# MCD: A Modified Community Diversity Approach for Detecting Influential Nodes
in Social Networks

Aaryan Gupta11footnotemark: 1
Department of Applied Mathematics
Delhi Technological University
New Delhi, India
<EMAIL_ADDRESS>
&Inder Khatri11footnotemark: 1
Biometric Research Laboratory
Delhi Technological University
New Delhi, India
<EMAIL_ADDRESS>
&Arjun Choudhry
Biometric Research Laboratory
Delhi Technological University
New Delhi, India
<EMAIL_ADDRESS>
&Sanjay Kumar
Department of Computer Science and Engineering
Delhi Technological University
New Delhi, India
<EMAIL_ADDRESS>

###### Abstract

Over the last couple of decades, Social Networks have connected people on the
web from across the globe and have become a crucial part of our daily life.
These networks have also rapidly grown as platforms for propagating products,
ideas, and opinions to target a wider audience. This calls for the need to
find influential nodes in a network for a variety of reasons, including the
curb of misinformation being spread across the networks, advertising products
efficiently, finding prominent protein structures in biological networks, etc.
In this paper, we propose Modified Community Diversity (MCD), a novel method
for finding influential nodes in a network by exploiting community detection
and a modified community diversity approach. We extend the concept of
community diversity to a two-hop scenario. This helps us evaluate a node’s
possible influence over a network more accurately and also avoids the
selection of seed nodes with an overlapping scope of influence. Experimental
results verify that MCD outperforms various other state-of-the-art approaches
on eight datasets cumulatively across three performance metrics.††*Equal
Contribution

_K_ eywords Community Diversity $\cdot$ Community Structure $\cdot$
Influential nodes $\cdot$ Independent Cascade Model $\cdot$ Social Networks
$\cdot$ Viral Marketing

##  1 Introduction

Nowadays, Social networks have become an integral part of our daily lives and
serve as an excellent medium for people to interact and propagate their views
and ideas. Due to the large number of users available on various social
networks across the globe, they act as expedient channels for advertisements
and promotion of the products (Mislove et al., 2007; Garton et al., 1997;
Kumar et al., 2006). Over the years, advertisements over social networks, also
known as “Viral Marketing” have seen a significant boost due to their cost-
effective nature and ability to propagate faster and wider. This approach,
also known as “word-of-mouth advertising,” exploits people’s trust in a known
or influential person’s suggestions over other traditional means of
advertisement (Ferguson, 2008; Zhou et al., 2019). Further, social networks
have been used for various other uses, including building collaborations,
friendships, and other social or professional relations. Hence, the problem of
finding nodes with wide influence over a network is a prominent area of
research in network analysis. This is further required for various tasks like
curbing rumors spread in social networks and identifying prominent proteins in
protein graphs and other graph-based networks.

The problem of finding influential nodes in a network can be delineated as the
task of identifying a set of $k$ influential nodes in a network whose
influence over the whole network is maximum across any set of $k$ nodes, where
$k$ is a small positive integral constant (Kempe et al., 2003a; Liu et al.,
2022). Each individual entity acts as a node in the network, whereas the
relationship between two nodes (friendship, follower-followee, co-authorship
between users, links between proteins, etc.) helps form edges in the network
to connect the nodes. For example, in social networks, the task entails
maximizing the spread of an opinion by communicating it across the maximum
number of nodes possible. Formally, the problem of finding the influential
nodes in a network, modelled as a graph $G(V,E)$ where $V$ is the set of all
nodes in the network, and $E$ is the set of edges connecting the nodes, can be
defined as shortlisting a small number of nodes ($k$) such that the spread of
influence can be maximized in the network. Mathematically, we can represent it
as follows.

$S=argmax\ \sigma(S)\,\ where\ S\in V,\ \|S\|=k,\|V\|\gg k$ (1)

Here, argmax $sigma(S)$ denotes the overall influence spread generated by a
set $S$ of $k$ number of nodes, and $k$ is a small and positive constant with
respect to the total number of nodes in the network. In general, selecting
influential nodes in a network consists of two main components: choosing a set
of $k$ spreaders or seed nodes and then computing the spread of influence
across the network originating from the seed nodes utilizing a suitable
information propagation model. While selecting the set of seed nodes, we first
ensure that the number of seed nodes is significantly lesser than the number
of nodes in the network to use the resources available optimally. We also need
to consider the influence or propagating capability of each node in the
network based on its locality and links within the network. Owing to the
presence of a community structuring system in social networks, involving
community-dependent features can be used to evaluate better the network
structure, as well as aid in accelerating the spread of the propaganda.

In this work, we propose the concept of Modified Community Diversity (MCD) for
finding influential nodes in a network. Our approach extends the idea of the
previously proposed Community-based Spreaders Ranking algorithm (CSR) (Kumar
et al., 2022a) for two-hop neighborhood information. The proposed work intends
to enable better utilization of available seed nodes for information spread.
We further propose the use of Extended Community Diversity for gathering
information about the neighborhood based upon a two-hop mechanism. This
information is further used to avoid the overlapping influence of influential
nodes in a network, thus improving the efficacy of finding influential nodes
in a network and providing an optimized spread of information or influence.
For further experimental analysis, we compare the performance of our proposed
approach with other state-of-the-art approaches for influential node detection
on eight datasets, seven of which are real-world datasets. We use the
Independent Cascade (IC) model (Goldenberg et al., 2001) to simulate the
influence propagation on all datasets. Our experimental evaluations over three
performance metrics validate the superiority of our proposed approach over the
other approaches across different domains and act as evidence of its efficacy
and widespread applicability. The main contributions of our paper can be
summarised as follows.

  * •

We introduce Modified Community Diversity (MCD) method using community
structure and extended community diversity to find influential nodes in social
networks.

  * •

The proposed work utilizes a two-hop concept which enables us to gather
information about the neighborhood of a node better to judge its influence
appropriately. This helps in avoiding the selection of influential nodes with
overlapping influence.

  * •

We verify the results of the proposed approach on eight network datasets
against seven other state-of-the-art methods. The obtained results reveal that
the proposed method outperforms different approaches cumulatively across three
metrics. Statistical tests verify the superiority of MCD over competing
approaches.

The breakdown of the remaining paper is as follows. Section 2 details the
related works and the existing research gaps. Section 3 elaborates on our
proposed methodology and algorithms. Section 4 consists of information about
the various experimental details used in our paper, including the datasets
used, the Information Propagation models, the performance metrics, and the
implementation environment. Section 5 contains our experimental results, their
analysis, and broad outcomes. Section 6 delineates the statistical tests we
have done to verify the efficacy of MCD for influence maximization. Section 7
concludes our paper.

##  2 Related Works

Due to the advent of the internet and the rise in the global usage of social
networks, the task of finding influential spreaders in a network as a research
problem has seen significant contributions from researchers. Researchers have
proposed various approaches to find the most influential nodes in a network,
to maximize the spread of information in a network (Li et al., 2018; Banerjee
et al., 2020). These approaches can be predominantly categorized as diffusion
process-based approaches, descriptive approaches, topological measures-based
approaches, statistical and stochastic approaches, data mining-based
approaches, and machine learning-based (Mendonça et al., 2022) approaches.
Experimental evaluations have found topological measures-based approaches to
be the most effective in identifying suitable spreader nodes (Kumar et al.,
2022b). Topological measures-based approaches utilize predefined centrality
measures for ranking nodes based on their efficacy in spreading information
across a network. These centrality measures aid in selecting suitable spreader
nodes with high influence on the other nodes in the network based on their
specific topological properties. Centrality measures-based approaches
generally consist of two main steps: generating a score for each node based on
a defined centrality measure and selecting the top _k_ nodes based on the
score obtained for the measure.

