Method for Characterizing Connectivity of Time-Varying Networks Based on Time-Expanded Graphs

Abstract

The present invention discloses a method for characterizing the connectivity of a time-varying network based on a time-expanded graph, mainly solving the problem that the traditional connectivity metrics of IP networks cannot accurately describe whether data can be transmitted between components in a time-varying network. The solution is as follows: defining the connectivity metric of the time-varying network; dividing the selected time interval into continuous time periods; using the time-expanded graph to characterize the time-period connectivity relationship and node storage capacity of the time-varying network; adding virtual nodes and virtual edges to the time-expanded graph to form a time-expanded graph containing virtual nodes; recording the number of connected virtual node pairs and the number of virtual node pairs for which the connectivity analysis has been completed; updating the number of virtual node pairs according to the connectivity of any pair of virtual nodes, and calculating the connectivity metric of the time-varying network; using this metric to reflect the quality of network connectivity. The present invention can accurately characterize the proportion of physical node pairs that can transmit data in a time-varying network within a given time range, and can be used in non-terrestrial networks, mobile Internet, and vehicle-to-everything networks.

Description

Technical Field

[0001] The present invention belongs to the field of communication technologies, and particularly relates to a method for characterizing the connectivity of a time-varying network based on a time-expanded graph, which can be used to measure the availability of typical time-varying networks such as non-terrestrial networks, mobile Internet, and vehicle-to-everything networks. Background Art

[0002] A time-varying network refers to a network in which the network topology or the bandwidth resources available for service transmission in the network change over time. Typical time-varying networks include non-terrestrial networks, mobile Internet, and vehicle-to-everything networks, etc., which provide services for communications worldwide.

[0003] Network connectivity is one of the basic metrics for measuring network availability. The IP Performance Metrics (IPPM) working group of the Internet Engineering Task Force (IETF) defines network connectivity as the ratio of the number of connected links to the total number of links, which characterizes the property of whether data can be transmitted between network components.

[0004] In a time-varying network, the dynamic topology and link available bandwidth resources will also cause the network connectivity to change over time. However, traditional network connectivity metric definitions and analysis methods regard the network as static and ignore the time-varying characteristics, which will result in the obtained network connectivity metrics not reflecting the actual availability of the time-varying network. For example, consider a non-terrestrial network scenario including four satellite nodes u1, u2, u3, and u4. Suppose that in the first time period, transmission links are only established between u1 and u2, and between u4 and u3. And in the second time period, the network topology changes, and transmission links are only established between u1 and u3, and between u4 and u2. Since there are no connected links between u1 and u4, and between u2 and u3 in the network topology at any time slot, according to the traditional IP network connectivity metric definition, the network connectivity in the 1st and 2nd time slots is both 1 / 3. In practice, with the storage resources of each node, cross-time-slot data transmission can be achieved. For example, u2 will host the data from u1 in the first time period to the second time period and forward it to u4, and u3 will host the data from u4 in the first time period to the second time period and forward it to u1, then mutual data transmission can be completed between u1 and u4. Similarly, data interaction can also be completed between u2 and u3 in the two time periods. The above cross-time-slot data transmission is common in time-varying networks, but it cannot be characterized by traditional IP network connectivity metrics, which will restrict the network availability.

[0005] Shu Jian, Jiang Shandong, Sun Limin et al. proposed a method for characterizing the connectivity of opportunistic sensing networks in the first issue of the Journal of Beijing University of Posts and Telecommunications. Aiming at opportunistic sensing networks with time evolution, it uses time snapshot graphs to model the time-varying topology of the network, and obtains the overall network connectivity by defining and calculating time paths, time distances, and connectivity efficiency, reflecting the connectivity of opportunistic sensing networks. Although this method considers the dynamics of the network topology, it still has the following deficiencies:

[0006] 1) Regarding the network as a time snapshot graph that is static in sub-periods, it does not characterize and utilize the storage resources of each node, restricting the search for cross-period connectivity paths between nodes and affecting the accurate measurement of network availability;

