Identifying risks in temporal supernetworks: an IO-SuperPageRank algorithm

Abstract

Complex systems pose risks characterized by factors such as uncertainty, nonlinearity, and diversity, making traditional risk measurement methods based on a probabilistic framework inadequate. Supernetworks can effectively model complex systems, and temporal supernetworks can capture the dynamic evolution of these systems. From the perspective of network stability, supernetworks can aid in risk identification for complex systems. In this paper, an IO-SuperPageRank algorithm is proposed based on the supernetwork topological structure. This algorithm reveals network instability by calculating changes in node importance, thereby helping to identify risks in complex systems. To validate the effectiveness of this algorithm, a four-layer supernetwork composed of scale-free networks is constructed. Simulated experiments are conducted to assess the impact of changes in intralayer edge numbers, intralayer node numbers, and interlayer superedge numbers on the risk indicator IO value. Linear regression and multiple tests were used to validate these relationships. The experiments show that changes in the three network topological indicators all bring about risks, with changes in intralayer node numbers having the most significant correlation with the risk indicator IO value. Compared to traditional measures of network node centrality and connectivity, this algorithm can more accurately predict the impact of node updates on network stability. Additionally, this paper collected trade data for crude oil, chemical light oil, man-made filaments and man-made staple fibers from the UN Comtrade Database. We constructed a man-made filaments and fibers supply chain temporal supernetwork, utilizing the algorithm to identify supply chain risks from December 2020 to October 2023. The study revealed that the algorithm effectively identified risks brought about by changes in international situations such as the Russia-Ukraine war, Israel–Hamas conflict, and the COVID-19 pandemic. This demonstrated the algorithm’s effectiveness in empirical analysis. In the future, we plan to further expand its application based on different scenarios, assess risks by analyzing changes in specific system elements, and implement effective risk intervention measures.

Introduction

The concept of risk has a long history. Risk refers to the likelihood of a particular loss occurring in a specific environment during a specific period. The definition of risk is not unique. The international standard definition of risk in different applications is the “effect of uncertainty on objectives.” A more general definition emphasizes that risk is a form of uncertainty, indicating that the outcome of risk can result in either loss, gain, or neither loss nor gain. Because uncertainty is a key concept in the definition and assessment of risk, it has been widely discussed by scholars from the early stages of risk assessment in the 1970s to the present (Aven, 2016; von Scheve and Lange, 2023). Probability analysis has consistently been the primary method used to address the uncertainty involved in risk analysis.

With the emergence of small-world networks (Watts and Strogatz, 1998) and scale-free networks (Barabási and Albert, 1999) at the end of the 20th century, research in the field of complex networks began to rapidly advance. Traditional methods for studying complex networks involve representing the components or basic units of each system as network nodes, representing interactions between units as edge connections, and using real numbers to denote the weights of these connections. However, this modeling approach places all network links of each unit on an equivalent basis, which may not fully capture real-world problems and could even lead to incorrect descriptions. These limitations led to the development of multilayer networks. Multilayer networks allow for the representation of different network layers within a system, accurately describing the interactions between different units in the network, which aligns more closely with real-world scenarios. For instance, a multilayer trade network can depict cross-border trade relationships between products. This nested structure can be utilized to predict a country’s economic growth and the collective evolution of the world economy (Ren et al., 2020). This paradigm shift in network modeling has led to various representations and nomenclature, one of which is supernetworks. The concept of supernetworks was first introduced by Sheffi in 1984 (Sheffi, 1984). Supernetworks gained widespread attention due to the in-depth research and applications by the Nagurney team. Nagurney defined supernetworks as networks beyond existing networks that are characterized by multiple layers, hierarchies, dimensions, and attributes (Nagurney and Dong, 2002; Nagurney and Wakolbinger, 2005). Some scholars have used supernetwork technology to study the cascading behavior of internet financial risk (Xu et al., 2020), credit risk prediction (Óskarsdóttir and Bravo, 2021), and sudden event risk prediction (Ma et al., 2022). Most of these studies are based on static multilayer networks and involve identifying and predicting risks by calculating network structural topological indicators, applying equilibrium models, and proposing superedge link prediction algorithms.

In the real world, most systems undergo dynamic changes over time and space. Hence, the development of temporal multilayer networks has become essential. In this paper, we construct a temporal supernetwork capable of describing the dynamic evolution of systems from multiple dimensions. Additionally, we introduce the IO-SuperPageRank algorithm, which, from a network topology perspective, can identify network instability and consequently recognize risks in complex systems. Next, we utilize global trade data for crude oil, chemical light oil, man-made filaments and man-made staple fibers from the UN Comtrade Database for the period of December 2020 to October 2023. We construct a man-made filaments and staple fibers supply chain temporal supernetwork and employed the algorithm to calculate changes in trade risk. By integrating the impact of international situations and policy changes of major nations on global trade, we validate the effectiveness of the algorithm in identifying global supply chain risks.

The remainder of the paper is organized as follows. The “Related work” section summarizes methods for risk identification and assessment, encompassing common approaches and models for supply chain risk management, as well as the theoretical development and applications of supernetworks. The “Temporal supernetwork and IO-SuperPageRank algorithm” section explains the modeling approach used for the temporal supernetwork in this paper and introduces the IO-SuperPageRank algorithm, which is proposed for identifying network risks. In the “Risk Assessment” section, the presence of risk is assessed based on the risk matrix obtained through the IO-SuperPageRank algorithm and stability testing. Subsequently, in the “Experimental study” section, we validate how IO-SuperPageRank values change with variations in network topology and explore interlayer impact relationships. Following that, the “Case study” section illustrates the construction of a man-made filaments and staple fibers supply chain temporal supernetwork. The algorithm is employed to identify global trade risks in this specific case. Finally, the study concludes with a “Conclusion” section that provides a concise overview of the key findings and contributions.

Related work

Methods of risk identification and assessment

Twenty-four hundred years ago, Athenians demonstrated the ability to conduct risk identification and assessment before making decisions (Bernstein, 1996). However, the development of risk management and risk assessment in the academic field has only occurred over the last three to four decades. The field of risk primarily focuses on two aspects: first, risks in various domains worldwide are addressed worldwide, and second, risk, encompassing theoretical frameworks, methods, principles, and models of risk assessment, is evaluated (Aven, 2016). Risk occurs in various decision-making contexts, with wide-ranging applications, including those in the financial sector (Nyman et al., 2021; Semieniuk et al., 2021; Zhao and Xu, 2023), environmental and natural disaster domains (Rathi et al., 2021; Terzi et al., 2019; Ward et al., 2020), health care (Anderson et al., 2000; Bohanec et al., 2000; Chakraborty and Ghosh, 2020), military (Crispim et al., 2020; Lincoln, 1985; Ryczyński and Tubis, 2021), cybersecurity (Cherdantseva et al. 2016; Fielder et al., 2018; Wu et al., 2015), global supply chain management (Fahimnia et al., 2015; Kraude et al., 2022; Tran et al., 2018; Xie et al., 2023), among others. Risks often do not occur in isolation; the emergence of risks in a specific domain typically triggers cascading effects, leading to systemic risks. For instance, in the realm of public opinion risk, the proliferation of rumors, social bots, and false information on social media brings about a series of systemic risks. The rapid dissemination of rumors and false information, especially during public health crises such as the COVID-19 pandemic, can provoke panic and jeopardize efforts in preparedness and control (Ball and Maxmen, 2020; Zhang et al., 2022). The widespread dissemination of social bots makes public opinion susceptible to manipulation, posing a potential threat to the ecosystem of social media discourse (Cai et al., 2023). This situation may result in a crisis of trust in information among the public, societal unrest, and even economic repercussions. Furthermore, the conflict between Russia and Ukraine significantly impacts global food security. This may increase the risk of supply chain disruptions for low-income countries heavily dependent on food imports (Liu et al., 2023). Hence, establishing effective mechanisms for risk identification and containment is crucial.

Risk identification is the process of discovering, confirming, and describing risks and typically involves the recognition of risk sources and events and their causes and potential consequences. Risk identification serves as the initial and often the most critical phase of risk management (Iqbal et al., 2015; Siraj and Fayek, 2019; Zayed et al., 2008). Traditional risk identification methods and techniques are quite diverse, encompassing scenario analysis, fault tree analysis, decision tree analysis, the analytic hierarchy process, Monte Carlo simulation, Bayesian analysis, SWOT analysis, risk matrices, and evidence-based methods such as checklist analysis and historical data analysis. Systematic team approaches, such as flowchart analysis, and inductive reasoning methods, such as hazard and operability studies (HAZOP), are also utilized. Furthermore, various supportive techniques such as brainstorming and the Delphi method can be employed to enhance the accuracy and comprehensiveness of risk identification (Carpenter, 2007; Dennis and Johnstone, 2016; Hogganvik and Stølen, 2006; Kıral et al., 2014; Markmann et al., 2013). Typically, a combination of these methods is used, with the specific techniques varying in different risk domains. For instance, in the construction industry, methods such as brainstorming and case analysis are frequently employed, and the primary output of the identification process is presented in a risk register (Lyons and Skitmore, 2004). When studying social network risks among users and risk spillover in financial institutions, network structure metrics are commonly utilized for risk identification and assessment (Wen et al., 2023). In the field of global trade and supply chain, researchers often employ methods such as mean-variance analysis, bayesian network modeling, analytic hierarchy process (AHP), and system dynamics for risk analysis and management (Choi et al. 2019; Ganguly and Kumar, 2019; Mehrjoo and Pasek, 2016; Ojha et al., 2018).