One of the first approaches using centrality scores for finding influential
nodes in a network was proposed by Freeman (1978). The approach used the
Degree Centrality measure, a local centrality measure that considers the
number of neighbors of a node to generate its score. Degree Centrality has
linear time complexity and was effective and efficient for small networks.
However, due to its reliance on the local immediate neighbors for a given
node, the effectiveness of Degree Centrality drops for nodes with few but
influential neighbors. To counter the issues faced with local centrality
measures, especially Degree Centrality, researchers proposed semi-local
centrality measures like LC (Chen et al., 2012) and LSC (Gao et al., 2014),
which go beyond the immediate scope of a node, but up to a limited range.
While LC ranks the nodes using one-hop and two-hop nearest neighbors, LSC also
considers the clustering coefficient between immediate neighbors.

These local centrality-based approaches do not consider a node’s global
location and only use the information corresponding to a certain locality,
thus reducing their efficacy in real-life large networks. To counter this,
researchers have also used global centrality measures like Betweenness
Centrality, Closeness Centrality, and K-shell Decomposition, among others, to
Influence Maximization. Freeman (1977) defined Betweenness for a specific node
as the ratio of the shortest path which passes through the node and the total
number of possible shortest paths in the network. Okamoto et al. (2008)
proposed Closeness for a node as the inverse of the sum of the shortest path
to all other nodes, thus determining the average closeness of a node to the
rest of the graph. Betweenness and Closeness suffer from high time complexity,
as they require the calculation of the shortest path between every node pair
of the network.

Some other global centrality measures successfully used for Influence
Maximization are Eigenvector Centrality (Bonacich, 2007) and Pagerank
Centrality measure (Brin and Page, 1998). They aid in ranking nodes based on
their importance among the neighborhood nodes. According to these measures,
the presence of important nodes around the central node makes it a viable
spreader node. Another global centrality-based measure, GLR (Salavati et al.,
2019), divides the graph into communities and then selects each community’s
local cores and gateway nodes. The nodes are ranked as influential spreaders
based on their shortest paths to the local cores and the gateway nodes.
Researchers have also proposed greedy search-based approaches (Kempe et al.,
2003b; Leskovec et al., 2007c) for ranking spreader nodes. These approaches
use the sub-modularity and monotony of the Independent Cascade (IC) model
along with Monte Carlo simulations to enhance the overall performance of the
approach. However, greedy approaches are generally inefficient due to many
simulations, and they require a lot of time to compute.

Some recent approaches also consider the bridge nodes, which connect two
different groups of nodes in a network, as influential nodes. Liu et al.
(2016) proposed the DIL centrality measure, where the importance of lines and
the degree value are used to highlight a bridge node. DIL also considers the
number of connected triangles when determining the relevance of a link, with
the bridge node taking the primary role in controlling information
propagation. Berahmand et al. (2019) introduced the DCL centrality measure,
which finds the bridge nodes using a node’s degree, clustering coefficient,
and the relationship between its one-hop neighbors. Wen and Deng (2019)
introduced LID, an approach where prominent influencers in complex networks
are extracted using quasi-local information, i.e., local structural
properties.

Recently, researchers have proposed various community structure-based
approaches (Kumar et al., 2021a; Chen et al., 2022) for spreader ranking,
which were found to be more effective in identifying suitable spreader nodes
in a network (Huang et al., 2019; Kumar et al., 2021b; Chen et al., 2021).
Some recent community-based algorithms used for Influence Maximization are IM-
ELPR (Kumar et al., 2021b), CFIN (Khomami et al., 2021), and CAOM (He et al.,
2020). CAOM divides the whole network into communities and selects only the
most significant ones. Certain candidate nodes are then selected from each
community using the one-hop measure. Hereafter, seed nodes are selected from
candidate nodes by eliminating those with influence overlap. CFIN follows a
similar approach, but the candidate nodes are selected using degree
centrality, and the seed nodes are selected using the clustering coefficient
measure. IM-ELPR uses node seeding, community detection, and label propagation
for influence maximization. The seed nodes are first identified using an
extended h-index, and then the communities are detected using label
propagation. The smaller, related communities are further merged into larger
communities, and then the top $k$ influential nodes are identified.

The approaches mentioned above utilized local structures to identify the
bridge nodes. However, unlike these approaches, the recently proposed
Community-based Spreaders Ranking algorithm (CSR) (Kumar et al., 2022a) for
influence maximization selects global bridge nodes by considering the
community bridge nodes as the most influential spreaders in the network. The
CSR algorithm utilized three measures to identify bridge nodes, i.e.,
community diversity, community modularity, and community density. However, CSR
didn’t consider the overlapping influence of the influential nodes in a
network. Our proposed approach (MCD) considers this factor and thus enables
better utilization of available seed nodes. Further, CSR considers only one-
hop neighbors to detect influential nodes. In contrast, MCD uses a two-hop
mechanism, which provides a better overview of the node’s neighborhood
provided to the algorithm. This leads to markedly superior results by our
approach as compared to CSR.

Figure 1: Flowchart representation of the steps involved for the calculation
of MCD for each node.

##  3 Proposed Methodology

In this section, we present our proposed methodology for finding influential
nodes in social networks using community detection and a modified community
diversity approach. First, we uncover the community structure in the input
network using Leiden’s algorithm. We consider the bridge nodes in the network
to play a vital role in spreading the information, and thus, we identify the
bridge nodes in the network using the Community Diversity (CD) score. To
incorporate the two-hop mechanism for gathering more information regarding a
node’s surroundings, we calculate the Extended Community Diversity (ECD)
score. Then we calculate the Modified Community Diversity score using ECD and
CD to ensure the proper distribution of the information. We rank the nodes
based on the MCD score and select the top $k$ nodes, where $k$ signifies the
constraint on the number of nodes that act as influencers. Finally, we
evaluate the influence of the selected $k$ seed nodes as influential nodes
over the entire network by using various evaluation parameters. The overall
proposed work can be understood using Fig. 1, which depicts the process flow
for calculating the MCD score for each node. The various phases of the
proposed work are discussed below.

###  3.1 Community Detection

A community can be illustrated as a subset of a graph with nodes representing
each member of the community (Yang et al., 2021; Chouchani and Abed, 2020; Ma
et al., 2017). The nodes have similar characteristics and are connected to
some of the other nodes by edges. These nodes can represent any entity in the
world, and their meaning is dependent on the scope of the community and, to a
major extent, the scope of the graph. Nodes present specifically in a
community are expected to have meaningful and substantial connections with
each other in comparison to the rest of the graph, thus ensuring a better
spread of information within the community as compared to outside the
community. In our proposed algorithm, we use Leiden’s algorithm (Traag et al.,
2019) to identify the communities in a network. Leiden’s algorithm is based on
greedy optimization techniques, which results in efficient processing and
quicker convergence. Leiden’s algorithm consists of three steps:

  * •

the local moving of the nodes.

  * •

refining the partition between communities.

  * •

accumulation of the nodes in the network on the basis of the refined
partition, with the non-refined partition creating an initial partition for
the accumulated network.

###  3.2 Community Diversity Score

It is often observed in real-world networks that influential nodes are usually
linked to multiple communities in the network. Thus, they have a significant
influence over the state of the network, as they can simultaneously
disseminate their influence to multiple communities. We consider these high-
degree nodes, which are connected to multiple communities, as influential
nodes by applying the concept of Community Diversity (Kumar et al., 2022a), as
they act as bridge nodes between communities. The community diversity score of
a node can be formally defined as the measure of the probability of finding a
certain community in the neighborhood of a node. Community diversity gives a
high score to a node that is connected to multiple communities, as well as
considers the number of communities to which the node transmits information.
The concept of community diversity revolves around the concept of Shannon’s
entropy (Lin, 1991) over the term of the probability of finding the $v^{th}$
node in the $c^{th}$ community in the neighborhood of a certain node. The
Community Diversity score for a node is computed as shown in Equation 2, where
$CDv_{i}$ is the community diversity value of the $i^{th}$ node, and
$CDv_{ic}$ is the contribution of $c^{th}$ community in Community Diversity of
$i^{th}$ node (Kumar et al., 2022a). The neighborhood contribution of each
community is shown in Equation 3, where $\eta_{in}$ represents the number of
nodes in the $c^{th}$ community in the neighborhood of node $i$, and
$\eta_{i}$ represents the number of nodes in the neighborhood of node $i$.

