Estimate Laplacian Spectral Properties of Large-Scale Networks by Random Walks and Graph Transformation

Abstract

For network graphs, numerous graph features are intimately linked to eigenvalues of the Laplacian matrix, such as connectivity and diameter. Thus, it is very important to solve eigenvalues of the Laplacian matrix for graphs. Similarly, for higher-order networks, eigenvalues of combinatorial Laplacian matrices are also important for invariants of graphs. However, for large-scale networks, it is difficult to calculate eigenvalues of the Laplacian matrix directly because it is either very difficult to obtain the whole network structure or requires a lot of computing resources. Therefore, this article makes the following contributions. Firstly, this paper proposes a random walk approach for estimating the bounds of the greatest eigenvalues of Laplacian matrices for large-scale networks. Considering the relationship between the spectral moments of the adjacency matrix and the closed paths in the network, we utilize the relationship between the adjacency matrix and the Laplacian matrix to establish the relationship between the Laplacian matrix and the closed paths. Then, we employ equiprobable random walks to sample the large graph to obtain the small graph. Through algebraic topology knowledge, we obtain the bounds of the largest eigenvalue of the Laplacian matrix of the large graph by using Laplacian spectral moments of the small graph. Secondly, for high-order networks, this paper proposes a method based on random walks and graph transformations. The graph transformation we propose mainly converts graphs with second-order simplices into ordinary weighted graphs, thereby transforming the problem of solving the spectral moments of the second-order combined Laplacian matrix into solving the spectral moments of the adjacency matrix. Then, we use the aforementioned random walk method to solve bounds of the greatest eigenvalue of the second-order combinatorial Laplacian matrix. Finally, by comparing the proposed method with existing algorithms in synthetic and real networks, its accuracy and superiority are demonstrated.

1. Introduction

In recent years, in the field of complex network research, people have focused on large-scale networks; that is, large networks with millions of nodes and tens of millions of connected edges. It is challenging to directly obtain the entire structure of a big network. Therefore, it is difficult to directly explore the topological properties of large-scale networks. The topological characteristics of large-scale networks include clustering coefficient, network density, spectral properties, and so on; among which the analysis of spectral properties has many applications in large-scale networks, such as web search index [1] or social network analysis [2]. Studying the characteristics of the graph’s adjacency matrix and Laplacian matrix is known as spectral property analysis. Numerous significant characteristics of graphs, such as network elasticity, community detection, and virus transmission speed, are demonstrated to be influenced by the matrix eigenvalues of graphs. The matrix eigenvalues of graphs have been used in chemistry, physics, and other applied sciences [3,4]. Solving the upper and lower bounds of the spectral radius of the Laplacian matrix in a network graph is of great significance; for example, the lower bound of the Laplacian matrix spectral radius is determined by the Cheeger constant, which measures the degree of connectivity. In practical application scenarios, if the lower bound of the spectral radius estimated by the algorithm in this article is greater than the Cheeger constant, it can be directly determined that the graph is connected and has strong connectivity. Moreover, estimating the Laplacian spectral radius can be used to estimate the density of a network graph. In data science, if the upper bound of the calculated Laplacian spectral radius is much smaller than the number of vertices, it can be considered a sparse graph and is suitable for efficient processing using spectral methods such as spectral clustering. Estimating the upper and lower bounds of the Laplacian spectral radius also plays an important role in network optimization and image segmentation.

In an ordinary network, an edge connects two nodes. The interaction relationship of nodes is binary, but, in real-world systems, the interaction relationship is non-binary and involves a joint nonlinear interaction of two more nodes. For example, in social systems [5], neuroscience [6,7], ecology [8], and biology [9], many connections take place at the group level between nodes of collective action rather than between pairs of nodes. We can better comprehend and forecast the dynamic behavior of a multi-component network by taking into account its higher-order structure. Higher-order networks can usually be used to describe the network relations of more than two nodes, and higher-order networks usually use hypergraphs [10] and simplicial complexes [11] to describe the interactions between nodes. For simplicial complexes, the interactions between nodes can be revealed by combinatorial Laplacian matrices. The eigenvalues of combinatorial Laplacian matrices for simplicial complex networks are of great importance in uncovering the properties of simplicial complex networks. For example, a representative ring of a particular homology (or cohomology) class can be extracted by the eigenvector associated with the zero eigenvalues of a combinatorial Laplacian matrix [12]. The combinatorial Laplacian operator has many applications. For example, it is a key operator in defining diffusion processes on simple complexes. Additionally, the combinational Laplacian operator can be applied to break down and analyze the evolution of simplex dynamic processes, so solving eigenvalues of combinational Laplacian matrices is very meaningful for exploring characteristics in simple complexes. The combinational Laplacian matrix is crucial in brain networks [13], game and traffic networks [14], data representation [15], machine learning [16], and other fields.

1.1. Previous Work

There are two kinds of analysis methods for spectral moments for the network. The first is to utilize an algebraic method to find the matrix’s eigenvalues and eigenvectors, and the second is to analyze the spectral moments of large-scale networks by the graphic method. For large-scale matrices, the calculation of eigenvalues requires a lot of computing resources and computing space, so direct calculation is not feasible. The eigenvalues must be obtained by other means. Among them, the iterative method forms the basis of most modern eigenvalue calculations. This method works by repeatedly refining the approximation of the eigenvector or eigenvalue and can terminate when the approximation reaches a suitable precision. The means include power iteration method [17], shift inverse iteration method [18], Rayleigh quotient method [19], simultaneous iteration method [20], and so on.

When only a series of local subgraphs in the network can be accessed, Preciado [21] was able to correlate connecting the adjacency matrix’s spectral moments with closed paths by studying the relationship between the structural characteristics of the network and the spectral moments of the adjacency matrices. To investigate the spectral characteristics of normalized Laplacian matrices of graphs, Wu [22] created a mathematical framework based on algebraic graph theory and convex optimization. Chu [23] proposed an algorithm to estimate the topological properties of large-scale networks by using extracted subgraphs to infer the key properties of the large graph. A sampling technique for determining the spectral radius of a big network adjacency matrix was presented by Abbas [24], who define a fractional value for the node and consider the sampling issue as an optimization issue. Unlike previous attempts to extract subgraphs of nodes with the most influence, Han [25] uses a straightforward random walk to select a small portion of a big network and calculates closed paths to determine the greatest eigenvalue of the adjacency matrix.

1.2. Contributions and Paper Organizations

In this research, we estimate the greatest eigenvalues of Laplacian matrices under big networks and estimate bounds of the greatest eigenvalues of combinatorial Laplacian matrices with simplex complexes. The main contributions of this paper are as follows:

• We propose a graph transformation method that converts solving spectral moments of the combinatorial Laplacian matrix into solving spectral moments of the adjacency matrix, addressing the issue that the combinatorial Laplacian matrix lacks practical physical significance.
• A random walk algorithm is proposed to estimate bounds of the greatest eigenvalues of the Laplacian matrix in large-scale networks. The correctness and superiority of the algorithm are proved by experimental simulation on real and synthetic networks.
• A method combining graph transformation and random walks is proposed to estimate the bounds of the greatest eigenvalues of the second-order combinatorial Laplacian matrix in big networks with higher-order structures. Simulation investigation on real and synthetic networks confirms the algorithm’s efficacy.

The remainder of this paper is organized as follows. Section 2 introduces symbolic expressions and preliminaries and further describes the problem addressed in the paper. Section 3 introduces an algorithm for estimating the bounds of the greatest eigenvalue of the Laplacian matrix based on random walks in big networks and further introduces an algorithm for estimating the bounds of the greatest eigenvalue of the second-order combinatorial Laplacian matrix based on random walks and graph transformations. Section 4 demonstrates the correctness and advantages of this algorithm by comparing it with the current optimal algorithm. Section 5 concludes the paper.

