Method for evaluating robustness of biomolecular network based on multi-layer high-order complex network

Abstract

The invention relates to the field of complex network cascade dynamics, and discloses a method for evaluating biomolecular network robustness based on a multi-layer high-order complex network, and the method comprises the steps: building a single-layer high-order hypergraph network model based on the original data and preset parameters of a biomolecular network, and obtaining the robustness of the biomolecular network by using the single-layer high-order hypergraph network model; expanding and constructing a multi-layer dependent hypergraph model, performing fault injection on the multi-layer dependent hypergraph model according to a preset initial fault parameter so as to enable part of nodes in the model to be invalid and form an initial fault state, and performing iterative simulation on the model in the initial fault state according to a preset fault propagation rule until the model reaches a steady state, and carrying out connectivity analysis on the model reaching the steady state to obtain a robustness evaluation result of the biomolecular network. According to the method, the structure of the biomolecular network is optimized, the robustness of the biomolecular network is improved, and stable operation of the biomolecular network in practical application is guaranteed.

Description

Technical Field

[0001] This application relates to the field of complex network cascade dynamics, and in particular to a method for evaluating the robustness of biomolecular networks based on multilayer high-order complex networks. Background Technology

[0002] Complex network theory has become an important tool for analyzing and optimizing complex systems, with wide applications in fields such as biology, sociology, and communication networks. Biomolecular networks are composed of biomolecules such as proteins and genes, and their interactions. In recent years, with in-depth research into biomolecular networks, it has been discovered that these networks are not merely composed of single-level interactions; in fact, multi-layered, interdependent complex network structures exist within biomolecular systems. Multi-layered interdependent networks can effectively enhance the stability and operational efficiency of a system. However, the interdependence between layers also increases the system's vulnerability, especially when a node or edge fails, as the failure can propagate between layers and potentially lead to the overall collapse of the system. Therefore, the robustness study of multi-layered interdependent networks has become an important direction in complex network research.

[0003] While significant progress has been made in complex network theory, existing techniques have significant limitations in robustness assessment of multilayer networks, particularly biomolecular networks. First, most current network analysis methods focus on binary interaction models, i.e., direct relationships between nodes, neglecting the multi-agent, high-order interactions prevalent in real biomolecular networks. In other words, current research on biomolecular networks typically relies on binary relation-based modeling methods, which can only describe one-to-one interactions between individual proteins and genes. However, in actual biological systems, multiple proteins and genes often interact collaboratively in a multi-agent manner, exhibiting high-order and complex relationships. Traditional binary relation models struggle to accurately characterize these multi-dimensional and multi-layered interaction features, resulting in insufficient analysis of the intrinsic mechanisms of biomolecular networks and failing to meet the needs of applications such as disease mechanism research and drug target discovery. Second, existing robustness assessment methods for multilayer interdependent networks are usually based on traditional percolation theory, but these methods fail to adequately consider the role of reinforcing small components within the network. In biomolecular networks, some nodes can effectively resist local failures through redundant connections or other hardening mechanisms. These hardening components have an important impact on the stability of the network, but traditional methods often fail to take this factor into account, resulting in inaccurate assessment of network robustness.

[0004] Therefore, in the study of multilayer complex networks, how to simultaneously consider higher-order interactions and the reinforcement effect on small components during the modeling process, thereby improving the overall robustness of biomolecular networks, especially protein-gene networks, has become an important problem that urgently needs to be solved in this field. Summary of the Invention

[0005] In view of this, embodiments of this application provide a method for evaluating the robustness of biomolecular networks based on multilayer high-order complex networks, which can effectively solve the problem in the prior art that it fails to accurately capture and evaluate the impact of high-order interactions and reinforcement of small components in multilayer interdependent networks on the robustness of biomolecular networks.

[0006] In a first aspect, embodiments of this application provide a method for evaluating the robustness of biomolecular networks based on multilayer high-order complex networks, including: Based on the original data and preset parameters of biomolecular networks, a single-layer high-order hypergraph network model is constructed. Using the single-layer high-order hypergraph network model, a multi-layer dependent hypergraph model is constructed. Fault injection is performed on the multi-layer dependent hypergraph model according to preset initial fault parameters, so that some nodes in the multi-layer dependent hypergraph model fail and form an initial fault state. The multi-layer interdependent network model in the initial fault state is simultaneously iteratively simulated according to a preset fault propagation rule until the multi-layer interdependent network model reaches a steady state. Connectivity analysis was performed on the multilayer interdependent network model that reached steady state to obtain the robustness evaluation results of the biomolecular network.

[0007] In some embodiments, the method further includes: The robustness evaluation results are compared with the preset robustness target, and at least one key parameter in the multilayer dependent hypergraph network model is fine-tuned based on the comparison results until the multilayer dependent hypergraph network model reaches the preset robustness target.

[0008] In some embodiments, constructing a single-layer high-order hypergraph network model based on the raw data and preset parameters of the biomolecular network includes: Extract biomolecular nodes and interaction information from the raw data of the biomolecular network; The biomolecule nodes and interaction information are standardized using the preset parameters to determine the properties and reinforcement status of the biomolecules. Based on the higher-order interaction rules between the biomolecular nodes, construct a set of hyperedges; Based on the properties of the biomolecular nodes, the reinforcement state, and the set of hyperedges, a single-layer high-order hypergraph network model is constructed.

[0009] In some embodiments, extending and constructing a multi-layer dependent hypergraph model using the single-layer high-order hypergraph network model includes: The single-layer high-order hypergraph network model is extended to generate multiple network layers. According to the preset inter-layer dependency rules, dependency relationships are assigned to the nodes of each network layer, and a set of inter-layer dependency edges is generated; The structure of each network layer is integrated with the corresponding set of inter-layer dependent edges to construct the multi-layer dependent hypergraph network model.