$CD_{i}=-\sum_{c=1}^{comm}CDv_{ic}=-\sum_{c=1}^{comm}P_{ic}*log(P_{ic})$ (2)

$P_{ic}=\frac{\eta_{ic}}{\eta_{i}}$ (3)

Algorithm 1 : Community Diversity Score (Kumar et al., 2022a)

1:$G=(V,E)$, where $n=\|V\|,e=\|E\|$

2:$Community\\_Diversity$

3:$Initialization$

4:$Communities\ \\{$c1, c2….cm$\\}:\ partitioned\ using\ Leiden^{\prime}s\
Algorithm$

5:$value\leftarrow 0$

6:$CD\leftarrow\\{\\}$

7:for $node$ $\in$ $\\{Nodes\ of\ Graph\ G\\}$ do

8: $sum\leftarrow 0$

9: $neigh\\_len\leftarrow number\ of\ neighbours\ of\ node$

10: for $c$ $\in$ $\\{$Communities$\\}$ do

11: $neigh\\_com\\_len\leftarrow number\ of\ neighbours\ of\ node\ belonging\
to\ c$

12: $ratio\leftarrow\frac{neigh\\_com\\_len}{neigh_{l}en}$

13: $value\leftarrow-ratio*log(ratio)$

14: $sum\leftarrow sum+value$

15: end for

16: $CD(node)\leftarrow sum$

17:end for

18:return $CD$

Algorithm 1 represents the procedure used to compute the Community Diversity
(CD) score for a graph $G$. It can be summarized as follows:

  * •

We divide the graph into communities using Leiden’s algorithm.

  * •

For each node, we calculate the number of neighbours. Further, for each
community, the number of neighbours of a node that are part of the respective
community is computed.

  * •

We evaluate the ratio of the total number of neighbours of the node and the
neighbour of the node in the respective community. This is further used to
calculate the entropy score, i.e., the Community Diversity score.

If a node is connected to multiple communities and the connections are found
to be uniformly distributed, then the community diversity score of that node
is observed to be higher. However, if a node has connections to only a few
communities or the distribution of connections is not uniform, then the
community diversity score is observed to be low. Hence, the concept of
community diversity can be utilized to find out the nodes capable of
efficiently spreading information to multiple parts of the network.

###  3.3 Extended Community Diversity Score

We formally propose the Extended Community Diversity score as the ratio of the
community diversity score of a node to that of its neighborhood. The concept
of Extended Community Diversity revolves around considering a node along with
its neighborhood for the prediction of its influence in a locality. The
neighbors of a node contribute to helping the algorithm understand what
relative position a node holds with respect to its neighborhood, which
ultimately tells how much a node will benefit from being made the seed node.
Extended Community Diversity gains information about the neighborhood of a
node by taking two hops, i.e., one hop from the node to its neighbors, and the
second hop from its neighbors to farther nodes. This provides more information
for the proposed algorithm to process and deduces which node is most
influential. The provided information is further used to avoid having many
influential nodes in the same region and hence avoid the wastage of useful
resources. Extended Community Diversity score for a node is computed as shown
in Equation 4 where $ECD_{v}$ is the Extended Community diversity value of
$v^{th}$ node, $neigh$ is the neighbours of node $v$ and $CD_{v}$ is the
community diversity score for node $v$ as computed in Equation 2.

$ECD_{v}=CD_{v}+\sum_{n=1}^{neigh}CD_{n}$ (4)

Algorithm 2 : Extended Community Diversity Score

1:$G=(V,E)$, where $n=\|V\|,\ e=\|E\|$

2:$Extended\\_Community\\_Diversity$

3:$Initialization$

4:$ECD\ \leftarrow\ \\{\\}$

5:$CD\ \leftarrow\ Community\ Diversity\ Score\ using\ algorithm\ 1$

6:$value\ \leftarrow\ 0$

7:for $v\in$ $\\{Nodes\ of\ Graph\ G\\}$ do

8: $value\ \leftarrow CD(v)$

9: for $nb$ $\in$ $\\{Neighbours\ of\ node\ v\\}$ do

10: $value\leftarrow value$ \+ $CD(nb)$

11: end for

12: $ECD(v)\leftarrow value$

13:end for

14:return $ECD$

Algorithm 2 represents the algorithmic procedure used to compute the Extended
Community Diversity (ECD) score for graph G. The summary of the algorithm is
as follows:

  * •

We calculate the previously defined Community Diversity score (CD) in
Algorithm 1.

  * •

Then, for each node, we calculate the summation of its Community Diversity
score, and its neighbor’s Community Diversity Score, which gives the Extended
Community Diversity score.

###  3.4 Modified Community Diversity Score

We formally define the Modified Community Diversity (MCD) score as
$MCD=-P*logP$, where $P$ is the ratio of the Community Diversity of a node to
its Extended Community Diversity. Modified Community Diversity uses the
limited available resources judiciously by ensuring that no two influential
nodes are in the same locality, which, if not taken care of, would result in
the wastage of valuable spreading power and overlapping of the region of
influence of influential nodes in the same locality. The problem of several
influential nodes existing in the same neighborhood is known as the Rich Club
effect. Our proposed approach tackles this problem by scoring the nodes based
on their strength relative to their neighboring nodes, thus ensuring that the
influential nodes are well spread throughout the network. The ratio of the
results of the previously calculated CD score and ECD score is used for
analyzing the MCD score. The concept of MCD maximizes when the distribution of
spreading power is uniform, which makes sense here since the algorithm not
only wants the respective node to be influential but also the neighborhood to
be sufficiently influential in carrying on the spread of information further.
This enables the node to maximize the spread of information in that locality
and beyond it. This results in the respective node being somewhat equivalent
to the whole neighborhood in terms of spreading power, thus making it the most
influential node in the locality. This mitigates the problem of the rich club
effect significantly by considering the overlap in the influence and handling
it in a balanced way to avoid wastage of useful spreading strength. This
results in an optimized spread of information with assured maximum throughput.
Modified Community diversity score for a node is computed as shown in Equation
6, where $P_{v}$ is as shown in Equation 6. In Equation 6, $CD_{v}$ is the
community diversity score of $v^{th}$ node, as shown in Equation 2, and
$ECD_{v}$ is the Extended Community diversity value of $v^{th}$ node, as
calculated in Equation 4.

$MCD_{v}=-P_{v}*log(P_{v})$ (5)

$P_{v}=\frac{CD_{v}}{ECD_{v}}$ (6)

Algorithm 3 : Modified Community Diversity Score

1:$G=(V,E)$, where $n=\|V\|,e=\|E\|$

2:$Modified\\_Community\\_Diversity$

3:$Initialization$

4:CD $\leftarrow\ $Community Diversity Score using algorithm 1

5:ECD $\leftarrow\ $Extended Community Diversity Score using algorithm 2

6:$MCD\leftarrow\\{\\}$

7:$value\leftarrow 0$

8:$p\leftarrow 0$

9:for $v$ $\in$ $\\{Nodes\ of\ Graph\ G\\}$ do

10: $p\leftarrow\frac{CD(v)}{ECD(v)}$

11: $value\leftarrow-p*log(p)$

12: $MCD(v)\leftarrow value$

13:end for

14:return $MCD$

Algorithm 3 represents the procedure used to calculate the Modified Community
Diversity (MCD) score for graph G. The summary of the algorithm is as follows:

  * •

The community diversity score (CD) is calculated using Algorithm 1, and the
extended community diversity score (ECD) is calculated using Algorithm 2.