2. Problem Formulation

2.1. Notation and Preliminaries

Consider a connected undirected simple graph $G=\{V,E\}$, where $V$ represents the set of nodes of the network, $E$ represents the set of connected edges of the network, and $v\in V$ is a node in the simple graph $G$. Let $N\left(v\right)$ represent the set of neighbors of node $v$, let $d\left(v\right)$ represent the degree of node $v$, let $D={\sum }_{v\in V}d\left(v\right)$ represent the sum of degrees of all nodes in the graph G, and let $△$ represent the degree matrix of the graph $G$. Let $A$ denote the adjacency matrix in graph $G$, whose size is $\left|V\right|\times \left|V\right|$. Since the graph $G$ is an undirected graph, the matrix A is symmetric and all its eigenvalues are real numbers, where ${\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_{\left|V\right|}$ represents the eigenvalue of the adjacency matrix A; let $Q=△-A$ represent the Laplacian matrix of graph $G$, whose size is |V|×|V|, where ${\mu }_1\ge {\mu }_2\ge \cdots \ge {\mu }_{\left|V\right|}$ represents the eigenvalue of the Laplacian matrix $Q$, where the spectral moment of the Laplacian matrix $Q$ of the graph $G$ is defined as follows:

$$m_k\left(Q\right)={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\mu }_i^k,$$

(1)

The graph $X$ is a simple complex network whose combinatorial Laplacian operators of order k can be expressed as $L_k$, and $k$ is the order of the highest simplex in the graph. $L_k$ is defined as follows:

$$L_k={B_k^{T}B}_k+B_{k+1}{B}_{k+1}^T,$$

(2)

$B_k$ can represent the relationship between a simplex of $k-1$ order and a simplex of $k$ order, and $B_{k+1}$ can represent the relationship between a simplex of $k$ order and a simplex of $k+1$ order. Formula (2) can be further expanded as follows:

$$\begin{array}{ll}{\left(L_k\right)}_{ij}& ={\left({B_k^{T}B}_k\right)}_{ij}+{\left(B_{k+1}{B}_{k+1}^T\right)}_{ij}\\ & =\left\{\begin{array}{ll}{deg}_u\left({\sigma }_i\right)+k+1& \mathrm{if} i=j,\\ \hspace{1em}\hspace{1em}\hspace{1em}1& \mathrm{if} i\ne j,{\sigma }_i⋔{\sigma }_j,\mathrm{and} {\sigma }_i\cup {\sigma }_j \mathrm{with} \mathrm{similar} \mathrm{orientation}\\ \hspace{1em}\hspace{1em} -1& \mathrm{if} i\ne j,{\sigma }_i⋔{\sigma }_j,\mathrm{and} {\sigma }_i\cup {\sigma }_j \mathrm{with} \mathrm{dissimilar} \mathrm{orientation}\\ \hspace{1em}\hspace{1em}\hspace{1em}0& \mathrm{if} i\ne j \mathrm{and} {\sigma }_i\cap {\sigma }_j \mathrm{or} {\sigma }_i\mathsf{\psi }{\sigma }_j \end{array},\right.\end{array}$$

(3)

Two k-simplex ${\sigma }_i$ and ${\sigma }_j$ of a simplicial complex X are upper adjacent if both are faces of a $k$+1-simplex $\mathsf{\epsilon }$ in X. This adjacency is denoted by ${\sigma }_i\cap {\sigma }_j$. If the orientations of ${\sigma }_i$ and ${\sigma }_j$ agree with the ones induced by $\mathsf{\epsilon }$, then ${\sigma }_i$, ${\sigma }_j$ are said to be similarly oriented with respect to $\mathsf{\epsilon }$. If not, we say that the two k-simplices are oriented differently. Two k-simplex ${\sigma }_i$ and ${\sigma }_j$ of a simplicial complex X are lower adjacent if both have a common face. This adjacency is denoted by ${\sigma }_i\cup {\sigma }_j$. The definition of similar orientation and dissimilar orientation are the same as before. The upper degree of a k-simplex, denoted ${deg}_u\left({\sigma }_i\right)$, is the number of $k$+1-simplices in X of which ${\sigma }_i$ is a face.

2.2. Random Walk

Random walk is a sampling method. There are different walking strategies in random walks, such as walking according to the equal probability of node neighbors and non-equal probability walks (the walks with a tendency; for example, they tend to walk to neighbors with large node degrees). This paper takes an equal probability walk; that is, a node walks from one node to the next node and selects a neighbor node from the neighbor node of the current node with an equal probability to walk. Such a walk mode can ensure that the walking path obtained can represent the entire network to the greatest extent.

In many cases, the selection of the initial node of the random walk is crucial to the whole random walk process. If the initial node is selected as an edge node (that is, a node with a small node degree), the selected node needs to go through more walks to represent the whole network. Therefore, before selecting the initial node, it is necessary to go through a sufficient number of walks to ensure the generality of the initial node. The state reached after a sufficient number of walks is also called stationary distribution, and the initial node $v_i$ selected after smooth distribution is only related to the degree of node $v_i$ itself. $P\left(v_i\right) \mathrm{i}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y} \mathrm{o}\mathrm{f} \mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{s} \mathrm{b}\mathrm{e}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{o}\mathrm{m}\mathrm{l}\mathrm{y} \mathrm{s}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{e}\mathrm{d}. P\left(v_i\right)$ is defined as follows:

$$P\left(v_i\right)={\displaystyle \frac{d\left(v_i\right)}{D}},$$

(4)

where D is the network diagram’s overall degree.

After the initial node is selected, the node starts to walk to its neighbors with equal probability and finally forms a series of node sequences, such as $X_i^{\left(k\right)}=\left(v_i,v_{i+1},\dots ,v_{i+k}\right)$, which represents the sequence of nodes obtained by node $v_i$ after $k$ steps in the random walk process, and $R^{\left(k\right)}$, which represents the set of wandering paths with all path lengths of $\mathrm{k}$ in the graph. The probability $P\left(X_i^{\left(k\right)}\right)$ that happens to pass through node sequence $X_i^{\left(k\right)}$ in the random walk process can be obtained as follows:

$$\begin{array}{ll}P\left(X_i^{\left(k\right)}\right)& ={\displaystyle \frac{d\left(v_i\right)}{D}}\ast {\displaystyle \frac{1}{d\left(v_i\right)}}\ast {\displaystyle \frac{1}{d\left(v_{i+1}\right)}}···{\displaystyle \frac{1}{d\left(v_{i+k-1}\right)}}\\ & =\left\{\begin{array}{c}{\displaystyle \frac{1}{D}}\prod _{j=1}^{k-1}{\displaystyle \frac{1}{d\left(v_{i+j}\right)}} k>1\\ {\displaystyle \frac{1}{D}} k=1\end{array}.\right.\end{array}$$

(5)

2.3. Problem Statement and Assumptions

It is quite challenging to directly determine the maximum eigenvalue of the Laplacian matrix for large networks, which requires a lot of computing resources and computing time. In many cases, it is impossible to obtain the complete topology of entire networks. Therefore, in order to estimate the border of the greatest eigenvalue of the Laplacian matrix, this research suggests an approach based on random walks. Additionally, this paper proposes an algorithm based on random walks and graph transformation to estimate the greatest eigenvalue boundary of the second-order combinatorial Laplacian operator.

The main problem studied in this article is based on the following assumptions: The network is an undirected simple graph; Network connectivity ensures the existence of the minimum non-zero eigenvalues of the Laplacian matrix; The edge weights are non negative real numbers and the weight values in high-order networks conform to the definition of the composite Laplacian matrix; The random walk process satisfies memorylessness, and the node transition probability depends only on the current node state.

2.4. The Symbol Table

Symbol Table	Meaning Explanation
$G=\{V,E\}$	Undirected simple graph, where V is the node set and E is the edge set
v	Any node in Figure G
$N\left(v\right)$	$\mathrm{the} \mathrm{set} \mathrm{of} \mathrm{neighbors} \mathrm{of} \mathrm{node} v$
$d\left(v\right)$	$\mathrm{the} \mathrm{degree} \mathrm{of} \mathrm{node} v$
$D$	the sum of degrees of all nodes
$△$	$\mathrm{the} \mathrm{degree} \mathrm{matrix} \mathrm{of} \mathrm{the} \mathrm{graph} G$
${\lambda }_i$	the i-th eigenvalue of the adjacency matrix A
$Q$	$\mathrm{the} \mathrm{Laplacian} \mathrm{matrix} \mathrm{of} \mathrm{graph} G$
${\mu }_i$	$\mathrm{the} \mathrm{i-th} \mathrm{eigenvalue} \mathrm{of} \mathrm{the} \mathrm{Laplacian} \mathrm{matrix} Q$
$m_k\left(Q\right)$	$\mathrm{the} \mathrm{k-th} \mathrm{spectral} \mathrm{moment} \mathrm{of} \mathrm{the} \mathrm{Laplacian} \mathrm{matrix} Q$
$X$	simple complex network
$L_k$	combinatorial Laplacian operators of order k
$B_k$	$\mathrm{Correlation} \mathrm{matrix}, \mathrm{representing} \mathrm{the} \mathrm{relationship} \mathrm{between} k-$1$\mathrm{order} \mathrm{and} \mathrm{a} \mathrm{simplex} \mathrm{of} k$ order
${\sigma }_k$	k-simplex
${deg}_u\left({\sigma }_i\right)$	$\mathrm{the} \mathrm{number} \mathrm{of} k$+1$-\mathrm{simplices} \mathrm{in} X \mathrm{of} \mathrm{which} {\sigma }_i$ is a face.
$X_i^{\left(k\right)}$	$\mathrm{the} \mathrm{sequence} \mathrm{of} \mathrm{nodes} \mathrm{obtained} \mathrm{by} \mathrm{node} v_i$ $\mathrm{after} k$ steps
$R^{\left(k\right)}$	$\mathrm{the} \mathrm{set} \mathrm{of} \mathrm{wandering} \mathrm{paths} \mathrm{with} \mathrm{all} \mathrm{path} \mathrm{lengths} \mathrm{of} \mathrm{k}$ in the graph.
$P\left(X_i^{\left(k\right)}\right)$	$\mathrm{The} \mathrm{probability} \mathrm{that} \mathrm{happens} \mathrm{to} \mathrm{pass} \mathrm{through} \mathrm{node} \mathrm{sequence} X_i^{\left(k\right)}$ in the random walk
$W\left(X_i^{\left(k\right)}\right)$	Path judgment function (take 1 when the path is a closed path, otherwise take 0)
$W$	a weighted adjacency matrix
$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\left(M\right)$	Trace of matrix M (sum of main diagonal elements)
${\mathrm{H}}_{2\mathrm{r}}\left(\mathrm{M}\right)$ ${\mathrm{H}}_{2\mathrm{r}+1}\left(\mathrm{M}\right)$	the Hankel matrix of M
${\rho }_G\left(\mu \right)$	$\mathrm{The} \mathrm{probability} \mathrm{density} \mathrm{function} \mathrm{of} \mathrm{the} \mathrm{eigenvalues} \mathrm{of} \mathrm{the} \mathrm{Laplacian} \mathrm{matrix} (\mathrm{defined} \mathrm{by} \mathrm{the} \mathrm{Dirac} \mathrm{function} \mathsf{\delta }$
Z	Second-order simplex (a triangular structure consisting of three nodes)
$\mathrm{N}\left(E_{ij}\right)$	$\mathrm{N}\left(E_{ij}\right)$ indicates the neighbor of $E_{ij}$
$\mathrm{W}(u,\mathrm{v})$	Edge weights between nodes u and v in a weighted graph
Var[⋅]	Variance of Random Variables
E[⋅]	Mathematical Expectations of Random Variables

