CN117544541A - Higher-order network cascading failure research method and system based on simplex complex - Google Patents
Higher-order network cascading failure research method and system based on simplex complex Download PDFInfo
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Abstract
The invention relates to the technical field of cascade fault analysis, in particular to a simplex-based high-order network cascade fault research method and system. And calculating connectivity of networks with different node capacity parameters, networks with different 1-simplex and 2-simplex weight coefficients and networks with different 2-simplex numbers, further evaluating robustness of the simplex high-order network, and performing cascading failure research on the simplex high-order network.
Description
Technical Field
The invention relates to the technical field of cascade fault analysis, in particular to a method and a system for researching a high-order network cascade fault based on a simplex complex.
Background
Cascading failure is a propagation diffusion phenomenon commonly found in infrastructure networks. Sudden natural disasters or artificial damages in the network may cause node failures in the network, further resulting in a breakdown of the entire network. The existing research results show that the cascade fault processes of networks with different structures are also greatly different. Most of the current research methods are developed on binary interaction networks, and simultaneous interaction of three or more nodes is not considered. However, with the development of socioeconomic performance, real-time interactions often involve multiple participants.
In an actual cascade failure occurrence, it is contemplated that load spreading of the failed node may occur in different ways, either through pairwise interactions or through group interactions, i.e., through higher order structures. The research on the cascade fault process of the simplex network can help technicians to fully recognize the cascade fault behavior occurring on the simplex network, and provides a reference for improving the robustness of a real network system. Therefore, the simplex cascade fault research method has important practical significance and reference value. The existing research method for the high-order network cascading failure behavior is only suitable for a common network or a super network, and the research method for the high-order network cascading failure behavior based on simple complex is lacking.
Disclosure of Invention
In view of the above, the present invention aims to provide a method and a system for researching a simplex-based high-order network cascading failure, so as to solve the problem that a research method for researching a simplex-based high-order network cascading failure behavior is lacking at present.
In order to achieve the above purpose, the invention adopts the following technical scheme:
in one aspect, the present application provides a method for researching a higher-order network cascade fault based on simplex, including:
s1, constructing a higher-order network which is based on a preference connection mechanism and only comprises 2-simplex according to a first algorithm;
s2, adding 1-simplex to the higher-order network only comprising 2-simplex according to a second algorithm to generate a simplex complex higher-order network;
s3, establishing a capacity-load cascade fault model;
s4, calculating the generalized degree of each node in the simplex higher-order network according to the number of 1-simplex and 2-simplex around each node in the simplex higher-order network through the capacity-load cascade fault model, and setting weight coefficients for the 1-simplex and the 2-simplex;
s5, calculating initial loads of all nodes in the simplex high-order network according to the generalized degree, the 1-simplex and the weight coefficient of the 2-simplex;
s6, calculating loads received by the 1-simplex node associated with the failed node and loads received by the 2-simplex node associated with the failed node when the node fails;
s7, calculating the total load of each node in the simplex higher-order network according to the load received by the 1-simplex node associated with the failed node and the load received by the 2-simplex node associated with the failed node;
s8, calculating the capacity of each node in the simplex complex higher-order network according to the capacity parameter;
s9, judging whether the nodes in the simplex high-order network fail or not according to the total load and capacity of each node in the simplex high-order network;
s10, if yes, returning to the step S6;
s11, if not, calculating connectivity of the simplex complex higher-order network;
s12, respectively changing node capacity parameters, weight coefficients of 1-simplex and 2-simplex and the number of 2-simplex in the simplex higher-order network;
s13, respectively calculating connectivity of the simplex complex higher-order network of different node capacity parameters, the simplex complex higher-order network of different 1-simplex and 2-simplex weight coefficients and the simplex complex higher-order network of different 2-simplex numbers through the capacity-load cascade fault model;
s14, evaluating the robustness of the simplex high-order network according to the connectivity of the simplex high-order network of the different node capacity parameters, the simplex high-order network of different 1-simplex and 2-simplex weight coefficients and the simplex high-order network of different 2-simplex numbers, and performing cascading failure research on the simplex high-order network.
Further, in the method described above, the step S1 of constructing a higher-order network based on a preferred connection mechanism and only including a 2-simplex according to a first algorithm includes:
at the initial moment, a network with only 3 nodes is established, and the 3 nodes are subjected to full connection operation to form a higher-order network containing a 2-simplex; wherein, the initial time is t=0;
adding a new node into a higher-order network formed at the previous moment when the next moment is reached, selecting an edge in the higher-order network formed at the previous moment according to the priority connection probability, and connecting two end points of the selected edge with the new node to generate a new 2-simplex;
at the time of T1, T1 nodes are newly added in the higher-order network of the 2-simplex, and the higher-order network only containing the 2-simplex with the total number of nodes being T1+3 is generated.