[0007] 2) It does not fully consider the time directionality of data transmission. It is considered that unidirectional communication between nodes means connectivity. In practice, two-way data interaction may not be possible. For example, for a certain node u i The data sent in the current period can be carried and forwarded by a mobile node (Ferry) and transmitted to another node u in a subsequent time slot j , but due to violating the time directionality, u j Cannot transmit data back to u through this time path i , so it should not be considered that u i Is connected to u j ;

[0008] 3) Calculating the overall network connectivity based on the adjacency matrix sequence corresponding to the time snapshot graph has a high computational complexity. Summary of the Invention

[0009] The purpose of the present invention is to provide a method for characterizing the connectivity of time-varying networks based on time-expanded graphs to reduce the computational complexity, accurately measure the network availability in the time dimension, and accurately describe the attribute of whether data can be transmitted between components in a time-varying network in view of the above deficiencies of the prior art. Its implementation steps are as follows:

[0010] (1) Define the connectivity index of the time-varying network as: the ratio of the number of node pairs with mutually connected paths in the network to the total number of node pairs within a certain time range;

[0011] (2) Select a time interval and divide it into continuous time periods;

[0012] (2a) Select a time interval T = [t0, t H ), where t0≥0 represents the start time, and t H >t0≥0 represents the end time;

[0013] (2b) According to the topology change times t1, t2,..., t h ,..., t H-1, divide the time interval T into a series of consecutive time periods τ1, τ2,..., τ h ,..., τ H , where τ h = [t h-1 , t h ) represents the h-th time period, t h-1 is the start time of τ h , t h is the end time of τ h , and H is the total number of time periods;

[0014] (3) Characterize the time-varying network:

[0015] (3a) Define the node set represents the i-th physical node within the time period τ h N is the total number of nodes;

[0016] (3b) Define the edge set E = E t ∪ E s , where represents the transmission edge set, represents to 's transmission link, is the j-th physical node u within the time period τ h j ; represents the storage edge set, represents to 's storage capacity, is the i-th physical node u within the time period τ h+1 i ;

[0017] (3c) Define the edge weight value set represents the weight value on the edge , is the j-th physical node u within the time period τ r j , and set to represent the hop count;

[0018] (3d) According to the above definitions of each set, use the original time-expanded graph G = (V, E, W) to characterize the time-varying network with N physical nodes, including the time period connectivity relationship and the node storage capacity;

[0019] (4) Add N virtual nodes and 2·N·H virtual edges​​​​ Obtain a time-expanded graph \(G'=\{V', E', W'\}\) containing virtual nodes, where \(V'\) is the set of expanded nodes, \(E'\) is the set of expanded edges, and \(W'\) is the set of expanded weight values, and set the weight values of each virtual edge to 0;

[0020] (5) Define the number \(C\) of connected virtual node pairs T , and initialize \(C\) T \(= 0\), which is used to record the number of physical node pairs with interconnected paths in the network within a given time interval \(T\);

[0021] (6) Define the number \(m\) of virtual node pairs for which the minimum-hop end-to-end path calculation has been completed, and initialize \(m = 0\);

[0022] (7) Calculate the total number of virtual node pairs

[0023] (8) Arbitrarily select the \(i\)-th virtual node and the \(j\)-th virtual node in the time-expanded graph \(G'\), and use the shortest path algorithm to calculate the minimum-hop end-to-end path \(P\) from to i,j in \(G'\), and the minimum-hop end-to-end path \(P\) from to j,i ;

[0024] (9) Increase the number of virtual node pairs for which the minimum-hop end-to-end path calculation has been completed by 1, that is, let \(m = m + 1\), and judge the connectivity of the virtual node pair and :

[0025] If both \(P\) i,j and \(P\) j,i exist, then is connected to , and increase the number \(C\) of connected virtual node pairs by 1, that is, let \(C\) T \(= C\) T \(+ 1\); T

[0026] Otherwise, is not connected to , and the value of \(C\) T is not updated;

[0027] (10) Compare the number \(m\) of virtual node pairs for which the minimum-hop end-to-end path calculation has been completed in the current time-expanded graph \(G'\) with the total number \(M\) of virtual node pairs:

[0028] If m < M, the connectivity judgment of all virtual node pairs is not completed. Return to step (8) and re-select the virtual node pairs whose connectivity has not been judged.

