# Temporal Graphs and Temporal Network Characteristics
for Bio-Inspired Networks During Optimization

N. DiBrita  Colgate University, Hamilton, NY 13346, USA  K. Eledlebi  Khalifa
University, Abu Dhabi 127788, UAE  H. Hildmann  Netherlands Organization for
Applied Scientific Research, 2597 AK, The Hague, NL  L. Culley  Colgate
University, Hamilton, NY 13346, USA  A. F. Isakovic  Colgate University,
Hamilton, NY 13346, USA  corresponding author email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Temporal network analysis and time evolution of network characteristics are
powerful tools in describing the changing topology of dynamic networks. This
paper uses such approaches to better visualize and provide analytical measures
for the changes in performance that we observed in Voronoi-type spatial
coverage, particularly for the example of time evolving networks with a
changing number of wireless sensors being deployed. Specifically, our analysis
focuses on the role different combinations of impenetrable obstacles and
environmental noise play in connectivity and overall network structure. It is
shown how the use of (i) temporal network graphs, and (ii) network centrality
and regularity measures illustrate the differences between various options
developed for the balancing act of energy and time efficiency in network
coverage. Lastly, we compare the outcome of these measures with the less
abstract classification variables, such as percent area covered, and
cumulative distance travelled.

Centrality Measures, Connectivity, Network Topology, Regularity, Temporal
Network Graphs

##  I Introduction, motivation, and research questions

Networks that evolve in time, such as infectious disease contact networks,
wireless sensor networks, and many others, have received considerable interest
in the past two decades [1, 2, 3]. Focus has been placed on understanding
their topology, interruptions (desirable and undesirable) within their
structure, their optimization and adaptive operation. Understanding networks
often relies on complex multi-parametric and/or multivariate settings. We
relied on harnessing the formalized mechanics of emergent behavior based on an
our own adaptation (cf. [4] for details) of the Voronoi tessellation [5] which
some consider a bio-inspired optimization technique [6].

In addition to a Voronoi-only approach, we used our own hybrid approach, with
Genetic Algorithm (GA) hybridized with Voronoi tessellation as the resulting
emergent self-organization behavior has been shown [5] to be promising for
solving coverage problems in realistic model situations. Our technique, termed
Bio-Inspired Self-Organizing Networks (BISON), leverages the converged
movement towards Voronoi cells’ centers with an intelligent node provisioning
algorithm to deliver a fully automated WSN that rapidly self-deploys itself
within any finite indoor environment without prior knowledge of the size and
structure of the target space [4, 5].

The BISON algorithm was merged with a localized Genetic Algorithm (GA) to push
the trade-off between the pace of space exploration and energy expense further
towards faster deployments, especially when faced with complex obstruction
structures. The mechanics of the approaches are omitted for reasons of brevity
but are summarized in SOM (Supporting Online Materials) - A and are available
(Open Access) in [4]. After considering the inherently dynamic nature of BISON
and its GA hybrid variant (GA + BISON), we relied on temporal network analysis
as a natural enhancement to the discovery and analysis of changes within the
networks generated by these algorithms [7]. Voronoi-type algorithms have been
utilized for similar purposes, with groups such as Wang et. al developing
different methods for node reallocation based on Voronoi edges [8], and Zou
et. al introducing a spreading algorithm (NSVA) to deploy a fixed number of
nodes within an obstacle free region [9].

Analyzing the behavior of animals in their environmental networks reveals that
they do not randomly interact with each other but rely on temporal properties
of their networks [10, 11]. Furthermore, interactions among different animal
species are inherently dynamic and change with time and the context of the
medium, such as reflections of the outside circumstances [12, 13, 14].

Thus, performing temporal network analysis on dynamic networks provides
information about individual members, the relationship between network nodes,
and allows to scale from a few-individuals behavior to a larger scale
population level [7]. Moreover, linking the dynamic patterns of connections in
time with the changing status of the members provides insight on the role of
the network structure for the specified application.

Centrality and correlations in temporal networks have previously been studied,
notably in [15, 16], where it is shown that correlations indicate the level of
connectivity in temporal networks. Naturally, there is a number of reasons
motivating the studies of temporal networks, ranging from the need to build
surrogate networks [17], examining the potential for the emergence of small
world networks [18, 19, 20], to studying networks as model system representing
phase transitions, such as [21] and [22].

In this paper, we demonstrate some of the advantages and trade-offs of BISON
(Voronoi-only) and GA-BISON (GA + Voronoi) in a sampling of different
environmental conditions, using a temporal network analysis framework
employing centrality measures and graph nodes statistics.

Specifically, we show that network theoretic characteristics and temporal
graphs offer a more thorough insight into time evolution of networks than
cumulative, application driven measures, such as cumulative distance traveled
(CDT), or percent area coverage (PAC) [4, 5] as well as the effect of
simulated noise on the network topology.

We are targeting the following research questions:

  1. 1.

How is the deployment process affecting the connection in the region of
interest?

  2. 2.

What is the temporal difference between the proposed algorithms in terms of
the connections achieved between the nodes, over the course of the overall
simulation time?

  3. 3.

How can we quantify the influence of the simulated noise in the on the
distribution and the connectivity of the nodes in the network?

  4. 4.

What do we learn from the time traces of network characteristics, such as
regularity and centrality?

  5. 5.

How do temporal characteristics compare with application-driven measures, such
as ADT, PAC?

The paper is organized as follows: section II briefly discusses our Voronoi-
only and GA + Voronoi approaches, section III provides an overview of recent
work related to temporal and static network analysis applied to WSN; section
IV deliberates the temporal network framework applied to Voronoi-only and GA +
BISON; section V and VI focus on the regularity and centrality measures
respectively, and section VII concludes the results.

##  II Overview of the BISON and GA-BISON approaches

We previously reported the progress in autonomous self-deployment of a WSN
into two-dimensional bounded target spaces of unknown geometry and topology
based on a variant of a Voronoi-based algorithm [5]. The proposed method
assumed entering the target space from the selected inlet (e.g., doors), and
triggering the sequential controlled and optimized release of vehicle or drone
carried WSN nodes, which autonomously spread and connect throughout the space
to rapidly form a blanket coverage network ready for delivering variety of
sensing, monitoring or communication services.

The sensor nodes autonomously move toward their range-dependent, partially
observable Voronoi cells’ centers, as shown in Figure 1, maintaining a stable
collision-free flow designed to rapidly explore and cover the whole target
space at the minimum possible time, using as few nodes as possible and
draining as little energy as possible, all without any prior knowledge about
the geometry of the space and the obstacles in it.

Extensive sets of simulated deployment experiments demonstrate convergence to
the stable near-full coverage network, achieved at a fraction of a deployment
cost and time, compared to competitive models reported in the literature,
which allow to consider BISON as a strong new approach to AI-flavored,
blanket-coverage WSN deployment achieved virtually without any human
intervention.

Figure 1: Illustration of the evolution of nodes using Voronoi-only and Ga +
Voronoi algorithms. The red lines indicate a suitable network connection for
information routing. Both approaches are also capable of expanding into
obstacle filled regions. Nodes are colored based on their eigenvector
centrality (see section VI for a more detailed discussion). Additional
examples of network evolution are offered in SOM - B.

In an attempt to further improve, simplify and generalize Voronoi-only
deployment, the core BISON algorithm is merged with the localized Genetic
Algorithm (GA) applied to push the trade-off between the pace of space
exploration and the energy expense further towards faster deployments,
especially when faced with complex obstruction structures. The GA with Voronoi
approach has an influence on the discovery process of the next best possible
local position of nodes in the network [23]. The opportunity of evaluating
several candidate solutions instead of Voronoi centroid allowed sensor nodes
to discover better locations, hence enhancing the network performance at
faster rates [24, 25, 26]. GA allowed us to control the randomization of the
candidate solutions to best fit our optimization problem, by tuning the
location and the rate of the generated solutions in every iteration.

The proposed GA-BISON (Conditional) approach allowed each sensor node to
decide whether to apply GA or to stay reliant on BISON to determine its next
position by checking its number of neighbors. If the sensor has between 1 and
3 neighbors, as shown in Figure 2 (lower) for node $n_{1}$, then there is
still a chance to move further and discover more regions by implementing a GA
approach.

If the sensor node has more than 4 neighbours as in the Figure 2 for node 9
(upper right panel) there is limited region to discover, unlike node 9 in the
lower right panel. Therefore, it is better for the sensor to optimize its
current location by moving towards the Voronoi centroid using BISON. This GA +
Voronoi approach improved the execution time and discovery rate of the
network, by discovering further locations instead of Voronoi centroids
implemented in BISON, but with the price of moderate energy expenditure
compared to the Voronoi-only approach. We also validated that GA-BISON
(Conditional)’s coverage performance is robust against the effects of noise.
Its performance is enhanced by noise but incurs an increased energy cost. From
these analyses, the efficiency that we can guarantee from the developed
approaches can cover several applications depending on their requirements and
abilities of WSN deployment.

Figure 2: Snapshots of a network generated by (upper) the Voronoi-only
algorithm in an obstacle-free environment, at different time steps and (lower)
the GA-BISON algorithm in an obstacle-free environment at comparable time
steps. ND signifies noise level (among options $0.01,0.05,0.1$). See Eq. 2.

##  III Review of related work

Temporal networks, also called time-ordered networks or dynamic networks, are
general network structures containing timing information about the actions of
and interactions between network members [27]. This added information can make
analysis significantly more difficult, however the added complexity makes
temporal networks natural choices for studying dynamic systems [1, 2].

Temporal network analysis has been used to study a wide variety of topics,
including anomalies in urban transportation, disease transmission, and video
super-resolution [28, 29, 30]. Temporal Network Graphs (TNGs) can be
constructed in several different ways. For our purposes here, it is convenient
to think of them as a sequence of unweighted, static graphs, each with an
accompanying adjacency matrix, defined in a standard manner for static
networks, where each element can be written:

$a_{ij}(t)=\begin{cases}1,&\text{if node }i\text{ is connected to }j\text{ at
time }t\\\ 0,&\text{otherwise}\end{cases}$ (1)

The TNG representation is powerful, as it allows us to utilize familiar static
network theoretic techniques on graphs within the sequence, specifically, for
each step of the sequence in time evolution of the network. Understanding the
interplay between the underlying dynamics and the temporal sampling rate is
critical to most temporal network analysis [1, 2, 3]. Both the Voronoi-only
approach and the GA + Voronoi operate at discrete time steps; giving us a
natural starting point for sampling times. We then construct TNGs for all
trials as the sequence of static graphs occurring at each time step.

Temporal network theory has been explored as a means of identifying shortest
time-respecting paths between nodes, making it useful in the development of
routing protocols for WSNs [31, 32, 33]. It also provides statistical analysis
on the average number of hops and members required from source to destination.
Temporal network analysis can also reveal members that are involved in
shortest paths, thus playing an important role in mediating the data flow in
the network [7]. Moreover, temporal and centrality measures are combined in
static WSN to decide their best placement based on their response to changes
in the surrounding environment [13, 34].