Further, in the method described above, the step S2 of adding 1-simplex to the higher-order network including only 2-simplex according to the second algorithm to generate a simplex higher-order network includes:
at the beginning time, adding a new node in the higher-order network only containing 2-simplex, selecting a node in the higher-order network only containing 2-simplex according to the priority connection probability, and connecting the selected node with the new node to generate a new 1-simplex; wherein the starting time is t=0;
adding a new node into a higher-order network formed at the previous moment when the next moment is reached, selecting a node in the higher-order network formed at the previous moment according to the priority connection probability, and connecting the selected node with the new node to generate a new 1-simplex;
at time T2, T2+1 nodes are newly added in the higher-order network only containing 2-simplex, and the simplex complex higher-order network only containing 1-simplex and 2-simplex with the total number of nodes being T1+T2+4 is generated.
Further, in the above method, the step S5 of calculating the initial load of each node in the simplex higher-order network according to the generalized degree and the weight coefficients of the 1-simplex and the 2-simplex includes:
according to the generalized degree, the 1-simplex and the weight coefficient of the 2-simplex, calculating the initial load of each node in the simplex higher-order network through an initial load calculation formula;
the initial load calculation formula is as follows:
R 1 is the weight coefficient of 1-simplex, R 2 Is a weighting coefficient of 2-simplex.
Further, the method described above, the step S6 of calculating, when a node fails, a load received by a 1-simplex node associated with the failed node and a load received by a 2-simplex node associated with the failed node, including:
when a node fails, respectively calculating the load received by the 1-simplex node associated with the failed node and the load received by the 2-simplex node associated with the failed node through a 1-simplex node load receiving calculation formula and a 2-simplex node load receiving calculation formula;
the 1-simplex node load receiving calculation formula is as follows:
wherein L is i (t) is the load of the node i at time t;
the 2-simplex node load receiving calculation formula is as follows:
further, in the method described above, the step S7 of calculating a total load of each node in the simplex higher-order network according to the load received by the 1-simplex node associated with the failed node and the load received by the 2-simplex node associated with the failed node includes:
according to the load received by the 1-simplex node associated with the failed node and the load received by the 2-simplex node associated with the failed node, calculating the total load of each node in the simplex higher-order network through a 1-simplex node total load calculation formula and a 2-simplex node total load calculation formula;
the 1-simplex node total load calculation formula is as follows:
L j (t)＝L j (t-1)+ΔL i→j (t)#(4)
wherein L is i (t-1) is the load of the node j at time t-1, ΔL i→j (t) is the load from the node i received by the node j at time t;
the 2-simplex node total load calculation formula is as follows:
L k (t)＝L k (t-1)+ΔL i→k (t)#(5)
wherein L is k (t-1) is the load of the node k at time t-1, ΔL i→k And (t) is the load from the node i received by the node k at the moment t.
Further, in the above method, the step S8 of calculating the capacity of each node in the simplex complex higher-order network according to the capacity parameter includes:
acquiring capacity parameters of each node in the simplex complex higher-order network;
according to the capacity parameter, calculating the capacity of each node in the simplex complex higher-order network through a node capacity calculation formula;
the node capacity calculation formula is as follows:
C j ＝(1+T)L j T≥0#(6)
wherein T is a capacity parameter, L j Is the load of node j.
Further, the above method is characterized in that the step S9 of determining whether the node in the simplex higher-order network fails according to the total load and capacity of each node in the simplex higher-order network includes:
acquiring the total load and capacity of each node in the simplex high-order network, and judging whether the node in the simplex high-order network fails or not according to a failure inequality and the total load and capacity of each node;
wherein the inequality is:
L j (t-1)+ΔL i→j (t)>C j #(8)。
on the other hand, the application also provides a high-order network cascading failure research system based on simple complex, which comprises a processor and a memory, wherein the processor is connected with the memory:
the processor is used for calling and executing the program stored in the memory;
the memory is used for storing the program, and the program is at least used for executing the high-order network cascading failure research method based on the simplex complex.
The beneficial effects of the invention are as follows:
in the method, a simplex high-order network with 1-simplex and 2-simplex is firstly established according to a preference mechanism, then a capacity-load cascade fault model is established, initial loads of all nodes in the simplex high-order network are calculated through the capacity-load cascade fault model, when the nodes fail, loads received by the 1-simplex nodes associated with the failed nodes, loads received by the 2-simplex nodes associated with the failed nodes and total loads of all the nodes in the simplex high-order network are calculated, capacity of all the nodes is calculated according to capacity parameters, whether all the nodes fail is judged through the total loads and the capacity of all the nodes, and when the nodes fail, the total loads of all the nodes are recalculated, whether the simplex high-order network fails again is judged until no node fails, and connectivity of the simplex high-order network is calculated. And calculating connectivity of simplex high-order networks of different node capacity parameters, simplex high-order networks of different 1-simplex and 2-simplex weight coefficients and simplex high-order networks of different 2-simplex numbers, further evaluating robustness of the simplex high-order networks, and carrying out cascade fault research on the simplex high-order networks.