In the field of risk decision-making, researchers have predominantly focused on the identification and management of negative risks, often neglecting opportunities (Hillson, 2002). Moreover, risk decision-making increasingly involves events and incidents characterized by significant uncertainty and unpredictability, rendering purely probabilistic analysis methods, such as Bayesian analysis, inadequate. Therefore, a key research focus is on transcending traditional static risk management frameworks to engage in dynamic risk assessment and management, addressing the challenges posed by events with high levels of uncertainty and sudden occurrences.

Theoretical development and application of supernetworks

As the field of network science and its applications have deepened, people have gradually realized the prevalence of intersecting networks in the real world, where networks are interconnected, interdependent, and mutually influential. These relationships are intricate and complex, and real-world characterization using a single network has limitations. Therefore, in recent years, supernetworks have emerged. The concept of supernetworks was initially introduced by Sheffi in 1984 (Sheffi, 1984). Currently, there are two main manifestations of supernetworks. The first manifestation is hypernetworks based on hypergraphs. Berge first introduced the concept of hypergraphs in 1973 (Berge, 1973), followed by the theory of undirected hypergraphs (Berge, 1989). Estrada and Rodriguez-Velasquez argued that networks built based on hypergraph topology are hypernetworks (Estrada and Rodríguez-Velázquez, 2006). The other type is graph-based supernetworks, often referred to as “networks of networks.” A notable scholar in the study of such multilayer supernetworks is Nagurney (Nagurney and Dong, 2002; Nagurney and Wakolbinger, 2005). Supernetworks can be viewed as collections of individual single-layer networks, with each layer having distinct functions or attributes. Complex interdependencies and relationships exist between layers. Examples include infrastructure supernetworks composed of networks such as power networks, transportation networks, and communication networks (Danziger and Barabási, 2022; Derrible, 2017). Complex brain supernetworks are made up of microscale neurons, mesoscale neural clusters, and large-scale brain regions (Bassett and Sporns, 2017; Feldt et al., 2011). Public opinion supernetworks consist of psychological states, key information, social networks, and keywords (Liu et al., 2014; Ma and Liu, 2014; Tian et al., 2015; G. Wang et al., 2015).

Delving deeper into the essential characteristics of supernetworks and constructing a comprehensive set of statistical metrics that accurately reflect the network structure are crucial aspects of supernetwork research. Notably, the measurement of node centrality in supernetworks is an important research area. Node centrality measures how central a node is within a network, and fundamental metrics for this purpose include the degree centrality, betweenness centrality, closeness centrality, and eigenvector centrality, among others. Additionally, many researchers have identified the importance of nodes in supernetworks by enhancing the PageRank algorithm. For example, Ma et al. introduced the SuperedgeRank algorithm to identify opinion leaders in public opinion supernetworks (Ma and Liu, 2014). Han et al. calculated user reputation in content-based social networks using an improved PageRank algorithm (Han et al., 2012). Zhang et al. proposed the Hypernet S-edgeRank algorithm to identify experts in the field of security (Zhang et al., 2022). Furthermore, some researchers have explored algorithms and applications based on supernetwork structures, including superlink prediction, community detection, evolutionary game theory, targeting attack and network resilience (Liu et al., 2014; Ma et al., 2022; G. Wang et al., 2021; Cheng et al., 2016; Pei, 2019; Zhou et al., 2023; Alvarez-Rodriguez et al., 2021; Civilini et al., 2021; Peng et al., 2022; Peng et al., 2022). These endeavors contribute to a deeper understanding of the complexities within supernetworks and their practical applications across various domains.

Compared to single-layer networks, supernetworks provide a closer and more realistic representation of the real world. However, the real world is in a constant state of change, and static supernetworks cannot precisely capture the dynamic evolution of systems. Some scholars have begun to investigate the temporal evolution of supernetworks, recognizing the importance of the temporal dimension (Fischer et al., 2021; Lyu et al., 2022; Neuhäuser et al., 2021). Wang and others have offered a more general definition of temporal multilayer networks (D. Wang et al., 2020). Temporal supernetworks enable a more precise description of the real world in both the temporal and spatial dimensions and allow for an intuitive representation of uncertainty in a system through network structure changes. Therefore, exploring risk source identification and risk transmission path mining through temporal supernetworks is a promising research direction that can provide valuable insights into the dynamics of complex systems.

Temporal supernetworks and the IO-SuperPageRank algorithm

Temporal supernetwork model

Before formally introducing the algorithm for temporal supernetwork risk measurement, we present a general mathematical model for temporal supernetworks in this section. In contrast to single-layer networks, supernetworks describe the multiple spatial topological relationships within complex systems.

Let $G=(\psi ,C)$ represent a supernetwork with N nodes and L layers. Here, $\psi =\{{G}_{\alpha }:\alpha \in \left\{\mathrm{1,2},\ldots ,L\right\}\}$ represents a collection of L single-layer networks, where ${V}_{\alpha }=\{{v}_{1},{v}_{2},\ldots ,{v}_{n}\},\{1\le \alpha \le L,1\le n < N\}$ denotes the node set of layer ${G}_{\alpha }$. Since the supernetwork captures different hierarchies and dimensions within a complex system, the meaning and quantity of nodes in different layers often differ. ${E}_{\alpha }$ represents the within-layer connections among nodes in layer ${G}_{\alpha }$. $C=\{{E}_{\alpha \beta }\subset {V}_{\alpha }\times {V}_{\beta };\alpha ,\beta \in \{\mathrm{1,2},\ldots ,L\}\}$ represents the interlayer node connections within this supernetwork, defined as the set of superedges. An example of the supernetwork is shown in Fig. 1.

Fig. 1: Example of supernetwork G.

A supernetwork with three layers, consisting of 5, 3, and 4 nodes in each layer, respectively, interconnected by superedges between layers.

Given node interaction data from ${t}_{1}$ to ${t}_{n}$ in a real supernetwork, a temporal supernetwork ${G}^{\tau }({t}_{1},{t}_{n})$ can be viewed as an ordered sequence of supernetworks $\{{G}_{1},{G}_{2},\ldots ,{G}_{T}\}$, which can be represented by ${G}_{t}=({\psi }_{t},{C}_{t})$. Specifically:

(1) τ represents the duration of each time window, expressed in time units (days, hours, minutes, etc.).

(2) $T=\frac{{t}_{n}-{t}_{1}}{\tau }=|{G}^{\tau }({t}_{1},{t}_{n})|$ represents the number of supernetworks in this time sequence.

For this temporal supernetwork representation, $\left\{T=1,L=1\right\}$, $\{T=1\}$ and $\left\{L=1\right\}$ correspond to a single-layer network, static supernetwork and regular temporal network, respectively. Therefore, this mathematical model provides a universal framework for describing real complex systems and integrating temporal dependencies of topological structures and spatial multiple relationships. An example of a temporal supernetwork is shown in Fig. 2.

Fig. 2: Example of temporal supernetwork.

A three-layer temporal supernetwork spanning n periods.

IO-PageRank algorithm

In complex systems modeled using a supernetwork, the stability of the network structure can characterize the system risk. Currently, research on the stability of complex networks is primarily based on network connectivity, including metrics derived from aspects such as network centrality, robustness, and community structure. However, in some complex systems, network structure stability is influenced not only by network connectivity but also by the importance of nodes. For example, in a public opinion system, opinion leaders acting as key nodes play a significant role in information dissemination within the network and can trigger public opinion risk. In complex networks, PageRank values can measure the importance of nodes, and the degree of change in PageRank value sequences between nodes in two consecutive temporal supernetworks can reveal the supernetwork’s risk. The degree of change in this sequence can be measured through the inversion order between the two consecutive sequences. Therefore, we design an algorithm based on node PageRank values and inversion order to measure the risk in temporal supernetworks between two consecutive periods.

In this section, we first construct the IO-PageRank algorithm based on the inversion order (IO_S) and node PageRank values in a single-layer network to measure risk in two consecutive single-layer temporal networks. The inversion order (IO_S) refers to the number of inversion pairs in a sequence S. An inversion pair refers to a situation in a sequence $S=\{{S}_{n}\}$ where if a number ${S}_{j}$ is preceded by a number ${S}_{i}$ that is greater than it, then these two numbers form an inversion pair. The inversion pair $I({S}_{i},{S}_{j})$ is defined as follows:

$${{IO}}_{S}=\sum _{\forall i < j}I\left({S}_{i},{S}_{j}\right),I\left(x,y\right)=\left\{\begin{array}{cc}1 & x\, >\, y\\ 0 & x\,\le\, y\end{array}\right.$$

(1)

In Formula 1, the IO_S reflects the degree of disorder from a completely ordered state $({S}_{1} < {S}_{2} < \cdots < {S}_{n})$ to a disordered state in sequence S. A higher IO_S value indicates a greater degree of disorder in the sequence. Figure 3 shows an example of measuring the degree of disorder in a sequence using the inversion order.

Fig. 3: Examples of inversion order.

Measuring the degree of disorder in a sequence using the inversion order.

The Google PageRank (PR) algorithm, also known as the webpage ranking algorithm, is an algorithm used by Google to rank web pages in its search engine results (Page et al., 1999). In a directed graph with n nodes, the PR value of the k-th node is defined as follows:

$${{PR}}_{k}=\alpha \sum _{i\in {U}_{k}}\frac{{{PR}}_{i}}{{D}_{i}}+\frac{1-\alpha }{n}$$

(2)

In Formula 2, U_k represents the set of all nodes connected to node k, and D_i represents the degree of the i-th node. A higher PR value indicates a higher importance of the node.