[0010] In some embodiments, the step of injecting faults into the multi-layer dependent hypergraph model according to preset initial fault parameters, so as to cause some nodes in the multi-layer dependent hypergraph model to fail and form an initial fault state, includes: Set the initial fault parameters; In the multi-layer dependent hypergraph model, some nodes are randomly selected according to the initial fault parameters, and these nodes are marked as failed, so as to form a multi-layer dependent hypergraph network model in which some of the nodes are in a failed state.

[0011] In some embodiments, the multilayer dependent network model of the initial fault state is iteratively simulated simultaneously according to a preset fault propagation rule until the multilayer dependent network model reaches a steady state, including: Based on the preset node-to-hyperedge, hyperedge-to-node, and cross-layer dependency fault propagation rules, the state of all nodes and hyperedges in the multi-layer dependent network model is updated. Check whether there are nodes or superedges that satisfy the fault propagation conditions in the updated multilayer interdependent network model, and change the state of the nodes or superedges that satisfy the fault propagation conditions. Repeat the state change and state check until the states of all nodes and hyperedges in the multi-layer dependent network model remain stable, confirming that the multi-layer dependent network model has reached a steady state.

[0012] In some embodiments, performing connectivity analysis on the multilayer interdependent network model that has reached a steady state to obtain network robustness evaluation results includes: The connection information of nodes and hyperedges in each layer are extracted from the multilayer interdependent network model that has reached a steady state; Based on the connection information, a percolation model is used to perform connectivity analysis, and the generalized maximum connected component and the maximum connected component in the multilayer interdependent network model are calculated. The generalized maximum connected component and the maximum connected component are used as the network robustness evaluation results.

[0013] The embodiments of this application have the following beneficial effects: This application presents a method for evaluating the robustness of biomolecular networks based on multi-layer high-order complex networks. By utilizing the original data and preset parameters of biomolecular networks, a single-layer high-order hypergraph network model is first constructed. Then, based on this model, a multi-layer dependent hypergraph network model is extended, fully characterizing the inter-layer dependencies and high-order interaction features. Next, a fault injection is performed on the multi-layer dependent hypergraph model using preset initial fault parameters to form an initial fault state. The fault state is then iteratively simulated according to preset fault propagation rules until the network reaches a steady state, thereby obtaining the generalized maximum connected component and its key robustness indicators. This method fully considers the actual characteristics of multi-agent high-order interactions, reinforced nodes, and cross-layer dependencies in biomolecular networks, effectively solving the problem of traditional robustness evaluation methods underestimating network stability. Simultaneously, through quantitative analysis of fault propagation and network connectivity, it can accurately reflect the recovery ability and overall stability of biomolecular networks under local fault disturbances, providing accurate and reliable quantitative basis for the optimization design and stability improvement of biomolecular networks, and possessing significant scientific research and engineering application value. Attached Figure Description

[0014] To more clearly illustrate the technical solutions of the embodiments of this application, the accompanying drawings used in the embodiments will be briefly introduced below. It should be understood that the following drawings only show some embodiments of this application and should not be regarded as a limitation of the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort.

[0015] Figure 1 A flowchart illustrating a method for evaluating the robustness of biomolecular networks based on multilayer high-order complex networks, according to an embodiment of this application, is shown. Figure 2 This illustration shows the dynamic changes of a multi-layer high-order network model during fault injection and propagation in a method for evaluating the robustness of biomolecular networks based on multi-layer high-order complex networks, according to an embodiment of this application. Detailed Implementation

[0016] The technical solutions in the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments.

[0017] The components of the embodiments of this application described and illustrated in the accompanying drawings can be arranged and designed in a variety of different configurations. Therefore, the following detailed description of the embodiments of this application provided in the drawings is not intended to limit the scope of the claimed application, but merely to illustrate selected embodiments of the application. All other embodiments obtained by those skilled in the art based on the embodiments of this application without inventive effort are within the scope of protection of this application.

[0018] In the following text, the terms "comprising," "having," and their cognates, which may be used in various embodiments of this application, are intended only to indicate a particular feature, number, step, operation, element, component, or combination thereof, and should not be construed as primarily excluding the presence of one or more other features, numbers, steps, operations, elements, components, or combinations thereof, or adding the possibility of one or more combinations thereof. Furthermore, the terms "first," "second," "third," etc., are used only for distinguishing descriptions and should not be construed as indicating or implying relative importance.

[0019] Unless otherwise specified, all terms used herein (including technical and scientific terms) shall have the same meaning as commonly understood by one of ordinary skill in the art to which the various embodiments of this application pertain. Terms (such as those defined in commonly used dictionaries) shall be interpreted as having the same meaning as in their contextual meaning in the relevant technical field and shall not be construed as having an idealized or overly formal meaning, unless clearly defined in the various embodiments of this application.

[0020] The following detailed description of some embodiments of this application is provided in conjunction with the accompanying drawings. Unless otherwise specified, the following embodiments and features can be combined with each other.

[0021] Considering that existing multilayer interdependent network models fail to effectively incorporate higher-order interaction structures and reinforcement node mechanisms in robustness assessment, leading to discrepancies between assessment results and actual biomolecular responses, this paper proposes a method for assessing the robustness of biomolecular networks based on multilayer high-order complex networks. This method constructs a single-layer high-order hypergraph network model based on original biomolecular network data and preset parameters, then expands it into a multilayer interdependent hypergraph network. Fault injection and iterative simulation are then performed until the biomolecular network reaches a steady state. Finally, connectivity analysis yields the assessment results of the biomolecular network's robustness. This method not only realistically reflects the dynamic response of biomolecular networks under fault propagation but also accurately quantifies the connectivity of biomolecular networks, providing a scientific basis for optimizing biomolecular networks and improving their stability.