Regarding TNG measures and statistics, we consider two broad categories: (a)
static measures recorded for individual graphs within the time-ordered
sequence, referred to here as time traces, and (b) temporal measures,
constructed by looking at a TNG as a single object. Given the added complexity
of TNGs, as well as the relative newness of the field, it has been difficult
to establish a consensus around the use and definition of measures of type (b)
[1, 3]. We chose to focus a bit more on measures of type (a), largely for the
reasons stated above, but recognize that future work utilizing measures of
type (b) has significant merit. This being said, we did extract the
distribution function of the length connections from the TNGs in the case of
our simulated physical situations. Overall, this type of work may also include
utilizing aggregate graphs [35, 36], supra-centrality measures [37, 38], and
phase transition analysis [39].

Static centrality measures are commonly used network measures for categorizing
and ranking nodes within a static network, and there exists a large variety of
possible measures that can be employed [40].

In WSNs, various centralities have been introduced to measure connectivity,
clustering/localization, data flow and energy expenditure [41, 31, 32, 33, 42,
43]. Still other centrality measures are used to show the robustness of the
network against errors, particularly in the face of node failure [44].
Choosing an applicable centrality measure largely depends on the specific
network being studied, its goals, and the environment it exists in.

Among the most frequent centrality measures discussed in literature is the
betweenness centrality, a simple measure that identifies nodes working as
important corridors for information flow within in the network [41, 45, 31,
32, 33, 44, 43]. This identification allows for increased control over this
information flow, as well as increased reinforcement of important nodes. The
limitations of this measure lie in its high computational cost, disregard for
the global structure of the network, and its inapplicability to choose the
shortest path for data transmission due to limited energy resources [8].

Another commonly employed centrality measure is the closeness centrality,
which measures the mean distance between a node and other nodes (also referred
to as the shortest geodesic distance from a node to all other nodes in the
network). Closeness centrality is useful in clustering and assigning energy-
saving sleep/wake schedules [41, 31, 32, 33, 44, 42, 43]. However, it does not
account for unreachable nodes and performs poorly in large networks, where it
cannot point to the main leaders [8].

We considered degree centrality [11, 46, 32, 44, 42, 43] and eigenvector
centrality [32, 43, 44, 47] to be used for WSNs when implemented in toxic-
leaks detection as described by Voronoi-type approaches [32]. These
centralities cover the two types of classifications: (i) local centrality
which is demonstrated through degree centrality by focusing on how nodes are
connected to their neighbors, and (ii) the global centrality demonstrated
through the eigenvector centrality that reflects how often can a node be
effective in transferring the data packets among the network [32]. Moreover,
in [44] the analysis of 4 different centrality measures (Degree, Betweenness,
Closeness, and Eigenvector) shows them identical in their analysis of
robustness against errors in the measurements (addition or removal of nodes)
and so choosing either one is adequate for network analysis. Additional
research has shown that eigenvector centrality performs better than other
centralities at measuring causal inference [48]. Complementing all of this is
its relative ease of computation.

##  IV Bison as a temporal network

Given the changing nature of BISON and its proposed target environments, we
believe temporal analysis is necessary to uncover the deeper structural
changes within the network in different environmental conditions [7, 1].

For our study, we consider four different environments (cf. Table 1). Here,
“moderate” noise is a “noise deviation” level of $0.05$ within a simple
Gaussian noise model:

$P(x)=\left(\frac{1}{\sigma\sqrt{2\pi}}\right)e^{\frac{(x-\mu)^{2}}{2\sigma^{2}}}$
(2)

Where $x$ is taken to be a random variable, $\mu$ is the mean, and $\sigma$ is
the standard deviation. Increasing $\sigma^{2}$ (given by
$\sigma^{2}=\frac{N_{o}}{2}$, with $N_{o}$ the noise power) thus detrimentally
impacts signal accuracy. “Scattering” obstacles refers to small obstacles
placed throughout the environment (see Figure 2 for the layout). We also limit
each algorithm to $40$ nodes, both for sake of comparison and to avoid large
computation times. It is straightforward to extend this study to $100s$ of
nodes and to further complexify the environmental conditions, such as number
of obstacles, wall penetrability, and the level of noise.

Table 1: We analyze each combination of the following parameters, giving us eight different cases to study. Algorithm | Noise-level | Obstacle arrangement
---|---|---
BISON | No Noise | No obstacles
GA-BISON | ND $=0.05$ | 10 scatterers

There are several ways of visually representing the information contained
within a TNG. A common representation is the edge-centric representation,
which lists the possible edge pairs belonging to the network along the
vertical axis, and the temporal dimension across the horizontal axis [27].
When an edge is active, a horizontal line is drawn from the first moment it is
active to the last. We generated such temporal network graphs in Figure 3, for
varied conditions of numerical experiments, in order to visualize the
connectivity actions influenced by various obstacles and noise variations.

In both the scatterer and the scatterer-free case, Voronoi-only reaches the
node cutoff much quicker when there is noise present. It is also immediately
apparent that the noise-free cases contain persistent, longer connections,
while the noisy cases are more sporadic. This graph representation is quite
sparse in the vertical dimension, as can be seen from the existence of
relatively wide “empty bands” of nodes whose connections never form. This
sparseness can be attributed to the unlikelihood of nodes encountering nodes
from a different time step of the simulation. That is, node $1$ is unlikely to
encounter node $40$, owing to the large amount of distance traversed by node
$1$ before node $40$ is inserted.

Figure 3: Edge-centric representation of Voronoi-only (BISON) and GA + Voronoi (GA-BISON) in both the obstacle-free environment and obstacle-rich environments. A given simulation has reached the cutoff condition at its rightmost edge. Notice that GA + Voronoi reaches cutoff after a fraction of the time required for Voronoi-only, and how Voronoi-only performs much more quickly in the presence of noise. Also note the sporadic nature of connections within GA + Voronoi, compared to the stability of Voronoi-only connections. a and b mark two of the significant empty bands, where connections never form. Table 2 lists the connections in a and b, we refer to SOM - C for a complete list of all missing pairs, for all panels of this figure. Table 2: : Edge pairs located within gaps a and b in Figure 3. These gaps indicate edges that never form. SOM - C contains all the missing pairs, for all four panels of this figure. a | b
---|---
(1,4) | (2,5), (2,6), (2, 7)
(1,6), (1,7), (1, 8) | (2,9)
(1,10), (1,11), (1, 12) | (2,11), (2,12), $\ldots$, (2, 40)
(1,14), (1,15), $\ldots$, (1, 19) |
(1,21), (1,22), $\ldots$, (1, 40) |

It is worth noting that this brings up a representation problem in Figure 3:
the graphs are significantly sparser at the bottom, not necessarily because of
lower clustering, but rather because there are more edges that start with low
numbers (no edge is double counted).

The networks resulting from GA + Voronoi are much more sporadic than any of
those seen in the Voronoi-only cases, a behavior consistent across various
environmental conditions. We see a similar vertical sparseness.

It is also worth noting that the required number of steps for node cutoff is
significantly less than any of the Voronoi-only runs; this is the case
regardless of noise.

Figure 4: Distribution of various lengths of connection for each of the cases
in Figure 3. We see a drastic qualitative difference between Voronoi-only in
noise-free and noisy environments, and smaller quantitative difference between
GA + Voronoi cases.

The comparison between different environments using Voronoi-only showed that
noise affects the simulation time steps required to achieve full coverage in
the network and affects the connectivity between the nodes together. In other
words, noise allowed nodes to be more distributed in the region, however,
created some disconnections between the nodes throughout the time steps,
affecting the flow of data packets between the nodes.

Among (almost) any network practical considerations are parameters like (i)
energy needed to operate and maintain network connections, (ii) the number of
the connections and their distribution in time. Such parameters are easily
accessible using the representation in Figure 4.

The behavior of GA + Voronoi showed that the connectivity is fluctuating more
often in obstacle-free than in obstacle-rich environments. This is because in
obstacle-free environments, sensor nodes are more freely to move and disperse,
losing by that their connectivity more easily than in obstacle-rich
environments. The temporal network figures showed significant differences in
the behavior between Voronoi-only and GA + Voronoi.

The first difference to discuss is the time steps. We can notice that in GA +
Voronoi, the number of time steps is not affected by noise, however in
Voronoi-only, the number of time steps decreases in the presence of noise.
This indicates that merging GA with Voronoi introduced a robustness
functionality to Voronoi-only against communication noise, at least in terms
of the execution time.

The second difference is regarding the number of connections established in
the network. We observed that the presence of noise in the environment allowed
more connections to be present between the sensor nodes compared to noise-free
environment. This behavior is observed in both approaches (Voronoi and GA +
Voronoi) with a slightly more connections available in Voronoi-only compared
to GA + Voronoi approach.

The third noticed difference is the connectivity duration among the agents in
the network. For GA + Voronoi, the system suffered from un-stabled connections
throughout the simulation, but with less effect in the presence of noise than
in noise-free situations. In Voronoi-only we observed the opposite behavior,
where in noise-free condition, the connections are established at earlier
stages of the simulation and are more stabled; while in noise-rich condition,
un-stabled connections are performed throughout the simulation which stabilize
as the number of nodes increase in the network over time.

Figure 5: Temporal evolution of the deviation from regularity for Voronoi-only
and GA + Voronoi approaches under different environmental conditions. The
measure $\Delta_{\text{reg}}$ is defined in Eq. 4, below.

##  V Regularity measure

Further observations of Voronoi-only and GA + Voronoi network-time evolution
motivates an elementary inquiry into the regularity of networks deployed.

A network is considered regular if [40]:

$\sum_{i=1}^{n}\lambda_{i}^{2}=n\lambda_{\text{max}}$ (3)

where $\lambda_{i}$ are the eigenvalues of the adjacency matrix,
$\lambda_{\text{max}}$ is the largest eigenvalue, and $n$ is the number of
nodes in the network, all of which are recorded at a particular time step.

We define the regularity difference at a given time as:

$\Delta_{\text{reg}}=\displaystyle\left\lvert\sum_{i=1}^{n}\lambda_{i}^{2}-n\lambda_{\text{max}}\right\rvert$
(4)

Figure 5 represents the results of the network regularity at different moments
in time and for several situations (obstructed environments and noise levels).

We can notice that for both approaches at earlier steps of the network
evolution, the network is closer to regularity than at further time steps.

Moreover, by looking at the moments of injecting new nodes, the regularity
difference ($\Delta$) increases in the network in most environments.
Regularity difference levels are dramatically different between the GA +
Voronoi and Voronoi-only cases. Furthermore, noise seemingly increases the
regularity difference ($\Delta$) in Vornoi-only networks, while the GA +
Voronoi networks maintain similar regularity difference levels in the face of
noise.

Below, we will discuss how the sharply different network irregularities
correlate with the practical performance measures, as well as energy
expenditure.

##  VI Centrality measure

The motivation of this work is driven by the need to understand and improve
the network coverage in the context of the spatial, temporal and energy
constraints. Specifically, implementations are often driven by:

  1. 1.

cumulative distance traveled (CDT) by the nodes

  2. 2.

percent area coverage (PAC) that nodes manage to cover by the signal

  3. 3.

minimal time to achieve target coverage

  4. 4.

optimized energy allowed for the network of drones

Figure 6: Eigenvector Centralities of the Voronoi-only approach in various
environments at different time steps; Vertical bar sign “$0.6$”, stacked
vertically, is common to all curves.