To measure the degree of change in a temporal network between adjacent time periods using the inverse order PageRank (IO-PageRank) algorithm, consider a temporal network $\left\{G\left(t\right)\right\}$ evolving over time $\left\{{t}_{1},{t}_{2},\cdots ,{t}_{m}\right\}$. In period t_i, the network contains N_i nodes, and in period t_i+1, it contains N_i+1 nodes. The PageRank values of nodes calculated through the PageRank algorithm in the period t_i are represented as ${{PR}}_{1}\left({t}_{i}\right),{{PR}}_{2}\left({t}_{i}\right),\cdots ,{{PR}}_{{N}_{i}}\left({t}_{i}\right)$. In the period t_i+1, the PageRank values (PR) of nodes are represented as ${{PR}}_{1}\left({t}_{i+1}\right),{{PR}}_{2}\left({t}_{i+1}\right),\cdots ,{{PR}}_{{N}_{i+1}}\left({t}_{i+1}\right)$ and can be expressed as vectors as follows:

$${\bf{PR}}\left({t}_{i}\right)=\left(\begin{array}{c}{{PR}}_{1}\left({t}_{i}\right)\\ \begin{array}{c}{{PR}}_{2}\left({t}_{i}\right)\\ \vdots \end{array}\\ {{PR}}_{{N}_{i}}\left({t}_{i}\right)\end{array}\right),{\bf{PR}}\left({t}_{i+1}\right)=\left(\begin{array}{c}{{PR}}_{1}\left({t}_{i+1}\right)\\ \begin{array}{c}{{PR}}_{2}\left({t}_{i+1}\right)\\ \vdots \end{array}\\ {{PR}}_{{N}_{i+1}}\left({t}_{i+1}\right)\end{array}\right)$$

(3)

By combining the vectors of the two periods from Formula 3 into the same matrix, we obtain:

$${\bf{PR}}\left({t}_{i},{t}_{i+1}\right)=\left(\begin{array}{cc}\begin{array}{c}{{PR}}_{1}\left({t}_{i}\right)\\ \begin{array}{c}{{PR}}_{2}\left({t}_{i}\right)\\ \vdots \end{array}\\ {{PR}}_{N}\left({t}_{i}\right)\end{array} & \begin{array}{c}{{PR}}_{1}\left({t}_{i+1}\right)\\ \begin{array}{c}{{PR}}_{2}\left({t}_{i+1}\right)\\ \vdots \end{array}\\ {{PR}}_{N}\left({t}_{i+1}\right)\end{array}\end{array}\right)$$

(4)

In Formula 4, $N=\max \left({N}_{i},{N}_{i+1}\right)$. The matrix ${\bf{PR}}\left({t}_{i},{t}_{i+1}\right)$ represents the PR values of the same node in the network for both time periods t_i and t_i+1. When a node k exists at t_i but does not exist at t_i+1, the k-th row of ${\bf{PR}}\left({t}_{i},{t}_{i+1}\right)$ is $\left(\begin{array}{cc}{{PR}}_{k}\left({t}_{i}\right) & 0\end{array}\right)$. When a node k does not exist at t_i but does exist at t_i+1, the k-th row of ${\bf{PR}}\left({t}_{i},{t}_{i+1}\right)$ is $\left(\begin{array}{cc}0 & {{PR}}_{k}\left({t}_{i+1}\right)\end{array}\right)$. The matrix ${\bf{PR}}\left({t}_{i},{t}_{i+1}\right)$ is rearranged and transformed based on the sorting of its first column elements in ascending order to obtain ${{\bf{PR}}}^{{IO}}\left({t}_{i},{t}_{i+1}\right)$, denoted as:

$${{\bf{PR}}}^{{IO}}\left({t}_{i},{t}_{i+1}\right)=\left(\begin{array}{cc}\begin{array}{c}{\widetilde{{PR}}}_{1}\left({t}_{i}\right)\\ \begin{array}{c}{\widetilde{{PR}}}_{2}\left({t}_{i}\right)\\ \vdots \end{array}\\ {\widetilde{{PR}}}_{N}\left({t}_{i}\right)\end{array} & \begin{array}{c}{\widetilde{{PR}}}_{1}\left({t}_{i+1}\right)\\ \begin{array}{c}{\widetilde{{PR}}}_{2}\left({t}_{i+1}\right)\\ \vdots \end{array}\\ {\widetilde{{PR}}}_{N}\left({t}_{i+1}\right)\end{array}\end{array}\right)$$

(5)

In Formula 5, ${\widetilde{{PR}}}_{1}\left({t}_{i}\right)\le {\widetilde{{PR}}}_{2}\left({t}_{i}\right)\le \cdots \le {\widetilde{{PR}}}_{N}\left({t}_{i}\right)$, i.e., the calculation of the inversion order for the second column of ${{\bf{PR}}}^{{IO}}\left({t}_{i},{t}_{i+1}\right)$, yields ${IO}\left({t}_{i},{t}_{i+1}\right)$. ${IO}\left({t}_{i},{t}_{i+1}\right)$ measures the instability of the network from period t_i to t_i+1, representing, to some extent, risk changes in a complex system. A higher value of ${IO}\left({t}_{i},{t}_{i+1}\right)$ indicates a higher degree of change in the importance order of network nodes from t_i to t_i+1, implying increased risk in the system from t_i to t_i+1. Figure 4 shows the IO-PageRank algorithm.

Fig. 4: IO-PageRank algorithm.

An algorithm for assessing the stability of a temporal single-layer network.

Layer-InfluenceRank algorithm

Due to the presence of multiple layers in the supernetwork, different layers are connected through superedges, and the meaning of nodes in each layer is different. Traditional single-layer network node centrality measurement methods (such as PageRank and degree centrality) cannot effectively measure the influence between nodes in different layers in a multilayer network. Therefore, we propose the Layer-InfluenceRank algorithm. The S-th layer and the T-th layer in the supernetwork G(A) are selected as examples. A represents the adjacency matrix of the supernetwork.

$${\bf{A}}=\left(\begin{array}{cc}\begin{array}{cc}{{\bf{A}}}_{11} & {{\bf{A}}}_{12}\\ {{\bf{A}}}_{21} & {{\bf{A}}}_{22}\end{array} & \begin{array}{cc}\cdots & {{\bf{A}}}_{1n}\\ \cdots & {{\bf{A}}}_{2n}\end{array}\\ \begin{array}{cc}\vdots & \vdots \\ {{\bf{A}}}_{n1} & {{\bf{A}}}_{n2}\end{array} & \begin{array}{cc}\ddots & \vdots \\ \cdots & {{\bf{A}}}_{{nn}}\end{array}\end{array}\right)$$

(6)

In Formula 6, A_ST(S ≠ T) represents the adjacency matrix composed of superedges from the S-th layer to the T-th layer. The S-th layer contains N_S nodes, and the T-th layer contains N_T nodes. The adjacency matrix is as follows:

$${{\bf{A}}}_{{ST}}=\left(\begin{array}{cc}\begin{array}{cc}{a}_{11} & {a}_{12}\\ {a}_{21} & {a}_{22}\end{array} & \begin{array}{cc}\cdots & {a}_{1{N}_{T}}\\ \cdots & {a}_{2{N}_{T}}\end{array}\\ \begin{array}{cc}\vdots & \vdots \\ {a}_{{N}_{S}1} & {a}_{{N}_{S}2}\end{array} & \begin{array}{cc}\ddots & \vdots \\ \cdots & {a}_{{N}_{S}{N}_{T}}\end{array}\end{array}\right)$$

(7)

In Formula 7:

$${a}_{{ij}}=\left\{\begin{array}{ll}\begin{array}{cc}1 & {\rm{Node}}\,i\,{\rm{in}}\,{\rm{layer}}\,S\,{\rm{is}}\,{\rm{connected}}\,{\rm{to}}\,{\rm{node}}\,j\,{\rm{in}}\,{\rm{layer}}\,T\,{\rm{through}}\,{\rm{a}}\,{\rm{su}}{\rm{peredge}}\end{array}\\ \begin{array}{cc}0 & {\rm{Node}}\,i\,{\rm{in}}\,{\rm{layer}}\,S\,{\rm{is}}\,{\rm{not}}\,{\rm{connected}}\,{\rm{to}}\,{\rm{node}}\,j\,{\rm{in}}\,{\rm{layer}}\,T\,{\rm{through}}\,{\rm{a}}\,{\rm{su}}{\rm{peredge}}\end{array}\end{array}\right.$$

The influence of nodes in the S-th layer on nodes in the T-th layer comprises two aspects:

(1) The importance of nodes within the S-th layer itself.

(2) The influence of nodes in the S-th layer on nodes in the T-th layer.

The importance of nodes within the S-th layer itself can be calculated using the PageRank algorithm. By placing the PageRank values of various nodes in the S-th layer into a single row vector, we obtain the ‘node importance vector’ for the S-th layer, represented as follows:

$${\bf{PRS}}=\left({{PRS}}_{1},{{PRS}}_{2},\cdots ,{{PRS}}_{{N}_{S}}\right)$$

(8)

In Formula 8, ${{PRS}}_{i}\left(i=\mathrm{1,2},\cdots {,N}_{S}\right)$ represents the PageRank value of the i-th node in the S-th layer.

