Algorithms for Densest Subgraphs of Vertex-Weighted Graphs

Abstract

Finding the densest subgraph has tremendous potential in computer vision and social network research, among other domains. In computer vision, it can demonstrate essential structures, and in social network research, it aids in identifying closely associated communities. The densest subgraph problem is finding a subgraph with maximum mean density. However, most densest subgraph-finding algorithms are based on edge-weighted graphs, where edge weights can only represent a single value dimension, whereas practical applications involve multiple dimensions. To resolve the challenge, we propose two algorithms for resolving the densest subgraph problem in a vertex-weighted graph. First, we present an exact algorithm that builds upon Goldberg’s original algorithm. Through theoretical exploration and analysis, we rigorously verify our proposed algorithm’s correctness and confirm that it can efficiently run in polynomial time O(n(n + m)log²n) is its temporal complexity. Our approach can be applied to identify closely related subgroups demonstrating the maximum average density in real-life situations. Additionally, we consistently offer an approximation algorithm that guarantees an accurate approximation ratio of 2. In conclusion, our contributions enrich theoretical foundations for addressing the densest subgraph problem.

1. Introduction

Data have exploded with the rapid development of computer technology and the Internet. In particular, relationships between people have become increasingly complex with the widely used social networking software. We can abstract entities in the world as vertices, such as people, social groups, and the environment. The relationship between entities can be abstracted to the edge. At last, the vertices and edges construct a network graph. The vertices in the graph are clustered together to form different dense subgraphs. They always contain valuable data, but it is challenging to analyze the network directly. Therefore, it is an effective method to analyze the densest subgraphs of the network.

Goldberg et al. [1] first proposed the idea of the maximum density subgraph of the undirected graph. Given an undirected graph G with n vertices, the density of a subgraph is the ratio of the sum of the edge weights to the number of vertices. The maximum density subgraph has the maximum density value. The task of identifying the maximum density subgraph is essentially the same as identifying an induced subgraph of maximum density in all subgraphs. Goldberg et. al used a polynomial time algorithm to solve this problem. At first, they constructed a network, then used the theory of the minimum cut to divide the network, and at last, solved the density subgraph problem using the binary method [1].

The problem of finding densest subgraphs has been the subject of extensive research. Many scientists have conducted numerous studies on it and have proposed a variety of polynomial time algorithms to solve this problem, as it is highly significant in research areas such as network analysis, web community inference, hypertext analysis, and the discovery of functions in complex biological networks [2,3,4,5,6]. Charikar gave a polynomial time algorithm based on linear programming (LP) and an approximation algorithm by iteratively deleting the vertex with the minimum degree in the remaining graph [7].

The dense k-subgraph problem is related to the densest subgraph problem, which is the problem of finding the densest subgraph with exact k vertices. The dense k-subgraph problem is well known as an NP-hard problem. Thus, some scientists proposed approximation algorithms for the dense k-subgraph problem. For example, Kortsarz and Peleg used a polynomial time approximation algorithm and showed its approximation ratio O(n^0.3885) [8]. Feige and Seltzer proposed an approximation algorithm based on semi-definite programming (SDP) [9]. Then, Srivastav and Wolf presented an approximation algorithm based on SDP [10]. Both have the same ratio n/k. Kannan and Vinay proposed an approximation algorithm with approximation ratio O(log n) in a directed graph [11]. In 2001, Feige gave another O(log n)-approximation algorithm to solve the dense k-subgraph problem [12]. Bhaskara et al. proposed an O(n^1/4)-approximation algorithm to detect high log densities [13].