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In order to more clearly illustrate the embodiments of the invention or the technical solutions in the prior art, the drawings that are required in the embodiments or the description of the prior art will be briefly described, it being obvious that the drawings in the following description are only some embodiments of the invention, and that other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a flow chart provided by one embodiment of a simplex-based high-order network cascading failure research method of the present invention;
FIG. 2 is a schematic diagram of a high-order network cascade fault research system based on simplex, according to an embodiment of the present invention;
FIG. 3 is a schematic diagram of a simplex structure provided by an embodiment of a simplex-based high-order network cascade fault study method according to the present invention;
fig. 4 is a schematic diagram of simulation results provided by an embodiment of a method for researching a high-order network cascading failure based on simplex.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the technical solutions of the present invention will be described in detail below. It will be apparent that the described embodiments are only some, but not all, embodiments of the invention. All other embodiments, based on the examples herein, which are within the scope of the invention as defined by the claims, will be within the scope of the invention as defined by the claims.
Fig. 1 is a flowchart provided by an embodiment of a simplex-based high-order network cascading failure research method. Referring to fig. 1, the present embodiment may include the following steps:
s1, constructing a higher-order network which is based on a preference connection mechanism and only comprises 2-simplex according to a first algorithm;
s2, adding 1-simplex to a higher-order network only containing 2-simplex according to a second algorithm to generate a simplex complex higher-order network;
s3, establishing a capacity-load cascade fault model;
s4, calculating the generalized degree of each node in the simplex high-order network according to the quantity of 1-simplex and 2-simplex around each node in the simplex high-order network through a capacity-load cascade fault model, and setting weight coefficients for the 1-simplex and 2-simplex;
s5, calculating initial loads of all nodes in the simplex higher-order network according to the generalized degree, the 1-simplex and the weight coefficient of the 2-simplex;
s6, calculating loads received by the 1-simplex node associated with the failed node and loads received by the 2-simplex node associated with the failed node when the node fails;
s7, calculating the total load of each node in the simplex higher-order network according to the load received by the 1-simplex node associated with the failed node and the load received by the 2-simplex node associated with the failed node;
s8, calculating the capacity of each node in the simplex complex higher-order network according to the capacity parameters;
s9, judging whether the nodes in the simplex high-order network fail or not according to the total load and capacity of each node in the simplex high-order network;
s10, if yes, returning to the step S6;
s11, if not, calculating connectivity of the simplex complex higher-order network;
s12, respectively changing node capacity parameters, weight coefficients of 1-simplex and 2-simplex and the number of 2-simplex in a simplex complex higher-order network;
s13, calculating connectivity of simplex high-order networks of different node capacity parameters, simplex high-order networks of different 1-simplex and 2-simplex weight coefficients and simplex high-order networks of different 2-simplex numbers respectively through a capacity-load cascade fault model;
s14, evaluating the robustness of the simplex high-order network according to the connectivity of the simplex high-order network of different node capacity parameters, the simplex high-order network of different 1-simplex and 2-simplex weight coefficients and the simplex high-order network of different 2-simplex numbers, and performing cascading failure research on the simplex high-order network.
It can be understood that in the application, a simplex high-order network with 1-simplex and 2-simplex is firstly established according to a preference mechanism, then a capacity-load cascade fault model is established, initial loads of all nodes in the simplex high-order network are calculated through the capacity-load cascade fault model, when a node fails, loads received by the 1-simplex node associated with the failed node, loads received by the 2-simplex node associated with the failed node and total loads of all nodes in the simplex high-order network are calculated, capacity of all the nodes is calculated according to capacity parameters, whether all the nodes fail is judged through the total loads and the capacity of all the nodes, and when the node fails, the total loads of all the nodes are recalculated, whether the simplex high-order network fails again until no node fails, and connectivity of the simplex high-order network is calculated. And calculating connectivity of simplex high-order networks of different node capacity parameters, simplex high-order networks of different 1-simplex and 2-simplex weight coefficients and simplex high-order networks of different 2-simplex numbers, further evaluating robustness of the simplex high-order networks, and carrying out cascade fault research on the simplex high-order networks.
It should be noted that, as shown in FIG. 3, FIG. 3 shows 0-simplex, 1-simplex, 2-simplex, and 3-simplex. Simplex: the d-dimensional simplex α is formed by a set of (d+1) interaction nodes. α= [ v0, v1, v2...vd]. Wherein the nodes are also 0-simplex, the connecting edge between two nodes is also called 1-simplex, and three nodesThe triangle formed by two points connected in pairs is 2-simplex, the tetrahedron formed by the nodes is 3-simplex, and so on. Simplex surface: the surface a' of the d-dimensional simplex a is a simplex formed by a suitable subset of nodes of the simplex a, e.g. a 2-simplex contains 3 0-simplex and 3 1-simplex, which are all surfaces of the 2-simplex. Simplex complex: simplex k is a structure formed by bonding a set of simplex. Dimension of simplex complex: the dimension of a simplex complex is the largest dimension that makes up its simplex. Generalized degree: generalized degree k of m-plane alpha in d-dimensional simplex complex d,m (α) is given by the number of d-dimensional simplex associated to the m-plane.
