature of the radio emission
mechanism.

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We thank the anonymous reviewer for several valuable comments that improved
the paper. The National Radio Astronomy Observatory is a facility of the
National Science Foundation operated under cooperative agreement by Associated
Universities, Inc. We acknowledge the support from DGAPA, UNAM (projects
IN112323, IN112417, IN112820 and IN108324) and CONAHCyT, México (projects
238631, 280775 and 263356).