In 2009, Andersen and Chellapilla [14] presented the densest at-least-k-subgraph problem (DalkS) and the densest at-most-k-subgraph problem (DamkS), which are about finding the densest subgraph of a size constraint undirected graph. They gave two approximation algorithms. One of the algorithms with an approximation ratio of 3 is the extension of Charikar’s greedy approximation algorithms. The other one with an approximation ratio of 2 is based on Gallo’s maximum flow algorithm [15]. Khuller and Saha also used an approximation algorithm based on the max-flow algorithm and another algorithm based on linear programming (LP) [16]. Both approximation algorithms are of an approximation ratio of 2. Chen et al. designed two approximation algorithms for DamkS [17], with approximation ratios of (n − 1)/(k − 1) and O(n^δ) for some δ < 1/3, respectively. Chen et al. proved that DalkS is still polynomial time-solvable when k is bounded by some constant c [17]. An improved algorithm with better time complexity was proposed in Chen et al.’s other paper [18], in which a greedy approximation algorithm was also proposed for the densest subgraph with a specified subset problem. More results about applying the densest subgraphs can be found in paper [19]. In 2023, Hanaka introduced algorithms to solve the densest k-subgraph problem by utilizing structural graph parameters [20]. They demonstrated that the problem becomes fixed-parameter tractable (FPT) when parameterized by neighborhood diversity, block deletion number, distance-hereditary deletion number, and cograph deletion number. Additionally, they presented a 2-approximation algorithm for the densest k-subgraph.

In 2022, Chekuri et al. explored fast algorithms and structural aspects of the densest subgraph problem (DSG) through the concept of supermodularity [21]. They presented a simple flow-based algorithm that offers a $1-\epsilon$-approximation in $\tilde{O}(\mathrm{m}/\epsilon )$ time for undirected graphs, where m is the number of edges. Additionally, they explored a peeling algorithm for the densest supermodular subset (DSS) problem. Finally, they extended the 2-approximation for the densest-at-least-k subgraph to the supermodular setting.

However, previous research has primarily focused on edge-weighted graphs, whereas real-world scenarios often involve multi-dimensional considerations. Therefore, this study introduces the problem of finding the densest subgraph in vertex-weighted graphs. How to find the densest subgraph in a vertex-weighted graph? In this paper, we introduce the densest subgraph problem of a vertex-weighted graph, and we propose a polynomial time algorithm for this problem. Our algorithm is an extension of Goldberg’s algorithm [1]. Here is an expanded description of our approach. First, given a vertex-weighted graph G, where D represents the density of an induced subgraph, we start by hypothesizing an initial guess g for D. Next, we construct a modified graph where edge weights are adjusted relative to g and utilize min-cut algorithms to partition it effectively. Subsequently, employing a binary search strategy, we identify the maximum density subgraph within G. Secondly, we introduce an additional contribution—a novel approximation algorithm with an approximation ratio of 2. This algorithm efficiently approximates the densest subgraph in vertex-weighted graphs, ensuring practical applicability in real-world scenarios.

2. Preliminaries

This paper refers to the main result as the “theorem” and the auxiliary result as the “lemma.” Lemmas are auxiliary results introduced to prove theorems. For example, Lemma 1 is used to prove Theorem 1.

In this section, we mainly introduce some basic concepts.

Definition 1.

The densest subgraph of a undirected graph.

Given an undirected graph G (V′) = (V′, E′), where V′ is the set of vertices and E′ is the set of edges. The induced subgraph G(V) = (V, E) is a subgraph that is induced from vertex V of G (V′). Let D(V) denote the density of G(V), which is equal to the number of edges over the number of vertices, i.e.,

$$D\left(V\right)=\frac{\left|E\right|}{\left|V\right|}$$

The problem of the densest subgraph in a undirected graph is a computational problem to finding the induced subgraph that is of the maximum density.

Definition 2.

The densest subgraph of a vertex-weighted undirected graph.

Give a vertex-weighted undirected graph G (V′) = (V′, E′, ω′), where V′ is the set of vertices, E′ is the set of edges, and ω′ is a weighted function, ω: V′→N, where N is a natural number set. For the induced subgraph G(V) of vertex V, its density D(V) is defined as the number of edges over the sum of the weight of the vertices, i.e.,

$$D\left(V\right)=\frac{\left|E\right|}{\left|{\displaystyle \sum _{i\in V}\omega (i)}\right|}$$

When the weight of vertices is 1, the value of D(V) is equal to the number of edges divided by the number of vertices.

The problem of the densest subgraph of vertex-weighted undirected graph is a computational problem to find the induced subgraph with the maximum density from a vertex-weighted undirected graph.

Definition 3.

Minimum Cut. Given the undirected graph $G=(V,E)$, where V is the vertex set and V = {v1, v2, …, vn}. $\left|V\right|$ is the number of vertices, $\left|V\right|=n$. E is the edge set. $\left|E\right|$ is the number of edges, $\left|E\right|=m$, and there is a capacity value c(u, v) for any edge $e=\left(u,v\right),\left(u,v\right)\in E$.