The influence of each node in the S-th layer on nodes in the T-th layer is reflected by the number of superedges associated with each node in the S-th layer and the T-th layer. Collecting the number of superedges connecting each node in the Sth layer to the Tth layer into a row vector can be represented as follows:

$${\bf{NSE}}=\left({N}_{{{SE}}_{1}},{N}_{{{SE}}_{2}},,{N}_{{{SE}}_{{N}_{S}}}\right)$$

(9)

In Formula 9, ${N}_{{{SE}}_{i}}\left(i=\mathrm{1,2},\cdots {,N}_{S}\right)$ represents the number of superedges connecting the i-th node in the S-th layer to nodes in the T-th layer. The ‘influence range’ of the i-th node in the S-th layer on the T-th layer network can be defined as the proportion of the i-th node in the S-th layer connected to nodes in the T-th layer relative to all the nodes in the T-th layer, denoted as follows:

$${I}_{{ST}}\left(i\right)=\frac{{N}_{{{SE}}_{i}}}{{N}_{T}}$$

(10)

Combining the ‘influence ranges’ of each point in the S-th layer into a single row vector yields the ‘influence range vector,’ denoted as follows:

$${\bf{I}}\left(S,T\right)=\left({I}_{{ST}}\left(1\right),{I}_{{ST}}\left(2\right),{\cdots ,I}_{{ST}}\left({N}_{S}\right)\right)=\frac{1}{{N}_{T}}{\boldsymbol{\bullet }}{\bf{NSE}}$$

(11)

The influence of the i-th node in the S-th layer on the T-th layer is defined as the product of the i-th node’s ‘importance’ in the S-th layer and its ‘influence range’ on the T-th layer, denoted as follows:

$${I}_{S}\left(i\right)={{PRS}}_{i}\cdot {I}_{{ST}}\left(i\right)$$

(12)

Combining the influence of each node in the S-th layer on the T-th layer into a single row vector yields the ‘influence vector’, denoted as follows:

$${\bf{PI}}\left(S,T\right)=\left({I}_{S}\left(1\right),{I}_{S}\left(2\right),{\cdots ,I}_{S}\left({N}_{S}\right)\right)={\bf{PRS}}{{\odot }}{\bf{I}}\left(S,T\right)$$

(13)

In Formula 13, $\odot$ represents the Hadamard product between vectors.

The influence of the j-th node in the T-th layer by the S-th layer can be defined as the sum of the influence of nodes in the S-th layer connected to the j-th node in the T-th layer, denoted as follows:

$${{IR}}_{T}^{S}\left(j\right)={\bf{PI}}\left(S,T\right)\cdot {{\bf{A}}}_{{ST}}\left(j\right)=\left({I}_{S}\left(1\right),{I}_{S}\left(2\right),{\cdots ,I}_{S}\left({N}_{S}\right)\right)\cdot \left(\begin{array}{c}\begin{array}{c}{a}_{1j}\\ {a}_{2j}\end{array}\\ \begin{array}{c}\vdots \\ {a}_{{N}_{S}j}\end{array}\end{array}\right)$$

(14)

Based on Formula 12 and Formula 14, we can conclude:

$${{IR}}_{T}^{S}\left(j\right)=\mathop{\sum }\limits_{i=1}^{{N}_{S}}\left({{a}_{{ij}}\cdot {PRS}}_{i}\cdot \frac{{N}_{{{SE}}_{i}}}{{N}_{S}}\right)$$

(15)

Since the T-th layer contains N_T nodes, combining the influence of each node in the T-th layer by the S-th layer into a single column vector yields the ‘influence-received vector’ for the T-th layer, denoted as follows:

$${{\bf{LIR}}}_{T}^{S}=\left(\begin{array}{c}\begin{array}{c}{{IR}}_{T}^{S}\left(1\right)\\ {{IR}}_{T}^{S}\left(2\right)\end{array}\\ \begin{array}{c}\vdots \\ {{IR}}_{T}^{S}\left({N}_{T}\right)\end{array}\end{array}\right)$$

(16)

The Layer-InfluenceRank algorithm is illustrated in Fig. 5.

Fig. 5: Layer-InfluenceRank algorithm.

An algorithm for measuring inter-layer influence in temporal supernetworks.

IO-SuperPageRank algorithm

By integrating the IO-PageRank and Layer-InfluenceRank algorithms, we develop the IO-SuperPageRank algorithm. This algorithm aims to identify risks through the measurement of temporal supernetwork stability. Assume a temporal supernetwork sequence $\left\{{G}_{t}\left({\bf{A}}(t)\right)\right\}$ evolves over a time series $\left\{{t}_{1},{t}_{2},\cdots ,{t}_{v},{t}_{v+1},\cdots ,{t}_{m}\right\}$, with $\left\{{G}_{t}\left({\bf{A}}(t)\right)\right\}$ comprising n layers, can be denoted as $\left\{{L}_{1},{L}_{2},\cdots ,{L}_{n}\right\}$.

The k-th layer network L_k is selected as an example. At time t_v, the influence on L_k from other layer networks can be obtained through the Layer-InfluenceRank algorithm as follows:

$${{\bf{LIR}}}_{k}^{r}\left({t}_{v}\right)=\left(\begin{array}{c}\begin{array}{c}{{IR}}_{k}^{r}\left({1}^{\left({t}_{v}\right)}\right)\\ {{IR}}_{k}^{r}\left({2}^{\left({t}_{v}\right)}\right)\end{array}\\ \begin{array}{c}\vdots \\ {{IR}}_{k}^{r}\left({{N}_{k}}^{\left({t}_{v}\right)}\right)\end{array}\end{array}\right)$$

(17)

In Formula 17, $r=1,\cdots ,n$, where $r\ne k$.

At time t_v, the influence on L_k from its own layer can be represented by the PageRank values of nodes within L_k. Nodes with higher PageRank values are more important within the local layer network, and they are also more susceptible to influence. Therefore, the influence on L_k from its own layer is defined as follows:

$${{\bf{PR}}}_{k}\left({t}_{v}\right)=\left(\begin{array}{c}\begin{array}{c}{{PR}}_{k}\left({1}^{\left({t}_{v}\right)}\right)\\ {{PR}}_{k}\left({2}^{\left({t}_{v}\right)}\right)\end{array}\\ \begin{array}{c}\vdots \\ {{PR}}_{k}\left({{N}_{k}}^{\left({t}_{v}\right)}\right)\end{array}\end{array}\right)$$

(18)

For the sake of consistency, we denote the influence on L_k from its own layer as follows:

$${{\bf{LIR}}}_{k}^{k}\left({t}_{v}\right)=\left(\begin{array}{c}\begin{array}{c}{{IR}}_{k}^{k}\left({1}^{\left({t}_{v}\right)}\right)\\ {{IR}}_{k}^{k}\left({2}^{\left({t}_{v}\right)}\right)\end{array}\\ \begin{array}{c}\vdots \\ {{IR}}_{k}^{k}\left({{N}_{k}}^{\left({t}_{v}\right)}\right)\end{array}\end{array}\right)$$

(19)

Therefore, the influence on L_k from all layers of the overall supernetwork at time t_v is represented as a row vector, denoted as follows:

$${{\bf{LIR}}}_{k}\left({t}_{v}\right)=\left({{\bf{LIR}}}_{k}^{1}\left({t}_{v}\right),{{\bf{LIR}}}_{k}^{2}\left({t}_{v}\right),\cdots ,{{\bf{LIR}}}_{k}^{k}\left({t}_{v}\right),\cdots ,{{\bf{LIR}}}_{k}^{n}\left({t}_{v}\right)\right)$$

(20)

At time t_v+1, the influence on L_k is as follows:

$${{\bf{LIR}}}_{k}\left({t}_{v+1}\right)=\left({{\bf{LIR}}}_{k}^{1}\left({t}_{v+1}\right),{{\bf{LIR}}}_{k}^{2}\left({t}_{v+1}\right),\cdots ,{{\bf{LIR}}}_{k}^{k}\left({t}_{v+1}\right),\cdots ,{{\bf{LIR}}}_{k}^{n}\left({t}_{v+1}\right)\right)$$

(21)

Thus, the influence on L_k from the r-th layer at time t_v and time t_v+1 is as follows:

$${{\bf{LIR}}}_{k}^{r}\left({t}_{v}\right)=\left(\begin{array}{c}\begin{array}{c}{{IR}}_{k}^{r}\left({1}^{\left({t}_{v}\right)}\right)\\ {{IR}}_{k}^{r}\left({2}^{\left({t}_{v}\right)}\right)\end{array}\\ \begin{array}{c}\vdots \\ {{IR}}_{k}^{r}\left({{N}_{k}}^{\left({t}_{v}\right)}\right)\end{array}\end{array}\right),{{\bf{LIR}}}_{k}^{r}\left({t}_{v+1}\right)=\left(\begin{array}{c}\begin{array}{c}{{IR}}_{k}^{r}\left({1}^{\left({t}_{v+1}\right)}\right)\\ {{IR}}_{k}^{r}\left({2}^{\left({t}_{v+1}\right)}\right)\end{array}\\ \begin{array}{c}\vdots \\ {{IR}}_{k}^{r}\left({{N}_{k}}^{\left({t}_{v+1}\right)}\right)\end{array}\end{array}\right)$$

(22)