The analysis so far illustrates how uneven the roles of most nodes are during
the deployment process. We also noted that as the deployment progresses, the
nodes that have few links can quickly gather many links, indicating that a
fair bit of temporal switching occurs within the networks. With the goal of
initial quantification of the role of individual nodes and the time-dependent
role they play, we determined eigenvector centrality (EC) [40], one of the
leading measures that help understands the evolving nature of temporal network
graphs.

We select eigenvector centrality as it is sufficiently general (typically
defined as a generalization of Katz index [40]), applicable to smaller size
networks (N $\approx$ 30-100 in most our cases), and is readily obtainable
from the adjacency matrices. It measures the importance of a node based on the
importance of its (connected) neighbors.

Comparing our values with [31] we notice that local centrality measures (our
degree centrality measures and their betweenness centrality measures) provide
high spike values for nodes that are more connected to their neighbors than
others, while global centralities (our eigenvector centrality measures and
their closeness centrality measures) have a smooth distribution of values
among the nodes. We also observed that the differences in the eigenvector
centralities of the nodes in the network at different time steps revealed the
uneven role of most of the nodes in the network during their deployment
process, which is considered an important finding to be used later to keep
track of the important nodes that can be relied on to transfer data packets
among the network members.

We present eigenvector centrality measurements for the various BISON cases in
Figure 6 for the selected environmental cases. Nodes with early injection time
immediately spike in importance, but drop off as deployment continues. By
contrast, nodes deployed later spike early as well, and then maintain some
non-zero importance throughout their lifetime. This initial surge is to be
expected, since nodes are injected from the same corner, resulting in an area
with a high node density.

As spreading continues, early nodes migrate to distant regions, and thus have
little opportunity to be deeply connected, while later nodes remain relatively
close to this highly dense region. These EC findings also clearly support our
previous insight that adding noise does, in fact, have a beneficial impact on
the performance of Voronoi-only generated networks with regard to deployment
speed. When comparing plots from simulations differing only in noise level, we
consistently see that the presence of noise smooths the EC time trace curve,
suggesting a more homogeneous distribution of the nodes. We see somewhat
similar spiking behavior in the GA + Voronoi trials featured in Figure 7,
although early deployment nodes are more likely to feature significant spikes
in EC later in their lifetimes. This is partly expected, due to the somewhat
randomized nature of the GA part of the algorithm.

Figure 7: Eigenvector Centralities of GA + Voronoi in various environments at
different time steps; Vertical bar sign “$0.6$” is common to all curves, which
are stacked vertically for clarity.

In neuroscience, EC has been found to correlate with a neuron’s firing rate
and in cellular biophysics with [49]. We conjecture an analogous phenomenon
could be exploited here; there may be an advantage in designing our WSN nodes
such that their message passing rate is highest immediately after their
injection, when a node can be expected to have high EC. After some time, this
transfer rate can drop to some consistent, energy-efficient base value. The
potential benefit that can be gained by this should be measurably higher for
environments where there is noise, since in such environments there are
consistently fewer - and lower – “flare-ups” of EC values as time progresses.

We have analyzed correlations between the eigenvector centrality time traces
from Figure 7, using the well known formula for Pearson’s correlation
coefficient, where we have taken the expectation values of our data to be the
arithmetic mean of our data sets. For two data sets $A$ and $B$, the
coefficient is

$C_{A,B}=\frac{\langle AB\rangle-\langle A\rangle\langle
B\rangle}{\sqrt{\langle A\rangle^{2}-\langle A^{2}\rangle}\sqrt{\langle
B\rangle^{2}-\langle B^{2}\rangle}}$ (5)

The outcome of that analysis is shown in Figure 8.

Figure 8: Time-lag correlation heat maps for time-dependent eigenvector
centralities from Figures 6 and 7. (selected node pairs shown only – more data
in SOM). (upper) Voronoi; (lower) GA + Voronoi; (both) ND $=0.05$ and 10
obstacles.

Having seen the overall shape and detailed behavior for a number of
eigenvector centralities, we decided to examine the time-lag correlations of
the time traces of eigenvector centralities for selected nodes. We see this as
an alternative method of presenting and analyzing node ranking. If one
imagines this work as implemented on board a number of (micro-)drones to make
their navigation and time-to-task efficient, having several approaches to
assessing time-evolving network functionality.

Keeping in mind that one way to think about eigenvector centrality is that it
serves as method for ranking nodes, it is understandable that (once enough
nodes are present in the network) increase in the ranking of node $k$, there
is a decrease in the ranking for one or more nodes $k-p$, $k+p$. Positive
correlation seems to arise when the nodes are close to becoming connected, are
connected, or have been connected, where negative correlation seems to suggest
the opposite.

Since the correlation itself depends on time, one can learn quite a bit by
reading out the map. It is clear there are “correlation bands” (of steadily
positive or negative correlation), often preceded by rapid switching between
positive and negative correlation. In the Figure 8 we have omitted quite a few
node-to-node EC correlations, for the sake of clarity. Supporting Online
Materials contain more examples, omitted from the main text in the interest of
space. Figure 8 contains “conditions rich” examples, with both, noise and
obstacles, for reaslism.

##  VII Conclusion

We demonstrated how to apply quantitative methods of temporal network analysis
to our Voronoi-like algorithms in order to compare and analyze the dynamics of
WSN coverage and robustness in the face of environmental noise and
obstructions. GA + Voronoi in particular demonstrated robustness in the face
of thee various environmental conditions. To summarize, we can provide the
following answers to our research questions in section I:

  1. 1.

With Voronoi-only, nodes move outwards during their deployment in a regular
fashion, and their importance to connectivity and information relaying drops
off with this outward movement. GA + Voronoi demonstrates similar behavior;
however, nodes are more likely to reestablish their importance later.

  2. 2.

We see that Voronoi-only in a noise-free environment maintains a small number
of long-lasting connections. Adding noise substantially redistributes those
connection lengths. By contrast, GA + Voronoi maintains qualitatively similar
connection length distributions through a variety of environments, featuring a
rapidly changing topology with short, frequent connections between nodes. In
all cases, nodes are highly likely to associate only with nodes sharing a
similar deployment time

  3. 3.

We see that noise increases the deviation from regularity, smooths out the EC
time trace, and impacts the connection length distribution in the Voronoi-only
cases. By contrast, noise has a much smaller impact on the same measures
applied to the GA + Voronoi cases. This supports previous work that had
indicated Voronoi-only changes substantially in the face of noise, while GA +
Voronoi is robust to such environmental changes.

  4. 4.

Utilizing temporal network characteristics allows us to measure and observe
the behavior of the network as a whole and if individual nodes.

  5. 5.

All this was possible without the explicit use of physical measurements of the
nodes’ coverage or their environment. These measures also suggest some kind of
deeper functional equivalence between Voronoi-only in noisy environments and
GA + Voronoi.

Such quantification measurements may prove useful in further refining our
algorithms, by allowing us to fine-tune communication routing protocols,
broadcasting strength, and other implementation details based on the expected
behavior.

While mapping these theoretical network measures onto applied performance
metrics can perhaps be challenging, we do believe there are some clear points
to take note of. When discussing regularity, we noted that regularity
difference rose in the noisy Voronoi-only cases and GA + Voronoi cases. This
irregularity seems to have some correspondence to speed of coverage, and we
posit that such highly irregular structures may be crucial in this rapid
deployment. By contrast, the lower regularity difference of Voronoi-only
likely corresponds to its slower but more consistent nature. This parallels
the behavior we see in measuring percent area covered (PAC), with GA + Voronoi
rapidly reaching high levels of PAC, while Voronoi-only slowly converges
towards an acceptable value.

We also discussed EC among nodes in the various cases. By looking at EC, we
see that nodes consistently move towards positions of lower importance,
especially in the Voronoi-only cases. Fluctuations following this are
indicative of a node relocating/reorganizing itself in the network and are
much more common in the application of GA + Voronoi. This fluctuating
corresponds to the cumulative distance traveled (CDT) measures. Assuming a
fluctuating EC value corresponds to a particularly mobile node, we would
expect the distance travel to be much higher for cases with fluctuating EC.
Indeed, our previous research indicated GA + Voronoi nodes travel
significantly longer distances, seemingly matching this explanation.

In addition to time evolution of irregularity (Section V) and time evolution
of eigenvector centrality (Section VI), we analyzed the nodes degree
distribution, where analogous observations could be made. In the interest of
brevity, this output is presented in SOM D, E.

###### Acknowledgements.

We acknowledge early support for this project under the UAE ICTFund grant
“Bio-inspired Selforganizing Networks”. KE is thankful for the Graduate
Students Fellowship at KU. A part of the work was performed while AFI was a
visiting scientist at Cornell University, the hospitality of which is greatly
acknowledged. NDB acknowledges the Colgate Undergraduate Research Fund. We
acknowledge early stimulating conversations with Dr. D. Ruta (KUST, Abu Dhabi,
UAE) and Dr. F. Saffre (VTT, Finland). NDB and KE contributed in equal parts
to the work reported in this paper.

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Supporting Online Materials (SOM)

## SOM – A: A Part of Voronoi Tessellation Process and Project Outline

### Illustrative description of the evolution of nodes in the unknown region

Illustrative description of the evolution of nodes in the unknown region, and
the generation of the Voronoi region for the next deployment step using our
version of Voronoi tessellation - BISON. (a) illustrates the disconnected
stage of the nodes, where the Voronoi region is the same as the sensing
region. (b) nodes establish perpendicular bisectors to identify their Voronoi
region and their next position. $R_{s}$ is the sensing range of the moving
sensors, while $R_{c}$ the center to center distance between two sensors.

### Organization of the paper

## SOM B

BISON (and GA-BISON) is adaptable to other kinds of environments. BISON is
shown here exploring regions containing different Obstacles. This figure is
related to Fig. 1 in the main text.