(1)

A cut refers to splitting the vertex set V into two sets V1 and V2. A cut edge set O, $O=\{\left(v_1,v_2\right)\in E:v_1\in V_1,v_2\in V_2\}$. The capacity of a cut is the sum of capacities of the edge set in this cut, $c\left(V_1,V_2\right)=\left\{{\displaystyle \sum c\left(v_1,v_2\right)}|v_1\in V_1,v_2\in V_2\right\}$.

(2)

Given two vertices $\mathrm{s},t\in V$, s-t cut is to divide V into two sets, and s and t belong to two different sets $V_1,V_2$, $\left\{(V_1,V_2)|\mathrm{s}\in V_1,t\in V_2\right\}$.

The minimum cut problem is to find the s-t cut with the minimum capacity.

3. The Algorithm of Finding the Densest Subgraph in a Vertex-Weighted Graph

In this section, we propose a polynomial time algorithm for the densest subgraph in a vertex-weighted graph. By extending Goldberg’s algorithm [1], we give a polynomial time algorithm. Our idea is as follows. Given a vertex-weighted graph G, let D be the maximum density of an induced subgraph. At first, we make a guess value g of D. Then, we try to construct a network whose edge weights relate to g and divide it using the min-cut algorithms. At last, we find the maximum density subgraph of G by means of a binary search method.

3.1. Undirected Graph Construction by a Guess Value

Given a vertex-weighted undirected graph G(V) = (V, E, ω), where ω is a weight function assigning a positive integer weight to each vertex: V→N, where N is the set of natural numbers. Assume that the degree of the vertex i is $d\left(i\right)$ and the weight of vertex is $\omega (i)$. Let |V| = n and |E| = m. Let D be the maximum density of the induced subgraphs. When we make a guess value g of D, we construct the graph G_g = (V_g, E_g) which is related to g. This is an extension of Goldberg’s algorithm [1].

We add source u and sink v to vertex sets of G. The capacity of edge sets is as follows. For any $\left(i,j\right)$, an undirected edge between vertex i and j of G, the capacity $c\left(i,j\right)$ of $\left(i,j\right)$ is 1. The capacity of edge $\left(u,i\right)$ is m, $c\left(u,i\right)=m,i\in V$. The capacity of edge $\left(i,v\right)$ is $m+2g\ast \omega \left(i\right)-d\left(i\right)$.

G_g = (V_g, E_g) is as follows.

$$\begin{array}{l}{V}_g=V\cup \{u,v\}\\ E_g=\left\{\left(i,j\right)|\left\{i,j\right\}\in E\right\}\cup \left\{\left(u,i\right)|i\in V\right\}\cup \left\{\left(i,v\right)|i\in V\right\}\\ c\left(i,j\right)=\begin{array}{cc}1& \left\{i,j\right\}\in E\end{array}\\ c\left(u,i\right)=\begin{array}{cc}m& i\in V\end{array}\\ c\left(i,v\right)=\begin{array}{cc}m+2g\times \omega \left(i\right)-d\left(i\right)& i\in V\end{array}\end{array}$$

3.2. Establish a Relationship with g and a Minimum Cut

We split the V_g into two sets S and T. If $u\in S,v\in T$, the cut (S, T) is an s-t cut. Let $V_1=S-\left\{u\right\},V_2=T-\left\{\mathrm{v}\right\}$. If |V1| = 0, the capacity of this cut $c\left(S,T\right)=m\left|V\right|$. Otherwise, the capacity of this cut is ${\displaystyle \sum c\left(i,j\right)}$, where i ∈ V₁, j ∈ V₂. Let D₁ be the density of the induced vertex-weighted subgraph from V₁, so we have the following conclusion:

$$c\left(S,T\right)=m\left|V\right|+2\left|V_1\right|(g-D_1)$$

The reason is as follows:

$$\begin{array}{l}c\left(S,T\right)={\displaystyle \sum _{i\in S,j\in T}c\left(i,j\right)}\\ ={\displaystyle \sum _{j\in V_2}c\left(u,j\right)}+{\displaystyle \sum _{i\in V_1}c\left(i,v\right)}+{\displaystyle \sum _{i\in V_1,j\in V_2}c\left(i,j\right)}\\ =m\left|V_2\right|+\left(m\left|V_1\right|+2g{\displaystyle \sum _{i\in V_1}\omega \left(i\right)}-{\displaystyle \sum _{i\in V1}d\left(i\right)}\right)+{\displaystyle \sum _{i\in V_1,j\in V_2}1}\\ =m\left|V\right|+\left|V_1\right|2\left(g-\left(\frac{{\displaystyle \sum _{i\in V1}d\left(i\right)}-{\displaystyle \sum _{i\in V_1,j\in V_2}1}}{2{\displaystyle \sum _{i\in V_1}\omega \left(i\right)}}\right)\right)\end{array}$$

${\displaystyle \sum _{i\in V_1}d\left(i\right)}-{\displaystyle \sum _{i\in V_1,j\in V_2}1}$ has twice the number of edges of the induced vertex-weighted subgraph from V₁, so:

$$D_1=\frac{{\displaystyle \sum _{i\in V1}d\left(i\right)}-{\displaystyle \sum _{i\in V_1,j\in V_2}1}}{2{\displaystyle \sum _{i\in V_1}\omega \left(i\right)}}$$

Hence: $c\left(S,T\right)=m\left|V\right|+2\left|V_1\right|(g-D_1)$. We have the following result according to the similar argument of Goldberg’s paper [1].

Lemma 1.

Let cut (S,T) be a min-cut. If $\left|V_1\right|\ne 0$, then $\mathrm{g}\le D$; if $V_1=0,\left(i.e.S=\left\{u\right\}\right)$, then $\mathrm{g}\ge D$.

Proof of Lemma 1.

Since $\left|V_1\right|\ne 0$, there is some flow or capacity contributing to the cut, and because (S, T) is a min-cut, the total flow or capacity g that crosses the cut cannot exceed capacity D. This is because if g were greater than D, it would imply that there exists a smaller cut with capacity less than g, contradicting the minimality of the cut (S, T). Therefore, $\mathrm{g}\le D$. Conversely, $V_1=0$; the capacity g that defines the min-cut (S, T) must be composed entirely of other edges’ capacities. Given the definition of a min-cut, capacity g must be at least D. If g were less than D, it would imply that there is a cut with a capacity smaller than the established minimal capacity D, which is a contradiction. Therefore, $\mathrm{g}\ge D$. □

3.3. Algorithm Description

Before describing the algorithm, let us prove the results below.

Lemma 2.

If G′ is a subgraph of the vertex-weighted undirected graph G and the density of G′ is D′, and the density of the other subgraph is more than or equal to $D^{\prime }+\frac{1}{N\left(N+1\right)}$, where N is the sum of all weights of the vertices of G, then G′ is the densest subgraph of G.

Proof of Lemma 2.

Suppose that the maximum density of an undirected graph G is D. D is the ratio of the number of edges and the sum of the weight of the vertices, so

$$D\in \left\{\frac{m^{\prime }}{n^{\prime }}|0\le m^{\prime }\le m,\min \left(\omega \left(i\right)\right)\le n^{\prime }\le N,N={\displaystyle \sum _{i=1}^n\omega \left(i\right)}\right\}$$

D is between 0 and $\frac{m}{\min \left(\omega \left(i\right)\right)}$. If the difference between two values of D is ∆,

$$\mathsf{\Delta }=\frac{m_1}{n_1}-\frac{m_2}{n_2}=\frac{m_1{n}_2-m_2{n}_1}{n_1{n}_2}$$

If ${\mathrm{n}}_1=n_2$, then $\left|\mathsf{\Delta }\right|\ge \frac{1}{N}$. Otherwise, ${\mathrm{n}}_1{\mathrm{n}}_2\le N\left(N-1\right)$, and then $\left|\mathsf{\Delta }\right|\ge \frac{1}{N\left(N-1\right)}$. So, the minimum difference between two values of D is $\frac{1}{N\left(N-1\right)}$. □

Our algorithm is given below.