[0022] Figure 1A flowchart illustrating a method for evaluating the robustness of biomolecular networks based on multilayer high-order complex networks, according to an embodiment of this application, is shown. Exemplarily, the method includes the following steps: Step S100: Based on the original data and preset parameters of the biomolecular network, construct a single-layer high-order hypergraph network model.

[0023] Among them, the raw data of biomolecular networks refers to biomolecular entities (such as proteins, RNA, genes, etc.) and their interactions from experimental detection, professional databases or literature records. They are usually represented in the form of nodes-edges to show the collaborative relationships between functional units in a biological system. The preset parameters include attribute classification rules, normalization standards, higher-order relationship determination criteria and reinforcement status identification thresholds, which are used to standardize the data processing flow and the construction logic of the graph model.

[0024] The biomolecular multilayer network in this embodiment is, for example: (1) Gene regulatory network data: a network constructed with genes as nodes (usually including transcription factors and their target genes) and regulatory relationships (such as activation or inhibition) as edges. It is mainly used to describe how transcription factors affect the expression of other genes through regulatory effects, thereby revealing the mechanism of gene expression regulation.

[0025] (2) Protein-Protein Interaction Network (PPI) data: A network constructed with proteins as nodes and physical or functional interactions between proteins as edges, designed to reflect the cooperative relationships between proteins.

[0026] As an example, nodes and their interaction information are extracted from raw biomolecular network data; the characteristics of nodes are normalized using preset parameters and it is determined whether each node is in a reinforced state; according to higher-order interaction rules, hyperedges that can reflect the complex interaction relationships between multiple nodes are established; and a single-layer higher-order hypergraph network model is generated by comprehensively considering the attributes of nodes, reinforced state and hyperedge structure.

[0027] This model can effectively express the complex high-order interactions between biomolecules, providing basic input and data support for subsequent multilayer network construction, fault simulation, and robustness analysis.

[0028] In an optional embodiment, step S100 includes the following steps: S101 extracts biomolecular nodes and interaction information from the raw data of the biomolecular network.

[0029] Among them, biomolecular nodes represent biomolecular units with independent functions in the network, such as proteins or genes; interaction information includes whether there are interaction relationships between these molecules and their interaction strength, direction and other characteristics.

[0030] Specifically, the raw biomolecular data undergoes formatted preprocessing. First, data cleaning is performed to remove incomplete records and significant outliers. Then, all valid molecular entities are extracted as a node set, and their corresponding adjacency information is constructed to generate a preliminary node set and inter-node interaction matrix, providing a clear data structure foundation for subsequent processing.

[0031] S102, using preset parameters to standardize the biomolecular nodes and interaction information, and determine the properties and reinforcement status of the biomolecules.

[0032] Among them, node attributes are used to describe the biological function or state of biomolecules, such as expression level, stability, number of binding sites, etc.; the hardening state refers to whether the node is identified as a key node with high robustness in the network due to structural or functional factors.

[0033] In detail, the extracted node attributes are normalized to bring their values ​​within a uniform scale range. Then, each node is classified according to the set rules, such as functional grouping or activity status classification. The attribute values ​​and structural indicators are combined to evaluate whether the node has a hardened status marker, thereby obtaining a node set with attribute identification and robust label.

[0034] S103, construct a hyperedge set based on the higher-order interaction rules between biomolecular nodes.

[0035] Among them, higher-order interaction rules refer to not only considering the one-to-one interaction between binary nodes, but also identifying the collaborative interaction patterns in which multiple nodes participate simultaneously, in order to reflect the multi-party linkage behavior in complex biological mechanisms such as signaling pathways and gene regulatory networks; hyperedges refer to edge structures that connect multiple nodes.

[0036] Specifically, based on preset higher-order rules, a combination and filtering process is performed on the standardized node set. For node groups that satisfy the rules, corresponding hyperedges are created and added to the hyperedge set. Each hyperedge retains the node identifier and rule source, forming a well-structured higher-order interaction mapping. The construction result can be expressed using a hyperedge-node mapping table.

[0037] S104. Based on the attributes and reinforcement status of biomolecular nodes and the set of hyperedges, a single-layer high-order hypergraph network model is constructed.

[0038] Among them, the high-order hypergraph network model is an extension of graph theory structure, where nodes represent individual molecular units and hyperedges represent complex interactive relationships between multiple nodes; the hardening state in the model is recorded in the node attributes, serving as an important reference factor for subsequent network perturbation or robustness simulation.

[0039] Specifically, the standardized node set and the constructed hyperedge set are input into the graph structure building module to execute the graph construction process. During construction, each hyperedge is bound to its associated nodes, and labels are added to indicate node attributes, reinforcement status, and interaction characteristics, ultimately forming a unified structure. This model structure not only preserves the interaction information of the original network but also enhances the ability to express multi-node collaborative relationships, serving as a crucial foundation for subsequent multi-layer dependency modeling, fault injection simulation, and connectivity assessment.

[0040] In other words, considering that interactions between biomolecules often involve the combined effects of more than two entities, this step combines the standardized nodes according to higher-order interaction rules and constructs a set of hyperedges reflecting multi-agent synergistic effects. Here, "hyperedge" not only represents the association between two nodes, but also integrates multiple nodes that satisfy specific interaction conditions to form a higher-order relational structure describing complex biological signal transduction or gene expression regulation. The obtained standardized node information, attributes, reinforcement states, and hyperedge sets will be integrated according to a predetermined logic. During the integration process, the data flow converges from individual node attributes and hyperedge information to the overall network structure, generating a complete single-layer higher-order hypergraph network model. This model is represented by a graph theory structure: nodes represent biomolecules, and hyperedges describe the complex relationships of multi-agent interactions. The integrated model not only retains the key information of the original data, but also, through standardization and rule construction, forms a network model with a clear structure and functional description.