Preferably, step S1 includes:
at the initial moment, a network with only 3 nodes is established, and the 3 nodes are fully connected to form a higher-order network containing a 2-simplex; wherein, the initial time is t=0;
adding a new node into the higher-order network formed at the previous moment when the next moment is reached, selecting an edge in the higher-order network formed at the previous moment according to the priority connection probability, and connecting two endpoints of the selected edge with the new node to generate a new 2-simplex;
at the time of T1, T1 nodes are newly added in the higher-order network of the 2-simplex, and a higher-order network which only contains the 2-simplex and has the total number of the nodes of T1+3 is generated.
Preferably, step S2 includes:
at the beginning time, adding a new node in the higher-order network only containing 2-simplex, selecting a node in the higher-order network only containing 2-simplex according to the priority connection probability, and connecting the selected node with the new node to generate a new 1-simplex; wherein the starting time is t=0;
adding a new node into the higher-order network formed at the previous moment when the next moment is reached, selecting a node in the higher-order network formed at the previous moment according to the priority connection probability, and connecting the selected node with the new node to generate a new 1-simplex;
at the time T2, a T2+1 node is newly added in the higher-order network only containing 2-simplex, and a simplex complex higher-order network only containing 1-simplex and 2-simplex, the total number of which is T1+T2+4, is generated.
In specific practice, a high-order network evolution algorithm based on a preference mechanism and comprising 1,2 simplex is specifically: constructing a higher-order network only comprising 2-simplex according to a first algorithm; adding 1-simplex to the network constructed by the first algorithm according to the second algorithm, and finally generating a higher-order network containing 1-simplex and 2-simplex.
The method comprises the steps of constructing a higher-order network only comprising 2-simplex according to a first algorithm, wherein the higher-order network comprises the following specific steps:
step1: at the initial time t=0, 3 nodes in the simplex complex network are set, and the 3 nodes are fully connected to form a higher-order network containing a 2-simplex;
step2: when t=1, adding a new node into the higher-order network generated in the previous step, and connecting the probability pi with the following priority ij Selecting one edge from the existing edges of the higher-order network, and connecting two end points of the one edge with newly added nodes to generate a new 2-simplex;
wherein k is 2,1 (ij) represents the number of 2-simplex involved in edge (i, j),representing the sum of the number of 2-simplex involved for all edges in the higher order network.
Step3: step2 is repeated to enable the total evolution time of the higher-order network to reach T1 (T1 > 1), namely T1 nodes are added, and finally, a higher-order network which is N=3+T1 in total number and only comprises 2-simplex is generated.
According to a second algorithm, adding 1-simplex in a higher-order network only comprising 2-simplex to generate a simplex higher-order network, wherein the method specifically comprises the following steps:
step1: at the initial time t=0, adding a new node to the higher-order network generated by the first algorithm, and connecting the probability pi with the following priority i Selecting an existing node from the higher-order network, and connecting the node with the newly added node to generate a new 1-simplex;
wherein k is 2,0 (i) Representing the number of 2-simplex involved in node i,representing the sum of the number of 2-simplex s involved by all nodes in the simplex network.
step2: when t=1, adding a new node to the higher-order network generated in the first step, and connecting the probability pi with the following priority i ' selecting a node from existing nodes of a higher-order network, and connecting the node with newly added nodes to generate a new 1-simplex;
wherein k is 2,0 (i) Representing the number, k, of 2-simplex involved in node i 1,0 (i) Representing the number of 1-simplex involved in node i,representing the sum of the numbers of 1-simplex and 2-simplex that all nodes participate in the simplex network.
step3: step2 is repeated to enable the total evolution time of the high-order network to reach T2 (T2 > 1), namely T2 nodes are additionally arranged, and finally a high-order network with the total number of nodes of N=4+T1+T2 and containing 1-simplex and 2-simplex is generated.
Preferably, step S5 preferably includes:
according to the generalized degree, the 1-simplex and the weight coefficient of the 2-simplex, calculating the initial load of each node in the simplex higher-order network through an initial load calculation formula;
the initial load calculation formula is as follows:
R 1 is the weight coefficient of 1-simplex, R 2 Is a weighting coefficient of 2-simplex.