  * •

Then, for each node, we calculate the ratio of the Community Diversity score
and Extended Community Diversity Score, which gives us the MCD score for the
respective node.

Figure 1 illustrates the process flow for calculating the MCD score for each
node. We first calculate the CD and ECD scores for each node. Using the
generated values of CD and ECD scores, we compute the MCD scores for each
node. We further rank the nodes on the basis of the MCD score and select the
top $k$ nodes, where $k$ signifies the constraint on the number of nodes that
act as influencers. We further evaluate the influence of the selected $k$
nodes over the entire network in Section 5.

##  4 Datasets, Information Propagation Model, & Performance Metrics

###  4.1 Dataset and Baseline Models

We evaluate our proposed approach on seven real-life datasets and one
synthetic dataset of varying sizes and types. Table 1 describes the different
datasets used during our evaluation. Further, we do a comparative analysis of
the proposed approach with seven other state-of-the-art approaches, namely CSR
(Kumar et al., 2022a), GLR (Salavati et al., 2019), DCL (Berahmand et al.,
2019), LID (Wen and Deng, 2019), DIL (Liu et al., 2016), H-Index (HI) (Lü et
al., 2016) and PageRank (PR) (Brin and Page, 1998), over three distinct
evaluation criteria. For the optimal performance for the baseline approaches,
we took the input parameters as reported in the original papers throughout all
our experiments.

  * •

Community Spreaders Ranking (CSR) (Kumar et al., 2022a): CSR measures a node’s
spreading ability with respect to different communities on the basis of its
connection to the communities.

  * •

Gateway Local Rank (GLR) (Salavati et al., 2019): GLR extends over the
closeness centrality measure and simplifies it by reducing the search set to
the local and gateway nodes. To get the optimal performance, we take
parameters $\alpha_{1}$ and $\alpha_{2}$ used for weighing the effect of core
and gateway nodes to be 1 as reported by (Salavati et al., 2019).

  * •

Degree and Clustering Coefficient and Location (DCL) (Berahmand et al., 2019):
DCL measures the spreading ability of nodes on the basis of a node’s degree,
its neighbour, and the clustering coefficient.

  * •

Local Information Dimensionality (LID) (Wen and Deng, 2019): LID measures the
spreading ability of a node by considering the quasilocal structure of a node.

  * •

Degree and Importance of Lines (DIL) (Liu et al., 2016): DIL ranks the nodes
on the basis of the degree and the importance of lines.

  * •

H-index (HI) (Lü et al., 2016): H-index is a node centrality measure that
considers the neighborhood measures, the quality, as well as the quantity of
the neighboring nodes simultaneously.

  * •

PageRank (PR) (Brin and Page, 1998): PageRank measures the significance of a
given node in the network based on its outgoing degree and damping factor.
Similar to the work Brin and Page (1998) and various other works, we take the
value of $d$ to be $0.85$ in all experiments to get optimal performance.

Table 1: The different datasets used for experimentation along with the various characteristics such as Number of Nodes, Edges, and Communities by Leiden Algorithm S.No. | Dataset | Nodes (n) | Edges (m) | Communities
---|---|---|---|---
1 | BA | 2,000 | 9,974 | 14
2 | PGP | 10,638 | 24,301 | 104
3 | Jazz Musicians | 198 | 2,742 | 4
4 | Ca-hepth | 7,126 | 35,324 | 481
5 | GR-QC | 5,242 | 14,496 | 394
6 | Dolphins | 62 | 159 | 5
7 | Erdos992 | 6,100 | 7,500 | 50
8 | p2p-Gnutella25 | 22,687 | 54,705 | 59

A brief description of different datasets is given below:

  * •

BA (Barabási and Albert, 1999): BA is a random graph generated using Barabási-
Albert preferential attachment model. A network of n nodes is generated by
adding new nodes, each having m edges that are preferentially coupled to
existing nodes with a high degree.

  * •

PGP (Boguñá et al., 2004): Pretty Good Privacy (PGP) is an encrypted
communication network. PGP algorithm is used for secure information
interchange. Here, the users are nodes, and the edges of this huge network are
the connectivity between users.

  * •

Jazz Musicians (GLEISER and DANON, 2003): The Jazz musicians dataset is a
real-world social network dataset, where each node represents a musician, and
each edge between two nodes indicates that the two musicians have played
together in a band.

  * •

Ca-hepth (Leskovec et al., 2007a); It represents a collaboration network of
authors who have collaborated on papers submitted to the High Energy Physics -
Theory Category.

  * •

General Relativity and Quantum Cosmology (GR-QC) (Leskovec et al., 2007b): GR-
QC dataset is a real-world social network dataset from arXiv and covers
scientific contributions by authors in General Relativity and Quantum
Cosmology. The authors are nodes, and if they have collaborated with another
author on a paper, then there is an undirected edge between them.

  * •

Dolphins dataset (Lusseau et al., 2003): The dolphins dataset is a social
network of bottlenose dolphins, where a node represents a dolphin, while an
edge represents frequent associations between dolphins.

  * •

Erdos992 (Batagelj and Mrvar, 2000): This dataset is about the collaborations
of Paul Erdős, a Hungarian Mathematician, with his co-authors. The co-authors
are nodes while collaborations between the nodes signify the edges.

  * •

p2p-Gnutella25 (Ripeanu et al., 2002): This dataset is one of the snapshots of
the Gnutella peer-to-peer file sharing network from August 2002. Nodes
represent hosts in the Gnutella network, while edges represent connections
between the Gnutella hosts.

###  4.2 Information Propagation Model

Information Propagation (IP) is described as the task of circulating
information in complex networks (Chen et al., 2013; Guille et al., 2013). IP
can be used to assess various spreader-ranking approaches. Generally, epidemic
models are used to model information propagation in a complex network. In
epidemic models, the population of users is represented as nodes, and the
requisite information is regarded as an infectious disease, which propagates
through the population upon the interaction between nodes.

While various epidemic models can be used for information propagation tasks,
like SIR (Hethcote, 2000), IC (Goldenberg et al., 2001) and LT (Granovetter,
1978), we employ the Independent Cascade (IC) model to assess the performance
of our proposed approach and evaluate it against other state-of-the-art
approaches. The nodes in an IC model can be allocated one out of two states:
infected node or susceptible node. The propagation of information in the
network begins with setting the state of a few nodes as infected. These nodes
further interact with their neighboring nodes and try to infect them. Further,
the newly infected nodes further interact with their neighbors to further
propagate the information to their neighbouring nodes until there are no new
nodes in the network that are infected. A probability ($P$) is associated with
these interactions between nodes, which elucidates the possibility of a
susceptible node getting infected upon interaction with an infected node. The
total influence of a seed node is calculated by taking the sum of all nodes
which get infected by the end of information propagation.

###  4.3 Performance Metrics

We use three performance metrics for comparing our approach with other state-
of-the-art approaches. They are as follows:

(i) Final Infected Scale: It is defined as the ratio of activated or infected
nodes in the network to the total number of nodes in the network at the end of
information propagation. The nodes only get activated when the activation
probability for a node is favourable for activation. The fraction of the total
nodes taken as the initial spreaders is called the spreader fraction, and it
plays a significant role in the final number of nodes that get infected.
Generally, a high spreader fraction leads to a high percentage of the total
nodes getting infected.

(ii) Average Shortest Path Length (ASPL): ASPL can be defined as the average
of the shortest paths between two influential nodes for all sets of
influential nodes. It enables us to evaluate the efficacy of information
propagation in a complex social network. The higher the value of ASPL, the
more even the spread of the influential nodes in the network. A high value of
ASPL further indicates that influential nodes cover a wider part of the
network instead of being concentrated in a small region, leading to the poor
spread of information in the network.