[0041] Step S200: Using a single-layer high-order hypergraph network model, extend and construct a multi-layer dependent hypergraph model.

[0042] Among them, the single-layer high-order hypergraph network model is a graph model that models the complex multi-agent relationships between biomolecules as a "node-hyperedge" structure, possessing the ability to express high-order interactions and local robustness. The multi-layer dependent hypergraph network model, on the other hand, introduces a mechanism of "multi-network layer expansion + cross-layer dependency connections" to simulate the coupling and dependency behavior between multiple functional subnetworks or time-series states in biological systems. By expanding the single-layer model into multiple independent network layers and establishing dependencies between nodes according to preset dependency rules, a well-defined, functionally cross-functional, and high-order multi-layer dependent hypergraph structure can be obtained, providing a more realistic network environment for subsequent fault propagation simulation and robustness assessment. Examples include random network models or scale-free network models.

[0043] In an optional embodiment, step S200 includes the following sub-steps: S201 extends the single-layer high-order hypergraph network model to generate multiple network layers.

[0044] In this context, a network layer refers to a graph structure unit with an independent set of nodes and hyperedges, used to represent the distribution of biomolecular networks under different functional states or temporal stages. The extension operation refers to generating multiple structurally equivalent network layer instances from the original single-layer network model, adhering to the principle of structure preservation.

[0045] Demonstratively, the model is extended according to predetermined rules to generate multiple independent but similar network layers. Each network layer retains the original structure of biomolecular nodes and hyperedges in the single-layer model, representing the component states of the biomolecular network under different functional regions, conditions, or time periods. This operation ensures consistent basic interaction relationships among multiple layers, laying the foundation for the subsequent introduction of inter-layer dependencies.

[0046] S202, according to the preset inter-layer dependency rules, assign dependency relationships to nodes of each network layer and generate a set of inter-layer dependency edges.

[0047] Inter-layer dependency rules refer to the strategies that specify which nodes in a network layer establish dependencies, including the proportion of dependent nodes, dependency directionality, and dependency matching strategies; the dependency edge set is used to represent the set of connections for such cross-layer node dependencies and is the key to building the coupling of multi-layer networks.

[0048] By way of example, nodes in each layer are analyzed according to these rules, and nodes with dependencies are "matched" between layers to generate a set of edges that reflect this cross-layer dependency. In this step, the allocation of dependency edges takes into account the synergy between biomolecules in different physiological functions or signal transduction processes, linking information from multiple layers to form a cross-layer dependency network.

[0049] For example, in hypergraph A, only the proportion is The nodes in graph A are randomly dependent on nodes in hypergraph B via dependency edges, while the proportion of autonomous nodes in hypergraph A is... The functionality of these autonomous nodes does not depend on the nodes in hypergraph B. Similarly, hypergraph B only has proportions. The nodes depend on the nodes of layer hypergraph A, and the proportion of autonomous nodes. The dependency ratio or the ratio of autonomous nodes can be between 0 and 1.

[0050] In hypergraphs A and B, a proportion is randomly selected respectively. and The nodes are designated as reinforcement nodes. For example, nodes from 0 to 1. Reinforcement nodes can support the normal operation of the hyperedge and its internal nodes, even if the hyperedge is separated from the large connected component of the hypergraph, the nodes inside the hyperedge can still function normally.

[0051] S203 integrates the structure of each network layer with the corresponding set of inter-layer dependency edges to construct a multi-layer dependent hypergraph network model.

[0052] Among them, the multilayer dependent hypergraph network model extends multiple network layers on the basis of the single-layer model. Each layer is a high-order hypergraph structure, and the dependencies between layers are established through preset rules, thereby reflecting the complex system model in which the layers are coupled with each other.

[0053] In detail, after completing the expansion of nodes at each layer and the construction of cross-layer dependency edges, the data structures and dependencies of all network layers are integrated. The resulting multi-layer dependent hypergraph network model retains the structural information built on high-order interactions within each network layer, and also includes the coupling relationships established between layers through dependency edges, thus comprehensively characterizing the interconnection characteristics of biomolecular networks at multiple levels and dimensions.

[0054] First, a multi-layered interdependent hypergraph network model reflecting biological regulatory mechanisms is constructed. The actual data used includes gene regulatory networks and protein-protein interaction networks, forming hypergraphs A and B respectively. After establishing cross-layer dependencies, a complete multi-layered network structure is formed. For example, a multi-layered high-order interdependent hypergraph network model is constructed using actual biological data. The Gene Regulatory Network (GRC) is used as hypergraph network A, with hyperedges constructed based on the regulatory or cooperative relationships between genes. The Protein-Protein Interaction Network (PPI network) is used as hypergraph network B, with hyperedges established through protein synergistic relationships. Between the two network layers, cross-layer dependencies are constructed by mapping protein-coding genes in the GRC network to their corresponding protein products in the PPI network, thus achieving network structure fusion. It should be noted that these protein products, especially key regulatory molecules such as transcription factors, can further regulate the behavior of gene layers by affecting the expression levels of target genes, thereby forming a bidirectional regulatory closed-loop structure in the biological system.

[0055] Step S300: Fault injection is performed on the multi-layer dependent hypergraph model according to the preset initial fault parameters, so that some nodes in the multi-layer dependent hypergraph model fail and form an initial fault state.

[0056] Fault injection refers to introducing a failure state into a manually set node range based on preset disturbance conditions, simulating the local network collapse process caused by factors such as attacks, damage, or functional loss in reality. The initial fault state represents the initial damaged structure formed by the injected fault before the network undergoes a dynamic propagation process. By setting the initial fault in the constructed multi-layer dependent hypergraph model, the external attack scenario suffered in the early stages of operation can be effectively recreated.