The inversion order ${{IO}}_{{kr}}\left({t}_{v},{t}_{v+1}\right)$ between ${{\bf{LIR}}}_{k}^{r}\left({t}_{v}\right)$ and ${{\bf{LIR}}}_{k}^{r}\left({t}_{v+1}\right)$ is calculated. Through ${{IO}}_{{kr}}\left({t}_{v},{t}_{v+1}\right)$, the degree of change in the influence of the r-th layer on the k-th layer from time t_v to t_v+1 is measured. The changes in the degree to which the k-th layer is influenced by each layer into a single row vector are calculated, and the ‘stability vector’ is calculated for the k-th layer network from time t_v to t_v+1, denoted as follows:

$${{\bf{IO}}}_{k}\left({t}_{v},{t}_{v+1}\right)=\left({{IO}}_{k1}\left({t}_{v},{t}_{v+1}\right),{{IO}}_{k2}\left({t}_{v},{t}_{v+1}\right),\cdots ,{{IO}}_{kk}\left({t}_{v},{t}_{v+1}\right),\cdots ,{{IO}}_{{kn}}\left({t}_{v},{t}_{v+1}\right)\right)$$

(23)

By combining the stability vectors for each layer network from time t_v to t_v+1 into a single column vector, we obtain the ‘risk measurement matrix’ for the overall supernetwork, denoted as follows:

$${\bf{MIO}}\left({t}_{v},{t}_{v+1}\right)=\left(\begin{array}{c}\begin{array}{c}{{\bf{IO}}}_{1}\left({t}_{v},{t}_{v+1}\right)\\ {{\bf{IO}}}_{2}\left({t}_{v},{t}_{v+1}\right)\end{array}\\ \begin{array}{c}\vdots \\ {{\bf{IO}}}_{n}\left({t}_{v},{t}_{v+1}\right)\end{array}\end{array}\right)$$

(24)

$${\bf{MIO}}\left({t}_{v},{t}_{v+1}\right)=\left(\begin{array}{ccccc}{{IO}}_{11}\left({t}_{v},{t}_{v+1}\right) & \cdots & {{IO}}_{1k}\left({t}_{v},{t}_{v+1}\right) & \cdots & {{IO}}_{1n}\left({t}_{v},{t}_{v+1}\right)\\ \vdots & \ddots & \vdots & \cdots & \vdots \\ {{IO}}_{k1}\left({t}_{v},{t}_{v+1}\right) & \cdots & {{IO}}_{{kk}}\left({t}_{v},{t}_{v+1}\right) & \cdots & {{IO}}_{{kn}}\left({t}_{v},{t}_{v+1}\right)\\ \vdots & \cdots & \vdots & \ddots & \vdots \\ {{IO}}_{n1}\left({t}_{v},{t}_{v+1}\right) & \cdots & {{IO}}_{{nk}1}\left({t}_{v},{t}_{v+1}\right) & \cdots & {{IO}}_{k1}\left({t}_{v},{t}_{v+1}\right)\end{array}\right)$$

(25)

The IO-SuperPageRank algorithm is illustrated in Fig. 6.

Fig. 6: IO-SuperPageRank algorithm.

An algorithm for measuring the stability of temporal supernetworks.

Risk assessment

Stability testing of the elements in the risk measurement matrix (MIO)

The changes in the temporal supernetwork from time period t_v to t_v+1 are measured using the risk measurement matrix ${\bf{MIO}}\left({t}_{v},{t}_{v+1}\right)$, where the elements ${{IO}}_{{kr}}\left({t}_{v},{t}_{v+1}\right)$ quantify the changes in the influence of the r-th layer network on the k-th layer network from time t_v to t_v+1. In a stable temporal supernetwork, each period’s individual changes and the influence between layers are not significant. Therefore, smaller values of ${{IO}}_{{kr}}\left({t}_{v},{t}_{v+1}\right)$ indicate higher network stability and lower risk. In the IO-SuperPageRank algorithm, the number of nodes in the k-th layer at time t_v is denoted as ${N}_{k}\left({t}_{v}\right)$, and the number of nodes in the network at time t_v+1 is ${N}_{k}\left({t}_{v+1}\right)$. To make the number of nodes consistent for both ${{\bf{LIR}}}_{k}^{r}\left({t}_{v}\right)$ and ${{\bf{LIR}}}_{k}^{r}\left({t}_{v+1}\right)$, these two period node counts are aligned to obtain $N({t}_{v},{t}_{v+1})=\max ({N}_{k}({t}_{v}),{N}_{k}({t}_{v+1}))$. Within the $N({t}_{v},{t}_{v+1})$ nodes, there exist ${C}_{N\left({t}_{v},{t}_{v+1}\right)}^{2}$ node pairs, each of which could either be an inversion pair or an ordered pair. Assuming the proportion of inversion pairs in node pairs is P, a random variable X_i$(i=\mathrm{1,2},\cdots ,{C}_{N\left({t}_{v},{t}_{v+1}\right)}^{2})$ can have the following values:

$${X}_{i}=\left\{\begin{array}{ll}1 & {\rm{The}}\,k-{\rm{th}}\; {\rm{pair}}\; {\rm{of}}\; {\rm{nodes}}\; {\rm{is}}\; {\rm{an}}\; {\rm{inversion}}\; {\rm{pair}}.\\ 0 & {\rm{The}}\,k-{\rm{th}}\; {\rm{pair}}\; {\rm{of}}\; {\rm{node}}\; {\rm{is}}\; {\rm{an}}\; {\rm{ordered}}\; {\rm{pair}}.\end{array}\right.$$

(26)

Therefore, X_i follows a Bernoulli distribution with a probability density function as follows:

$$f\left({X}_{i}\right)=\left\{\begin{array}{cc}P & {X}_{i}\,=\,1\\ 0 & {X}_{i}\,=\,0\end{array}\right.$$

(27)

Then, ${{IO}}_{{kr}}\left({t}_{v},{t}_{v+1}\right)$ is as follows:

$${{IO}}_{{kr}}\left({t}_{v},{t}_{v+1}\right)=\mathop{\sum }\limits_{i=1}^{{C}_{N\left({t}_{v},{t}_{v+1}\right)}^{2}}{X}_{i}$$

(28)

${{IO}}_{{kr}}\left({t}_{v},{t}_{v+1}\right)$ follows a binomial distribution $B({C}_{N({t}_{v},{t}_{v+1})}^{2},P)$, where P is the proportion of inversion pairs. Therefore, as P approaches 0, the network becomes more stable. For the hypothesis testing of P, a subjectively acceptable proportion of inversion pairs is denoted as P₀. The null hypothesis and the alternative hypothesis for the hypothesis test are as follows:

$$H0:P \,<\, P0,H1\!:P\,\ge\, {P}_{0}$$

The p value of the test statistic is calculated, and if

$$\mathop{\sum }\limits_{i={NP}}^{{C}_{N\left({t}_{v},{t}_{v+1}\right)}^{2}}{C}_{{C}_{N}^{2}}^{i}{P}_{0}^{i}{\left(1-{P}_{0}\right)}^{{C}_{N}^{2}-i} < 0.05$$

(29)

then reject the null hypothesis. In this case, the network is considered statistically unstable.

When the parameter n of the binomial distribution is greater than 30, it falls into the category of large samples, i.e., ${C}_{N}^{2} > 30$, and N > 9. In complex networks, the number of nodes is usually much larger, making the sample size large. Therefore, under the conditions of a large sample, a normality test is applied using the following method:

$$H0\!:P \,<\, P0,H1\!:P\,\ge\, {P}_{0}$$

The test statistic is as follows:

$${Z}_{{kr}}\left({t}_{v},{t}_{v+1}\right)=\frac{{{IO}}_{{kr}}\left({t}_{v},{t}_{v+1}\right)-{P}_{0}{C}_{N\left({t}_{v},{t}_{v+1}\right)}^{2}}{\sqrt{{P}_{0}\left(1-{P}_{0}\right){C}_{N\left({t}_{v},{t}_{v+1}\right)}^{2}}}$$

(30)

The null hypothesis is rejected when ${Z}_{{kr}}\left({t}_{v},{t}_{v+1}\right)\, >\, {Z}_{0.95}$.

Risk assessment of the two-period temporal supernetworks

Because the changes in temporal supernetworks from time period t_v to t_v+1 are assessed using the risk measurement matrix ${\bf{MIO}}\left({t}_{v},{t}_{v+1}\right)$, multiple tests are involved, including multiple ${{IO}}_{{kr}}\left({t}_{v},{t}_{v+1}\right)$ tests. In total, there are $n\times n={n}^{2}$ tests. To control the overall error rate, it is necessary to employ multiple testing methods, such as the Bonferroni method, Holm method, and Benjamini‒Hochberg (BH) method.

The diagonal elements of the matrix ${\bf{MIO}}({t}_{v},{t}_{v+1})$ measure the stability within each layer network. As there are n layer networks, it is necessary to perform multiple tests on the n elements ${{IO}}_{{kk}}({t}_{v},{t}_{v+1})$, where k = 1, 2…, n. The number of tests that pass the examination is denoted as ${n}_{{pass}}^{{diag}}$. Based on the results of this test, an intralayer stability metric is proposed as follows:

$${H}_{{in}-{layer}}=\frac{n-{n}_{{pass}}^{{diag}}}{n}=1-\frac{{n}_{{pass}}^{{diag}}}{n}$$

(31)

In Formula 31, ${H}_{{in}-{layer}}$ measures the intralayer stability of the temporal supernetwork from time period t_v to t_v+1. A higher ${H}_{{in}-{layer}}$ value indicates greater intralayer changes in the supernetwork from t_v to t_v+1, signifying higher risk.