(iii) Execution Time: Execution Time: We evaluate our approach against other
approaches based on the time taken to rank nodes based on their ability to
propagate information in a network. The time consumption for all centralities
was assessed in a Google Colaboratory environment. The code was run on an
Intel Xeon CPU with one core, with 13GB of RAM available for use.

##  5 Experimental Results and Analysis

This section presents the experimental results and analysis performed on the
eight datasets using three performance metrics as mentioned in Section 4. We
conducted a comparative analysis of the proposed approach with seven other
state-of-the-art approaches for finding influential nodes, namely, H-Index
(HI) (Lü et al., 2016), PageRank (PR) (Brin and Page, 1998), Community-based
Spreader Ranking (CSR) (Kumar et al., 2022a), Gateway Local Ranking (GLR)
(Salavati et al., 2019), Degree and Clustering Coefficient and Location (DCL)
(Berahmand et al., 2019), Local Information Dimensionality (LID) (Wen and
Deng, 2019), and Degree and Importance of Lines (DIL) (Liu et al., 2016). We
utilized the Information Cascade (IC) model to simulate information
propagation. Our analysis of the results obtained over three metrics,
illustrated in Figures 2, 3, and 4, is elaborated below.

###  5.1 Final Infected State

The Final Infected Scale illustrates the final degree of propagation or
_infection_ in the network at the end of the simulation, for varying values of
fraction of nodes taken as initial spreaders. We evaluated our proposed
approach against other competing approaches using the IC simulation model for
100 test runs and plotted the average results in Figure 2. For small datasets,
i.e., datasets with the number of nodes less than 2000, we take the values of
the initial spreader fraction from the set 0.02, 0.03, 0.04, 0.05, 0.06, 0.07,
0.08, 0.09, 0.1, while for larger datasets with the number of nodes greater
than 2000 nodes, we take the values of the initial spreader fraction from the
set 0.005, 0.01, 0.015, 0.02, 0.025, 0.03, 0.035, 0.04. We have taken larger
values of the initial spreader fraction for smaller datasets, as the absolute
number of initial spreaders will be small for smaller values of the initial
spreader fraction in smaller datasets. We have taken the _infection
probability_ , i.e., the probability of influencing a susceptible node as 0.1
for all combinations of datasets and approaches.

The graphs for Final Infected Scale with variations in Spreader Fraction for
all approaches and datasets are illustrated in Figure 2. We observed that the
final infected scale generally increased with an increase in spreader
fraction, irrespective of the dataset. We observed that our proposed approach,
MCD, outperformed competing approaches significantly across all datasets,
except the Dolphins dataset, where PR outperforms our approach. We found DCL
and PR to perform reasonably well across the majority of the datasets too. HI
was consistently among the worst-performing approaches of the lot for most
datasets. Thus, we conclude that our approach handily outperforms other
approaches in maximizing information propagation in a network for a wide set
of values of the initial spreader fraction.

Figure 2: Final infected scale plots for various values for inital spreader
fraction on the (a) PGP dataset (b) BA dataset (c) Ca-hepth dataset (d) GR-QC
dataset (e) Jazz Musicians dataset (f) Dolphins dataset (g) Erdos992 dataset
(h) p2p-Gnutella25 dataset. The results are averaged for 100 independent
simulations of the Information Cascade model with an activation probability
(P) equal to 0.1. Figure 3: Average distance between spreaders with respect to
different initial spreaders fraction values for (a) PGP dataset (b) BA dataset
(c) Ca-hepth dataset (d) GR-QC dataset (e) Jazz Musicians dataset (f) Dolphins
dataset (g) Erdos992 dataset (h) p2p-Gnutella25 dataset. The results are
averaged for 100 independent simulations of the Information Cascade model with
an activation probability (P) equal to 0.1. Figure 4: The execution time plots
for different methods for ranking the spreaders for (a) PGP dataset (b) BA
dataset (c) Ca-hepth dataset (d) GR-QC dataset (e) Jazz Musicians dataset (f)
Dolphins dataset (g) Erdos992 dataset (h) p2p-Gnutella25 dataset.

###  5.2 Average Distance between Spreaders

It is desirable for the initial spreaders to have the least overlapping area
of influence so that a wider portion of the network can be activated more
efficiently. Thus, it is preferable that the average distance between
spreaders is maximized. To compare our proposed MCD approach against other
approaches, we calculated the average shortest distance between the initial
spreaders ($L_{s}$) for a wide set of values of the spreader fraction.

The graphs for the Average Distance between Spreaders vs. Initial Spreader
Fraction for all approaches and datasets are illustrated in Figure 3. We
observed that MCD significantly outperforms other approaches across datasets
for almost all values of the initial spreader fraction. This ensures that MCD
selects nodes that are far apart, thus avoiding overlapping of influence as
stated in the manuscript. It verifies that MCD is able to cater to the rich
club effect, which other competing approaches are unable to counter. The
significant gap in the figure between our approach and competing approaches
indicates that MCD selects nodes that are different from the ones selected by
other approaches, and significantly farther apart. This further ensures nodes
selected by MCD may not belong to the same region, thus having a wider spread
of influence and lesser overlap between the regions of influence for each seed
node. Further, we observe that on increasing the spreaders fraction, the
difference between seed nodes decreases due to the increased absolute number
of seed nodes for the whole network. This is observed in the case of all
approaches, as increasing the absolute number of spreaders in a graph is bound
to decrease the average distance between the spreaders. However, MCD still has
a higher average spreader distance as compared to other approaches, even for
greater values of Spreader Fraction, verifying MCD’s superiority over
competing approaches. Among the other approaches, we found DCL and PR to
perform reasonably well. However, there was still a significant gap between
these approaches and MCD. We found GLR consistently performs the worst in this
metric across almost all datasets. Thus, we conclude that MCD suffers from a
significantly lesser _rich club_ effect and outperforms all other approaches
significantly.

###  5.3 Execution Time

We compared our proposed approach, MCD, against other competing approaches
based on the execution time required by each approach for ranking the nodes
for all datasets. Figure 4 illustrates the line graph comparing MCD with other
approaches based on the execution time for all datasets. DIL performed well on
larger datasets but was relatively more computationally expensive for smaller
datasets. We found LID and PR to be the least expensive computationally, while
GLR was consistently the most expensive. Our proposed approach was
computationally the least expensive for smaller datasets, while it was more
expensive than local hybrid centrality-based approaches like LID for larger
datasets. Thus, we observe that MCD performs well in terms of execution time
when compared to other approaches, especially when considering the improved
performance in other metrics.

##  6 Statistical Testing

To further evaluate the performance of MCD using statistical tests, we have
conducted the Friedman test (Friedman, 1937), which is a non-parametric
statistical test used for multiple comparisons (more than two given methods).
We have performed this test to determine whether the performance of our
proposed method is significantly different than the other approaches for two
or more data sets. The Friedman test attempts to detect significant
differences based on the ranking of the methods, rather than their errors. It
consists of two hypotheses: the null ($H_{0}$) and the alternate hypothesis
($H_{1}$). The former states that there are no prominent differences between
these algorithms (equality of medians) and the latter insists that the
algorithms have significant differences in their median of the population,
thus negating the null hypothesis. We have performed the following test on the
Final Infected Scale metric as well as the Average Distance between Spreaders
metric.

In order to perform the Friedman Statistical Test, we execute the following
procedure:

  * •

Gather the generated results for each problem pair.

  * •

Rank the values in ascending order from 1 (best value) to $n$ (worst value)
for each problem $i$ for a particular algorithm $j$.