[0057] As an example, for a multi-layer dependent hypergraph network model, a fault injection operation is performed on the network using preset initial fault parameters, thereby setting some nodes of the originally fully running multi-layer dependent hypergraph network to a failed state, forming an initial fault state.

[0058] In an optional embodiment, step S300 includes the following sub-steps: Set initial fault parameters, randomly select some nodes in the multi-layer dependent hypergraph model according to the initial fault parameters, and mark some nodes as failed to form a multi-layer dependent hypergraph network model with some nodes in a failed state.

[0059] The "initial fault parameters" refer to the indicators pre-set before fault injection, including the proportion of nodes that fail in each network layer, the failure triggering conditions, and the corresponding thresholds, which are used to guide the fault injection operation. For example, the proportion of target nodes in the initial attack, the specific node selection strategy for the attack (random or specific nodes), and the "failure state" indicates that the node is in a state that cannot operate normally, which is a key indicator for simulating the network collapse process.

[0060] As an example, initial fault parameters are set. These parameters explicitly define the proportion of nodes that need to fail in each layer of the multilayer network and the specific conditions that trigger fault failure, such as node attributes, coupling relationships, or random probabilities. Then, in the already constructed multilayer dependent hypergraph network model, a subset of nodes is randomly selected according to the aforementioned initial fault parameters. This operation, based on a random sampling method, selects nodes within a preset proportion from each network layer, ensuring that the selected nodes meet the fault triggering conditions. Next, the selected nodes are uniformly marked as in a failed state. That is, the state attributes of these nodes are converted into "failed" identifiers, and the node state information in the multilayer dependent hypergraph network model is updated. This step causes a partial malfunction in the originally fully functioning network model, generating a new state where some nodes are failed. The updated network model represents the initial fault state after fault injection.

[0061] Step S400: Simultaneously perform iterative simulation on the multi-layer dependent network model of the initial fault state according to the preset fault propagation rules until the multi-layer dependent network model reaches a steady state.

[0062] The "fault propagation rule" defines how a fault spreads from a node to a hyperedge and then from the hyperedge to other nodes in a multi-layer dependent hypergraph model, achieving inter-layer state transfer through cross-layer dependency mechanisms. This propagation process simulates the spread and chain reaction of faults in real-world systems, helping to quantify the stability and robustness limits of the network under initial perturbations. By performing multiple rounds of state updates and checks on the multi-layer network model until the states of all nodes and hyperedges in the network remain unchanged, the network is considered to have reached a "steady state," meaning that further fault propagation no longer occurs.

[0063] Demonstratively, a multilayer interdependent network model under an initial fault state is iteratively simulated by applying pre-defined fault propagation rules. The network state is updated after each iteration until the states of all nodes and hyperedges remain unchanged, confirming that the network has reached a steady state. The core principle is to progressively update the states of each node and hyperedge in each iteration, and by checking whether the network is stable after each update, the response and repair process of a real biomolecular network under local faults is simulated. Finally, after multiple iterations, the network reaches a steady state, providing a stable data foundation for subsequent evaluation of network robustness.

[0064] In an optional embodiment, step S400 includes the following sub-steps: S401, based on the preset node-to-hyperedge, hyperedge-to-node, and cross-layer dependency fault propagation rules, update the state of all nodes and hyperedges in the multi-layer dependent network model.

[0065] Exemplary, the "node-to-hyperedge failure propagation rule" refers to the fact that when the state of a node changes (e.g., it fails), the hyperedge it participates in (representing higher-order interactions between multiple nodes) will also be affected, potentially leading to a state change; that is, when the proportion of failed nodes within a hyperedge exceeds a threshold... When the number of failed nodes in a hyperedge reaches a certain proportion, the entire hyperedge will fail to function properly.

[0066] The “Hyperedge-to-Node Failure Propagation Rule” states that when a hyperedge's state changes, the nodes contained within that hyperedge may fail due to the loss of key interaction relationships. However, if these nodes also belong to other still-valid hyperedges, their state may remain normal. In other words, when a hyperedge fails, some nodes within that hyperedge will fail because they are removed from the generalized massively connected component. However, if these nodes also belong to other functional hyperedges, they will not fail due to the failure of the current hyperedge. This means that the activity level of a node depends on whether the other hyperedges to which it belongs are still valid.

[0067] The "cross-layer dependency failure propagation rule" addresses the coupling effect established between different network layers in a multi-layer network model through dependencies. When a node in one layer fails, a node in another layer that depends on it will also fail. In other words, if a node in hypergraph A depends on a node in layer B, and the dependent node in hypergraph B fails, then the dependent node in hypergraph A will also immediately fail, and vice versa. This indicates that the dependencies between two hypergraph layers are reciprocal; the failure of a node in one layer may trigger the failure of a node in another layer that depends on it.

[0068] S402, check whether there are nodes or superedges that satisfy the fault propagation conditions in the updated multi-layer dependent network model, and change the state of the nodes or superedges that satisfy the fault propagation conditions.

[0069] "Meeting the propagation conditions" means that the nodes or hyperedges structurally conform to the propagation rules, such as: the failure rate of the hyperedges in which the node participates exceeds a threshold, or the state of the nodes it depends on changes to failure. The goal of this step is to identify key units that may still propagate the fault and avoid missing propagation paths.

[0070] As an example, the updated network is checked to see if any nodes or hyperedges still meet the failure propagation conditions. For instance, if the proportion of failed nodes in a hyperedge exceeds a set threshold, the hyperedge will be considered failed. Similarly, in inter-layer dependencies, if the failure of a node in one layer causes an abnormal state in a dependent node in another layer, these nodes will also be updated to be failed. At this point, the state of the parts that meet the conditions will be changed again.