The off-diagonal elements of the matrix ${\bf{MIO}}\left({t}_{v},{t}_{v+1}\right)$ measure the interlayer stability among the various layer networks. As there are n layer networks, there are a total of ${n}^{2}-n$ elements ${{IO}}_{kr}({t}_{v},{t}_{v+1})$ where k ≠ r. Multiple tests are conducted on these elements, and the number of tests that pass the examination is denoted as ${n}_{{pass}}^{{no}-{diag}}$. Based on the test results, an interlayer stability metric is proposed as follows:

$${H}_{{between}-{layer}}=\frac{{n}^{2}-n-{n}_{{pass}}^{{no}-{diag}}}{{n}^{2}-n}\,=\,1-\frac{{n}_{{pass}}^{{no}-{diag}}}{{n}^{2}-n}$$

(32)

In Formula 32, ${H}_{{between}-{layer}}$ measures the interlayer stability of the temporal supernetwork from time period t_v to t_v+1. A higher ${H}_{{between}-{layer}}$ value indicates greater interlayer changes in the supernetwork from ${H}_{{between}-{layer}}$, signifying higher risk.

In addition to the assessment metrics for the two aspects mentioned above, the matrix ${\bf{MIO}}({t}_{v},{t}_{v+1})$ is also influenced by extreme values. In the evaluation of stability, it is important to consider cases where the number of unpassed ${{IO}}_{kr}({t}_{v},{t}_{v+1})$ tests is low, but the extreme values within ${{IO}}_{kr}({t}_{v},{t}_{v+1})$ are large. Therefore, we propose ${H}_{{extreme}}$ as follows:

$${H}_{{extreme}}=\max \left({{IO}}_{kr}\left({t}_{v},{t}_{v+1}\right)\right)$$

(33)

The stability assessment of the temporal supernetwork from time period t_v to t_v+1 requires considering all three aspects mentioned above, namely, the stability metrics ${H}_{{in}-{layer}}$, ${H}_{{between}-{layer}}$ and ${H}_{{extreme}}$. The stability metric for the temporal supernetwork, ${H}_{{supernet}}$, is a multivariate function of these three metrics:

$${H}_{{supernet}}=f\left({H}_{{in}-{layer}},{H}_{{between}-{layer}},{H}_{{extreme}}\right)$$

(34)

For temporal supernetworks with different structures, the specific analytical expression of $f({H}_{{in}-{layer}},{H}_{{between}-{layer}},{H}_{{extreme}})$ may vary. For example, in temporal multiplex networks, node and connection changes mainly occur within individual layers, while in multidimensional superpernetworks (e.g., opinion supernetworks), changes are primarily manifested in interlayer superedges. In the process of constructing ${H}_{{supernet}}$, although the analytical expressions may differ, certain common principles need to be followed.

Monotonicity principle

In the process of constructing the analytical expression for ${H}_{{supernet}}$, the higher the values of ${H}_{{in}-{layer}}$, ${H}_{{between}-{layer}}$, and ${H}_{{extreme}}$ are, the greater the instability of the temporal supernetwork from time period t_v to t_v+1. The mathematical expression is as follows:

$$\frac{\partial {H}_{{supernet}}}{\partial {H}_{{in}-{layer}}}\ge 0,\frac{\partial {H}_{{supernet}}}{\partial {H}_{{between}-{layer}}}\ge 0,\frac{\partial {H}_{{supernet}}}{\partial {H}_{{extreme}}}\ge 0$$

(35)

Complementarity principle

In the process of constructing the analytical expression for ${H}_{{supernet}}$, under the condition where the same ${H}_{{supernet}}$ metric values are present, there exists a complementary substitution relationship between ${H}_{{in}-{layer}}$, ${H}_{{between}-{layer}}$, and ${H}_{{extreme}}$. That is, under the condition where

$$f\left({H}_{{in}-{layer}}^{1},{H}_{{between}-{layer}}^{1},{H}_{{extreme}}^{1}\right)=f\left({H}_{{in}-{layer}}^{2},{H}_{{between}-{layer}}^{2},{H}_{{extreme}}^{2}\right)$$

(36)

the following does not exist:

$$\left\{\begin{array}{c}{H}_{{in}-{layer}}^{1} \,<\, {H}_{{in}-{layer}}^{2}\\ {H}_{{between}-{layer}}^{1}\, <\, {H}_{{between}-{layer}}^{2}\\ {H}_{{extreme}}^{1} \,<\, {H}_{{extreme}}^{2}\end{array}\right.{or}\left\{\begin{array}{c}{H}_{{in}-{layer}}^{1} \,>\, {H}_{{in}-{layer}}^{2}\\ {H}_{{between}-{layer}}^{1}\, >\, {H}_{{between}-{layer}}^{2}\\ {H}_{{extreme}}^{1} \,>\, {H}_{{extreme}}^{2}\end{array}\right.$$

(37)

Concavity principle

In the process of constructing the analytical expression for ${H}_{{supernet}}$, we aim to simultaneously consider the influences of ${H}_{{in}-{layer}}$, ${H}_{{between}-{layer}}$, and ${H}_{{extreme}}$. From a risk control perspective, the risk associated with single-dimensional instability should be lower than the systemic risk resulting from instability in multiple dimensions. This can be expressed in mathematical terms as follows:

$$\begin{array}{c}f\left(\sum _{i}{w}_{i}{H}_{{in}-{layer}}^{i},\sum _{i}{{w}_{i}H}_{{between}-{layer}}^{i},\sum _{i}{{w}_{i}H}_{{extreme}}^{i}\right)\\ \ge \sum _{i}{w}_{i}f\left({H}_{{in}-{layer}}^{i},{H}_{{between}-{layer}}^{i},{H}_{{extreme}}^{i}\right)\end{array}$$

(38)

where $\sum _{i}{w}_{i}=1$. When $f\left({H}_{{in}-{layer}},{H}_{{between}-{layer}},{H}_{{extreme}}\right)$ is twice differentiable, the Hessian matrix H_supernet is represented as:

$$\left(\begin{array}{ccc}\frac{{\partial }^{2}{H}_{{supernet}}}{{\partial }^{2}{H}_{{in}-{layer}}} & \frac{{\partial }^{2}{H}_{{supernet}}}{\partial {H}_{{between}-{layer}}\partial {H}_{{in}-{layer}}} & \frac{{\partial }^{2}{H}_{{supernet}}}{\partial {H}_{{extreme}}\partial {H}_{{in}-{layer}}}\\ \frac{{\partial }^{2}{H}_{{supernet}}}{\partial {H}_{{in}-{layer}}\partial {H}_{{between}-{layer}}} & \frac{{\partial }^{2}{H}_{{supernet}}}{{\partial }^{2}{H}_{{between}-{layer}}} & \frac{{\partial }^{2}{H}_{{supernet}}}{\partial {H}_{{extreme}}\partial {H}_{{between}-{layer}}}\\ \frac{{\partial }^{2}{H}_{{supernet}}}{\partial {H}_{{in}-{layer}}\partial {H}_{{extreme}}} & \frac{{\partial }^{2}{H}_{{supernet}}}{\partial {H}_{{between}-{layer}}\partial {H}_{{extreme}}} & \frac{{\partial }^{2}{H}_{{supernet}}}{{\partial }^{2}{H}_{{extreme}}}\end{array}\right)$$

(39)

When concavity is satisfied, the eigenvalues of H_supernet are negative.

Risk assessment of temporal supernetwork sequences

The stability metric ${H}_{{supernet}}({t}_{v},{t}_{v+1})$ introduced earlier for two consecutive periods is used to construct a sequence stability evaluation metric for the time series of supernetworks. If the subjective acceptable proportion for each period is denoted as H_sub, then

$$V\left({t}_{v},{t}_{v+1}\right)=\left\{\begin{array}{cc}1 & {H}_{{supernet}}\left({t}_{v},{t}_{v+1}\right) \,<\, {H}_{{sub}}\\ 0 & {H}_{{supernet}}\left({t}_{v},{t}_{v+1}\right) \,>\, {H}_{{sub}}\end{array}\right.$$

(40)

$${m}_{{pass}}=\mathop{\sum }\limits_{v=1}^{m-1}V\left({t}_{v},{t}_{v+1}\right)$$

(41)

$$V=\frac{{m}_{{pass}}}{m-1}$$

(42)

Assuming an overall subjective acceptable proportion V₀, if $V\le {V}_{0}$, the temporal supernetwork $\left\{G\left(t\right)\right\}$ is considered stable over the time points $\left\{{t}_{1},{t}_{2},\cdots ,{t}_{v},{t}_{v+1},\cdots ,{t}_{m}\right\}$. Otherwise, it is considered unstable, indicating a higher level of risk likelihood.

Simulation experiment

Experimental settings

To validate the effectiveness of the IO-SuperPageRank algorithm for risk identification in temporal supernetworks, a four-layer supernetwork model was constructed with different layer sizes: 10, 20, 30, and 40 nodes in each layer. Considering that in practical applications, scale-free networks are the most common (Barabási and Albert, 1999), and various networks such as social networks, biological networks, and trade networks exhibit scale-free characteristics, the initial construction of the four-layer supernetworks in this study involved creating random scale-free networks for each layer. The superedges between layers were generated randomly (illustrated in Fig. 7).

Fig. 7: The initial four-layer supernetwork model.

Randomly generate a four-layer supernetwork as the initialization for a temporal supernetwork.