3. Algorithm Description

3.1. Spectral Moment Based on Random Walks

The spectral moment of the adjacency matrix has actual physical significance in the network, while the spectral moment of the Laplacian matrix has no actual physical significance. We explore the relationship between the spectral moment of the Laplacian matrix and the spectral moment of the adjacency matrix, and the problem of solving the spectral moment of the Laplacian matrix is transformed into the problem of solving the spectral moment of the adjacency matrix. Therefore, the following lemmas are given:

Lemma 1 

[26]. The trace of the $k$ power of the Laplacian matrix $Q$ of graph $G$ is equal to the $k$-orderspectral moment of $Q$ multiplied by the total number of nodes.

$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\left(Q^k\right)=\sum _{i=1}^{\left|V\right|}{\mu }_i^k=\left|V\right|\ast m_k\left(Q\right),$$

(6)

Lemma 2 

[26]. The number of closed walking paths of length $k$ in graph $G$ is equal to the $k$ power of the adjacency matrix, i.e., the trace of $A^k$.

$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\left(A^k\right)=\sum _{i=1}^{\left|V\right|}{\lambda }_i^k=\sum _{X_i^{\left(k\right)}\in R^{\left(k\right)}}W\left(X_i^{\left(k\right)}\right),$$

(7)

among them, $W\left(X_i^{\left(k\right)}\right)$ is used to judge whether the node sequence $X_i^{\left(k\right)}$ formed by the walking path is closed. If the path is closed, $W\left(X_i^{\left(k\right)}\right)=1$; otherwise, $W\left(X_i^{\left(k\right)}\right)=0$.

Lemma 3 

[26]. For the $kth$ power of the degree matrix and adjacency matrix of the graph $G$ , only the diagonal elements are non-zero, and all other elements are zero.

Theorem 1. 

The $k$-order spectral moment $m_k\left(Q\right)$ of the Laplacian matrix $Q$ is expressed as follows:

$$m_k\left(Q\right)={\displaystyle \frac{1}{\left|V\right|}}\sum _{i=1}^{\left|V\right|}{\mu }_i^k={\displaystyle \frac{1}{\left|V\right|}}trace\left(Q^k\right),$$

(8)

From Theorem 1, the following formula can be obtained:

$$m_k\left(Q\right)={\displaystyle \frac{1}{\left|V\right|}}\sum _{m=0}^k\left(\begin{array}{c}k\\ m\end{array}\right){\left(-1\right)}^{k-m}\sum _{l=1}^N{d\left(v_l\right)}^m{\left(A^{k-m}\right)}_{ll},$$

(9)

Proof. 

Since the Laplacian matrix of graph $G$ is equal to the degree matrix $△$ of graph $G$ minus the adjacency matrix $A$ of graph $G$, the following formula can be obtained:

$$trace\left(Q^k\right)=trace{\left(△-A\right)}^k,$$

(10)

Using the binomial theorem to expand ${\left(△-A\right)}^k$: for two matrices X, Y, then the binomial theorem can be directly extended to matrix power. ${\left(X+Y\right)}^k={\sum }_{m=0}^k\left(\begin{array}{c}k\\ m\end{array}\right)X^m{Y}^{k-m}$. The linear property of trace $trace\left(cM+N\right)=ctrace\left(M\right)+trace\left(N\right)$. Thus, we can obtain:

$$\begin{array}{ll}trace\left(Q^k\right)& =trace{\left(△-A\right)}^k\\ & =trace({\displaystyle \sum _{m=0}^k}\left(\begin{array}{c}k\\ m\end{array}\right){△}^m{(-A)}^{k-m})\\ & ={\displaystyle \sum _{m=0}^k}\left(\begin{array}{c}k\\ m\end{array}\right)trace\left({△}^m{(-A)}^{k-m}\right)\\ & ={\displaystyle \sum _{m=0}^k}\left(\begin{array}{c}k\\ m\end{array}\right){\left(-1\right)}^{k-m}trace\left({△}^m{A}^{k-m}\right),\end{array}$$

(11)

According to the definition of the trace of the matrix, we can obtain:

$$trace\left({△}^m{A}^{k-m}\right)=\sum _{l=1}^{\left|V\right|}{\left({△}^m{A}^{k-m}\right)}_{ll},$$

(12)

From the matrix multiplication theorem, we can obtain:

$${\left[AB\right]}_{ll}=\sum _{q=1}^{\left|V\right|}{\left(A\right)}_{lq}{\left(B\right)}_{ql},$$

(13)

Therefore, according to Formula (13), we can obtain:

$$\sum _{\mathrm{l}=1}^{\left|\mathrm{V}\right|}{\left({△}^{\mathrm{m}}{\mathrm{A}}^{\mathrm{k}-\mathrm{m}}\right)}_{\mathrm{l}\mathrm{l}}=\sum _{\mathrm{l}=1}^{\left|\mathrm{V}\right|}\sum _{\mathrm{q}=1}^{\left|\mathrm{V}\right|}{\left({△}^{\mathrm{m}}\right)}_{\mathrm{l}\mathrm{q}}{{(\mathrm{A}}^{\mathrm{k}-\mathrm{m}})}_{\mathrm{q}\mathrm{l}},$$

(14)

According to Lemma 3, the following formula can be deduced:

$$\begin{array}{ll}{\sum }_{l=1}^{\left|V\right|}{\left({△}^m{A}^{k-m}\right)}_{ll}& ={\displaystyle \sum _{l=1}^{\left|V\right|}}{\displaystyle \sum _{q=1}^{\left|V\right|}}{\left({△}^m\right)}_{lq}{{(A}^{k-m})}_{ql}\\ & ={\displaystyle \sum _{l=1}^{\left|V\right|}}{\left({△}^m\right)}_{ll}{{(A}^{k-m})}_{ll}\\ & ={\displaystyle \sum _{l=1}^{\left|V\right|}}{d\left(v_l\right)}^m{{(A}^{k-m})}_{ll} ,\end{array}$$

(15)

By combining Formulas (11) and (15), spectral moments of the Laplacian matrix can be successfully converted into equations related to the adjacency matrix

$$\begin{array}{ll}{m}_k\left(Q\right)& ={\displaystyle \frac{1}{\left|V\right|}}trace\left(Q^k\right)\\ & ={\displaystyle \frac{1}{\left|V\right|}}{\displaystyle \sum _{m=0}^k}\left(\begin{array}{c}k\\ m\end{array}\right){\left(-1\right)}^{k-m}trace\left({△}^m{A}^{k-m}\right)\\ & ={\displaystyle \frac{1}{\left|V\right|}}{\displaystyle \sum _{m=0}^k}\left(\begin{array}{c}k\\ m\end{array}\right){\left(-1\right)}^{k-m}{\displaystyle \sum _{l=1}^N}{d\left(v_l\right)}^m{\left(A^{k-m}\right)}_{ll}, \end{array}$$

(16)

which completes the proof. □

Theorem 2. 