## SOM C

BISON
---
No Obstacles | 10 Scatterers
No Noise | Noisy | No Noise | Noisy
(1,4) | (1,6) | (1,5) | (1,4)
(1,6)$\ldots$(1,8) | (1,7) | (1,7)$\ldots$(1,9) | (1,5)
(1,10)$\ldots$(1,12) | (1,9)$\ldots$(1,40) | (1,12)$\ldots$(1,20) | (1,7)$\ldots$(1,40)
(1,14)$\ldots$(1,19) | (2,5) | (1,22)$\ldots$(1,24) | (2,6)
(1,21)$\ldots$(1,40) | (2,8)$\ldots$(2,16) | (1,26)$\ldots$(1,40) | (2,7)
(2,5)$\ldots$(2,7) | (2,18)$\ldots$(2,40) | (2,7)$\ldots$(2,9) | (2,9)$\ldots$(2,17)
(2,9) | (3,6)$\ldots$(3,8) | (2,12)$\ldots$(2,40) | (2,19)$\ldots$(2,40)
(2,11)$\ldots$(2,40) | (3,10) | (3,6) | (3,8)
(3,6) | (3,13)$\ldots$(3,40) | (3,8)$\ldots$(3,40) | (3,10)$\ldots$(3,40)
(3,8)$\ldots$(3,40) | (4,9)$\ldots$(4,13) | (4,6) | (4,9)
(4,7) | (4,15)$\ldots$(4,40) | (4,7) | (4,11)$\ldots$(4,17)
(4,9)$\ldots$(4,11) | (5,10) | (4,9)$\ldots$(4,40) | (4,19)
(4,13)$\ldots$(4,20) | (5,12)$\ldots$(5,14) | (5,8) | (4,22)
(4,22)$\ldots$(4,40) | (5,16)$\ldots$(5,40) | (5,10)$\ldots$(5,40) | (4,23)
(5,8) | (6,9) | (6,9) | (4,25)$\ldots$(4,40)
(5,10) | (6,11) | (6,11)$\ldots$(6,40) | (5,9)
(5,12) | (6,12) | (7,10) | (5,11)$\ldots$(5,40)
(5,14)$\ldots$(5,40) | (6,14)$\ldots$(6,16) | (7,11) | (6,10)
(6,9) | (6,18)$\ldots$(6,21) | (7,13)$\ldots$(7,16) | (6,12)
(6,11)$\ldots$(6,40) | (6,23)$\ldots$(6,40) | (7,18) | (6,14)
(7,12)$\ldots$(7,40) | (7,9) | (7,19) | (6,16)
(8,11) | (7,11) | (7,21) | (6,18)
(8,13)$\ldots$(8,15) | (7,12) | (7,23)$\ldots$(7,40) | (6,20)$\ldots$(6,40)
(8,17)$\ldots$(8,20) | (7,14)$\ldots$(7,21) | (8,12) | (7,10)
(8,22) | (7,23)$\ldots$(7,40) | (8,14) | (7,14)$\ldots$(7,40)
(8,24)$\ldots$(8,40) | (8,12) | (8,15) | (8,13)$\ldots$(8,15)
(9,12) | (8,16)$\ldots$(8,40) | (8,18) | (8,17)
(9,14) | (9,13)$\ldots$(9,17) | (8,19) | (8,19)$\ldots$(8,40)
(9,16)$\ldots$(9,40) | (9,19)$\ldots$(9,40) | (8,21) | (9,14)$\ldots$(9,16)
(10,13) | (10,15)$\ldots$(10,40) | (8,23)$\ldots$(8,40) | (9,18)$\ldots$(9,40)
(10,15) | (11,13) | (9,13)$\ldots$(9,16) | (10,13)
(10,17)$\ldots$(10,40) | (11,17)$\ldots$(11,20) | (9,18) | (10,15)
(11,15)$\ldots$(11,17) | (11,22)$\ldots$(11,40) | (9,19) | (10,17)$\ldots$(10,40)
(11,19)$\ldots$(11,40) | (12,17) | (9,21) | (11,14)
(12,15) | (12,19)$\ldots$(12,40) | (9,23)$\ldots$(9,27) | (11,17)
(12,18)$\ldots$(12,20) | (13,18) | (9,29)$\ldots$(9,40) | (11,19)
(12,22)$\ldots$(12,40) | (13,21) | (10,12) | (11,20)
(13,16) | (13,23)$\ldots$(13,40) | (10,15) | (11,24)$\ldots$(11,40)
(13,17) | (14,16)$\ldots$(14,18) | (10,17) | (12,19)
(13,19) | (14,20) | (10,19) | (12,20)
(13,21)$\ldots$(13,40) | (14,22)$\ldots$(14,25) | (10,20) | (12,22)$\ldots$(12,40)
(14,17) | (14,27)$\ldots$(14,40) | (10,22)$\ldots$(10,40) | (13,18)$\ldots$(13,40)
(14,20)$\ldots$(14,40) | (15,20) | (11,15)$\ldots$(11,40) | (14,19)
(15,19) | (15,22)$\ldots$(15,40) | (12,16) | (14,21)$\ldots$(14,26)
(15,21)$\ldots$(15,23) | (16,19) | (12,18) | (14,28)$\ldots$(14,40)
(15,25)$\ldots$(15,40) | (16,20) | (12,19) | (15,24)
(16,18) | (16,22) | (12,21) | (15,26)$\ldots$(15,31)
(16,20) | (16,23) | (12,23) | (15,33)$\ldots$(15,40)
(16,22) | (16,26)$\ldots$(16,40) | (12,25)$\ldots$(12,27) | (16,17)
(16,24)$\ldots$(16,40) | (17,21) | (12,29)$\ldots$(12,40) | (16,19)
(17,20) | (17,24)$\ldots$(17,40) | (13,17)$\ldots$(13,19) | (16,22)
(17,22)$\ldots$(17,24) | (18,23) | (13,23) | (16,26)
(17,26)$\ldots$(17,40) | (18,26)$\ldots$(18,40) | (13,25) | (16,28)
(18,21) | (19,24) | (13,27) | (16,30)
(18,23)$\ldots$(18,40) | (19,25) | (13,28) | (16,31)
(19,24)$\ldots$(19,26) | (19,27) | (13,30)$\ldots$(13,40) | (16,33)
(19,29)$\ldots$(19,40) | (19,29)$\ldots$(19,40) | (14,17) | (16,35)$\ldots$(16,40)
(20,23) | (20,24)$\ldots$(20,26) | (14,20)$\ldots$(14,40) | (17,21)
(20,25)$\ldots$(20,40) | (20,28)$\ldots$(20,40) | (15,18) | (17,23)$\ldots$(17,25)
(21,24) | (21,23) | (15,19) | (17,27)$\ldots$(17,40)
(21,26)$\ldots$(21,40) | (21,25) | (15,23) | (18,22)
(22,25) | (21,27)$\ldots$(21,40) | (15,25) | (18,23)
(22,29)$\ldots$(22,40) | (22,27)$\ldots$(22,40) | (15,27) | (18,25)$\ldots$(18,40)
(23,26) | (23,30)$\ldots$(23,40) | (15,30) | (19,24)
(23,28) | (24,27) | (15,34) | (19,25)
(23,31)$\ldots$(23,40) | (24,31)$\ldots$(24,40) | (15,36)$\ldots$(15,40) | (19,27)$\ldots$(19,29)
(24,27)$\ldots$(24,30) | (25,29) | (16,20) | (19,31)$\ldots$(19,40)
(24,32)$\ldots$(24,40) | (25,32)$\ldots$(25,40) | (16,22)$\ldots$(16,25) | (20,26)
(25,28) | (26,30)$\ldots$(26,40) | (16,27) | (20,28)
(25,30) | (27,33)$\ldots$(27,40) | (16,28) | (20,30)$\ldots$(20,40)
(25,31) | (28,34)$\ldots$(28,40) | (16,30)$\ldots$(16,40) | (21,26)$\ldots$(21,40)
(25,33)$\ldots$(25,40) | (29,35)$\ldots$(29,40) | (17,21)$\ldots$(17,40) | (22,27)
(26,29) | (30,35)$\ldots$(30,40) | (18,20) | (22,29)
(26,30) | (31,35) | (18,22) | (22,31)$\ldots$(22,40)
(26,32) | (31,37)$\ldots$(31,40) | (18,24) | (23,26)
(26,34)$\ldots$(26,40) | (32,34) | (18,28) | (23,27)
(27,31)$\ldots$(27,40) | (32,36)$\ldots$(32,40) | (18,31) | (23,29)$\ldots$(23,31)
(28,32) | (33,39) | (18,33)$\ldots$(18,40) | (23,33)$\ldots$(23,40)
(28,34)$\ldots$(28,40) | (33,40) | (19,22) | (24,33)$\ldots$(24,40)
(29,31) | (34,38)$\ldots$(34,40) | (19,24)$\ldots$(19,40) | (25,30)
(29,33)$\ldots$(29,40) | (35,39) | (20,23)$\ldots$(20,40) | (25,31)
(30,35) | (35,40) | (21,27)$\ldots$(21,40) | (25,34)$\ldots$(25,40)
(30,36) | (36,40) | (22,25)$\ldots$(22,40) | (26,31)$\ldots$(26,40)
(30,38)$\ldots$(30,40) |  | (23,26) | (27,31)$\ldots$(27,40)
(31,34) |  | (23,28)$\ldots$(23,40) | (28,33)
(31,36)$\ldots$(31,40) |  | (24,27) | (28,34)
(32,35) |  | (24,29) | (28,37)$\ldots$(28,40)
(32,36) |  | (24,30) | (29,35)$\ldots$(29,40)
(32,38)$\ldots$(32,40) |  | (24,32)$\ldots$(24,40) | (30,33)$\ldots$(30,35)
(33,36)$\ldots$(33,40) |  | (25,28)$\ldots$(25,40) | (30,37)$\ldots$(30,40)
(34,40) |  | (26,31)$\ldots$(26,40) | (31,37)$\ldots$(31,40)
(35,37)$\ldots$(35,40) |  | (27,31)$\ldots$(27,40) | (32,37)
(37,39) |  | (28,30) | (32,39)
(37,40) |  | (28,32)$\ldots$(28,40) | (32,40)
|  | (29,34)$\ldots$(29,40) | (33,39)
|  | (30,33) | (33,40)
|  | (30,35)$\ldots$(30,40) | (34,39)
|  | (31,34)$\ldots$(31,40) | (34,40)
|  | (32,37)$\ldots$(32,40) | (36,40)
|  | (33,36) |
|  | (33,38)$\ldots$(33,40) |
|  | (34,39) |
|  | (34,40) |
|  | (35,40) |
|  | (36,40) |
|  | (37,40) |