In temporal supernetworks, risk primarily stems from the instability of temporal changes in the network structure. Changes in the network structure mainly manifest in three aspects: variations in the numbers of intralayer edges, intralayer nodes, and interlayer superedges. To validate the effectiveness of the IO-SuperPageRank algorithm, each of the initial four layers in the supernetwork underwent a monotonic linear change in the numbers of intralayer edges, intralayer nodes, and interlayer superedges. These changes were made with the understanding that any single nonlinear increase could be viewed as a superposition of multiple linear changes over several periods. The risk measurement matrix MIO was then observed and tested for its correlation with the three aforementioned dimensions. Due to the varying numbers of nodes and edges in the initial supernetwork layers, the number of simulation periods proportionally adjusted (Tables 1 to 3).

Table 1 The range of changes in the number of intralayer edges for each layer in the temporal supernetwork.

Table 2 The range of changes in the number of intralayer nodes for each layer in the temporal supernetwork.

Table 3 The range of changes in the number of interlayer superedges in the temporal supernetwork.

The MIO risk measurement matrix is calculated by assessing the changes in node importance relative to the initial supernetwork for each period in the temporal supernetwork. We then observed the correlation between the elements at corresponding positions in the risk measurement matrix and the changes in the network structure. To provide a more precise measurement of the relationship between network structure changes and the IO risk metric, we conducted one-variable linear regression using the data from the three sets of experiments. We then applied various methods for multiple testing corrections, such as the Bonferroni method, Sidak method, Holm method, and Benjamini‒Hochberg (BH) method, to further validate the relationship between IO values and network structure.

Furthermore, to realistically simulate the changes in complex systems, we conducted a set of simulation experiments involving overall random changes in the supernetwork. The number of periods for these changes was set to 3000, and in each period, the numbers of intralayer nodes, intralayer edges, and interlayer superedges underwent random increases and decreases. We computed the risk measurement matrix MIO for each period relative to the initial period. By applying various multiple testing methods to the MIO matrix under different subjective risk acceptance ratios (${P}_{0}$), we obtained intralayer risk indicators (${H}_{{in}-{layer}}$) and interlayer risk indicators (${H}_{{between}-{layer}}$). We observed the changing trends of (${H}_{{in}-{layer}}$ and ${H}_{{between}-{layer}}$ as the subjective risk acceptance ratio increased. Through these experiments, we were able to validate the effectiveness of the IO-SuperPageRank algorithm for risk identification.

Results and dicussion

Changes occurred in the numbers of intralayer edges, intralayer nodes, and interlayer superedges independently

Overall, we can observe that as the numbers of intralayer edges, intralayer nodes, and interlayer superedges increase, the IO risk metric generally tends to increase as well. This indicates a positive correlation between risk induced by changes in network structure and the IO value. The IO-SuperPageRank algorithm effectively identifies and measures risk. Figures 8–10 illustrate the changes in each element of the risk measurement matrix (MIO matrix) concerning alterations in network structure metrics.

Fig. 8: The impact of the number of intralayer edges on the risk metric.

a–d represent the changes in IO values for each layer as the number of edges changes monotonically in the first layer. e–h represent the second layer, (i–l) represent the third layer, and (m–p) represent the fourth layer.

Fig. 9: The impact of the number of intralayer nodes on the risk metric.

a–d represent the changes in IO values for each layer as the number of nodes changes monotonically in the first layer. e–h represent the second layer, (i–l) represent the third layer, and (m–p) represent the fourth layer.

Fig. 10: The impact of the number of interlayer superedges on the risk metric.

a–f represent the changes in IO values for each layer as the number of interlayer superedges changes monotonically between layers 1 and 2, 1 and 3, 1 and 4, 2 and 3, 2 and 4, and 3 and 4, respectively.

From Fig. 10, it is evident that when the number of interlayer superedges changes, the network layer with fewer nodes has a greater impact on the risk of the layer with more nodes (e.g., IO[2][1] > IO[1][2], IO[3][2] > IO[2][3]). It is worth noting that because the first layer has the fewest nodes, the change in IO[k][1] (k = 2,3,4) is less pronounced. However, this also reflects a limitation of the IO-SuperPageRank algorithm in that it may not be highly sensitive to risk identification in network layers with fewer nodes.

We further conducted one-variable linear regression experiments on the changes in IO values and network structure metrics for these three sets of experiments (Fig. 11). Subsequently, we applied one-variable linear regression to multiple testing corrections (Fig. 12). Based on the distribution of p values, it is evident that IO values are significantly correlated with changes in the numbers of intralayer edges, intralayer nodes, and interlayer superedges. Furthermore, we can observe that the most significant correlation exists between the number of intralayer nodes and IO values, with over 90% of p values being less than 1.00E-10. The primary reason is that changes in the number of intralayer nodes encompass changes in intralayer edges simultaneously. This conclusion aligns well with the real-world scenario, where the entry or exit of new elements into a complex system has the most significant impact on the system’s risk. For instance, in a public opinion field, the entry of new opinion leaders may directly influence the direction of public sentiment, thereby introducing public opinion risks.

Fig. 11: The distribution of p values.

The significance (p values) distribution of one-variable linear regression for intralayer edges, intralayer nodes, and interlayer superedges concerning the risk metric (IO values).

Fig. 12: The distribution of p values after multiple testing corrections.

The significance (p values) distribution of the multiple testing corrections for the one-variable linear regression coefficients of intralayer edges (a), intralayer nodes (b), and interlayer superedges (c) concerning the risk metric (IO values).

The entire supernetwork underwent random changes

When the entire network underwent random changes, we calculated the MIO matrix for each period compared to that of the initial period. By selecting an appropriate subjective risk acceptance rate (${P}_{0}$) under different multiple testing correction methods (Bonferroni, Holm, Sidak, BH, BY, Hommel), we calculated the average values of the intralayer risk index (${H}_{{in}-{layer}}$) and the interlayer risk index (${H}_{{between}-{layer}}$). By letting the subjective risk acceptance rate (${P}_{0}$) vary from 10% to 90% as the ratio of changes in the ranks, where a higher ratio indicates higher tolerance for risk, the results (Fig. 13) show that as the subjective acceptance rate increases, the average values of ${H}_{{in}-{layer}}$ and ${H}_{{between}-{layer}}$ decrease. This intuitively aligns with the practical requirements of risk assessment, meaning that for different assessors, while measuring the same risk value, assessors with higher risk tolerance have relatively low risk.

Fig. 13: The distributions of the average values of the intralayer stability index and interlayer stability index calculated using different multiple testing correction methods at different subjective risk acceptance rates (P₀).

a, b represents the distribution of the mean value of the intralayer stability index (${H}_{{in}-{layer}}$) and the interlayer stability index (${H}_{{between}-{layer}}$), respectively.

Discussion

The IO-SuperPageRank algorithm proposed in this paper can identify risks by tracking the structural changes in temporal supernetworks. From the experimental results, it is evident that changes in intralayer edges, intralayer nodes, and interlayer superedges in temporal supernetworks lead to an increase in the risk indicator IO, indicating an elevated likelihood of risk occurrence. However, there are still some specific situations in which the algorithm may not perform well in risk identification. For instance, in a single network, there may be cases where edge changes occur but the importance of nodes remains relatively stable (as shown in Fig. 14). In such cases, the algorithm computes a risk indicator IO of 0, but in reality, an increase in edges implies more coupling relationships between nodes, which can increase network instability and, consequently, risk. Additionally, the algorithm is based on changes in node importance to represent the risk situation, which is generally reasonable because uncertainty itself is a form of risk. This uncertainty can bring both opportunities and potential losses. However, in practical applications, if more emphasis is placed on negative risks, it is necessary to consider specific node attributes and the physical meaning of edges.

Fig. 14: Specific situation.

An example in which intralayer edges change but the importance of nodes remains unchanged.

Compared with traditional metrics measuring network node stability, the proposed risk metric IO effectively reflects the influence of node updates on network stability. In the comparison of network stability metrics under three scenarios, as shown in Fig. 15, including the difference in the average degree between time periods ${t}_{v}$ and ${t}_{v+1}$, the difference in closeness centrality (here, we use the harmonic centrality for disconnected graphs), the difference in clustering coefficient, the difference in robustness index R, and the IO value between two periods. In ${Situation\_}1$, all nodes are updated, but the connectivity and node centrality metrics do not change between the two periods. However, the IO value can measure changes in the network structure, thus identifying risks. Furthermore, from an intuitive perspective, compared to ${Situation\_}2$ and ${Situation\_}3$, ${Situation\_}1$ should exhibit the most significant changes in risk indicators because of the complete node updates. However, metrics such as the difference in average degree, closeness centrality, clustering coefficient, and robustness index do not reflect that the risk in ${Situation\_}1$ is higher than that in ${Situation\_}2$ and ${Situation\_}3$. In contrast, the IO value effectively detects changes in risk (Table 4).

Fig. 15: Examples of node updates.

Node updating strategies in three different situations.

Table 4 Comparison of the IO value and the absolute difference of traditional network structure indices between two periods.

Case study

Case introduction

With the rapid development of global economic globalization, the liberalization of international trade, the swift advancement of information technology, and the improvement of logistics and transportation, the global supply chain has gradually emerged since the latter half of the 20th century. Particularly in the energy sector, the uneven distribution of energy resources in different regions globally is bound to stimulate the global trade of energy raw materials to final products. However, in recent years, the world has witnessed a series of profoundly impactful global events. Factors such as the COVID-19 pandemic, the crisis in Ukraine, and escalating geopolitical tensions have resulted in global or regional supply chain crises. Simultaneously, supply chains are closely tied to major national strategies, and identifying global supply chain risks is crucial for effectively addressing supply chain crises. Therefore, this article takes the supply chain of man-made filaments and staple fibers as an example, constructing a simplified supply chain temporal supernetwork using raw materials, intermediate products, and final products. The variation in supply chain risk is measured by calculating the MIO matrix at each stage of the temporal supernetwork.