The relationship between k-order Laplacian spectral moments and walking paths is as follows:

$$m_k\left(Q\right)=\left\{\begin{array}{c}{\displaystyle \frac{D}{\left|V\right|}}{\displaystyle \sum _{m=0}^k}\left(\begin{array}{c}k\\ m\end{array}\right){\left(-1\right)}^{k-m}E\left[w\left(X_i^{\left(k-m\right)}\right)\ast {\displaystyle \prod _{j=1}^{k-m-1}}d\left(v_{i+j}\right){d\left(v_i\right)}^m\right] k>1\\ {\displaystyle \frac{D}{\left|V\right|}}{\displaystyle \sum _{m=0}^k}\left(\begin{array}{c}k\\ m\end{array}\right){\left(-1\right)}^{k-m}\ast E\left[w\left(X_i^{\left(k-m\right)}\right)\ast {d\left(v_i\right)}^m \right] k=1\end{array}\right.$$

(17)

Proof. 

The full expectation formula is:

$$\mathrm{E}\left(\mathrm{X}\right)=\mathrm{E}\left(\mathrm{E}\left(\mathrm{X}|\mathrm{Y}\right)\right),$$

(18)

For any sequence of nodes whose path length is $\mathrm{k}$ passing through node $v_i$, $X_i^{\left(k\right)}=\left(v_i,v_{i+1},\dots ,v_{i+k}\right)$, $R^{\left(k\right)}$ represents the sequence of nodes formed by all paths of length $k$. Define function as follows:

$$\begin{array}{ll}f\left(X_i^{\left(k\right)}\right)& ={\displaystyle \frac{{d\left(v_i\right)}^m}{P\left(X_i^{\left(k\right)}\right)D}}\\ & =\left\{\begin{array}{c}{\displaystyle \prod _{j=1}^{k-1}}d\left(v_{i+j}\right){d\left(v_i\right)}^m k>1\\ {d\left(v_i\right)}^m k=1\end{array}\right., \end{array}$$

(19)

where m is a constant.

Any path of length k in a network forms a succession of nodes $P_i^{\left(k\right)}$, where $P_i^{\left(k\right)}\in R^{\left(k\right)}$. $X^{\left(k\right)}$ represents a node sequence formed by path of length k, where $X^{\left(k\right)}\in R^{\left(k\right)}$. $R_i^{\left(k\right)}$ represents the set of all paths of length $k$ passing through node $v_i$. Solving for the expected value of $W\left(P_i^{\left(k\right)}\right)f\left(P_i^{\left(k\right)}\right)$ and using Formula $\left(19\right)$, we obtain:

$$\begin{array}{ll}\mathrm{E}\left[W\left(P_i^{\left(k\right)}\right)\ast f\left(P_i^{\left(k\right)}\right)\right]& ={\displaystyle \sum _{X^{\left(k\right)}\in R^{\left(k\right)}}}\mathrm{E}\left[W\left(P_i^{\left(k\right)}\right)\ast f\left(P_i^{\left(k\right)}\right)|P_i^{\left(k\right)}=X^{\left(k\right)}\right]P\left(P_i^{\left(k\right)}=X^{\left(k\right)}\right)\\ & ={\displaystyle \sum _{X^{\left(k\right)}\in R^{\left(k\right)}}}W\left(X^{\left(k\right)}\right)\ast f\left(X^{\left(k\right)}\right)\ast P\left(X^{\left(k\right)}\right)\\ & ={\displaystyle \sum _{i=1}^{\left|V\right|}}{\displaystyle \sum _{X_i^{\left(k\right)}\in R_i^{\left(k\right)}}}W\left(X_i^{\left(k\right)}\right)\ast f\left(X_i^{\left(k\right)}\right)\ast P\left(X_i^{\left(k\right)}\right)\\ & ={\displaystyle \sum _{i=1}^{\left|V\right|}}{\displaystyle \sum _{X_i^{\left(k\right)}\in R_i^{\left(k\right)}}}W\left(X_i^{\left(k\right)}\right)\ast {\displaystyle \frac{{d\left(v_i\right)}^m}{P\left(X_i^{\left(k\right)}\right)D}}\ast P\left(X_i^{\left(k\right)}\right)\\ & ={\displaystyle \frac{1}{D}}{\displaystyle \sum _{i=1}^{\left|V\right|}}{\displaystyle \sum _{X_i^{\left(k\right)}\in R_i^{\left(k\right)}}}{d\left(v_i\right)}^m\ast W\left(X_i^{\left(k\right)}\right),\end{array}$$

(20)

Formula (21) can be obtained from Formula (20).

$$\sum _{i=1}^{\left|V\right|}\sum _{X_i^{\left(k\right)}\in R_i^{\left(k\right)}}{d\left(v_i\right)}^m\ast W\left(X_i^{\left(k\right)}\right)=D\ast E\left[W\left(P_i^{\left(k\right)}\right)\ast f\left(P_i^{\left(k\right)}\right)\right],$$

(21)

From Lemma 2, we can obtain the following:

$$\sum _{i=1}^{\left|V\right|}\sum _{X_i^{\left(k\right)}\in R_i^{\left(k\right)}}{d\left(v_i\right)}^m\ast W\left(X_i^{\left(k\right)}\right)=\sum _{i=1}^{\left|V\right|}{d\left(v_i\right)}^m\ast {\left(A^k\right)}_{ii},$$

(22)

From Formulas (21) and (22), the following formula can be obtained:

$$\sum _{l=1}^N{d\left(v_l\right)}^m{\left(A^{k-m}\right)}_{ll}=D\ast E\left[W\left(P_i^{\left(k-m\right)}\right)\ast f\left(P_i^{\left(k-m\right)}\right)\right],$$

(23)

where the $f\left(X_i^{\left(k\right)}\right)$ expression is as shown in Formula (19). Therefore, Formula $\left(24\right)$ can be obtained as follows:

$$\begin{array}{ll}{m}_k\left(Q\right)& ={\displaystyle \frac{1}{\left|V\right|}}{\displaystyle \sum _{m=0}^k}\left(\begin{array}{c}k\\ m\end{array}\right){\left(-1\right)}^{k-m}{\displaystyle \sum _{l=1}^N}{d\left(v_l\right)}^m{\left(A^{k-m}\right)}_{ll}\\ & ={\displaystyle \frac{D}{\left|V\right|}}{\displaystyle \sum _{m=0}^k}\left(\begin{array}{c}k\\ m\end{array}\right){\left(-1\right)}^{k-m}\ast E\left[W\left(X_i^{\left(k-m\right)}\right)\ast f\left(X_i^{\left(k-m\right)}\right)\right] \\ & =\left\{\begin{array}{c}{\displaystyle \frac{D}{\left|V\right|}}{\displaystyle \sum _{m=0}^k}\left(\begin{array}{c}k\\ m\end{array}\right){\left(-1\right)}^{k-m}\ast E\left[W\left(X_i^{\left(k-m\right)}\right)\ast {\displaystyle \prod _{j=1}^{k-m-1}}d\left(v_{i+j}\right){d\left(v_i\right)}^m\right]k>1\\ {\displaystyle \frac{D}{\left|V\right|}}{\displaystyle \sum _{m=0}^k}\left(\begin{array}{c}k\\ m\end{array}\right){\left(-1\right)}^{k-m}\ast E\left[W\left(X_i^{\left(k-m\right)}\right)\ast {d\left(v_i\right)}^m \right] k=1\end{array}\right.\end{array}$$

(24)

If $X_i^{\left(k-m\right)}$ is a closed path, $W\left(X_i^{\left(k-m\right)}\right)$ is one; otherwise, it is zero.

Which completes the proof. □

To better illustrate the above process, we take k = 3 as an example. According to the definition of spectral moments, we can deduce that $m_3\left(Q\right)={\displaystyle \frac{1}{\left|V\right|}}{\sum }_{i=1}^{\left|V\right|}{\mu }_i^3={\displaystyle \frac{1}{\left|V\right|}}trace\left(Q^3\right)$; according to the definition of the Laplacian matrix and the binomial theorem, we can obtain $trace\left(Q^3\right)={\sum }_{m=0}^3\left(\begin{array}{c}3\\ m\end{array}\right){\left(-1\right)}^{3-m}trace\left({△}^m{A}^{3-m}\right)$, then use the definition of trace and the principle of random walk to obtain $trace\left({△}^m{A}^{3-m}\right)=D\ast E\left[W\left(X_i^{\left(3-m\right)}\right)\ast {d\left(v_i\right)}^m \right]$. In the end, we obtained Theorem 2.