  * •

For algorithm $j$, the average rank for each problem $i$ is calculated using
the Eq. 7:

$R_{j}=\frac{1}{n}\sum_{i=1}^{n}(r_{i}^{j})$ (7)

where $r^{j}$ is the rank ($1<j<k$) and $R_{j}$ is the average rank.

  * •

Now that all algorithms are ranked according to their priority, compute the
Friedman Statistic $F_{f}$ using the following equation. $F_{f}$ is based on a
Chi-square distribution, with $k-1$ degree of freedom. $F_{f}$ is computed as
shown in Eq. 8.

$F_{f}=\frac{12n}{k(k+1)}\left[\sum_{j=1}^{k}R_{j}^{2}-\frac{k(k+1)^{2}}{4}\right]$
(8)

  * •

The previous statistic produced relatively conservative results, which were
not desired. Thus, (Iman and Davenport, 1980) introduced a new statistic
$F_{id}$, as shown below, which follows the $F-distribution$ with the degree
of freedom as $k-1$ and $(n-1)(k-1)$, given in Eq. 9:

$F_{id}=\frac{(n-1)\chi_{F}^{2}}{n(k-1)-\chi_{F}^{2}}$ (9)

Table 2 and Table 3 show the average ranking calculated from the Friedman test
for all the approaches used in this paper. MCD clearly has a better average
ranking than the other competing approaches.

Table 2: Average ranking of algorithms using the Friedman test based on Final Infected State | Method | Average Rank
---|---|---
1 | MCD | 1.567
2 | PR | 2.983
3 | DCL | 3.467
4 | CSR | 4.675
5 | LID | 5.033
6 | DIL | 5.225
7 | GLR | 6.517
8 | HI | 6.533
Table 3: Average ranking of algorithms using the Friedman test based on Average Distance between Spreaders | Approach | Average Rank
---|---|---
1 | MCD | 1.067
2 | PR | 2.983
3 | DCL | 3.517
4 | LID | 4.675
5 | HI | 5.3
6 | CSR | 5.367
7 | DIL | 5.508
8 | GLR | 7.583

The unadjusted P-value or Holm P-value, obtained from the Iman-Davenport
statistic, corresponding to the performance of MCD, advocates the rejection of
the null hypothesis $H_{0}$. All of the computed P-values are less than the
standard significance level $\alpha=0.05$. This indicates that there is a
significant difference between the performance of the baseline approaches and
our proposed approach. Hence, the obtained P-values conclude the negation of
null hypothesis $H_{0}$ but are not appropriate for comparison with different
methods. In order to compare these approaches with each other, we must
calculate their respective adjusted P-values keeping MCD as the control
algorithm.

Hence, we use adjusted P-values (APVs) for evaluation in order to perform a
comparative study between the control method and benchmark methods on a
statistical ground. Adjusted P-value provides the correct correlation between
these algorithms, taking into account the accumulated family error with
respect to the MCD control algorithm. Moreover, the adjusted P-values can be
directed compared with the significance level $\alpha$, which is equivalent to
0.05. In order to calculate the adjusted P-values, a few post-hoc procedures
need to be defined. Various post-hoc procedures such as (Dunn, 1961) and
(Holland and Copenhaver, 1987) differ as they adjust the value of $\alpha$ to
compensate for multiple comparisons for multiple methods. In this paper, we
have used the common Holm’s procedure (Holm, 1979) to evaluate the respective
APVs. The values of Holm P-values and APVs are always sorted in ascending
order. The equation for the same is given below, where indices $i$ and $j$
refers to the main hypothesis whose APVs are being computed and different
hypothesis in the set respectively. $P_{j}$ is the P-value for the $j^{th}$
hypothesis. Holm APV is computed as shown in Eq. 10.

$HolmAPV_{i}=\min\\{v,1\\},\text{where}\,v=\max\\{(k-j)p_{j}:1\leq j\leq i\\}$
(10)

Table 4: Adjusted P-values (APVs) using Holm procedure for Final Infected State | Approach | Z-score | P-value | APV
---|---|---|---|---
1 | HI | -11.11 | 5.88E-29 | 4.11E-28
2 | GLR | -11.06 | 8.91E-29 | 5.35E-28
3 | DIL | -8.18 | 1.42E-16 | 7.08E-16
4 | LID | -7.75 | 4.53E-15 | 1.81E-14
5 | CSR | -6.95 | 1.82E-12 | 5.46E-12
6 | DCL | -4.24 | 1.08E-05 | 2.15E-05
7 | PR | -3.16 | 7.68E-04 | 7.68E-04
Table 5: Adjusted P-values (APVs) using Holm procedure for Average Distance between Spreaders | Approach | Z-score | P-value | APV
---|---|---|---|---
1 | GLR | -14.57 | 2.13E-48 | 1.49E-47
2 | DIL | -9.93 | 1.51E-23 | 9.08E-23
3 | CSR | -9.61 | 3.45E-22 | 1.73E-21
4 | HI | -9.46 | 1.45E-21 | 5.81E-21
5 | LID | -8.06 | 3.56E-16 | 1.07E-15
6 | DCL | -5.47 | 2.15E-08 | 4.29E-08
7 | PR | -4.28 | 9.10E-06 | 9.10E-06

The results of the APVs using Holm’s procedure are shown in Table 4 and Table
5. The intended APVs are less than the significance level, thus rejecting the
null hypothesis. It is clearly observed from the results obtained that MCD is
significantly better than the competing approaches on these statistical tests
as well.

##  7 Conclusion

Finding influential nodes in complex networks like social networks play a
vital role in achieving maximum information spread across the network. It has
found numerous applications in business, marketing, and generating awareness
for a social cause. In this work, we proposed a modified Community Diversity
(MCD) method for finding influential nodes. First, we introduced the concept
of Extended Community Diversity, which represents the community diversity
score of the extended neighborhood of a node. Next, we introduced Modified
Community Diversity, built upon the concepts of Community Diversity and
Extended Community Diversity. To verify the efficacy of our approach in
determining the most effective influential nodes in a network, we compared MCD
with seven recent state-of-the-art methods over eight datasets. We found that
MCD cumulatively outperformed the competing approaches across three metrics.
Further, we observed that seed nodes selected by MCD were reasonably close
together for extended activation due to a node’s influence but far enough to
not significantly overlap with each other’s scope of influence, thus
mitigating the Rich Club effect significantly.

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[2]NedaDarbeheshti [1]ElaheMoradi 1]Department of Electrical Engineering,
Yadegar-e-Imam Khomeini (RAH) Shahre Rey Branch, Islamic Azad University,
Tehran, Iran

Elahe Moradi<EMAIL_ADDRESS><EMAIL_ADDRESS>
# LSTM-Based Forecasting Model for GRACE Accelerometer Data

###### Abstract

The Gravity Recovery and Climate Experiment (GRACE) satellite mission,
spanning from 2002 to 2017, has provided a valuable dataset for monitoring
variations in Earth’s gravity field, enabling diverse applications in
geophysics and hydrology. The mission was followed by GRACE Follow-On in 2018,
continuing data collection efforts. The monthly Earth gravity field, derived
from the integration different instruments onboard satellites, has shown
inconsistencies due to various factors, including gaps in observations for
certain instruments since the beginning of the GRACE mission.

With over two decades of GRACE and GRACE Follow-On data now available, this
paper proposes an approach to fill the data gaps and forecast GRACE
accelerometer data. Specifically, we focus on accelerometer data and employ
Long Short-Term Memory (LSTM) networks to train a model capable of predicting
accelerometer data for all three axes.

In this study, we describe the methodology used to preprocess the
accelerometer data, prepare it for LSTM training, and evaluate the model’s
performance. Through experimentation and validation, we assess the model’s
accuracy and its ability to predict accelerometer data for the three axes. Our
results demonstrate the effectiveness of the LSTM forecasting model in filling
gaps and forecasting GRACE accelerometer data.