[0071] S403, Repeat the state change and state check until the states of all nodes and hyperedges in the multi-layer dependent network model remain stable, confirming that the multi-layer dependent network model has reached a steady state.

[0072] "Repeated execution" means that the system continues to perform the next round of propagation simulation before the aforementioned update-check process reaches the termination condition; "State remains stable" means that no new nodes or changes in the state of superedges are added in the network between two consecutive rounds of state updates; "Steady state confirmation" is the sign of the termination of iteration, indicating that the propagation has ended.

[0073] As an example, the state update and check process is carried out in a cyclical iterative manner. The multilayer interdependent network model is confirmed to have reached a steady state only when the states of all nodes and hyperedges are stable and no component satisfies the conditions for continued propagation.

[0074] Step S500: Perform connectivity analysis on the multilayer interdependent network model that has reached steady state to obtain the robustness evaluation results of the biomolecular network.

[0075] Among them, "connectivity analysis" refers to evaluating the remaining structural connectivity of the entire system after a fault impact by calculating the connection relationships between nodes and hyperedges in the network under steady-state structure; "robustness evaluation results" are quantitative indicators that measure the network's ability to maintain information transmission and functional integrity under node failure or structural damage. "Multi-layer dependent network model" represents a multi-layer coupled structure system composed of multiple high-order hyperlayers through node dependencies, and its topology under steady-state reflects the remaining connectivity after fault propagation.

[0076] By example, by extracting the connectivity information between nodes and hyperedges in the steady-state structure and using percolation theory, the generalized maximum connected component and the maximum connected component are calculated, serving as a measure of the fault tolerance of biomolecular networks. This process not only identifies the connectivity of the network's residual structure but also reflects the support effect of reinforcement nodes and dependency mechanisms on connectivity, making it a key step in network robustness modeling and analysis.

[0077] In an optional embodiment, step S500 includes the following sub-steps: S501: Extract the connection information of nodes and hyperedges in each layer from the multilayer dependent network model that has reached steady state. Based on the connection information, use the percolation model to perform connectivity analysis and calculate the generalized maximum connected component and the maximum connected component in the multilayer dependent network model.

[0078] The percolation model is a connectivity analysis method based on percolation theory, used to study the connectivity changes of nodes and edges (or superedges) in a network under random failures or attacks, and to determine the scale of the network's connected components.

[0079] The largest connected component is the part of a network that is strictly interconnected among all nodes, and it is a traditional metric used to measure network connectivity.

[0080] The generalized maximum connected component is a connected component that, while not constituting the traditional global maximum connected component, still maintains connectivity in function, based on hardened nodes and local redundancy mechanisms. It can more realistically reflect the actual robustness of the network under fault conditions.

[0081] Exemplarily, from a steady-state multilayer dependent network model, each network layer is traversed to extract the connectivity relationships between nodes and hyperedges within each layer and across layers. This step ensures that the acquired data covers all nodes, hyperedges, and their dependencies, forming a complete connectivity information dataset. The extracted connectivity information is then input into a connectivity algorithm based on percolation theory to perform network connectivity analysis. Under simulated network random failure conditions, this algorithm calculates and determines the traditionally defined maximum connected component and the generalized maximum connected component including locally reinforced nodes by statistically analyzing the connection probabilities and dependencies between nodes. During this process, the algorithm progressively simulates the network structure evolution after failure propagation based on the connection probabilities and dependencies between nodes and hyperedges, and outputs the number or proportion of nodes in each component.

[0082] S502 uses the generalized maximum connected component and the maximum connected component as the network robustness evaluation results.

[0083] Among them, the "robustness assessment results" reflect the proportion of effective structure and connectivity that the network retains after node failure through the above two connectivity indicators, which are usually expressed in the form of node percentage, connectivity rate or structural integrity indicators.

[0084] Specifically, based on the two types of connected components obtained from connectivity calculations, the number of nodes contained in each component and their proportion to the total number of nodes in the overall network are statistically analyzed, and this ratio is output as a robustness evaluation index. The proportion of the largest connected component reflects the network's usable structure under the most conservative estimate, while the generalized largest connected component reflects the maximum functional structural range after considering reinforcement. Combining these two approaches provides robustness analysis results from a dual perspective.

[0085] In one possible implementation, the steady-state connectivity in a multilayer dependent network model can be modeled and analyzed based on the theory of generating functions and self-consistent equations. A numerical analytical scheme can then be used to solve for the generalized maximum connected component and the maximum connected component, thereby evaluating the network's robustness. (1) For the node connection structure in a multi-layer high-order hypergraph network, the generating function method is used to model the superdegree distribution and residual superdegree distribution of the nodes: Generating functions are tools in combinatorics used to characterize the distribution properties of sequences. In network modeling, they can be used to represent the hyperdegree distribution of nodes as an algebraic expression. In this embodiment, the hyperdegree distribution and residual hyperdegree distribution of nodes in a multi-layer high-order hypergraph network model are described by the following generating functions:

[0086] in: and Let represent the degree distribution generating functions of nodes in hypergraph A (or hypergraph B), respectively; This represents the probability of a node's hyperedge connection state. (or The number of hyperedges to which a node in hypergraph A (or hypergraph B) is attached represents the degree of that node.

[0087] or( The degree of a node in hypergraph A (or hypergraph B) is 1. (or The probability of () is the superdegree distribution of hypergraph A (or hypergraph B); here or The average extent of hypergraph A (or hypergraph B).

[0088] and Let represent the residual hyperdegree distribution generating functions of nodes in hypergraph A (or hypergraph B), respectively.

[0089] or Let be the residual extent distribution of hypergraph A (or hypergraph B).