Thus, it can be seen from Formula $\left(24\right)$ that the Laplacian spectral moment is ultimately transformed into the process of searching the closed path. The steps are as follows: first, let $m=\mathrm{0,1},2,\dots ,k$. For each m, walk several times and observe whether a closed path of length $k-m$ is formed during each walk. If so, the value of $\prod _{j=1}^{k-m-1}d\left(v_{i+j}\right){d\left(v_i\right)}^m$ on the closed path can be solved. The expected value of $\prod _{j=1}^{k-m-1}d\left(v_{i+j}\right){d\left(v_i\right)}^m$ is calculated by the sum of $\prod _{j=1}^{k-m-1}d\left(v_{i+j}\right){d\left(v_i\right)}^m$ of the closed path in the walk divided by the total number of walking steps. Finally, $m_k\left(Q\right)$ can be derived from Formula $\left(24\right)$.

In the above solution process, if the total degrees $D$ of all nodes in graph $G$ is unknown, $D$ needs to be estimated by random walk, which can be obtained by Formula (25):

$$\mathrm{E}\left[{\displaystyle \frac{1}{d\left(r_i\right)}}\right]={\sum }_{v\in V}{\displaystyle \frac{d\left(v\right)}{D}}{\displaystyle \frac{1}{d\left(v\right)}}={\displaystyle \frac{\left|V\right|}{D}},$$

(25)

Then, the expression of $D$ can be obtained as follows:

$$D={\displaystyle \frac{\left|V\right|}{\mathrm{E}\left[\frac{1}{d\left(r_i\right)}\right]}},$$

(26)

It is evident that Formula (26) can be used to determine the total degrees $D$ of all nodes in the network. The above is the process of solving the spectral moment of the Laplacian matrix by using random walks. Formula (26) is heavily biased for short walks. When there are few wandering nodes, serious errors can occur. Only when the number of sampling nodes reaches a certain level can the overall network degree be accurately estimated.

3.2. Spectral Moment Based on Second-Order Graph Transformation and Random Walks

In this section, a graph transformation method is proposed that converts the spectral moment of the second-order combinatorial Laplacian operator into the spectral moment of the adjacency matrix in a weighted graph. According to the definition of the combinatorial Laplacian operator in Formula ($3$), the specific transformation method can be obtained as follows:

(1)

Now there are two graphs. One is the network with simplicial complexes before transformation, and the other is the weighted network after transformation. First, each connected edge in the network with simplicial complexes is transformed into a node of the weighted network.

(2)

Add a self-ring to nodes of the weighted network. The weight of the self-ring is equal to ${deg}_u\left({\sigma }_i\right)+k+1$, where k represents the order of ${\sigma }_i$ in a simple complex. ${deg}_u\left({\sigma }_i\right)$ represents the number of simplex ${\sigma }_i$ belonging to a higher order simplex. If the simplex ${\sigma }_i$ in the simplex complex does not belong to any of the simplex structures of order $k+1$, then ${deg}_u\left({\sigma }_i\right)$ is 0.

(3)

Observe the relationship between the different simplex ${\sigma }_i$ in the simple complex. When studying the second-order combinatorial Laplacian matrix, observe the adjacent connected edges in the simple complex network. There are no linked connections between the respective nodes in the weighted network if the two connected edges are in the same second-order simplex. If the adjacent connected edges are not in the same second-order simplex, it is judged whether there are common nodes in the two connected edges. If there are common nodes, we continue to judge whether the direction of the connected edges is the same for the common nodes. If so, the connected edge weight between the nodes corresponding to the weighted network is 1. The edge weight between the nodes that correspond to the weighted network is −1 if they differ.

To better illustrate the aforementioned steps, Figure 1 is taken as an example.

(1)

Treat each edge as a node, as indicated in Figure 1.

(2)

Add a self-loop to each converted node, with a weight of ${deg}_u\left({\sigma }_i\right)+k+1$. ${deg}_u\left({\sigma }_i\right)$ is determined by whether the edge before conversion is in a second-order simplex. For example, if the edge is in a second-order simplex, then the weight of the self-loop of the converted node is 1 + 1 + 1, which is 3. If the edge before conversion does not belong to any second-order simplex, then the weight of the self-loop of the converted node is 0 + 1 + 1, which is 2.

(3)

Check to see if the edge and its neighboring edge in the left image share a node. If there is a common node, then determine whether the edge directions are the same for the common point. If they are the same, then the weight between the nodes corresponding to the edges in the right image is 1. If they are not the same, then the weight between the nodes that correspond to the edges in the right image is −1.

(4)

Determine whether the edge and its adjacent edge in the left image belong to the same second-order simplex. If they belong to the same second-order simplex, then there will be no edge between the corresponding nodes after conversion.

According to the above method, the left image in Figure 1 has been transformed into the right image.

As we can see, on the left side of Figure 1 is the network with second-order simplex, and on the right side is the weighted graph obtained according to the above steps. The gray area in the left figure forms a second-order simplex. Firstly, transform each connected edge in the left image into each node in the right image, and then add self loops with different weights based on whether the connected edge is in the second-order simplex. Then, based on whether the connected edges are in the same second-order simplex, if they are in the same second-order simplex, such as connected edges 3 and 5, then there is no connected edge between nodes 3 and 5 in the corresponding graph on the right; If the edges are not connected in the same second-order simplex, and the directions are consistent, the weight between the corresponding nodes is 1. Otherwise, it is −1. As shown in the left figure, both edge 1 and edge 2 are emitted from the same node, and their corresponding weight between node 1 and node 2 in the right figure is 1; Similarly, in the right figure node 2 and node 6 have different directions, so their edge weights are −1.

From the above operations, solving the spectral moment of the second-order combinatorial Laplacian matrix is transformed into solving the spectral moment of the adjacency matrix in the weighted graph. The above steps can be simplified into the following Algorithm 1 flowchart.

Algorithm 1 Graph Transformation

Require: $\mathrm{Simplex} \mathrm{complex} \mathrm{graph} \overline{G}$$, \mathrm{weighted} \mathrm{graph} \tilde{G}$, second-order simplex $\mathrm{Z}=\{{\mathrm{v}}_{\mathrm{i}},{\mathrm{v}}_{\mathrm{j}},{\mathrm{v}}_{\mathrm{k}}\}$, $\mathrm{N}\left(E_{ij}\right)$ indicates the neighbor of $E_{ij}$

$1: \mathrm{Each} \mathrm{edge} \mathrm{in} \mathrm{graph} \overline{G}$ $\mathrm{corresponds} \mathrm{to} \mathrm{each} \mathrm{node} \mathrm{in} \mathrm{graph} \tilde{G}$$, E_{ij}$ in $\overline{G}$$\mathrm{corresponding} \mathrm{to} \mathrm{node} v_{E_{ij}}$ in graph $\tilde{G}$

2: $\mathrm{Purpose}: \mathrm{Convert} \mathrm{graph} \overline{G}$$\mathrm{into} \mathrm{graph} \tilde{G}$

3: Step 1: edges in simplex are converted to nodes

4: for $E_{ij}$$\mathrm{in} \overline{G}$ do

$5: \mathbf{if} E_{ij}ϵ Z$then

$6: \mathrm{the} \mathrm{self-loop} \mathrm{of} \mathrm{node} v_{E_{ij}}$ corresponding to the edge $E_{ij}$ is added, and the weight of the self-loop is 3

7:  else

$8: \mathrm{the} \mathrm{self-loop} \mathrm{of} \mathrm{node} v_{E_{ij}}$ corresponding to the edge $E_{ij}$ is added, and the weight is 2.

9:  end if

10: Step 2: add weights to edges

$11: \mathbf{for} e$$\mathrm{in} \mathrm{N}\left(E_{ij}\right)$ do

$12:$ if $E_{ij}ϵ Z$ and $eϵ Z$ then

$13: \mathrm{W}(v_{E_{ij}},v_e)=0$

$14: \mathbf{else} \mathbf{if} \mathrm{the} \mathrm{direction} \mathrm{of} \mathrm{the} \mathrm{edges} \mathrm{is} \mathrm{the} \mathrm{same} \mathrm{for} \mathrm{the} \mathrm{common} \mathrm{nodes} \mathrm{of} E_{ij}$ and e then

$15: \mathrm{W}(v_{E_{ij}},v_e)=1$

$16: \mathbf{else} \mathbf{if} \mathrm{the} \mathrm{direction} \mathrm{of} \mathrm{the} \mathrm{edges} \mathrm{is} \mathrm{the} \mathrm{opposite} \mathrm{for} \mathrm{the} \mathrm{common} \mathrm{nodes} \mathrm{of} E_{ij}$ and e then