GA-BISON
---
No Obstacles | 10 Scatterers
No Noise | Noisy | No Noise | Noisy
(1,5) | (1,9) | (1,5) | (1,8)
(1,6) | (1,14) | (1,7) | (1,10)
(1,9) | (1,15) | (1,9)$\ldots$(1,12) | (1,11)
(1,10) | (1,17) | (1,14) | (1,18)
(1,12)$\ldots$(1,14) | (1,21) | (1,15) | (1,20)
(1,17)$\ldots$(1,20) | (1,25)$\ldots$(1,40) | (1,19)$\ldots$(1,21) | (1,21)
(1,22)$\ldots$(1,40) | (2,5) | (1,23)$\ldots$(1,26) | (1,23)$\ldots$(1,25)
(2,5) | (2,9)$\ldots$(2,11) | (1,28)$\ldots$(1,40) | (1,27)$\ldots$(1,40)
(2,6) | (2,15) | (2,6) | (2,8)
(2,9) | (2,17) | (2,8) | (2,10)$\ldots$(2,18)
(2,10) | (2,19) | (2,11) | (2,20)
(2,12)$\ldots$(2,15) | (2,27)$\ldots$(2,29) | (2,13) | (2,21)
(2,17)$\ldots$(2,19) | (2,31)$\ldots$(2,40) | (2,16)$\ldots$(2,22) | (2,23)$\ldots$(2,29)
(2,22)$\ldots$(2,25) | (3,9) | (2,24)$\ldots$(2,40) | (2,31)$\ldots$(2,40)
(2,27)$\ldots$(2,29) | (3,15) | (3,6) | (3,4)
(2,31)$\ldots$(2,40) | (3,21) | (3,11) | (3,9)
(3,9)$\ldots$(3,15) | (3,24)$\ldots$(3,40) | (3,13) | (3,14)
(3,17) | (4,7) | (3,15) | (3,16)$\ldots$(3,40)
(3,19) | (4,9) | (3,16) | (4,8)
(3,22)$\ldots$(3,24) | (4,15)$\ldots$(4,17) | (3,18)$\ldots$(3,40) | (4,10)
(3,27)$\ldots$(3,29) | (4,24) | (4,6) | (4,11)
(3,31)$\ldots$(3,40) | (4,26)$\ldots$(4,28) | (4,8) | (4,13)
(4,17) | (4,30) | (4,16) | (4,14)
(4,22) | (4,32)$\ldots$(4,34) | (4,18)$\ldots$(4,22) | (4,17)
(4,24) | (4,36)$\ldots$(4,40) | (4,25)$\ldots$(4,40) | (4,18)
(4,26) | (5,10) | (5,11) | (4,20)
(4,27) | (5,12)$\ldots$(5,14) | (5,12) | (4,23)$\ldots$(4,27)
(4,30) | (5,16) | (5,16)$\ldots$(5,19) | (4,29)$\ldots$(4,40)
(4,32)$\ldots$(4,40) | (5,18)$\ldots$(5,23) | (5,22) | (5,14)
(5,16) | (5,25) | (5,25)$\ldots$(5,40) | (5,17)
(5,18)$\ldots$(5,21) | (5,26) | (6,12)$\ldots$(6,15) | (5,18)
(5,25) | (5,28)$\ldots$(5,40) | (6,21) | (5,20)$\ldots$(5,40)
(5,26) | (6,9)$\ldots$(6,11) | (6,24) | (6,11)
(5,28) | (6,15) | (6,25) | (6,16)$\ldots$(6,40)
(5,30)$\ldots$(5,32) | (6,17)$\ldots$(6,22) | (6,28) | (7,14)
(5,34)$\ldots$(5,40) | (6,25)$\ldots$(6,40) | (6,30)$\ldots$(6,40) | (7,17)
(6,11) | (7,10) | (7,8) | (7,18)
(6,14) | (7,14) | (7,11) | (7,20)
(6,15) | (7,17)$\ldots$(7,21) | (7,13) | (7,21)
(6,17) | (7,23) | (7,16) | (7,23)$\ldots$(7,40)
(6,18) | (7,25)$\ldots$(7,40) | (7,18) | (8,12)
(6,20)$\ldots$(6,40) | (8,13) | (7,19) | (8,14)
(7,11) | (8,14) | (7,22) | (8,16)
(7,14) | (8,18)$\ldots$(8,20) | (7,25)$\ldots$(7,40) | (8,19)$\ldots$(8,23)
(7,22) | (8,22) | (8,10) | (8,25)$\ldots$(8,40)
(7,24) | (8,23) | (8,11) | (9,18)
(7,26)$\ldots$(7,40) | (8,25) | (8,14) | (9,20)
(8,11) | (8,26) | (8,15) | (9,21)
(8,20)$\ldots$(8,22) | (8,28)$\ldots$(8,40) | (8,17)$\ldots$(8,21) | (9,25)
(8,24)$\ldots$(8,40) | (9,13) | (8,23)$\ldots$(8,26) | (9,27)
(9,16) | (9,14) | (8,29)$\ldots$(8,40) | (9,29)$\ldots$(9,40)
(9,20) | (9,18)$\ldots$(9,26) | (9,15) | (10,19)$\ldots$(10,21)
(9,21) | (9,29)$\ldots$(9,33) | (9,21) | (10,23)
(9,26) | (9,35)$\ldots$(9,40) | (9,22) | (10,25)$\ldots$(10,33)
(9,29)$\ldots$(9,40) | (10,13) | (9,24)$\ldots$(9,26) | (10,35)$\ldots$(10,40)
(10,18)$\ldots$(10,21) | (10,15)$\ldots$(10,17) | (9,28)$\ldots$(9,40) | (11,19)
(10,23)$\ldots$(10,26) | (10,20) | (10,16) | (11,21)$\ldots$(11,23)
(10,28)$\ldots$(10,40) | (10,24) | (10,18) | (11,25)$\ldots$(11,33)
(11,18) | (10,25) | (10,19) | (11,35)$\ldots$(11,40)
(11,22)$\ldots$(11,24) | (10,27) | (10,22) | (12,18)
(11,27)$\ldots$(11,29) | (10,28) | (10,25)$\ldots$(10,40) | (12,20)
(11,33)$\ldots$(11,40) | (10,30)$\ldots$(10,40) | (11,29) | (12,21)
(12,21) | (11,13) | (11,31)$\ldots$(11,40) | (12,25)$\ldots$(12,40)
(12,25) | (11,28) | (12,15) | (13,19)$\ldots$(13,23)
(12,27) | (11,30)$\ldots$(11,33) | (12,18) | (13,25)$\ldots$(13,27)
(12,38)$\ldots$(12,40) | (11,35)$\ldots$(11,40) | (12,19) | (13,29)$\ldots$(13,40)
(13,20)$\ldots$(13,40) | (12,17) | (12,21)$\ldots$(12,23) | (14,29)
(14,20) | (12,20) | (12,25)$\ldots$(12,40) | (14,32)
(14,21) | (12,21) | (13,20)$\ldots$(13,26) | (14,34)$\ldots$(14,40)
(14,30)$\ldots$(14,32) | (12,24)$\ldots$(12,28) | (13,28)$\ldots$(13,40) | (15,19)
(14,34)$\ldots$(14,40) | (12,30)$\ldots$(12,40) | (14,16) | (15,21)
(15,18) | (13,17)$\ldots$(13,21) | (14,19)$\ldots$(14,40) | (15,27)
(15,20) | (13,25)$\ldots$(13,40) | (15,22) | (15,29)$\ldots$(15,40)
(15,21) | (14,21) | (15,25)$\ldots$(15,28) | (16,23)
(15,23) | (14,24) | (15,30) | (16,25)
(15,25) | (14,25) | (15,34)$\ldots$(15,40) | (16,27)
(15,26) | (14,27) | (16,21)$\ldots$(16,26) | (16,29)$\ldots$(16,40)
(15,28)$\ldots$(15,40) | (14,28) | (16,28)$\ldots$(16,40) | (17,27)
(16,22) | (14,30)$\ldots$(14,40) | (17,22)$\ldots$(17,40) | (17,30)
(16,24)$\ldots$(16,40) | (15,29)$\ldots$(15,40) | (18,23)$\ldots$(18,26) | (17,31)
(17,21) | (16,25) | (18,28)$\ldots$(18,40) | (17,33)
(17,22) | (16,27) | (19,28) | (17,35)$\ldots$(17,40)
(17,24) | (16,29)$\ldots$(16,32) | (19,30) | (18,21)
(17,26) | (16,34)$\ldots$(16,40) | (19,31) | (18,26)$\ldots$(18,28)
(17,27) | (17,29)$\ldots$(17,40) | (19,33)$\ldots$(19,40) | (18,30)
(17,29)$\ldots$(17,40) | (18,21) | (20,25) | (18,31)
(18,27)$\ldots$(18,40) | (18,24) | (20,26) | (18,33)
(19,22) | (18,28) | (20,28) | (18,35)$\ldots$(18,40)
(19,24)$\ldots$(19,40) | (18,30)$\ldots$(18,34) | (20,30) | (19,21)
(20,27) | (18,36)$\ldots$(18,40) | (20,31) | (19,23)$\ldots$(19,27)
(20,29) | (19,27)$\ldots$(19,40) | (20,33)$\ldots$(20,40) | (19,29)$\ldots$(19,40)
(20,30) | (20,27) | (21,30) | (20,26)
(20,32)$\ldots$(20,40) | (20,28) | (21,34)$\ldots$(21,40) | (20,27)
(21,24) | (20,35) | (22,29)$\ldots$(22,40) | (20,30)
(21,27)$\ldots$(21,29) | (20,37) | (23,27) | (20,37)$\ldots$(20,40)
(21,32)$\ldots$(21,40) | (20,39) | (23,28) | (21,29)
(22,25) | (20,40) | (23,30) | (21,32)
(22,26) | (21,35)$\ldots$(21,40) | (23,31) | (21,34)$\ldots$(21,40)
(22,28) | (22,25)$\ldots$(22,40) | (23,33)$\ldots$(23,40) | (22,27)$\ldots$(22,29)
(22,30)$\ldots$(22,32) | (23,28)$\ldots$(23,40) | (24,30) | (22,31)$\ldots$(22,40)
(22,34) | (24,29) | (24,33)$\ldots$(24,40) | (23,31)$\ldots$(23,40)
(22,36)$\ldots$(22,40) | (24,31)$\ldots$(24,40) | (25,40) | (24,30)$\ldots$(24,40)
(23,29)$\ldots$(23,40) | (25,32) | (26,33) | (25,31)
(24,30) | (25,36)$\ldots$(25,40) | (26,34) | (25,33)$\ldots$(25,40)
(24,32) | (26,31) | (26,36)$\ldots$(26,40) | (26,32)
(24,34) | (26,35)$\ldots$(26,40) | (27,30)$\ldots$(27,40) | (26,34)$\ldots$(26,36)
(24,36)$\ldots$(24,40) | (27,32) | (28,33)$\ldots$(28,40) | (26,38)$\ldots$(26,40)
(25,30) | (27,33) | (29,33)$\ldots$(29,40) | (27,36)
(25,32)$\ldots$(25,40) | (27,35)$\ldots$(27,40) | (30,36)$\ldots$(30,40) | (27,38)$\ldots$(27,40)
(26,29) | (28,35)$\ldots$(28,40) | (31,34) | (28,33)$\ldots$(28,40)
(26,33)$\ldots$(26,40) | (29,35)$\ldots$(29,40) | (31,35) | (29,35)
(27,31) | (30,40) | (31,37)$\ldots$(31,40) | (29,37)
(27,34)$\ldots$(27,40) | (31,37)$\ldots$(31,40) | (32,35)$\ldots$(32,40) | (29,39)
(28,32)$\ldots$(28,40) | (32,39) | (33,39) | (29,40)
(29,36)$\ldots$(29,40) | (32,40) | (33,40) | (30,34)$\ldots$(30,40)
(30,33)$\ldots$(30,40) | (33,39) | (35,40) | (31,40)
(31,33) | (33,40) |  | (32,36)$\ldots$(32,40)
(31,35)$\ldots$(31,40) | (34,37)$\ldots$(34,40) |  | (33,40)
(32,35) | (35,39) |  | (34,39)
(32,37)$\ldots$(32,40) | (35,40) |  | (34,40)
(33,36) | (36,40) |  | (37,40)
(33,38)$\ldots$(33,40) |  |  |
(35,38)$\ldots$(35,40) |  |  |
(36,39) |  |  |
(36,40) |  |  |
(37,40) |  |  |

Significant gaps are present in the edge-centric representation of our
temporal network graphs. This “banding” is inherently due to the associative
nature of the two algorithms; in both cases, nodes are far more likely to
connect with nodes sharing similar deployment times, due to the spatial
connectivity limitations. In other words, nodes stay close to their initial
neighbors, and rarely do nodes “sneak” into other parts of the network.
Investigating the gaps also highlights an issue with this representation:
since edges are not double counted ((1,40) can be found on the y axis, while
(40,1) is omitted), the gaps are biased towards being narrower towards the top
of the figure.

## SOM D

### BISON-Voronoi, No Obstacles, No Noise

### BISON-Voronoi, No Obstacles, Noisy

### BISON-Voronoi, 10 Obstacles, No Noise

### BISON-Voronoi, 10 Obstacles, Noisy

### GA-Voronoi, No Obstacles, No Noise

### GA-Voronoi, No Obstacles, Noisy

### GA-Voronoi, 10 Obstacles, No Noise

### GA-Voronoi, 10 Obstacles, Noisy

These above snapshots illustrate how the degree distributions evolve in time
for the various environmental cases. We can see that the average degree
typically increases with time as the network fills the space. We can also see
that the distribution becomes wider over time, corresponding to the increasing
regularity difference we also observed.