In the supply chain of man-made filaments and staple fibers, the main raw material is crude oil. Crude oil is refined to produce an intermediate product called chemical light oil. This chemical light oil undergoes various steps of transformation and synthesis to eventually form polymers. Subsequently, through spinning and processing, chemical fiber products are created. In our case, the import network of crude oil among countries constitutes the raw material trading layer of the supernetwork. The export network of man-made filaments and staple fibers among countries forms the product trading layer of the supernetwork. The export of the intermediate commodity, chemical light oil, serves as inter-layer superedges to connect the raw material layer with the final products layer, forming the supply chain supernetwork (Fig. 16).

Fig. 16: Example of supply chain supernetwork.

Schematic diagram of the man-made filaments and staple fibres supply chain supernetwork.

Data description

We constructed a man-made filaments and staple fibers supply chain temporal supernetwork using monthly trade data from December 2020 to October 2023 among 131 countries, sourced from the United Nations Comtrade Database. Each layer of the supernetwork is a weighted directed graph. The data selected for crude oil corresponds to HS code 2709, specifically “Petroleum oils and oils obtained from bituminous minerals”. Chemical light oil is represented by data with HS code 2709, denoting “light oils and preparations”. Data for man-made filaments and man-made staple fibers are selected with HS codes 54 and 55, respectively. The network nodes in the raw material trading layer represent 131 countries globally involved in crude oil trade, with edge weights indicating the volume of crude oil imports between countries. Similarly, the nodes in the product trading layer also represent the same 131 countries. As the product trading layer encompasses two types of goods, for the sake of standardization, we set the edge weights as the total export value of goods with HS codes 54 and 55 between countries. The connection between the raw material layer and the product layer is established by intermediate goods, chemical light oil, exported from countries in the raw material layer to countries in the product layer. These chemical light oil serve as superedges, and the weight of each superedge is determined by the volume of chemical light oil exported. The matrix for the constructed man-made filaments and staple fibers supply chain temporal supernetwork is presented in Formula 44, and the data characteristics of intra-layer edges and inter-layer superedges are depicted in Fig. 17.

$${\boldsymbol{G}}\left(t\right)=\left(\begin{array}{cc}\left(\begin{array}{cccc}0 & {w}_{{AND},{AZE}}^{C{rude}} & \cdots & {w}_{{AND},{GMB}}^{C{rude}}\\ {w}_{{AZE},A{ND}}^{C{rude}} & 0 & \cdots & {w}_{A{ZE},{GMB}}^{C{rude}}\\ \vdots & \vdots & \ddots & \vdots \\ {w}_{{GMB},A{ND}}^{C{rude}} & {w}_{{GMB},{AZE}}^{C{rude}} & \cdots & 0\end{array}\right) & \left(\begin{array}{cccc}0 & {w}_{{AND},{AZE}}^{{Light}\,{oil}} & \cdots & {w}_{{AND},{GMB}}^{{Light}\,{oil}}\\ {w}_{A{ZE},A{ND}}^{{Light}\,{oil}} & 0 & \cdots & {w}_{A{ZE},{GMB}}^{{Light}\,{oil}}\\ \vdots & \vdots & \ddots & \vdots \\ {w}_{G{MB},A{ND}}^{{Light}\,{oil}} & {w}_{G{MB},{AZE}}^{{Light}\,{oil}} & \cdots & 0\end{array}\right)\\ {\left(\begin{array}{ccccccc}0 & 0 & \cdots & 0 & 0 & \cdots & 0\\ 0 & 0 & \cdots & 0 & 0 & \cdots & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 & 0 & \cdots & 0\\ 0 & 0 & \cdots & 0 & 0 & \cdots & 0\end{array}\right)}_{131\times 131} & \left(\begin{array}{cccc}0 & {w}_{{AND},{AZE}}^{{Fabric}} & \cdots & {w}_{{AND},{GMB}}^{{Fabric}}\\ {w}_{A{ZE},A{ND}}^{{Fabric}} & 0 & \cdots & {w}_{A{ZE},{GMB}}^{{Fabric}}\\ \vdots & \vdots & \ddots & \vdots \\ {w}_{G{MB},A{ND}}^{{Fabric}} & {w}_{G{MB},A{ND}}^{{Fabric}} & \cdots & 0\end{array}\right)\end{array}\right)$$

(43)

Fig. 17: Data characteristics.

The interconnection data characteristics of the man-made filaments and staple fibres supply chain temporal supernetwork.

The results

We employed the IO-SuperPageRank algorithm to compute data for a total of 35 months from December 2020 to October 2023, resulting in 34 MIO risk measurement matrices. The structure of the MIO matrix is depicted in Formula 45. In this representation, IO₁₁ represents the temporal variation in risk in the raw material trading layer’s crude oil import network, IO₂₂ signifies the temporal variation in risk in the product trading layer’s man-made filaments and staple fibers export network. Since the impact of the raw material trading network on the product trading network is unidirectional, IO₂₁ represents the temporal variation in risk in the product trading layer influenced by the raw material trading layer, while IO₁₂ is set to zero matrix.

$${\bf{MIO}}\left({t}_{v},{t}_{v+1}\right)=\left(\begin{array}{cc}{{IO}}_{11} & 0\\ {{IO}}_{21} & {{IO}}_{22}\end{array}\right)$$

(44)

From the computational results, the MIO risk measurement matrix can comprehensively measure the changes in risk in the supply chain of man-made filaments and staple fibers across multiple dimensions and layers. Figure 18a illustrates the variation in the intra-layer risk indicator IO₁₁ for the crude oil trading layer network. The graph indicates high crude oil trading risk in November to December 2021 and August to September 2023. In December 2021, Russia deployed over 100,000 troops along the Russia-Ukraine border, marking the largest military mobilization in Europe since World War II. The tense war situation between Russia and Ukraine was likely to impact global crude oil trade. Additionally, the uncertainty of crude oil imports was intensified by the ongoing COVID-19 pandemic. In September 2023, escalating tensions between Israel and Hamas heightened the risk, with the imminent possibility of war between the two nations contributing to international instability, thereby affecting crude oil trade. The risk variation in the trading layer of man-made filaments and staple fibers stems from two aspects. Firstly, the risk in the crude oil trading network is transmitted to man-made filaments and staple fibers trading network through the intermediate product, chemical light oil. Comparing Fig. 18a, b, it can be observed that the variation in the risk indicator IO₂₁ has a high correlation with the risk indicator IO₁₁ in the crude oil layer network. This illustrates that the trading layer of man-made filaments and staple fibers is influenced by the crude oil trading layer, with higher risk indicators during the Russia-Ukraine war and the Israel-Hamas conflicts. On the other hand, the risk in the trading layer of man-made filaments and staple fibers is also influenced by the supply and demand relationship of the products themselves. The variation in the risk indicator IO₂₂ indicates higher risk from December 2022 to February 2023. During this period, the Chinese government ended the policy of dynamic zeroing for COVID-19, leading to a significant decrease in the trade volume of medical products such as masks. This indirectly resulted in changes in the supply and demand for man-made filaments and staple fibers, heightening global trade risks.

Fig. 18: The variation of IO values in the MIO matrix for each period in the man-made filaments and staple fibers supply chain temporal supernetwork.

a, b respectively depict the temporal changes of IO values in the raw material trading layer and product trading layer within the MIO matrix.

This case demonstrates that the risk indicators calculated from the MIO matrices for various periods based on the temporal supernetwork exhibit a strong correlation with the potential risks in reality. It effectively enables risk identification. Moreover, the different IO values in the MIO matrix categorize risks into multiple dimensions and layers, allowing for a more comprehensive measurement of various dimensions of risk and the transmission effects of risk. For supply chains, the variations in different IO values within and between layers can assess the risk situations at various stages of the supply chain. This approach also provides insights into the depth and breadth of risk transmission.

Conclusion

One of the significant contributions of this study is the introduction of a new algorithm named IO-SuperPageRank, which measures the changes in node importance in a time-varying supernetwork. This algorithm enables risk assessment in complex systems. Moreover, based on the risk metrics obtained from the algorithm, we have developed methods and indicators for risk assessment. Through simulation experiments, we discovered a significant correlation between the risk indicator IO values and intralayer edge, intralayer node, and interlayer superedge changes. Therefore, when there are structural changes in the network, this algorithm can precisely measure and identify risks. Additionally, we conducted experiments to simulate random changes in the overall temporal supernetwork, and we found that changes in intralayer node counts have the most substantial impact on the entire system. Compared to traditional measures based on degree centrality and network connectivity, the IO-SuperPageRank algorithm can more accurately identify the impact of node updates on the entire network. However, it is essential to note that the algorithm has certain limitations, such as its inability to identify certain risk situations where the importance of nodes in the network remains unchanged. Next, we applied this algorithm to conduct empirical analysis on the global supply chain of man-made filaments and fibers. The results indicated that the IO values could effectively reflect potential risks present in the real world, demonstrating the efficacy of this algorithm in addressing real-world issues.This algorithm offers substantial room for extension and generalization. It can be tailored and enhanced for specific application scenarios by making targeted modifications to accommodate different network structures. In addition, the algorithm identifies a broad spectrum of risks, encompassing both losses and opportunities. In future empirical studies applying this algorithm, specific risk scenarios can be identified by analyzing the attributes of important nodes causing network changes. Subsequently, suitable intervention methods can be employed for risk management.