Gravimetry satellite missions, exemplified by the Gravity Recovery And Climate
Experiment (GRACE, Tapley et al. (2004)) and GRACE Follow-On (GRACE-FO, Tapley
et al. (2019)), play a pivotal role in advancing our understanding of Earth’s
gravity field (Wahr et al., 1998). These missions employ a suite of
instruments, including GPS, accelerometers, star cameras, and the KBR (K-Band
Ranging) instrument, to meticulously capture crucial data that underpins the
accurate recovery of gravitational perturbations.

However, the success of such missions is often hindered by data gaps that
arise due to technical and operational constraints. Table 1 provides an
overview of available and missing data throughout the lifetimes of the GRACE
and GRACE-FO missions (Ahi and Cekim, 2021). Notably, missing months denote
periods devoid of Level-2 products or monthly gravity field data. The
available gravity solutions can be accessed from the International Center for
Global Earth Models website (ICGEM, (Ince et al., 2019)) at http://icgem.gfz-
potsdam.de/series/03_other/AIUB/AIUB-G3P. The gaps, highlighted in red and
white, present critical challenges that need innovative solutions for having a
consistent time series of Earth’s gravity field.

Table 1: Availability of GRACE (blue) and GRACE Follow-On (green) Level-2 product. Pink color indicates missing months. The information presented is adapted from Ahi and Cekim (2021) and based on the most recent update available as of August 2023 from the ICGEM website. |  Jan |  Feb |  Mar |  Apr |  May |  Jun |  Jul |  Aug |  Sep |  Oct |  Nov |  Dec
---|---|---|---|---|---|---|---|---|---|---|---|---
2002 |  |  |  |  |  |  |  |  |  |  |  |
2003 |  |  |  |  |  |  |  |  |  |  |  |
2004 |  |  |  |  |  |  |  |  |  |  |  |
2005 |  |  |  |  |  |  |  |  |  |  |  |
2006 |  |  |  |  |  |  |  |  |  |  |  |
2007 |  |  |  |  |  |  |  |  |  |  |  |
2008 |  |  |  |  |  |  |  |  |  |  |  |
2009 |  |  |  |  |  |  |  |  |  |  |  |
2010 |  |  |  |  |  |  |  |  |  |  |  |
2011 |  |  |  |  |  |  |  |  |  |  |  |
2012 |  |  |  |  |  |  |  |  |  |  |  |
2013 |  |  |  |  |  |  |  |  |  |  |  |
2014 |  |  |  |  |  |  |  |  |  |  |  |
2015 |  |  |  |  |  |  |  |  |  |  |  |
2016 |  |  |  |  |  |  |  |  |  |  |  |
2017 |  |  |  |  |  |  |  |  |  |  |  |
2018 |  |  |  |  |  |  |  |  |  |  |  |
2019 |  |  |  |  |  |  |  |  |  |  |  |
2020 |  |  |  |  |  |  |  |  |  |  |  |
2021 |  |  |  |  |  |  |  |  |  |  |  |
2022 |  |  |  |  |  |  |  |  |  |  |  |
2023 |  |  |  |  |  |  |  |  |  |  |  |

The presented research takes a specific focus on accelerometer data. While all
instruments contribute substantially, the distinctive attributes of
accelerometer data make it an optimal starting point (Darbeheshti et al.,
2017, 2023). Accelerometers record non-gravitational forces affecting
satellite’s motion, such as atmospheric drag, solar radiation pressure, and
albedo. Correctly disentangling these non-gravitational influences from the
desired gravitational signals is of paramount importance, as the quality of
accelerometer data, referred to as ACC products, influences the precision of
monthly gravity field models (Klinger, 2018; Klinger and Mayer Gürr, 2016). In
the latter stages of the GRACE mission, operational limitations necessitated
the deactivation of the on-board accelerometer in GRACE-B. Synthetic
accelerometer data, referred to as transplant data, were generated by adapting
GRACE-A ACC data to fill the void left by the missing measurements. A parallel
approach was adopted during the GRACE-FO mission to mitigate degraded GRACE-D
ACC data (Bandikova et al., 2019; Behzadpour et al., 2021).

Our study delves into the potential of utilizing Long Short-Term Memory (LSTM)
networks for forecasting GRACE accelerometer data (GRACE ACC). The choice of
LSTM (Chollet, 2017) is underpinned by its proven efficacy in handling time
series data, a characteristic intrinsic to instrument observations.
Importantly, LSTM’s applicability in geodesy and related fields further
bolsters its suitability for our objectives (Kaselimi, 2021; Gou et al.,
2023).

In the following sections, we present our methodology, experimental setup, and
results.

##  1 Data and Methods

Figure 1 displays a representative day of GRACE accelerometer data. Access to
GRACE daily accelerometer data is freely available. Similar to other GRACE
instrument data, each file contains one day of data, with accelerometer data
sampled at one-second intervals, resulting in 86400 samples along each of the
three axes. For the convenience of the machine learning community, the GRACE
data used in this paper has been downloaded and made accessible on Kaggle
(https://www.kaggle.com/datasets/nedadarbeheshti/grace-satellites-asc-files).

Figure 1: Accelerations for 30 May 2005 for GRACE A and GRACE B.

###  1.1 Data Preparation

As shown in Figure 1, some of these axes exhibit outliers, data points that
deviate significantly from the overall pattern. Outliers can distort
statistical measures and adversely affect the performance of machine learning
models during training. Therefore, as a preprocessing step prior to training,
we apply an outlier removal procedure to ensure the integrity of our data.

In this procedure, we first identify the quartiles, specifically the first
quartile ($Q1$) and the third quartile ($Q3$), which divide the data into four
equal parts. The interquartile range ($IQR$) is then calculated as the
difference between $Q3$ and $Q1$. Based on the $IQR$, we determine acceptable
upper and lower limits for data points that should not be considered outliers.
These limits are obtained by adding and subtracting a multiple (usually 1.5)
of the $IQR$ from the quartiles.

Any data points that fall beyond these limits are considered outliers and are
subsequently removed from the dataset. This process helps us mitigate the
impact of extreme values on our analysis and ensures that our subsequent
machine learning models are more robust and accurate. The function
$remove\\_outliers(axis)$ implements this outlier removal process.

    def remove_outliers(axis):
        q1 = np.percentile(axis, 25)
        q3 = np.percentile(axis, 75)
        IQR = q3 - q1
        max_limit = q3 + (1.5 * IQR)
        min_limit = q1 - (1.5 * IQR)
        outliers = np.logical_or(axis > max_limit, axis < min_limit)
        cleaned_data = axis[~outliers]
        return cleaned_data

Given a time series along a specific axis, the function calculates the
quartiles, determines the acceptable limits, and filters out any data points
that lie beyond these limits. The resulting ’cleaned’ time series, shown in
the Figure 2, allows us to train our models on more reliable and meaningful
data, ultimately leading to improved results during the analysis phase.

Figure 2: Accelerations after outliers removed for 30 May 2005 for GRACE A and
GRACE B.

The next step is to normalize the data. We normalize each time series
individually using the sklearn.preprocessing module. We tested two
normalization methods, namely MinMaxScaler and RobustScaler.

MinMaxScaler: This normalization technique transforms each feature
individually by scaling it to a given range, usually between 0 and 1. It works
by subtracting the minimum value of the feature and then dividing by the range
of the feature values. This method is sensitive to outliers, as extreme values
can affect the scaling of the entire feature. MinMaxScaler is particularly
useful when features have a known range and need to be transformed to a common
scale, making them suitable for machine learning algorithms that rely on
feature scaling.