$17: \mathrm{W}(v_{E_{ij}},v_e)=-1$

18:    end if

19:  end for

20: end for

$21: \mathbf{Output}\mathbf{:} \mathrm{weighted} \mathrm{graph} \tilde{G}$s

The above process can be summarized as follows: given a simplex complex graph $\overline{\mathrm{G}}$, we convert each connected edge in $\overline{\mathrm{G}}$ into a node in weighted graph $\tilde{\mathrm{G}}$. For any two nodes ${\mathrm{v}}_{{\mathrm{E}}_{\mathrm{i}\mathrm{j}}}$ and ${\mathrm{v}}_{\mathrm{e}}$ in graph $\tilde{\mathrm{G}}$, the weight values of the edges between them are as follows:

$$\mathrm{W}(v_{E_{ij}},v_e)=\begin{array}{ll}{deg}_u\left({\sigma }_i\right)+k+1& \mathrm{if} v_{E_{ij}}=v_e,\\ \hspace{1em}\hspace{1em}\hspace{1em}1& \mathrm{if} v_{E_{ij}}\ne v_e,{\sigma }_{v_{E_{ij}}}⋔{\sigma }_{v_e},\mathrm{and} {\sigma }_{v_{E_{ij}}}\cup {\sigma }_{v_e} \mathrm{with} \mathrm{similar} \mathrm{orientation}\\ \hspace{1em}\hspace{1em} -1& \mathrm{if} v_{E_{ij}}\ne v_e,{\sigma }_{v_{E_{ij}}}⋔{\sigma }_{v_e},\mathrm{and} {\sigma }_{v_{E_{ij}}}\cup {\sigma }_{v_e}\mathrm{with} \mathrm{dissimilar} \mathrm{orientation}\\ \hspace{1em}\hspace{1em}\hspace{1em}0& \mathrm{if} v_{E_{ij}}\ne v_e\mathrm{and}{\sigma }_{v_{E_{ij}}}⋔{\sigma }_{v_e} \mathrm{or} {\sigma }_{v_{E_{ij}}}\mathsf{\psi }{\sigma }_{v_e} \end{array}$$

Lemma 4 

[26]. The trace of the k power of a weighted adjacency matrix $W$ is equal to the sum of the k powers of its eigenvalues.

$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\left(W^k\right)=\sum _{i=1}^{\left|V\right|}{\lambda }_i^k$$

(27)

Lemma 5 

[26]. The number of closed walking paths of length $k$ in weighted graph $\mathrm{H}$ is equal to the $\mathrm{k}$ power of the weighted adjacency matrix, i.e., the trace of $W^k$

$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\left(W^k\right)=\sum _{i=1}^{\left|V\right|}{\lambda }_i^k=\sum _{X_i^{\left(k\right)}\in R^{\left(k\right)}}W\left(X_i^{\left(k\right)}\right),$$

(28)

where $X_i^{\left(k\right)}=\left(v_i,v_{i+1},\dots ,v_{i+k}\right)$ represents the node sequence obtained by node $v_i$ after $k$ steps in the random walk process. $R^{\left(k\right)}$ represents all the sequences of nodes in graph $H$ whose path length is k. $W\left(X_i^{\left(k\right)}\right)$ is defined as follows:

$$W\left(X_i^{\left(k\right)}\right)=\left\{\begin{array}{cc}0& X_i^{\left(k\right)}\mathrm{is} \mathrm{not} \mathrm{a} \mathrm{closed} \mathrm{path}\\ W_{i,i+1}{W}_{i+1,i+2}\dots W_{i+k-1,i}& X_i^{\left(k\right)}\mathrm{is} \mathrm{a} \mathrm{closed} \mathrm{path}\end{array}\right.,$$

(29)

By combining Formulas $\left(27\right)$ and $\left(28\right)$ , we can deduce:

$$m_k\left(W\right)={\displaystyle \frac{1}{\left|V\right|}}\sum _{X_i^{\left(k\right)}\in R^{\left(k\right)}}W\left(X_i^{\left(k\right)}\right),$$

(30)

Let

$$f\left(X_i^{\left(k\right)}\right)={\displaystyle \frac{1}{P\left(X_i^{\left(k\right)}\right)D}}=\left\{\begin{array}{ll}{\displaystyle \prod _{j=1}^{k-1}}d\left(r_{i+j}\right)& k>1\\ \hspace{1em}\hspace{1em}1& k=1\end{array}\right.,$$

(31)

From the full expectation formula, we can deduce:

$$\begin{array}{ll}\mathrm{E}\left[W\left(R_i^{\left(k\right)}\right)\ast f\left(R_i^{\left(k\right)}\right)\right]& ={\displaystyle \sum _{{\mathrm{X}}_{\mathrm{i}}^{\left(\mathrm{k}\right)}\in R^{\left(k\right)}}}\mathrm{E}\left[W\left(R_i^{\left(k\right)}\right)\ast f\left(R_i^{\left(k\right)}\right)|R_i^{\left(k\right)}=X_i^{\left(k\right)}\right]P\left(R_i^{\left(k\right)}=X_i^{\left(k\right)}\right)\\ & ={\displaystyle \sum _{X_i^{\left(k\right)}\in R^{\left(k\right)}}}W\left(X_i^{\left(k\right)}\right)\ast f\left(X_i^{\left(k\right)}\right)\ast P\left(X_i^{\left(k\right)}\right)\\ & ={\displaystyle \sum _{X_i^{\left(k\right)}\in R^{\left(k\right)}}}W\left(X_i^{\left(k\right)}\right)\ast {\displaystyle \frac{1}{P\left(X_i^{\left(k\right)}\right)D}}\ast P\left(X_i^{\left(k\right)}\right)\\ & ={\displaystyle \frac{1}{D}}{\displaystyle \sum _{X_i^{\left(k\right)}\in R^{\left(k\right)}}}W\left(X_i^{\left(k\right)}\right),\end{array}$$

(32)

According to Formula $\left(32\right),we can reduce$

$$\begin{array}{ll}{\displaystyle \sum _{X_i^{\left(k\right)}\in R^{\left(k\right)}}}W\left(X_i^{\left(k\right)}\right)& =\mathrm{D}\ast \mathrm{E}\left[w\left(R_i^{\left(k\right)}\right)\ast f\left(R_i^{\left(k\right)}\right)\right]\\ & =\left\{\begin{array}{ll}D\ast E\left[W\left(R_i^{\left(k\right)}\right)\ast {\displaystyle \prod _{j=1}^{k-1}}d\left(v_{i+j}\right)\right],& k>1\\ \hspace{1em}\hspace{1em}D\ast E\left[W\left(R_i^{\left(k\right)}\right)\right],& k=1\end{array}\right.,\end{array}$$

(33)

The expression formula of $W\left(R_i^{\left(k\right)}\right)$ is Formula $\left(33\right)$ , which can be deduced finally:

$$\begin{array}{ll}{m}_k\left(W\right)& ={\displaystyle \frac{1}{\left|V\right|}}{\displaystyle \sum _{X_i^{\left(k\right)}\in R^{\left(k\right)}}}W\left(X_i^{\left(k\right)}\right)\\ & =\left\{\begin{array}{ll}D\ast E\left[W\left(R_i^{\left(k\right)}\right)\ast {\displaystyle \prod _{j=1}^{k-1}}d\left(v_{i+j}\right)\right],& k>1\\ \hspace{1em}\hspace{1em}D\ast E\left[W\left(R_i^{\left(k\right)}\right)\right],& k=1\end{array}\right.,\end{array}$$

(34)

This section introduces a method based on graph transformation to transform the problem of solving spectral moments of a second-order combinatorial Laplacian matrix into solving spectral moments of a weighted adjacency matrix. Then, based on the previously mentioned random walk algorithm, the spectral moment of the second-order combinatorial Laplacian matrix is successfully derived from Formula $\left(34\right)$.

3.3. Error Analysis

According to Formula $\left(33\right)$, we can obtain the variance of estimated values $\sum _{X_i^{\left(k\right)}\in R^{\left(k\right)}}W\left(X_i^{\left(k\right)}\right)$ as follows:

$$\begin{array}{ll}Var\left[DW\left(R_i^{\left(k\right)}\right)\ast f\left(R_i^{\left(k\right)}\right)\right]& =E\left[{(DW\left(R_i^{\left(k\right)}\right)\ast f\left(R_i^{\left(k\right)}\right))}^2\right]-{E[DW\left(R_i^{\left(k\right)}\right)\ast f\left(R_i^{\left(k\right)}\right)]}^2\\ & ={\displaystyle \sum _{X_i^{\left(k\right)}\in R^{\left(k\right)}}}W\left(X_i^{\left(k\right)}\right)\ast ({\displaystyle \frac{1}{P\left(X_i^{\left(k\right)}\right)}}-1)\end{array}$$

(35)

The above process demonstrates that, as k increases, the estimated values of spectral moments also increase. Let $\tilde{\mathrm{M}}$ represent the average value of spectral moment estimation. According to the Central limit Theorem, the 95% confidence interval for $\tilde{\mathrm{M}}$ is ($\tilde{\mathrm{M}}-1.96{\displaystyle \frac{\mathsf{\sigma }}{\mathrm{n}}}$, $\tilde{\mathrm{M}}+1.96{\displaystyle \frac{\mathsf{\sigma }}{\mathrm{n}}}$), where ${\mathsf{\sigma }}^2$ is the variance of estimated values, and n is the number of samples used. Therefore, according to the above equation, we must reduce ${\displaystyle \frac{\mathsf{\sigma }}{\mathrm{n}}}$ by increasing the number of sampling nodes n to improve the accuracy of the estimator.