Preferably, step S6 includes:
when a node fails, respectively calculating the load received by the 1-simplex node associated with the failed node and the load received by the 2-simplex node associated with the failed node through a 1-simplex node load receiving calculation formula and a 2-simplex node load receiving calculation formula;
the 1-simplex node load receiving calculation formula is as follows:
wherein L is i (t) is the load of the node i at time t;
the 2-simplex node load receiving calculation formula is as follows:
preferably, step S7 includes:
according to the load received by the 1-simplex node associated with the failed node and the load received by the 2-simplex node associated with the failed node, calculating the total load of each node in the simplex higher-order network through a 1-simplex node total load calculation formula and a 2-simplex node total load calculation formula;
the 1-simplex node total load calculation formula is as follows:
L j (t)＝L j (t-1)+ΔL i→j (t)#(4)
wherein L is i (t-1) is the load of the node j at time t-1, ΔL i→j (t) is the load from the node i received by the node j at time t;
the 2-simplex node total load calculation formula is as follows:
L k (t)＝L k (t-1)+ΔL i→k (t)#(5)
wherein L is k (t-1) is the load of the node k at time t-1, ΔL i→k And (t) is the load from the node i received by the node k at the moment t.
Preferably, step S8 includes:
acquiring capacity parameters of each node in the simplex complex higher-order network;
according to the capacity parameter, calculating the capacity of each node in the simplex complex higher-order network through a node capacity calculation formula;
the node capacity calculation formula is as follows:
C j ＝(1+T)L j T≥0#(6)
wherein T is a capacity parameter, L j Is the load of node j.
Preferably, step S9 includes:
acquiring the total load and capacity of each node in the simplex high-order network, and judging whether the node in the simplex high-order network fails or not according to a failure inequality and the total load and capacity of each node;
wherein the inequality is:
L j (t-1)+ΔL i→j (t)>C j #(8)。
it will be appreciated that the capacity-load cascade fault model includes:
at the initial moment, the calculation formula of the node initial load:
R 1 is the weight coefficient of 1-simplex, R 2 Weight coefficients that are 2-simplex;
at time t, when the node i fails, the load calculation formula received by the node j in all the non-failed 1-simplex related to the node i is as follows:
at time t, when the node i fails, the load calculation formula received by the node k in all the non-failed 2-simplex related to the node i is as follows:
if the node j in the 1-simplex associated with the node i at the time t is not invalid, the calculation formula of the load of the node j is as follows:
L j (t)＝L j (t-1)+ΔL i→j (t)#(4)
wherein L is i (t-1) is the load of node j at time t-1, ΔL i→j (t) is the load from node i received by node j at time t;
if node k in the 2-simplex associated with node i at time t is not invalid, the calculation formula of the load of node k is as follows:
L k (t)＝L k (t-1)+ΔL i→k (t)#(5)
wherein L is k (t-1) is the load of node k at time t-1, ΔL i→k (t) is the load from node i received by node k at time t;
capacity C of node j j The method comprises the following steps:
C j ＝(1+T)L j T≥0#(6)
wherein T is a capacity parameter, L j Is the load of node j;
capacity C of node k k The method comprises the following steps:
C k ＝(1+T)L k T≥0#(7)
wherein L is k Is the load of node k.
If node j fails after receiving the load from node i, the following inequality needs to be satisfied:
L j (t-1)+ΔL i→j (t)>C j #(8)
if node k fails after receiving the load from node i, the following inequality needs to be satisfied:
after the network, in order to better describe the influence of the higher-order network structure on the robustness thereof, based on a classical capacity-load model, a new capacity-load model is proposed herein for analyzing the influence of simplex of different dimensions on the robustness of the higher-order network and for analyzing the relation between the higher-order network structure and the robustness thereof. Initial load of node: initial load of node i and generalized degree k of the node 1,0 (i)、k 2,0 (i) Concerning, its initial load L i (0) As shown in formula (1). Whereas the generalized degree k of the node 1,0 (i)、k 2,0 (i) Specifically the number of 1-simplex and 2-simplex connected to a node. In order to control the initial load of the node i, let α be the load parameter and β be the adjustable parameter. From (1), it is known that the node initial load and the node generalized degree k 1,0 (i)、k 2,0 (i) Proportional to the ratio.
Let the numbers of 1-simplex and 2-simplex around node i be k, respectively 1,0 (i)、k 2,0 (i) A. The invention relates to a method for producing a fibre-reinforced plastic composite To analyze the impact of the importance of 1-simplex, 2-simplex on network robustness, a weighting coefficient of 1-simplex is defined as R 1 The weighting coefficient of the 2-simplex is defined as R 2 。
At time t, when node i fails due to failure, it is first loaded with L i Will follow the weighted generalized degree R of the failed node i 1 *k 1,0 (i)+R 2 *k 2,0 (i) Aliquoting and distributing the aliquoted load to all non-failed 1-simplex and 2-simplex associated therewith, node j of all non-failed 1-simplex associated with node i is connectedThe received load is shown in formula (2). The load received by each node k in all non-failed 2-simplex associated with node i is shown in equation (3).