[0090] (2) Construct and solve the self-consistent equations to obtain the scales of HMGGCC and HMRGCC: (2.1) A generalized huge connected component refers to a functional connected region formed by a finite small component containing a hardened node and a huge connected component. Its probabilistic expression is:

[0091] in: and These represent the probabilities of randomly selecting a node from hypergraph A (or hypergraph B) that belongs to a generalized giant connected component (HMGGCC); Indicates the initial attack hypergraph The probability of retaining nodes after the proportional nodes; (or ) represent the probabilities that a node in hypergraph A (or hypergraph B) is a reinforced node; (or ) represents the coupling strength, indicating the number of couplings in hypergraph A (or hypergraph B). (or The proportion of non-autonomous nodes depends on nodes in hypergraph B (or hypergraph A).

[0092] and Let represent the probabilities that a hyperedge in hypergraph A (or hypergraph B) reaches HMGGCC through a random node.

[0093] (2.2) In order to solve for the auxiliary probability representing that nodes in hypergraphs A and B are connected to the generalized maximum connected component HMGGCC through random hyperedges. and A self-consistent equation is constructed based on partial dependency conditions, hyperedge survival conditions, and connectivity conditions: First, under partial dependency conditions, if a node in hypergraph A is a non-consistent node, then when the corresponding node in the hypergraph B it depends on belongs to HMGGCC, that node can be considered still valid. Based on this, the following function is defined:

[0094] in, , These represent the proportions of reinforced nodes in hypergraphs A and B, respectively. and This is a function for generating the transcendence.

[0095] Secondly, under the condition of hyperedge survival, let the size of the hyperedge be m and the number of failed nodes in the hyperedge be n. When The superedge is considered still alive at this point. Nodes within the superedge exhibit the following four scenarios and their probabilities of being active: E3 node: Not subjected to the initial attack and is an autonomous node, with a probability of [missing value]. ; E4 node: Not subjected to the initial attack, not autonomous, but its dependent partner in graph B is connected to HMGGCC with a probability of [value missing]. ; E5 node: Subject to the initial attack, is an autonomous node, with a probability of [missing value]. ; E6 node: Subject to initial attack, non-autonomous, and dependent nodes fail, with a probability of [missing value]. .

[0096] Combining the states and probabilities of various nodes, the probability expression for the activity of a hyperedge in hypergraph A can be obtained as follows:

[0097] Similarly, the probability expression for the activity of a hyperedge in hypergraph B is:

[0098] in, It is a response function, if ,but ;like ,but .

[0099] Furthermore, if there are active nodes in a hyperedge, then at least one of those nodes must be connected to HMGGCC, and the connectivity probability is controlled by the residual hyperdegree distribution function: The probability that a node in hypergraph A is connected to HMGGCC is:

[0100] Similarly, in hypergraph B, the expression is:

[0101] Ultimately, self-consistent parameters and The expression is as follows:

[0102] By iteratively solving this set of self-consistent equations, the connectivity probability under steady state can be obtained. and And further used for network robustness evaluation and optimization.

[0103] Similarly, define This question asks for the probability that a randomly selected node in hypergraph A (hypergraph B) belongs to a massive connected component (HMRGCC) of enhanced inter-graph interconnections in hypergraph A (hypergraph B). The solution is to calculate... Introducing auxiliary parameters ,here Let represent the probability that the hyperedge reached from a random node in hypergraph A (hypergraph B) belongs to the HMRGCC of hypergraph A (hypergraph B). Therefore, analogous to... The calculation method, It can be represented as:

[0104] As can be seen from the definition, It also needs to satisfy partial dependency conditions, hyperedge survival conditions, and connectivity conditions. Its expression is as follows:

[0105] (3) Numerical simulation: After completing the self-consistent equation modeling, numerical methods were used for multiple iterative solutions. Simulations showed that as the value of p increases, the retention probability of nodes also increases, and more nodes in the network maintain normal operation. The generalized maximum connected component and the maximum connected component at steady state also increase, indicating that the network's robustness is enhanced.

[0106] When the value of p is less than a critical value, the network will not have a maximum connected component when it eventually reaches steady state. This means that after a severe attack, the network is unable to maintain basic connectivity and crashes.

[0107] When p is greater than or equal to the critical value, the maximum connected component starts to increase continuously from 0. The larger the maximum connected component, the stronger the robustness of the network. The network is better able to resist attacks or failures and maintain its functionality.

[0108] It is worth noting that coupling strength includes and B. When the coupling strength decreases, the retention probability of nodes in the dependency layer increases, which leads to an increase in the maximum connected component when the network reaches steady state, making the network system more robust.

[0109] In an optional embodiment, the method for evaluating the robustness of biomolecular networks based on multilayer high-order complex networks further includes: The robustness evaluation results are compared with the preset robustness target, and at least one key parameter in the multilayer dependent hypergraph network model is fine-tuned based on the comparison results until the multilayer dependent hypergraph network model reaches the preset robustness target.

[0110] In a demonstrative manner, the robustness assessment results obtained through percolation model analysis are compared with the preset robustness target to determine whether the current network's robustness level meets the design requirements. If the assessment result is lower than the target value, the key parameters in the model are adjusted according to the influence of various parameters on network sensitivity in the preset fault propagation mechanism. For example, the fault tolerance of some nodes or hyperedges during fault propagation can be enhanced by increasing the proportion of reinforced nodes or adjusting the allocation strategy of inter-layer dependent edges. The adjustment operation changes the model parameter input, thereby reconstructing and simulating the multilayer dependent hypergraph network model, which improves the connectivity of the network structure. Subsequently, after new fault injection, state update, and connectivity analysis, a new robustness assessment result is output; this result is then compared with the preset target value. If the target is still not met, the key parameters are fine-tuned, and the above closed-loop process is repeated. Finally, through continuous iteration, until all key parameters in the model are adjusted and the network robustness assessment result meets or exceeds the preset robustness target, the model is ensured to achieve the expected stability and connectivity level under fault conditions.