3.4. Estimate the Upper and Lower Bounds Based on Spectral Moments

Definition 1. 

Given a series of spectral moments $\overline{M_{2r+1}}=(M_0,M_1,\dots ,M_{2r+1})$, let the Hankel matrix $H_{2r}\left(\overline{M_{2r+1}}\right)$ and $H_{2r+1}\left(\overline{M_{2r+1}}\right)\in {\mathbb{R}}^{(r+1)\times (r+1)}$ be defined as follows:

$${\left[{\mathrm{H}}_{2\mathrm{r}}\right]}_{\mathrm{i}\mathrm{j}}={\mathrm{M}}_{\mathrm{i}+\mathrm{j}-2}$$

(36)

$${\left[{\mathrm{H}}_{2\mathrm{r}+1}\right]}_{\mathrm{i}\mathrm{j}}={\mathrm{M}}_{\mathrm{i}+\mathrm{j}-1}$$

(37)

Lasserre [27] proposed a method that can be used to solve the boundary of the spectral radius when given only a set of spectral moments. Although Lasserre’s SDP bounds are utilized for biased empirical moments without error correction, Lasserre’s SDP bound can be used to estimate the boundary of the maximum eigenvalue. Compared to Chu using the Lasserre’s SDP bounds to estimate the maximum eigenvalue boundary, the maximum eigenvalue boundary estimated by the Lasserre’s SDP bounds in this paper is closer to the true value and requires fewer sampling points. Lasserre proved Theorem 3.

Theorem 3 

[27]. Given that $\mathsf{\rho }$ is a probability density function with respect to $\mathrm{x}$ , $\mathrm{M}={\left({\mathrm{M}}_{\mathrm{k}}\right)}_{\mathrm{k}=1}^{2\mathrm{s}+1}$ is the spectral moment associated with $\mathsf{\rho }$ , which can be deduced as follows:

$$\begin{array}{c}a\le {\alpha }_s\left(M\right)=\underset{x}{\mathit{max}}\left\{x:H_s\left(x,M\right)\succcurlyeq 0\right\}, \end{array}$$

(38)

$$\begin{array}{c}b\ge {\beta }_s\left(M\right)=\underset{x}{\mathit{min}}\left\{x:-H_s\left(x,M\right)\succcurlyeq 0\right\},\end{array}$$

(39)

where$[a,b]$ is the minimum gap of the support set of probability density function $\rho$, and the correlation between $M$ and $\rho$ is shown in Formula $\left(40\right)$.

$$\begin{array}{c}b\ge {\beta }_s\left(M\right)=\underset{x}{\mathit{min}}\left\{x:-H_s\left(x,M\right)\succcurlyeq 0\right\},\end{array}$$

(40)

Definition 2. 

Given that $L_H$ is a weighted undirected Laplacian matrix whose Laplacian eigenvalue is ${\left\{{\mu }_i\right\}}_{\mathrm{i}=1}^{\mathrm{n}}$ , define an important probability density function as follows:

$${\rho }_G\left(\mu \right)={\displaystyle \frac{1}{n}}\sum _{i=1}^n\delta \left(x-{\mu }_i\right),$$

(41)

where $\mathsf{\delta }$ is the Dirac function.

From the definition of the support set, it can be seen that the support set of ${\rho }_G\left(\mu \right)$ is ${\left\{{\mu }_i\right\}}_{i=1}^n$, which can be obtained from Theorem 3, where the expression of $M$ closely related to ${\rho }_G\left(\mu \right)$ is:

$$\begin{array}{c}M=E_{{\mu }_{L_H}}\left(X^k\right)={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{x_i}^k=m_k\left(L_H\right), \end{array}$$

(42)

As can be seen from Formula $\left(42\right)$, $\mathrm{M}$ at this time is exactly equal to the spectral moment $m_k\left(L_H\right)$ of the Laplace matrix $L_H$.

According to Theorem 3, $[a,b]$ is the minimum gap of the support set of the probability density function ${\rho }_G\left(\mu \right)$, and because the support set of ${\rho }_G\left(\mu \right)$ is $[{\mu }_n,{\mu }_1]$, it can be deduced that:

$$\begin{array}{c}{\mu }_1\ge b, \end{array}$$

(43)

where b of Formula $\left(44\right)$ can be obtained from Formula (43) [27].

$$\begin{array}{c}b\ge {\beta }_s\left(M\right)=\underset{x}{\mathit{min}}\left\{x:-H_s\left(x,M\right)\succcurlyeq 0\right\}, \end{array}$$

(44)

Formula $\left(42\right)$ can be expanded as follows:

$$\begin{gathered} {\beta }_r^{\ast }\left(M\right)={min}_x x \\ \mathrm{s}.\mathrm{t}. H_{2r}\ge 0 \\ x{H}_{2r}-H_{2r+1}\ge 0 \end{gathered}$$

(45)

where $H_{2r}$, $H_{2r+1}$ are referred to in Definition 1.

The lower bound of the greatest eigenvalue of the Laplace matrix can be obtained from Formula $\left(45\right)$. The upper bound of the greatest eigenvalue of the Laplacian matrix can be obtained by the dual transformation of Formulas $\left(44\right)$ and $\left(45\right)$.

The upper bound of the greatest eigenvalue of the Laplace matrix is shown in Formulas $\left(46\right)$ and $\left(47\right)$,

$${\mu }_1\le {\delta }_r^{\ast }\left(m_{2r+1},n\right),$$

(46)

where

$$\begin{gathered} {\mathsf{\delta }}_{\mathrm{r}}^{\ast }\left({\mathrm{m}}_{2\mathrm{r}+1},\mathrm{n}\right)={\mathrm{m}\mathrm{a}\mathrm{x}}_{\mathrm{y}} \\ \mathrm{y}\mathrm{s}.\mathrm{t}.{\mathrm{T}}_{2\mathrm{r}}\ge 0, \\ \mathrm{y}{\mathrm{T}}_{2\mathrm{r}}-{\mathrm{T}}_{2\mathrm{r}+1}\ge 0, \\ {\mathrm{T}}_{2\mathrm{r}+1}\ge 0. \end{gathered}$$

(47)

Among them:

$$\begin{array}{c}{\left[{\mathrm{T}}_{2\mathrm{r}}\right]}_{\mathrm{i}\mathrm{j}}={\displaystyle \frac{\mathrm{n}}{\mathrm{n}-1}}{\mathrm{m}}_{\mathrm{i}+\mathrm{j}-2}-{\displaystyle \frac{1}{\mathrm{n}-1}}{\mathrm{y}}^{\mathrm{i}+\mathrm{j}-2} \end{array}$$

(48)

$${\left[{\mathrm{T}}_{2\mathrm{r}+1}\right]}_{\mathrm{i}\mathrm{j}}={\displaystyle \frac{\mathrm{n}}{\mathrm{n}-1}}{\mathrm{m}}_{\mathrm{i}+\mathrm{j}-1}-{\displaystyle \frac{1}{\mathrm{n}-1}}{\mathrm{y}}^{\mathrm{i}+\mathrm{j}-1}$$

(49)

In this section, we employ algebraic topology methods to transform the boundary of the largest eigenvalue into an algebraic problem of optimization. Ultimately, we can obtain the boundary of the largest eigenvalue using Laplacian spectral moments through the aforementioned formula.

4. Results

To more accurately compare with the sampling algorithm suggested by Wu [22], the proposed algorithm is verified by simulation in a synthetic network and a real network.

4.1. Experimental Setup

For the synthetic network, we selected an ER network [28] and a BA network [29]. The ER network is a classic random network model, characterized by completely random connections of edges, and is suitable for simulating scenarios where the probability of connections between nodes is equal (such as early communication networks and simple interactive systems). The BA network is a classic model of a scale-free network, which is based on a “growth mechanism” (nodes gradually join) and “priority connection” (new nodes tend to connect to high-degree nodes), and is suitable for simulating the real network (such as a social network, Internet topology, or biological network). The ER network and BA network consist of 10,000 nodes, which form edges according to corresponding rules with a probability of 0.1. Walk step setting: Take 50 values at intervals of 200 to 2000 nodes. Number of walks per value: 10. The average value of bounds of the spectral radius calculated by 10 experiments is taken as bounds of the spectral radius of the final result.

For the real network, we selected Euro-Email network [30], which is made up of 183,831 edges connecting 36,692 nodes. The other is the Com-Youtube network [30]. The Com-Youtube network is an undirected network made up of 2,987,624 edges connecting 1,134,890 nodes. A number of nodes are successively selected from the 5000 nodes for walking and sampling. Walk step setting: Take 50 values at intervals of 100 to 600 nodes. Number of walks per value: 10. The average value of bounds of the spectral radius calculated by 10 experiments is taken as the bounds of the spectral radius of the final result.