$$\begin{array}{l}x\leftarrow 0;y\leftarrow \frac{m}{\min \left(\mathsf{\omega }\left(\mathrm{i}\right)\right)};V_1\leftarrow empty;\\ while\left(y-x\ge \frac{1}{N\left(N-1\right)}\right)\\ \{\\ \hspace{1em}\hspace{1em}g\leftarrow \frac{x+y}{2}\\ \hspace{1em}\hspace{1em}construct {{\displaystyle G}}_{\mathrm{g}}=\left(V_g,E_g\right);\\ \hspace{1em}\hspace{1em}find \min \text{-}cut\left(S,T\right);\\ \hspace{1em}\hspace{1em}if\left(S=\left\{s\right\}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}y\leftarrow g;\\ \hspace{1em}\hspace{1em}else \{\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}x\leftarrow g;\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{V}_1\leftarrow S-\left\{s\right\};\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\}\\ \}\\ return \left(densest subgraph of G induced by V_1\right)\end{array}$$

G′ is a subgraph of undirected graph G; V₁ is a set of vertices of G′; D′ is the density of G′, given two guess values, x and y, which are the highest and the lowest values of D′, i.e., $D^{\prime }\in \left[x,y\right]$. g is a guess value of D′. N is the sum of the vertex weights of graph G. Lemma 2 tell us that if $y\ge x+\frac{1}{N\left(N-1\right)}$, then G′ is the densest subgraph. So it is a loop condition for a while loop. In the while loop, we first assign the value $\frac{x+y}{2}$ to g, and next construct an undirected graph G_g, typically used to represent a graph with nodes and edges, where each edge has a specific capacity. Nodes represent the entities or vertices in the graph. Edges represent the connections or links between nodes. Then, we divide N and find the Min-Cut (S, T), using the Stoer–Wagner algorithm [22] to get the maximum flow, and then derive the min-cut from the residual graph. Then, we use Lemma 1 to determine the value of g. Finally, the return is the densest subgraph of G.

Theorem 1.

The running time of the algorithm is O(n(n + m)log² n).

Proof of Theorem 1.

In the algorithm, the time of the while loop is $O\left(\log n\right)$. In the loop, we define a function K(v,e) as the time to find the min-cut in undirected graph with v vertices and e edges. The graph has n + 2 vertices and 2(m + n) edges. In the time complexity, we can ignore constants, because they do not affect the growth rate of the function. So the time of finding min-cut is O(K(n, n + m)log n). Hence, the running time of the algorithm is $O\left(K\left(n,n+m\right)\log n\right)$. The time of Sleator and Tarjan’s algorithm [23] for the minimum-cut problem is O(|V||E|log |V|). Sleator and Tarjan’s algorithm refers to the Dynamic Trees data structure, also known as Link/Cut Trees. This data structure is used to efficiently maintain and manipulate a forest of trees. So our algorithm time complexity is O(n(n + m)log² n). □

4. An Approximation Algorithm for DS in a Vertex Weighted Graph

In [7], Charikar used a greedy algorithm to compute an approximate solution of the densest subgraph, in which he repeatedly removed a vertex of minimum weighted degree. Similarly, we use a greedy strategy to compute an approximate solution for DS as follows (Algorithm 1).

Algorithm 1: The approximation algorithm for DS in a vertex weighted graph

Input: $G\left(V,E,\omega \right)$, x, y, g

Output: The approximation of the densest subgraph

The process is as follows:

(1) Let G = (V, E) be a vertex weighted graph, |V| = n. Let G1 = G;

(2) For i = 1 to n {

(3)   Find a vertex $v_i\in V$ such that $\frac{\omega \left(G,v_i\right)}{c_i}$ is the minimum, then deleting vi;

(4)   Let Gi + 1 = Gi − {vi};

(5)  };

(6)  From the graphs G1, …, Gn, find a graph Gi such that D(Gi) is the maximum, output Gi.

In the following, we show the approximation ratio of the algorithm.

We assign each edge (i, j) to either i or j. For a vertex, let:

$$\mathrm{r}\left(i\right)=\frac{{\displaystyle \sum _{edges\begin{array}{cccc}& e& assigned& to\begin{array}{cc}& i\end{array}\end{array}}\omega \left(e\right)}}{c_i}$$

and ${\mathrm{r}}^{\max }={\max }_i\left\{r\left(i\right)\right\}$.

We show d(S) ≤ ${\mathrm{r}}^{\max }$ as follows.