[0111] like Figure 2 As shown, a specific example is used to further explain the dynamic changes of the multi-layer high-order network model in the fault injection and propagation process of the method for evaluating the robustness of biomolecular networks based on multi-layer high-order complex networks in this application: Figure 2 (a) shows the initial state of the network, in which some nodes are rendered inoperable due to the initial attack, showing the location of the attack and the initially inoperable nodes. Figure 2 (b) and Figure 2Figure (c) shows the state after the fault begins to propagate. Some nodes and hyperedges fail due to the fault, and the hardened nodes prevent the fault from propagating further to some extent. Figure 2 Figure (d) shows the state of the network after it has reached steady state, where the maximum connected component (HMRGCC) is identified, showing the parts of the network that remain connected. Figure 2 (e) further analyzes the generalized maximum connected components (HMGGCC), including reinforced nodes and their supported hyperedges, highlighting the role of reinforced nodes in maintaining network connectivity. Figure 2 Figure (f) shows the final state of the network after fault propagation, and identifies HMRGCC and HMGGCC, demonstrating the network's robustness and connectivity. Figure 2 It visually demonstrates the dynamic changes and resilience of a network after an attack, highlighting the importance of hardened nodes in improving network robustness.

[0112] The above description is merely a specific embodiment of this application, but the scope of protection of this application is not limited thereto. Any changes or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application.

Claims

1. A method for evaluating the robustness of biomolecular networks based on multilayer high-order complex networks, characterized in that, The method includes: Based on the original data and preset parameters of biomolecular networks, a single-layer high-order hypergraph network model is constructed. Using the single-layer high-order hypergraph network model, a multi-layer dependent hypergraph model is constructed. Fault injection is performed on the multi-layer dependent hypergraph model according to preset initial fault parameters, so that some nodes in the multi-layer dependent hypergraph model fail and form an initial fault state. The multi-layer interdependent network model in the initial fault state is simultaneously iteratively simulated according to a preset fault propagation rule until the multi-layer interdependent network model reaches a steady state. Connectivity analysis was performed on the multilayer interdependent network model that reached steady state to obtain the robustness evaluation results of the biomolecular network.

2. The method for evaluating the robustness of biomolecular networks based on multilayer high-order complex networks according to claim 1, characterized in that, The method further includes: The robustness evaluation results are compared with the preset robustness target, and at least one key parameter in the multilayer dependent hypergraph network model is fine-tuned based on the comparison results until the multilayer dependent hypergraph network model reaches the preset robustness target.

3. The method for evaluating the robustness of biomolecular networks based on multilayer high-order complex networks according to claim 1, characterized in that, The construction of a single-layer high-order hypergraph network model based on the original data and preset parameters of biomolecular networks includes: Extract biomolecular nodes and interaction information from the raw data of the biomolecular network; The biomolecule nodes and interaction information are standardized using the preset parameters to determine the properties and reinforcement status of the biomolecules. Based on the higher-order interaction rules between the biomolecular nodes, construct a set of hyperedges; Based on the properties of the biomolecular nodes, the reinforcement state, and the set of hyperedges, a single-layer high-order hypergraph network model is constructed.

4. The method for evaluating the robustness of biomolecular networks based on multilayer high-order complex networks according to claim 1, characterized in that, The process of extending and constructing a multi-layer dependent hypergraph model using the single-layer high-order hypergraph network model includes: The single-layer high-order hypergraph network model is extended to generate multiple network layers. According to the preset inter-layer dependency rules, dependency relationships are assigned to the nodes of each network layer, and a set of inter-layer dependency edges is generated; The structure of each network layer is integrated with the corresponding set of inter-layer dependent edges to construct the multi-layer dependent hypergraph network model.

5. The method for evaluating the robustness of biomolecular networks based on multilayer high-order complex networks according to claim 1, characterized in that, The step of injecting faults into the multi-layer dependent hypergraph model according to preset initial fault parameters, so as to cause some nodes in the multi-layer dependent hypergraph model to fail and form an initial fault state, includes: Set the initial fault parameters; In the multi-layer dependent hypergraph model, some nodes are randomly selected according to the initial fault parameters, and these nodes are marked as failed, so as to form a multi-layer dependent hypergraph network model in which some of the nodes are in a failed state.

6. The method for evaluating the robustness of biomolecular networks based on multilayer high-order complex networks according to claim 1, characterized in that, The multilayer interdependent network model of the initial fault state is iteratively simulated simultaneously according to a preset fault propagation rule until the multilayer interdependent network model reaches a steady state, including: Based on the preset node-to-hyperedge, hyperedge-to-node, and cross-layer dependency fault propagation rules, the state of all nodes and hyperedges in the multi-layer dependent network model is updated. Check whether there are nodes or superedges that satisfy the fault propagation conditions in the updated multilayer interdependent network model, and change the state of the nodes or superedges that satisfy the fault propagation conditions. Repeat the state change and state check until the states of all nodes and hyperedges in the multi-layer dependent network model remain stable, confirming that the multi-layer dependent network model has reached a steady state.

7. The method for evaluating the robustness of biomolecular networks based on multilayer high-order complex networks according to claim 1, characterized in that, The connectivity analysis performed on the multilayer interdependent network model that has reached a steady state yields network robustness evaluation results, including: The connection information of nodes and hyperedges in each layer are extracted from the multilayer interdependent network model that has reached a steady state; Based on the connection information, a percolation model is used to perform connectivity analysis, and the generalized maximum connected component and the maximum connected component in the multilayer interdependent network model are calculated. The generalized maximum connected component and the maximum connected component are used as the network robustness evaluation results.

Images