RobustScaler: In contrast to MinMaxScaler, RobustScaler is robust to outliers
in the data. It scales features using statistics that are robust to outliers,
specifically the median and the interquartile range (IQR). The median is
subtracted from each data point, and the result is divided by the IQR,
ensuring that the scaling is less affected by extreme values. RobustScaler is
a good choice when dealing with datasets that contain outliers, as it provides
a more stable scaling that is less influenced by these anomalies.

In our study, we compared the performance of both normalization techniques,
MinMaxScaler and RobustScaler, on the GRACE accelerometer data. Through
experimentation, we found that MinMaxScaler produced better results for our
training process, likely due to the characteristics of our dataset and the
nature of the accelerometer measurements. As a result, our subsequent analysis
and model training are based on data normalized using MinMaxScaler.

The final step of data preparation involves downsampling the data. Our
motivation for downsampling is to expedite the training process, considering
that for other instruments such as KBR, the data sampling rate is 5 seconds.
Data sampling of 10 seconds aligns with what we use for all data inputs to
gravity field recovery. Therefore, the sampling rate of the accelerometer data
along each axis entering the training is 10 seconds, resulting in
approximately 8640 samples along each axis.

###  1.2 Data Splitting

The data splitting process involves dividing the daily time series into
training and testing subsets. For this purpose, we use a train-test split
ratio of $70\%$, meaning only the first $70\%$ of the daily time series is
utilized for training our models. This partitioning strategy helps ensure that
the models are trained on a sufficient portion of the data while reserving a
subset for unbiased evaluation.

To facilitate this data splitting process, we employ the function
create_dataset. This function is responsible for generating the input-output
pairs (features and labels) required for training and testing. Specifically,
it takes the daily time series data as input and constructs the appropriate
sequences of past observations as inputs (X) and the subsequent observation as
the corresponding output (Y). This sequential arrangement allows the models to
learn temporal dependencies and patterns present in the accelerometer data.

As a result of this process, we obtain four datasets: trainX and trainY for
training, and testX and testY for testing. The training datasets, trainX and
trainY, are utilized to optimize the model’s parameters during the training
phase. The testing datasets, testX and testY, remain unseen by the model
during training and are employed to assess the model’s generalization
performance on unseen data.

By adhering to this data splitting approach, we ensure that our models are
effectively trained and evaluated, and we are able to draw meaningful
conclusions about their performance in forecasting accelerometer data.

###  1.3 LSTM model

Long Short-Term Memory (LSTM) networks are a type of recurrent neural network
(RNN) architecture that excel in capturing temporal dependencies and patterns
in sequential data. In this section, we present the configuration and training
of our LSTM model for forecasting GRACE accelerometer data.

The LSTM model is defined using the Keras framework. The model architecture
consists of several layers that collectively learn and represent complex
relationships within the input data. Here is a breakdown of the model
structure:

  * •

A look-back window of size 15 is used to consider the previous 15 time steps
as input features.

  * •

The model begins with an LSTM layer with 8 units. The input shape is set to
(1, look_back), reflecting the univariate nature of the time series.

  * •

Following the LSTM layer, the model includes three dense layers. The first
dense layer has 10 units, the second has 32 units, and the final output layer
has 1 unit, which predicts the next value in the time series.

  * •

The model is compiled using the Adam optimizer with a learning rate of 0.001.
The mean squared error (MSE) is chosen as the loss function, and the mean
absolute error (MAE) is used as a metric to evaluate the model’s performance
during training.

  * •

The training data (trainX and trainY) is used to train the model for 300
epochs. The training data is divided into batches of size 8 for efficient
optimization. A validation split of $15\%$ is applied, where a portion of the
training data is used for validation during each epoch.

The provided Python code snippet illustrates the implementation of the LSTM
model:

    look_back = 15
    model = Sequential()
    model.add(LSTM(8, input_shape=(1, look_back)))
    model.add(Dense(10))
    model.add(Dense(32))
    model.add(Dense(1))
    model.compile(optimizer=Adam(learning_rate=0.001), loss=’mse’, metrics=["mae"])
    history = model.fit(trainX, trainY, epochs=300, batch_size=8, verbose=0,
                        validation_split=0.15)

Throughout the training process, the model learns to capture the temporal
relationships and dependencies present in the accelerometer data. The
validation results monitored by the MAE metric provide insights into the
model’s ability to generalize to unseen data. The history object (history)
stores information about the training progress, allowing for further analysis
and visualization of performance trends. Figure 3 presents the LSTM network.

Figure 3: Diagram of the LSTM network.

##  2 Results

Figure 4 illustrates the loss and prediction plots for each axis of the
accelerometer data. Across all axes, the training loss consistently remains
lower than the validation loss. This observed trend underscores the efficacy
of the employed outlier removal procedure, demonstrating successful mitigation
of overfitting. Notably, the loss plot demonstrates a more favorable
convergence for the $x$ axis, evident in both satellites. This convergence
occurs at a level below 0.01 MAE for both training and validation datasets.

It is worth highlighting that for the $y$ and $z$ axes of GRACE B, discernible
disparities emerge between the training and validation loss. This distinction
is further underscored by the divergent RMSE scores, as indicated in Table 2
and visually depicted in Figure 5. The presence of differences between the
RMSE scores for training and testing reinforces the noteworthy variability
observed in the y and z axes of GRACE B.

Figure 4: LSTM loss and prediction plots for each axis per satellite.

Table 2 furnishes additional details concerning the total number of data
points subsequent to the outlier removal process. It is noteworthy that the
root mean square error (RMSE) scores for both training and testing phases
exhibit a remarkable degree of uniformity across each axis. This uniformity
signifies that the executed outlier removal procedure adeptly preserves
equilibrium between the training and test subsets, further affirming the
procedure’s efficacy in facilitating a balanced distribution of data points.

Table 2: The total number of data points post outlier removal and RMSE scores
for each axis.

GRACE A
---
Axis | Size | Train Score ($10^{-6}$ RMSE) | Test Score ($10^{-6}$ RMSE)
$x$ | (8548) | $1$ | $1$
$y$ | (8618) | $3.29$ | $2.45$
$z$ | (8608) | $2$ | $2$

GRACE B
---
Axis | Size | Train Score ($10^{-6}$ RMSE) | Test Score ($10^{-6}$ RMSE)
$x$ | (8545) | $0.53$ | $0.50$
$y$ | (8634) | $2$ | $3$
$z$ | (8613) | $2$ | $3$

Figure 5: Bar plot of RMSE scores for each axis.

The comparable performance between training and testing scores across axes
further affirms the robustness of our approach. It indicates that the LSTM
model has learned the underlying patterns and dependencies in the
accelerometer data without excessively memorizing the noise or anomalies.

##  3 Conclusion and Future Directions

In the context of this paper, we have embarked on an the application of Long
Short-Term Memory (LSTM) networks for forecasting GRACE accelerometer data, a
concept with implications for the broader realm of GRACE instruments. The
success of this endeavor lays the groundwork for extending this approach to
other key instruments within the GRACE mission, including GPS, KBR (K-Band
Ranging), and star cameras.

Moving forward, our next phase involves implementing this approach to create
time series for each instrument over a month’s span. This study aims to assess
the feasibility of utilizing LSTM networks in terms of computational resources
and time efficiency to forecast data gaps across various instruments.

This study is a significant step toward the development of a specialized
Generative Pre-trained Transformer (GPT) tailored specifically for the GRACE
type missions. This holds the potential to generate accurate time series data
for each instrument within defined temporal boundaries.

Data and code are available at
https://www.kaggle.com/datasets/nedadarbeheshti/grace-satellites-asc-files and
https://github.com/Darbeheshti/LSTM-based-Analysis-for-GRACE-Accelerometers

Elahe Moradi reviewed the LSTM code and the article. Neda Darbeheshti
performed the GRACE data processing and developed LSTM code and prepared the
manuscript with contributions from Elahe Moradi.

The authors declare that they have no conflict of interest.

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