As can be seen from the formula (2), the extra load delta L received by the non-failure node j in the 1-simplex at the time t i→j Generalized degree k of failure node i between (t) and t 1,0 (i)、k 2,0 (i) 1-simplex weight coefficient R1, 2-simplex weight coefficient R2, and the initial load of node i. As can be seen from the formula (3), the extra load DeltaL received by the non-failure node k in the 2-simplex at the time t i→k Generalized degree k of failure node i between (t) and t 1,0 (i)、k 2,0 (i) 1-simplex weight coefficient R1, 2-simplex weight coefficient R2, and the initial load of node i.
If the node j in the 1-simplex associated with the node i at the time t is not failed, the load of the node j at the time t-1 is added with the load redistributed from the failed node i, as shown in the formula (4), and if the node k in the 2-simplex associated with the node i at the time t is not failed, the load of the node k at the time t-1 is added with the load redistributed from the failed node i, as shown in the formula (5).
In a real higher order network, the capacity is the maximum value of the load that a node or an overtravel can handle, and is proportional to the initial load of the node. Let the capacity of node j be C j Let the capacity of node k be C k Represented by formulas (6) and (7).
Where T is a capacity parameter, the larger the value of T, the larger the capacity of the node, the stronger the capability of resisting faults, but the higher the resisting cost. The critical threshold TC is the minimum capacity value that avoids the higher order network from global crashes. When T > TC, no global crash occurs throughout the higher order network. When T < TC, a global crash of the entire higher order network may occur. Therefore, the critical threshold TC of T is an important indicator for measuring the robustness of the higher order network. Clearly, the smaller the TC, the more robust the higher order network.
If node j, node k fails after obtaining the additional load, it should satisfy equations (8) and (9). If the formulas (8) and (9) are satisfied, the nodes j and k will be overloaded and fail, and after the loads of the nodes j and k are redistributed, other nodes may fail.
In specific practice, to analyze the impact of the importance of a 1-simplex, 2-simplex on the robustness of a higher order network, wherein the 1-simplex weight coefficient is R 1 The 2-simplex weight coefficient is R 2 . R1=r2, R1 are performed separately>R2、R1<Experiment of R2. The parameters r1=1, r2=0.2, r2=0.5, r2=1, r2=2, r2=5 were chosen in the experiment. The experimental results are shown in FIG. 4.
As can be seen from fig. 4, when the parameter r1 > R2, i.e. r1=1, r2=0.2, r2=0.5, the robustness of the higher order network is stronger, when the parameter r1=r2, i.e. r1=1, r2=1, the robustness of the higher order network is moderate, and when R1< R2, i.e. r2=2, r2=5, the robustness of the higher order network is worse, and as the 2-simplex weight coefficient decreases, the higher order network becomes more robust.
From this, the more important the 1-simplex in the higher order network, the more robust the higher order network, the more important the 2-simplex in the higher order network, and the more fragile the higher order network. Therefore, the robustness of the higher-order network can be improved by setting the parameter R2 to be less than or equal to R1, and if R2 is more than R1, the difference between R2 and R1 is reduced as much as possible, so that the robustness of the higher-order network is improved.
As is known from equation (2), the larger the higher order network capacity parameter T, the more capable each node is to handle the load. However, in a real higher order network, the capacity of each node to handle the load is generally limited by cost, so the capacity parameter T cannot be infinitely increased.
The embodiment generates a higher-order network comprising 1-simplex and 2-simplex according to a preference mechanism, and sets the scale of the higher-order network as N. Meanwhile, by using the method of the embodiment, the cascading failure process of the high-order network constructed by the invention is analyzed through Python software simulation. In simulation analysis, analyzing the change of the proportion of fault nodes along with the capacity parameter T under different scales of the high-order network; under different attack modes, the proportion of fault nodes changes along with the change of the capacity parameter T; under different parameters R1 and R2, the proportion of fault nodes changes along with the change of the capacity parameter T; the proportion of failed nodes varies with the capacity parameter T at different numbers of 2-simplex.
As shown by simulation experiment results, the lower the number of 2-simplex in the higher-order network constructed by the embodiment is, the more robust the higher-order network is; the more robust the higher order network is when the weight coefficient of the 2-simplex is smaller than the weight coefficient of the 1-simplex. Furthermore, the larger the scale of the higher order network, the better its robustness to random attacks and intentional attacks.
The invention also provides a high-order network cascading failure research system based on the simplex complex, which is used for realizing the method embodiment. Fig. 2 is a schematic structural diagram of a high-order network cascading failure research system based on simplex complex according to an embodiment of the present invention. As shown in fig. 2, the simplex-based high-order network cascade fault research system of the present embodiment includes a processor 21 and a memory 22, where the processor 21 is connected to the memory 22. Wherein the processor 21 is used for calling and executing the program stored in the memory 22; the memory 22 is used to store the program at least for executing the simplex-based high-order network cascade fault study method in the above embodiment.
Specific implementation manners of the simplex-based high-order network cascading failure research system provided in this application may refer to implementation manners of the simplex-based high-order network cascading failure research method in any of the above embodiments, and are not described herein again.