4.2. Synthetic Network

For the synthetic network, we selected the BA network and ER network. The experimental data were set as Section 4.1. Meanwhile, to illustrate the algorithm’s supremacy, the algorithm proposed in this paper is compared with the sampling algorithm proposed by Wu [22]. The sampling algorithm proposed by Wu is selected, with the same BA network and the same ER network. The calculation results of the two algorithms for bounds of the greatest eigenvalue of the Laplace matrix are shown in Figure 2:

Figure 2 and Figure 3 demonstrate that the estimation performance of the bounds of the spectral radius based on random walk is superior to that based on the sampling approach. The estimation of the bounds of the spectral radius based on random walk is nearer to the spectral radius’s actual value. The upper bound convergence of the spectral-radius-based random walk method is better than that of the sampling method, and the upper bound of the largest eigenvalue obtained by the random walk algorithm can converge to a stationary state faster. Moreover, the bounds calculated by the method based on random walks are closer to the true values than those calculated by the method based on sampling.

4.3. Real Network

Two real networks are used for the experimental data. One real network is the data of Euro-Email network [30], which is made up of 183,831 edges connecting 36,692 nodes. The other is the Com-Youtube network [30]. The Com-Youtube network is an undirected network which is made up of 2,987,624 edges connecting 1,134,890 nodes.

To better verify the accuracy of the algorithm presented in this paper, we need to clearly calculate the largest eigenvalues of the Laplacian matrix of the network graph. Therefore, we extracted 5000 nodes from the real network as our research network. These 5000 nodes were randomly selected using BFS and are representative. The experimental data were set as Section 4.1. The results are as follows:

It can be seen from Figure 3 that, in real networks, the algorithm suggested by this paper is more accurate in estimating bounds of the greatest eigenvalues of the Laplacian matrix. Compared with the algorithm proposed by Wu, the method proposed in this paper has a better performance and higher estimation accuracy. The upper and lower bounds of the spectral radius are closer to the true value, and the upper bounds of the spectral radius converge to the stationary state faster.

4.4. Synthetic Network for the Higher-Order Network

We conducted simulation verification of the algorithm in two synthetic networks, namely the BA network and the ER network. We used 10,000 nodes to form a scale-free BA network and conducted 500 random samplings. Each sampling selected three nodes, which form a second-order simplex complex structure. The network is formed according to the above steps. Then, we start to walk in the network, the walking steps are set to be between 200 and 2000 at equal intervals, and we perform 10 samplings for each walk step. The average value of the 10 samplings was taken as the final estimate, and from the bounds of the combinatorial Laplacian matrix, the greatest eigenvalue could be obtained. The relationship between walking steps and the estimated value of the combinatorial Laplacian matrix is shown in Figure 4:

As illustrated in Figure 4, when the steps of random walks exceed a certain threshold, the method based on random walks can accurately estimate the spectral radius of the combinatorial Laplacian matrix, and the greatest eigenvalue of the combinatorial Laplacian matrix falls completely within the estimated bounds. Similarly, we can also observe that as the steps of random walks increase, the upper bound of the greatest eigenvalue of the combinatorial Laplacian matrix becomes increasingly relaxed until it stabilizes, while the lower bound of the spectral radius of the combinatorial Laplacian matrix slowly contracts towards the true value of the greatest eigenvalue.

To quantitatively study the accuracy of estimating upper and lower bounds, this paper further investigates the relationship between different walking steps and estimation accuracy. The estimation accuracy is defined as the probability of correct estimation among 10 samplings. Figure 5 displays the outcomes. In order to better observe the impact of wandering nodes on the estimated values, we use a distribution histogram to observe their statistical characteristics. The result is shown in Figure 6:

As can be seen from Figure 5, as the walking steps increase, the estimation accuracy of the spectral radius of the Laplacian matrix becomes higher. To further analyze the impact of the second-order simplex complex structure on estimating the greatest eigenvalue of the combinatorial Laplacian matrix, we investigate the relationship between the number of simplices traversed and bounds of the greatest eigenvalue of the combinatorial Laplacian matrix. According to the above results, when the number of nodes traversed during the random walk is 1500, the accuracy of estimating the spectral radius of the combinatorial Laplacian matrix is 1. Therefore, in this experiment, we conducted random walks on 1500 nodes. This experiment was conducted 50 times, and we observed the number of second-order simplices traversed each time. If the number of simplices traversed was the same, the average of the estimated values was taken. The results are shown in Figure 6:

4.5. Real Network for the Higher-Order Network

In this section, the estimation of the spectral radius of the combinatorial Laplacian matrix for the higher-order network is verified in the real network. In the experiment, this paper uses the same Euro-Email network and Com-Youtube network as before.

To better verify the accuracy of the algorithm presented in this paper, we need to clearly calculate the largest eigenvalues of the Laplacian matrix of the network graph. Therefore, we extracted 5000 nodes from the real network as our research network. These 5000 nodes were randomly selected using BFS and are representative. The subject of the investigation will be a subgraph with 5000 nodes. A total of 250 random selections were made from 5000 nodes, and 3 nodes were selected from each to form a second-order simplex structure. Then, 50 values were selected as the walking steps. We selected 50 values at medium intervals from 100 to 600 as walking steps. The average value of 10 times can be taken to obtain bounds of the greatest eigenvalue of the second-order combinatorial Laplacian matrix. Figure 7 shows the relationship between walking steps and the estimated bounds of the spectral radius of the second-order combinatorial Laplacian matrix in the Euro-Email network and Com-Youtube network.

With the increase in walking steps, the random walk proposed in this paper can correctly estimate the bounds of the spectral radius of the second-order combinatorial Laplacian matrix, and the spectral radius of the second-order combinatorial Laplacian matrix completely falls within the estimated upper and lower boundaries. Moreover, with the increase in walking steps, the upper bound of the spectral radius of the second-order combinatorial Laplacian matrix becomes more and more relaxed until it becomes stable, and the lower bound of the second-order combinatorial Laplacian matrix oscillates slowly.

Similarly, to quantitatively study the accuracy of estimating upper and lower bounds, this paper further investigates the relationship between different walking steps and estimation accuracy for real networks. The results are as follows in Figure 8.

We can see, as the walking steps increase, the estimation accuracy of the spectral radius of the combinatorial Laplacian matrix becomes higher. To explore the impact of the second-order simplex on the combinatorial Laplacian spectral moments, we conducted the same experiments in real networks as we did in synthetic networks. The experimental results are as follows:

According to Figure 9, we can observe that, as the number of second-order simplices traversed increases, the estimations of the upper and lower bounds of the spectral radius of the combined Laplacian matrix become more accurate. From this, we can analyze that when traversing a second-order simplex, the node contains more information, which is more helpful for obtaining the upper and lower bounds of the spectral radius of the Laplacian matrix.

5. Conclusions and Future Directions

The spectral characteristics of large networks are examined in this research. Large-scale networks have many nodes and many connected edges, so it is difficult to obtain the complete structure. To analyze the Laplacian matrices of networks, it is required to extract the valid information from large-scale networks and estimate features of the networks. Firstly, considering the relationship between the adjacency matrix and closed paths of the network, we obtain the closed path using random walk. By utilizing the relationship between the Laplacian matrix and adjacency matrix, we ultimately succeeded in establishing the connection between closed paths and the Laplacian matrix, and the boundary estimate of the greatest eigenvalue of the Laplacian matrix is obtained by solving the spectral moment with algebraic knowledge. In addition to studying the spectral properties of ordinary large-scale networks, this paper also explores how to solve the boundary associated with the greatest eigenvalues of the second-order combinatorial Laplacian matrices and proposes an algorithm based on graph transformation and random walk, which can estimate bounds of the spectral radius of combinatorial Laplacian matrices with simplicial complex.

Although this paper proposes estimates for bounds of the greatest eigenvalue of the Laplacian matrix for big networks, there are still some unresolved issues that may serve as directions for future research. Firstly, it is difficult to solve the combinatorial Laplacian matrix with simplex complex. The graph transformation for solving the eigenvalues of the combinatorial Laplacian matrix with high-order simplex complex is very complex. How to perform graph transformation on high-order complexes and solve the eigenvalue boundary is a problem worthy of study. Secondly, in the exploration of high-order networks for big networks, subgraphs obtained through different sampling methods exhibit different topological characteristics from the large graph. It is highly meaningful to investigate the specific topological characteristics of high-order networks, which requires appropriate subgraph sampling to extract the specific subgraphs to represent the large graph in terms of specific topological properties. Thirdly, it is difficult but highly meaningful to explore a specific sampling method that can directly solve the greatest eigenvalue of the Laplacian matrix in a big network, rather than solving for its boundary.