Lemma 3.

$\underset{\mathrm{S}\subseteq \mathrm{V}}{\max } d\left(S\right)\le r^{\max }$

Proof of Lemma 3.

Suppose S is the set such that f(S) is maximum, where $f\left(S\right)=\frac{\left|E\left(S\right)\right|}{\left|S\right|}$. Since each edge must be assigned to one vertex, $W\left(G\left(S\right)\right)={\displaystyle \sum _{v_i\in S}\omega \left(G\left(S\right),v_i\right)}$. So,

$$\begin{array}{ll}d\left(S\right)& =\frac{W\left(G\left(S\right)\right)}{{\displaystyle \sum _{v_i\in S}{c}_i}}\le \frac{{\displaystyle \sum _{v_i\in S}\omega \left(G\left(S\right),v_i\right)}}{{\displaystyle \sum _{v_i\in S}{c}_i}}\\ & \le \frac{{\displaystyle \sum _{v_i\in S}\omega \left(G,v_i\right)}}{{\displaystyle \sum _{v_i\in S}{c}_i}}\le \frac{{\displaystyle \sum _{v_i\in S}{r}^{\max }\ast c_i}}{{\displaystyle \sum _{v_i\in S}{c}_i}}\\ & \le {\mathrm{r}}^{\max }\end{array}$$

Hence, $\max \underset{S\subseteq V}{d}(S)\le r^{\max }$. □

During the execution of the greedy algorithm, when a vertex is removed from S, the vertex is assigned all edges that go from the vertex to the rest of the vertices in S. Let ${\mathrm{r}}^{\max }$ be defined as before. Then, we can get the following conclusion.

Lemma 4.

If ${\mathrm{v}}^{\max }$ is the maximum value of d(S) for all sets S obtained in this process of greedy algorithm, then $r^{\max }\le 2v^{\max }$.

Proof of Lemma 4.

Consider a single iteration of the greedy algorithm. Suppose S is the vertex set $\left\{v_{i_1}\dots v_{i_s}\right\}$. $v_{\min }$ is the selected vertex such that $\frac{\omega \left(G\left(S\right),v_{\min }\right)}{c_{\min }}$ is the minimum. Then,

$$\begin{array}{ll}\frac{\omega \left(G\left(S\right),v_{\min }\right)}{c_{\min }}& \le \frac{{\displaystyle \sum _{v_i\in S}\omega \left(G\left(S\right),v_i\right)}}{{\displaystyle \sum _{v_i\in S}{c}_i}}\\ & =\frac{2E\left(G\left(S\right)\right)}{{\displaystyle \sum _{v_i\in S}{c}_i}}\\ & =2d(S)\le 2v^{\max }\end{array}$$

Hence, $r^{\max }\le 2v^{\max }$. □

In the following, we get the approximation ratio.

Theorem 2.

The approximation algorithm has an approximation ratio of 2.

Theorem 3.

The running time of the approximation algorithm is O(n + m).

Proof of Theorem 3.

Assume that G is of n vertices and the number of edges is m. At step (3), its time of finding vi and deleting vi is O(n). Using the adjacency list, when deleting vi, the degrees of its adjacency vertices are updated. For each updating, its time is O(m). Hence, the total time is O(n + m). □

5. Conclusions

This paper presents the notion of the densest subgraph in undirected graphs and vertex-weighted graphs. This introduction provides a foundation for gaining a more profound comprehension of graph structures, particularly within the framework of weighted networks. Furthermore, we introduce an exact algorithm that expands upon Goldberg’s algorithm to identify the densest subgraph within a graph with weighted vertices. This development is significant as it enables a more precise identification of critical subgraphs in complex networks, taking into account vertex weights. Additionally, we suggest an approximation algorithm as an alternative solution that balances computational efficiency and solution accuracy. This algorithm ensures an approximation ratio of 2, making it suitable for limited computational resources or requiring rapid approximations.

Researchers and practitioners can obtain valuable information about the network’s most influential or closely connected organizations by detecting the most compact subgraph. This knowledge is vital for comprehending network dynamics, forecasting actions, and making well-informed judgments. Our future work will focus on applying these algorithms to directed graphs and examining their integration with deep-learning approaches to enhance graph analysis capabilities.