It is to be understood that the same or similar parts in the above embodiments may be referred to each other, and that in some embodiments, the same or similar parts in other embodiments may be referred to.
It should be noted that in the description of the present invention, the terms "first," "second," and the like are used for descriptive purposes only and are not to be construed as indicating or implying relative importance. Furthermore, in the description of the present invention, unless otherwise indicated, the meaning of "plurality" means at least two.
Any process or method descriptions in flow charts or otherwise described herein may be understood as representing modules, segments, or portions of code which include one or more executable instructions for implementing specific logical functions or steps of the process, and further implementations are included within the scope of the preferred embodiment of the present invention in which functions may be executed out of order from that shown or discussed, including substantially concurrently or in reverse order, depending on the functionality involved, as would be understood by those reasonably skilled in the art of the present invention.
It is to be understood that portions of the present invention may be implemented in hardware, software, firmware, or a combination thereof. In the above-described embodiments, the various steps or methods may be implemented in software or firmware stored in a memory and executed by a suitable instruction execution system. For example, if implemented in hardware, as in another embodiment, may be implemented using any one or combination of the following techniques, as is well known in the art: discrete logic circuits having logic gates for implementing logic functions on data signals, application specific integrated circuits having suitable combinational logic gates, programmable Gate Arrays (PGAs), field Programmable Gate Arrays (FPGAs), and the like.
Those of ordinary skill in the art will appreciate that all or a portion of the steps carried out in the method of the above-described embodiments may be implemented by a program to instruct related hardware, where the program may be stored in a computer readable storage medium, and where the program, when executed, includes one or a combination of the steps of the method embodiments.
In addition, each functional unit in the embodiments of the present invention may be integrated in one processing module, or each unit may exist alone physically, or two or more units may be integrated in one module. The integrated modules may be implemented in hardware or in software functional modules. The integrated modules may also be stored in a computer readable storage medium if implemented in the form of software functional modules and sold or used as a stand-alone product.
The above-mentioned storage medium may be a read-only memory, a magnetic disk or an optical disk, or the like.
In the description of the present specification, a description referring to terms "one embodiment," "some embodiments," "examples," "specific examples," or "some examples," etc., means that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the present invention. In this specification, schematic representations of the above terms do not necessarily refer to the same embodiments or examples. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.
While embodiments of the present invention have been shown and described above, it will be understood that the above embodiments are illustrative and not to be construed as limiting the invention, and that variations, modifications, alternatives and variations may be made to the above embodiments by one of ordinary skill in the art within the scope of the invention.
Claims (9)
1. The method for researching the cascade faults of the high-order network based on the simplex complex is characterized by comprising the following steps of:
s1, constructing a higher-order network which is based on a preference connection mechanism and only comprises 2-simplex according to a first algorithm;
s2, adding 1-simplex to the higher-order network only comprising 2-simplex according to a second algorithm to generate a simplex complex higher-order network;
s3, establishing a capacity-load cascade fault model;
s4, calculating the generalized degree of each node in the simplex higher-order network according to the number of 1-simplex and 2-simplex around each node in the simplex higher-order network through the capacity-load cascade fault model, and setting weight coefficients for the 1-simplex and the 2-simplex;
s5, calculating initial loads of all nodes in the simplex high-order network according to the generalized degree, the 1-simplex and the weight coefficient of the 2-simplex;
s6, calculating loads received by the 1-simplex node associated with the failed node and loads received by the 2-simplex node associated with the failed node when the node fails;
s7, calculating the total load of each node in the simplex higher-order network according to the load received by the 1-simplex node associated with the failed node and the load received by the 2-simplex node associated with the failed node;
s8, calculating the capacity of each node in the simplex complex higher-order network according to the capacity parameter;
s9, judging whether the nodes in the simplex high-order network fail or not according to the total load and capacity of each node in the simplex high-order network;
s10, if yes, returning to the step S6;
s11, if not, calculating connectivity of the simplex complex higher-order network;
s12, respectively changing node capacity parameters, weight coefficients of 1-simplex and 2-simplex and the number of 2-simplex in the simplex higher-order network;
s13, respectively calculating connectivity of the simplex complex higher-order network of different node capacity parameters, the simplex complex higher-order network of different 1-simplex and 2-simplex weight coefficients and the simplex complex higher-order network of different 2-simplex numbers through the capacity-load cascade fault model;
s14, evaluating the robustness of the simplex high-order network according to the connectivity of the simplex high-order network of the different node capacity parameters, the simplex high-order network of different 1-simplex and 2-simplex weight coefficients and the simplex high-order network of different 2-simplex numbers, and performing cascading failure research on the simplex high-order network.