## SOM E

We are displaying these here to show the variation in eigenvector centrality
time traces amongst close neighbor nodes. The time trace for three nodes are
plotted in each graph, and a variety of environmental conditions were used.
These plots are extensions of the ones featured in Figures 6 and 7 of the main
text.

### Nodes 1, 2, and 3

### Nodes 4, 5, and 6

### Nodes 11, 12, and 13

### Nodes 19, 20, and 21

Nodes with similar deployment times exhibit qualitatively similar behavior in
their eigenvector centrality scores. For BISON-Voronoi, this typically means a
sharp spike when the node enters the network, followed by a fluctuating decay.
These fluctuations are more pronounced in the GA-Voronoi cases, and nodes are
more likely to find positions of increased importance later in their lifetimes
than their BISON counterparts.


# Deep Learning-based Data-aided Activity Detection with Extraction Network in
Grant-free Sparse Code Multiple Access Systems

Minsig Han, Ameha T. Abebe, and Chung G. Kang M. Han and C. Kang are with the
School of Electrical Engineering, Korea University, Seoul 136-701, South Korea
(e-mail<EMAIL_ADDRESS>ccgkang@korea.ac.kr). Ameha T. Abebe is with
Samsung Research, Samsung Electronics, Seoul 06762, South Korea (e-mail:
amehat.abebe@samsung.com).

###### Abstract

This letter proposes a deep learning-based data-aided active user detection
network (D-AUDN) for grant-free sparse code multiple access (SCMA) systems
that leverages both SCMA codebook and Zadoff-Chu preamble for activity
detection. Due to disparate data and preamble distribution as well as codebook
collision, existing D-AUDNs experience performance degradation when multiple
preambles are associated with each codebook. To address this, a user activity
extraction network (UAEN) is integrated within the D-AUDN to extract a-priori
activity information from the codebook, improving activity detection of the
associated preambles. Additionally, efficient SCMA codebook design and Zadoff-
Chu preamble association are considered to further enhance performance.

###### Index Terms:

Grant-free random access, sparse code multiple access, active user detection,
deep learning, extraction network

##  I Introduction

Grant-free random access (GF-RA) is useful for massive machine type
communication (mMTC) in IoT devices, allowing for sporadic uplink traffic.
Grant-free non-orthogonal multiple access (GF-NOMA) uses code-domain non-
orthogonal multiple access (CD-NOMA) to spread user data with non-orthogonal
signatures for better overloading performance [1, 2, 3]. Active users transmit
a preamble and spread data using a randomly selected signature without
scheduling, and receivers estimate each user’s channel and detect transmitted
data using the associated signature as long as active user preambles are
detected [2].

While it is possible to associate a unique preamble to an individual
signature, i.e., one-to-one association, when there are many active users, a
sufficient number of preambles is required to reduce a preamble collision
rate. Due to the limited number of signatures as compared to the number of
preambles, however, each signature may be associated with multiple preambles.
This many-to-one association can lead to unavoidable performance degradation
caused by signature collisions even when different preambles are selected [2,
3]. However, if the channel of the active user is accurately estimated and
sufficiently separated, data can still be detected through the user-specific
channel, implying that accurately estimating the channel of active users is
crucial for data detection in GF-NOMA [3]. On the other hand, accurate channel
estimation (CE) heavily relies on the performance of the active user detection
(AUD) procedure [4, 5, 6, 7, 8, 9], which conventionally uses compressive
sensing (CS) algorithms to exploit sparsity in user activity. However, these
methods’ performance deteriorates significantly with higher user activity
rates [4]. Recent deep learning (DL)-based AUD schemes have been proposed to
overcome this challenge, outperforming CS-based approaches while maintaining
lower complexity [5, 6, 7, 8, 9].

DL-based AUD networks (AUDNs) can be categorized as preamble-based (P-AUDN)
[5, 6] or data-aided (D-AUDN) [7, 8, 9]. D-AUDN improves upon P-AUDN by using
both preamble and data for activity detection. However, current
implementations of D-AUDN only consider one-to-one preamble-signature
association scenarios, and there is no existing implementation for many-to-one
association scenarios. Using a single deep neural network (DNN) architecture
for both preamble and signature activity detection can result in reduced
training efficiency of D-AUDN [7, 8]. An alternative approach is to use a
separate DNN structure for preamble and signature activity detection, which
has been shown to improve training efficiency [9]. However, signature
collision may result in a significant deterioration of D-AUDN’s performance.

This letter proposes a novel D-AUDN structure for many-to-one preamble-
signature association scenarios in GF-RA, using SCMA as a representative CD-
NOMA scheme. SCMA’s optimized low-density codebook (CB) allows for low-
complexity multi-user detection through a message passing algorithm (MPA) [10,
11, 12]. Inspired by the extraction network in [13] that derives channel
information from $M$-ary data, we propose a user activity extraction network
(UAEN) that extracts user activity information from the spread and superposed
data, improving activity detection of associated preambles in the long short-
term memory (LSTM)-based AUDN. Our current design focuses on the receiver side
for a fading channel, unlike the joint preamble and UAEN design in [14, 15]
which is subject to additive white Gaussian noise channel. Additionally, we
propose efficient SCMA CB structures and preamble association methods to
enhance performance. Simulation results show that our proposed approach
outperforms state-of-the-art AUDNs in GF-SCMA systems in terms of active
users, signal-to-noise ratio (SNR), and activity detection error rate (ADER).

Figure 1: Transceiver model for GF-SCMA system

##  II System Model

###  II-A Grant-free SCMA

A total of ${{N}_{u}}$ users who have the potential to transmit, identified by
their indices $n\in\\{0,1,\ldots,{{N}_{u}}-1\\}$, are restricted to a specific
time slot, which is shared subject to collision among them. Ultimately, only a
subset of ${{N}_{u}}$ users will become active and transmit a preamble before
sending a data burst to the time-synchronized base station. Assume that the
activity of each user follows a Bernoulli distribution. More specifically, let
${{\delta}_{n}}\in\\{0,1\\}$ denote activity indicator of user $n$, i.e.,
${{\delta}_{n}}$ = 1 indicating that user $n$ is active with a probability of
${{p}_{n}}$, and ${{\delta}_{n}}$ = 0 with a probability of ($1-{{p}_{n}}$).
The activity indicators for all users constitute a random activity vector,
represented as
$\bm{\updelta}={{[{{\delta}_{0}},{{\delta}_{1}},\ldots,{{\delta}_{{{N}_{u}}-1}}]}^{T}}$.

For SCMA-based grant-free access with $J$ different CBs, let
$\mathcal{C}{{}_{j}}$ denote the $j$-th $M$-ary CB, which is represented as
${{\mathcal{C}}_{j}}=\\{\mathbf{c}_{0}^{(j)},\mathbf{c}_{1}^{(j)},\ldots,\mathbf{c}_{M-1}^{(j)}\\}$,
$j=1,2,\cdots,J$ where $\mathbf{c}_{m}^{(j)}\in{{\mathbb{C}}^{K}}$ denotes its
$m$-th $K$-dimensional codeword (CW), e.g., employing $K$ subcarriers. Note
that each CW is a sparse vector with $N$ non-zero elements ($N<K$). The $j$-th
CB has an average power of
${{P}_{j}}=\left({}^{1}/{}_{M}\right)\cdot\sum\nolimits_{m=0}^{M-1}{\left\|\mathbf{c}_{m}^{(j)}\right\|}_{2}^{2}$
and the total power of all CBs is $P=\sum\nolimits_{j=0}^{J-1}{{{P}_{j}}}$. In
this context, there are two design options for SCMA CBs: the power-balanced CB
(PB-CB), which distributes power evenly across all CBs [10, 11], and the power
imbalanced CB (PI-CB), which distributes power unevenly among CBs [12].

Meanwhile, we consider non-orthogonal Zadoff-Chu (ZC) sequence-based
preambles. A ZC sequence with length ${{N}_{ZC}}$, assuming ${{N}_{ZC}}$ is a
prime number, and a root $u=\\{1,2,\ldots,{{N}_{ZC}}-1\\}$ is given as

$\small z{}_{u}(k)=\exp\left({}^{-i\pi
uk(k+1)}/{}_{{{N}_{ZC}}}\right)\,,k\text{=1,2,}\ldots\text{,}{{N}_{\text{ZC}}}$
(1)

From a single root, ${{N}_{ZC}}$ orthogonal preambles can be generated through
non-zero cyclic shift up to ${{N}_{ZC}}$. When more than ${{N}_{ZC}}$
preambles are required to maintain an acceptable level of preamble collision,
ones from the different roots can be generated with their cross-correlation of
$1/\sqrt{{{N}_{ZC}}}$. Accordingly, a total of ${{N}_{ZC}}({{N}_{ZC}}-1)$ non-
orthogonal ZC preambles can be generated with a prime length ${{N}_{ZC}}$.

Assume that each CB is associated $L$ ZC preamble sequences. Therefore, we are
considering a total of ${{N}_{R}}=J\cdot L$ contention transmission units
(CTUs) [2]. In mMTC, the number of users is much larger than the number of
available preambles, ${{N}_{R}}\ll{{N}_{u}}$. Therefore, in order to minimize
the probability of preamble collisions, it is advantageous to associate as
many preambles as possible [2].

###  II-B Virtual Transceiver Model for DL-based AUD Design

Due to a massive number of users, it is challenging to optimize AUD for all
users while taking potential collisions into account. As in the existing DL-
based AUD designs, therefore, we adopt a similar design approach optimized for
all possible ${{N}_{R}}$ preambles in the absence of collisions [5, 6, 7, 8,
9]. Toward this end, we consider a virtual transceiver model as shown in Fig.
1, which deals with ${{N}_{R}}$ simultaneously users, indexed by
$n\in\\{0,1,\ldots,{{N}_{R}}-1\\}$. Let
${{\mathbf{p}}_{n}}\in{{\mathbb{C}}^{{{N}_{ZC}}}}$ represent a ZC preamble
assigned to user $n$. Assigning a unique non-orthogonal ZC preamble to each
user, it intends to train DL-based AUD subject to the highly overloaded
situation.

As $J<{{N}_{R}}$, CBs must be reused for many-to-one preamble-CB association,
suffering from multi-user interference. We assume that ${{N}_{R}}$ users
employ a means of round-robin CB selection, i.e., user $n$ to select a CB with
index $\nu(n)=\bmod(n,J)$. Then, let ${{\mathcal{C}}_{\nu(n)}}$ denote a CB
for user $n$ where $\nu(n)=\bmod(n,J)$. To average out a CB collision rate,
homogeneous user activity is assumed by
${{p}_{0}}={{p}_{1}}=\ldots={{p}_{{{N}_{R}}-1}}=\bar{p}$ [14]. Then the
expected number of active users within a time slot, denoted by ${{N}_{a}}$, is
given as ${{N}_{a}}={{N}_{u}}\bar{p}$.