2. The method according to claim 1, wherein S1 builds a higher order network based on a preferred connection mechanism and comprising only 2-simplex according to a first algorithm, comprising:
at the initial moment, a network with only 3 nodes is established, and the 3 nodes are subjected to full connection operation to form a higher-order network containing a 2-simplex; wherein, the initial time is t=0;
adding a new node into a higher-order network formed at the previous moment when the next moment is reached, selecting an edge in the higher-order network formed at the previous moment according to the priority connection probability, and connecting two end points of the selected edge with the new node to generate a new 2-simplex;
at the time of T1, T1 nodes are newly added in the higher-order network of the 2-simplex, and the higher-order network only containing the 2-simplex with the total number of nodes being T1+3 is generated.
3. The method according to claim 2, wherein the S2 adding 1-simplex to the higher order network containing only 2-simplex according to the second algorithm to generate a simplex higher order network comprises:
at the beginning time, adding a new node in the higher-order network only containing 2-simplex, selecting a node in the higher-order network only containing 2-simplex according to the priority connection probability, and connecting the selected node with the new node to generate a new 1-simplex; wherein the starting time is t=0;
adding a new node into a higher-order network formed at the previous moment when the next moment is reached, selecting a node in the higher-order network formed at the previous moment according to the priority connection probability, and connecting the selected node with the new node to generate a new 1-simplex;
at time T2, T2+1 nodes are newly added in the higher-order network only containing 2-simplex, and the simplex complex higher-order network only containing 1-simplex and 2-simplex with the total number of nodes being T1+T2+4 is generated.
4. A method according to claim 3, wherein S5 calculates an initial load of each node in the simplex higher-order network according to the generalized degree and the weight coefficients of the 1-simplex and 2-simplex, comprising:
according to the generalized degree, the 1-simplex and the weight coefficient of the 2-simplex, calculating the initial load of each node in the simplex higher-order network through an initial load calculation formula;
the initial load calculation formula is as follows:
L i (0)＝α(R 1 *k 1,0 (i)+R 2 *k 2,0 (i)) β i＝1,2,...,N,α≥1，β≥1#(1)
wherein k is 1,0 (i) And k 2,0 (i) The generalized degree of the node i is that alpha is a load parameter, beta is an adjustable parameter, R 1 Is the weight coefficient of 1-simplex, R 2 Is a weighting coefficient of 2-simplex.
5. The method of claim 4, wherein the S6 computation of the load received by the 1-simplex node associated with the failed node and the load received by the 2-simplex node associated with the failed node when the node fails comprises:
when a node fails, respectively calculating the load received by the 1-simplex node associated with the failed node and the load received by the 2-simplex node associated with the failed node through a 1-simplex node load receiving calculation formula and a 2-simplex node load receiving calculation formula;
the 1-simplex node load receiving calculation formula is as follows:
wherein L is i (t) is the load of the node i at time t;
the 2-simplex node load receiving calculation formula is as follows:
6. the method of claim 5, wherein S7 calculates a total load for each node in the simplex higher-order network based on the load received by the 1-simplex node associated with the failed node and the load received by the 2-simplex node associated with the failed node, comprising:
according to the load received by the 1-simplex node associated with the failed node and the load received by the 2-simplex node associated with the failed node, calculating the total load of each node in the simplex higher-order network through a 1-simplex node total load calculation formula and a 2-simplex node total load calculation formula;
the 1-simplex node total load calculation formula is as follows:
L j (t)＝L j (t-1)+ΔL i→j (t)#(4)
wherein L is i (t-1) is the load of the node j at time t-1, ΔL i→j (t) is the load from the node i received by the node j at time t;
the 2-simplex node total load calculation formula is as follows:
L k (t)＝L k (t-1)+ΔL i→k (t)#(5)
wherein L is k (t-1) is the load of the node k at time t-1, ΔL i→k And (t) is the load from the node i received by the node k at the moment t.
7. The method according to claim 6, wherein the step of S8 calculating the capacity of each node in the simplex higher-order network according to the capacity parameter includes:
acquiring capacity parameters of each node in the simplex complex higher-order network;
according to the capacity parameter, calculating the capacity of each node in the simplex complex higher-order network through a node capacity calculation formula;
the node capacity calculation formula is as follows:
C j ＝(1+T)L j T≥0#(6)
wherein T is a capacity parameter, L j Is the load of node j.
8. The method according to claim 7, wherein the step S9 of determining whether the node in the simplex higher-order network fails according to the total load and capacity of each node in the simplex higher-order network includes:
acquiring the total load and capacity of each node in the simplex high-order network, and judging whether the node in the simplex high-order network fails or not according to a failure inequality and the total load and capacity of each node;
wherein the inequality is:
L j (t-1)+ΔL i→j (t)>C j #(8)。
9. the high-order network cascading failure research system based on the simplex complex is characterized by comprising a processor and a memory, wherein the processor is connected with the memory:
the processor is used for calling and executing the program stored in the memory;
the memory is used for storing the program, and the program is at least used for executing the simplex-based high-order network cascading failure research method according to any one of claims 1-8.
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