A bit sequence of active user is divided into ${{N}_{d}}$ blocks, each with
${{\log}_{2}}M$ bits. Then, each block is encoded with its own preassigned CB.
Let $\mathbf{b}_{i}^{(n)}\in{{\mathbb{B}}^{{{\log}_{2}}M}}$ denote the $i$-th
block of user $n$, which is encoded with a CW, denoted as
$\mathbf{w}_{i}^{\nu(n)}\in{{\mathcal{C}}_{\nu(n)}}$. Assuming that CTU
transmissions are experiencing a flat fading channel, let ${{h}_{n}}$ denote a
channel coefficient for user $n$, thus constituting a channel vector for all
users, represented as
$\mathbf{h}={{[{{h}_{0}},{{h}_{1}},\ldots,{{h}_{{{N}_{R}}-1}}]}^{T}}$. Note
that a flat fading channel model is commonly accepted for the mMTC users with
no movement [4, 5, 6, 7, 8, 9]. Consequently, the received signals,
${{\mathbf{y}}^{(p)}}$ and $\mathbf{y}_{i}^{(d)}$, for the preamble and the
$i$-th CW superposed over the given resources are expressed respectively as

$\small{{\mathbf{y}}^{(p)}}=\sum\nolimits_{n=0}^{{{N}_{R}}-1}{{{\delta}_{n}}{{h}_{n}}{{\mathbf{p}}^{(n)}}}+{{\mathbf{n}}^{(p)}},$
(2)

and

$\small\mathbf{y}_{i}^{(d)}=\sum\nolimits_{n=0}^{N-1}{{{\delta}_{n}}{{h}_{n}}\mathbf{w}_{i}^{\nu(n)}}+\mathbf{n}_{i}^{(d)},\text{
}i\text{=}0,\ldots,{{N}_{d}}-1$ (3)

where
${{\mathbf{n}}^{(p)}}\sim\mathcal{C}\mathcal{N}(0,{{\sigma}^{2}}{{\mathbf{I}}_{{{K}^{(p)}}}})$
and
$\mathbf{n}_{i}^{(d)}\sim\mathcal{C}\mathcal{N}(0,{{\sigma}^{2}}{{\mathbf{I}}_{K}})$
are the additive white Gaussian noise (AWGN) vector with $M\times M$ identity
matrix ${{\mathbf{I}}_{M}}$. In Fig. 1, conventional CS-based AUD can be
performed only using the superposed preamble ${{\mathbf{y}}^{(p)}}$ to detect
an activity vector, represented as
$\bm{\hat{\updelta}}={{[{{\hat{\delta}}_{0}},{{\hat{\delta}}_{1}},\ldots,{{\hat{\delta}}_{{{N}_{R}}-1}}]}^{T}}$.
After the AUD process, CE and multi-user detection (MUD) for active users can
be performed using a linear minimum mean square estimator and MPA,
respectively [2].

The objective of the DL-based AUDs is to optimize an AUD network (AUDN),
denoted as non-linear mapping $g(\cdot)$, in terms of its trainable parameters
[5, 6, 7, 8, 9]. More specifically, the AUDN must be designed to minimize the
ADER, which can be formulated as the following optimization problem:

$\small{{g}^{*}}=\underset{g}{\mathop{\arg\min}}\,{{\left\|\bm{\updelta}-g(\cdot;{{\bm{\uptheta}}^{(g)}})\right\|}_{0}}$
(4)

where ${{\bm{\uptheta}}^{(g)}}$ is a vector of weight and bias in AUDN and
${{\left\|\cdot\right\|}_{0}}$ denotes L-0 norm. Depending on whether it is
preamble-based AUDN (P-AUDN) [5, 6] or data-aided (D-AUDN) [7, 8, 9], DNN is
represented as
$g(\cdot;{{\bm{\uptheta}}^{(g)}})={{g}_{p}}({{\mathbf{y}}^{(p)}};{{\bm{\uptheta}}^{({{g}_{p}})}})$
or
$g(\cdot;{{\bm{\uptheta}}^{(g)}})={{g}_{d}}({{\mathbf{y}}^{(p)}},\\{\mathbf{y}_{i}^{(d)}\\}_{i=0}^{{{N}_{d}}-1};{{\bm{\uptheta}}^{({{g}_{d}})}})$,
respectively, where ${{\bm{\uptheta}}^{({{g}_{k}})}}$ is a vector of weight
and bias of ${{g}_{k}}(\cdot)$. Both
${{g}_{p}}({{\mathbf{y}}^{(p)}};{{\bm{\uptheta}}^{({{g}_{p}})}})$ and
${{g}_{d}}({{\mathbf{y}}^{(p)}},\\{\mathbf{y}_{i}^{(d)}\\}_{i=0}^{{{N}_{d}}-1};{{\bm{\uptheta}}^{({{g}_{d}})}})$
are trained to detect user activity $\bm{\hat{\updelta}}$, i.e.,
$\bm{\hat{\updelta}}={{g}_{p}}({{\mathbf{y}}^{(p)}};{{\bm{\uptheta}}^{({{g}_{p}})}})$
or
$\bm{\hat{\updelta}}={{g}_{d}}({{\mathbf{y}}^{(p)}},\\{\mathbf{y}_{i}^{(d)}\\}_{i=0}^{{{N}_{d}}-1};{{\bm{\uptheta}}^{({{g}_{d}})}})$.
In GF-SCMA, since ${{\mathbf{y}}^{(p)}}$ and
$\\{\mathbf{y}_{i}^{(d)}\\}_{i=0}^{{{N}_{d}}-1}$ over the same resources are
received from the same active users, their joint distributions can be
exploited to improve the AUDN performance.

##  III Proposed Design of Data-aided AUD Network (D-AUDN)

###  III-A Overall Structure

Fig. 2(a) shows the proposed architecture for D-AUDN with a user activity
extraction network (UAEN) that extracts activity information jointly from
$\\{\mathbf{y}_{i}^{(d)}\\}_{i=0}^{{{N}_{d}}-1}$ and ${{\mathbf{y}}^{(p)}}$.
In order to exploit both ${{\mathbf{y}}^{(p)}}$ and
$\\{\mathbf{y}_{i}^{(d)}\\}_{i=0}^{{{N}_{d}}-1}$ as input for the AUDN, it
needs to be transformed into a double-length real vector by concatenating its
real and imaginary parts. The UAEN is designed to generate a priori
probabilities of user activity for individual users, denoted as a vector
$\bm{\upalpha}=[{{\alpha}_{0}},{{\alpha}_{1}},\cdots,{{\alpha}_{{{N}_{R}}-1}}]$,
where ${{\alpha}_{n}}$ represents a probability that user $n$ is active given
$\\{\mathbf{y}_{i}^{(d)}\\}_{i=0}^{{{N}_{d}}-1}$. More specifically, the UAEN
is represented by the following nonlinear mapping
$h(\\{\mathbf{y}_{i}^{(d)}\\}_{i=0}^{{{N}_{d}}-1};{{\bm{\uptheta}}^{(h)}})=\bm{\upalpha}$
where ${{\bm{\uptheta}}^{(h)}}$ is a vector of weight and bias in the network.
The detailed design approach is discussed in the sequel.

As $\bm{\upalpha}$ is exploited as a priori information for our proposed
D-AUDN, the proposed AUDN can be represented by the following nonlinear
mapping
${{g}_{\alpha}}(\bm{\upalpha},{{\mathbf{y}}^{(p)}};{{\bm{\uptheta}}^{({{g}_{\alpha}})}})=\bm{\hat{\upeta}}$
where ${{\bm{\uptheta}}^{({{g}_{\alpha}})}}$ is a vector of weight and bias in
the proposed D-AUDN and
$\bm{\hat{\upeta}}=[{{\hat{\eta}}_{0}},{{\hat{\eta}}_{1}},\cdots,{{\hat{\eta}}_{{{N}_{R}}-1}}]$.
The user activity $\bm{\hat{\updelta}}$ is finally determined by a threshold
criterion, i.e., user $n$ is declared active if ${{\hat{\eta}}_{n}}>\gamma$,
where $\gamma$ denotes an activity threshold [5, 6, 7, 8, 9]. A serially
concatenated vector of $\bm{\upalpha}$ and ${{\mathbf{y}}^{(p)}}$ is taken as
an input to the D-AUDN ${{g}_{\alpha}}(\cdot)$. As $\bm{\upalpha}$ is user
activity information extracted from
$\\{\mathbf{y}_{i}^{(d)}\\}_{i=0}^{{{N}_{d}}-1}$, the proposed D-AUDN is
expected to improve the performance through activity information jointly
extracted from both data and preamble.

(a) Overall Receiver Structure

(b) Detailed Structure of LSTM-based AUDN

Figure 2: Structure of Proposed D-AUDN

As shown in Fig. 2(b), ${{g}_{\alpha}}(\cdot)$ utilizes the LSTM-based AUDN
which was adopted in [8]. It uses the input and forget gates of the LSTM
network to select and reject active and inactive users from NOMA measurements,
which results in better convergence and generalization performance compared to
residual network (ResNet)-based AUDNs [6, 7]. This is particularly true when
input sparsity varies dynamically over time. The LSTM-based AUDN structure is
detailed in the sequel.

The $s$-th LSTM cell consists of cell-state $l_{c}^{s}$ and three gates, i.e.,
forget gate $l_{f}^{s}$, input gate $l_{i}^{s}$, and output gate $l_{o}^{s}$.
Let $\mathbf{z}^{s}$ and $\mathbf{o}^{s}$ denote the input and output of the
$s$-th cell. These gating operations are defined as
$l_{x}^{s}={{\sigma}_{g}}({{\mathbf{W}}_{x}}{{\mathbf{z}}^{s}}+{{\mathbf{U}}_{x}}{{\mathbf{o}}^{s-1}}+{{\mathbf{b}}_{x}})$,
$x=f,i$ and $o$, where ${{\sigma}_{g}}(x)=[1/(1+{{e}^{-x}})]$ denotes sigmoid
function, ${{\mathbf{W}}_{x}}\in{{\mathbb{R}}^{10{{N}_{R}}\times
10{{N}_{R}}}}$ and ${{\mathbf{U}}_{x}}\in{{\mathbb{R}}^{10{{N}_{R}}\times
10{{N}_{R}}}}$ are the weights associated with ${{\mathbf{z}}^{s}}$ and
$\mathbf{o}^{s-1}$, respectively, while
${{\mathbf{b}}_{x}}\in{{\mathbb{R}}^{10{{N}_{R}}\times 1}}$ is the bias. Then,
a cell-state $l_{c}^{s}$ of the $s$-th cell is given by
$l_{c}^{s}=l_{f}^{s}\circ l_{c}^{s-1}+l_{i}^{s}\circ\bar{l}_{c}^{s}$ where
$\bar{l}_{c}^{s}=\tanh({{\mathbf{W}}_{c}}{{\mathbf{z}}^{s}}+{{\mathbf{U}}_{c}}{{\mathbf{o}}^{s-1}}+{{\mathbf{b}}_{c}})$
while $\circ$ denotes the Hadamard product. Meanwhile, the output of the
$s$-th LSTM cell is given by
${{\mathbf{o}}^{s}}=l_{o}^{s}\circ\tanh(l_{c}^{s})$ [8].

Ideally, it is possible to jointly train the AUDN and UAEN only by BCE loss
function in [5, 6, 7, 8], which is given as

$\small\mathcal{L}(\bm{\updelta},\bm{\hat{\upeta}})=-\sum\nolimits_{n=0}^{N-1}{\\{{{\delta}_{n}}\log({{{\hat{\eta}}}_{n}})+(1-{{\delta}_{n}})\log(1-{{{\hat{\eta}}}_{n}})\\}}.$
(5)

However, the deep location of the UAEN component within AUDN causes gradient
vanishing during backpropagation, leading to poor convergence performance when
using equation (5). To address this issue, a pre-training scheme and a
specific UAEN structure are proposed in the following subsections.

###  III-B Design of User Activity Extraction Network (UAEN)

Processing $\\{\mathbf{y}_{i}^{(d)}\\}_{i=0}^{{{N}_{d}}-1}$ jointly with UAEN
requires expanding the size of UAEN, which may lead to the vanishing gradient
problem and high computational complexity. To address these issues, one
possible solution is to reduce the number of data samples by splitting the
processing of $\\{\mathbf{y}_{i}^{(d)}\\}_{i=0}^{{{N}_{d}}-1}$ into
independent smaller networks. The outputs of these smaller networks can then
be combined later in an ensemble network. This approach can help manage the
computational complexity and mitigate the vanishing gradient problem.

Fig. 3 shows the proposed UAEN employs two 1D-convolutional layers to produce
an intermediate activity vector. The first convolutional layer is made of
$N_{\text{kernel}}^{(1)}$ kernels with a size of $2K$. Let
${{\bm{{\alpha}^{\prime\prime}}}_{i}}\in{{\mathbb{R}}^{N_{\text{kernel}}^{(1)}}}$
represent its output vector for the first kernels from $\mathbf{y}_{i}^{(d)}$.
As there tends to be too many trainable parameters for the dense network of
combining the intermediate activity vectors, we consider another convolutional
layer that can reduce the number of trainable parameters. The second
convolutional layer to reduce
$\\{{{\bm{{\alpha}^{\prime\prime}}}_{i}}\\}_{i=0}^{{{N}_{d}}-1}$ is made of
$N_{\text{kernel}}^{(2)}$ kernels with a size of $N_{\text{kernel}}^{(1)}$,
where $N_{\text{kernel}}^{(1)}>N_{\text{kernel}}^{(2)}$. Let
${{\bm{{\alpha}^{\prime}}}_{i}}\in{{\mathbb{R}}^{N_{\text{kernel}}^{(2)}}}$
represent its output vector for
${{\bm{{\alpha}^{\prime\prime}}}_{i}}\in{{\mathbb{R}}^{N_{\text{kernel}}^{(1)}}}$,
forming the intermediate activity vector. After constructing
$\\{{{\bm{{\alpha}^{\prime}}}_{i}}\\}_{i=0}^{{{N}_{d}}-1}$, a single dense
layer is applied to the UAEN with ${{N}_{R}}$ nodes to ensemble them. Finally,
another dense layer with sigmoid function ${{\sigma}_{g}}(\cdot)$ is adopted
in the output to produce the activity vector
$\bm{\upalpha}\in{{\mathbb{R}}^{{{N}_{R}}}}$. Note that batch normalization
and ReLU activation function are used after each layer for the improved
convergence performance.

Figure 3: Detailed Structure of UAEN

###  III-C Progressive Training for proposed D-AUDN

D-AUDN is deeper than existing DL-based AUDNs as it includes an additional
network for extracting user-activity information, but may suffer from the
vanishing gradient problem if only joint training with the loss function in
(5) is used. To address this, a progressive training approach is employed
where the UAEN component is pre-trained before joint UAEN and AUDN training.

A self-supervised loss function is employed for pre-training the UAEN in such
a way that its output $\bm{\upalpha}$ becomes close to the activity vector
$\bm{\updelta}$ as follows:

$\small{{\mathcal{L}}_{PT}}(\bm{\updelta},\bm{\upalpha})=-\sum\nolimits_{n=0}^{N-1}{\\{{{\delta}_{n}}\log({{\alpha}_{n}})+(1-{{\delta}_{n}})\log(1-{{\alpha}_{n}})\\}}.$
(6)

To comprehensively incorporate the AUDN, fine-tuning of the UAEN through joint
training with equation (5) is necessary, as using equation (6) for pre-
training alone does not consider the AUDN. This joint training incorporates
$\bm{\upalpha}$ as the priori user activity information for AUDN to minimize
ADER. A two-step training approach is employed that combines pre-training and
joint training, as described in [14].

##  IV CTU Design for Proposed Data-aided AUDN

###  IV-A Optimizing CB Design for D-AUDN

Detecting CB activity directly from superposed data in fading channels is a
challenging task, especially for the original SCMA CB design [10] that employs
constellation rotation. This design is vulnerable to phase changes in the
fading channel, leading to potential loss of CB activity information. Resource
sparsity may differentiate CBs, but it becomes harder with more active users.

To overcome this challenge, we have explored the use of CBs with asymmetric
constellations in the same resource, such as PB-CB [11] and PI-CB [12]. Among
these, PI-CB offers two significant advantages over PB-CB for D-AUDN. Firstly,
it provides a greater minimum Euclidean distance, aiding the D-AUDN in
distinguishing CB activity. Secondly, the asymmetry of the constellation,
induced by different average CB power levels, is useful for the UAEN in
extracting CB activity in a fading channel. Asymmetric constellation points
have also been shown to be useful in estimating time-varying fading channels
without pilots in [13].

###  IV-B Root-separated Zadoff-Chu Association

To ensure an effective CB-preamble association for CTU design, both CB
distance and preamble distance should be considered together to maximize the
minimum distance between CTUs. The distance between preambles can be measured
using cross-correlation, with higher cross-correlation indicating a closer
distance [4]. As CBs with the same index have a closer distance, two CTUs with
the same CB must reply on the preambles with a greater distance, i.e.,
associating the same CB with the preambles of the farther distance (lower
cross-correlation), as discussed in [15]. Toward this end, each CB is assigned
to a distinct set of orthogonal ZC preambles generated from the same root
sequence, with the different roots evenly distributed among all CBs. This
root-separated (RS) ZC preamble association improves a distance between active
CTUs belonging to the same CB.

##  V Simulation Results

We consider a GF-SCMA system with $J=6$, $K=4$, and $N=2$. Two SCMA
constellation designs, i.e., PB-CB in [11] or PI-CB in [12], are considered
for the data transmission. The number of CWs used for data transmission is set
to ${{N}_{d}}=16$ [14]. Considering two different lengths of the ZC preambles,
i.e., ${{N}_{ZC}}=7$ or ${{N}_{ZC}}=13$, orthogonal preambles from the six
roots, i.e., $u=\\{1,2,\ldots,6\\}$, are utilized to generate a total number
of $6{{N}_{ZC}}$ non-orthogonal preambles. Those ZC preambles are associated
with $J=6$ SCMA CBs, either randomly or using RS ZC association in Section
IV-B.

We assume independent and identically distributed Rayleigh flat-fading channel
with the SNR of 20dB for training data set. We assume that the number of
active users, $N_{a}$, is uniformly distributed over $[1,6]$. Data samples are
constructed by activity vectors $\bm{\updelta}$ for the given number of active
users. We generate $5\cdot 10^{5}$ activity vectors in the pre-training step,
while $5\cdot 10^{6}$ activity vectors are used in the joint training step.
The learning rates of 0.001 and 0.0001 for pre-training and joint training,
respectively, and the training epoch is set to 50. We employ an ADAM optimizer
with a batch size of 1000 [8]. Since it is verified that DL-based AUDs have
superior ADER performance and lower computational complexity compared to
conventional CS-based AUDs, we will focus on how much performance gain can be
achieved compared to DL-based AUD designs, i.e., P-AUDN in [6] and D-AUDN in
[8]. The number of LSTM cells for AUDN and activity threshold are set to
$S=10$ and $\gamma=0.5$, respectively [8]. We selected sufficiently large
hyperparameters for UAEN, i.e., $N_{\text{kernel}}^{(1)}=10{{N}_{R}}$ and
$N_{\text{kernel}}^{(2)}=2{{N}_{R}}$, through hyperparameter tuning in
Rayleigh fading channel. The number of trainable parameters in the UAEN are
only $2.43\%$ and $4.67\%$ of those in AUDN for ${{N}_{R}}=42$ and
${{N}_{ZC}}=78$, respectively.

(a) ADER vs. SNR

(b) ADER vs. $N_{a}$

Figure 4: ADER performance comparison with existing AUDNs

Fig.’s 4(a) and 4(b) compare the ADER performance of the proposed AUDN with
that of the existing DL-based AUDN in [6] and [8], considering the PB-CB and
random ZC association. In Fig. 4(a), ADER performance is presented as varying
the SNRs in a flat-fading channel with ZC preamble length ${{N}_{ZC}}=7$. The
proposed D-AUDN outperforms both P-AUDN and conventional D-AUDN in a wide
range of SNR in both ${{N}_{a}}=3$ and ${{N}_{a}}=6$. Meanwhile, Fig. 4(b)
shows the ADER as varying the number of active users at 20dB SNR. Note that
the proposed D-AUDN is trained with $N_{a}=[1,6]$, while the ADER is evaluated
with a wider range of $N_{a}=[2,12]$. It has been observed that the proposed
D-AUDN can robustly support additional active users compared to P-AUDN at ADER
of 0.05 when using short ZC sequences with a length of 7 and 13, respectively,
while outperforming the conventional D-AUDN approach. This suggests that a
robust data-aided gain can be achieved from the proposed D-AUDN structure.

In Fig. 5, we examined the impact of proposed CTU design on D-AUDN performance
in ${{N}_{a}}=6$, comparing two different CB designs, PB-CB and PI-CB,
respectively. In addition, we evaluate the efficiency of RS ZC preamble
association proposed in Section IV-B by comparing it with random ZC preamble
association. Specifically, in RS ZC preamble association, preambles generated
from the 6 roots are separately associated with one of the six CBs. Fig. 5
indicates that using PI-CB results in around 2dB gain at ADER of 0.1 and 0.05,
compared to PB-CB in ${{N}_{ZC}}=7$ and 13, respectively. Furthermore,
utilizing RS ZC preamble association can provide up to 1dB and 0.5dB
performance gain over the random ZC preamble association in ${{N}_{ZC}}=7$ and
13, respectively. Note that the shorter ZC preambles result in higher non-
orthogonality, causing a smaller minimum CTU distance when randomly
associated. As a result, the performance gain from RS ZC association is more
significant with shorter ZC sequences.

(a) $N_{ZC}=7$

(b) $N_{ZC}=13$

Figure 5: Impact of CTU Design for proposed D-AUDN: $N_{a}=6$

##  VI Conclusions

We proposed a DL-based approach for D-AUDN in GF-SCMA systems, utilizing a
user activity extraction network that efficiently exploits SCMA CBs and
Zadoff-Chu preamble for activity detection. Additionally, we used power-
imbalanced SCMA CB and a root-separated ZC preamble association method to
further improve D-AUDN performance. Our proposed structure can also be applied
to other types of GF-NOMA systems requiring many-to-one association between
preamble and signature. Future research could extend our proposed architecture
to integrate with CE and MUD in highly overloaded mMTC systems, where
challenges such as preamble non-orthogonality and signature collisions are
present.

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