[0029] Otherwise, the connectivity judgment of all virtual node pairs has been completed. Calculate and output the connectivity index value of the time-varying network. Execute step (11).

[0030] (11) Use the connectivity index value 0 ≤ α of the time-varying network. G,T ≤ 100% to characterize the connectivity of the network:

[0031] If α G,T approaches 100%, the connectivity of the time-varying network is good.

[0032] If α G,T approaches 0, the connectivity of the time-varying network is poor.

[0033] The present invention has the following advantages:

[0034] 1. When defining the connectivity index of the time-varying network, the present invention fully considers the time-varying characteristics of the time-varying network topology and resources, and defines the index as the proportion of physical node pairs that can transmit data to each other in the network within a given time range. Therefore, it can reveal the availability of the time-varying network.

[0035] 2. When calculating the connectivity index of the time-varying network, the present invention uses a time-expanded graph to model the time-varying network, jointly considers the link transmission and node storage capabilities in different time periods, supports cross-time-period storage and data transmission, and improves the reachable connectivity of the time-varying network. At the same time, since the shortest path algorithm is used to judge whether there is a connected path between any nodes, it ensures that mutual data transmission can be completed between connected nodes, and the computational complexity is low. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 is the implementation flowchart of the present invention;

[0037] Figure 2 is the existing time-expanded graph used in the present invention;

[0038] Figure 3 is the time-expanded graph obtained by adding virtual nodes and virtual edges to the existing time-expanded graph in the present invention;

[0039] Figure 4 is the pair of virtual nodes in the present invention and the end-to-end path graph of the minimum number of hops between;

[0040] Figure 5 is the pair of virtual nodes in the present invention and Minimum-hop end-to-end path graph between;

[0041] Figure 6 is the minimum-hop end-to-end path graph between virtual node pairs in the present invention and Minimum-hop end-to-end path graph between;

[0042] Figure 7 is the minimum-hop end-to-end path graph between virtual node pairs in the present invention and Minimum-hop end-to-end path graph between;

[0043] Figure 8 is the minimum-hop end-to-end path graph between virtual node pairs in the present invention and Minimum-hop end-to-end path graph between;

[0044] Figure 9 is the minimum-hop end-to-end path graph between virtual node pairs in the present invention and Minimum-hop end-to-end path graph between. Detailed implementation manners

[0045] The embodiments of the present invention will be further described in detail below with reference to the accompanying drawings.

[0046] In order to accurately characterize the connectivity of a time-varying network and reflect the attribute of whether each network node can communicate with each other, it is necessary to accurately define the connectivity index of the time-varying network. In this embodiment, fully considering the dynamic characteristics of the given time-varying network, the proportion of physically connected node pairs within a period of time is used as a measure of the quality of connectivity. Based on the definition of the index, it is necessary to determine whether there is a connected path between any two physical nodes in the given time-varying network. Use the time-expanded graph to model the topology, links, and storage resources of the given time-varying network in different time periods to support the search for cross-time-period transmission paths; in addition, by adding virtual nodes and virtual edges to the time-expanded graph, a time-expanded graph is obtained, and the Dijkstra algorithm is applied to quickly calculate the minimum-hop end-to-end path between any two virtual nodes. Only when both the two minimum-hop end-to-end paths in both directions exist can it be determined that the physical node pair corresponding to the virtual node pair is connected. Finally, by calculating the proportion of the total number of connected virtual node pairs to the total number of all virtual node pairs, the connectivity index of the given time-varying network can be obtained to judge the quality of the connectivity performance.

[0047] Refer to Figure 1 , the implementation steps of this example are as follows:

[0048] Step 1, Definition of the connectivity index of the time-varying network

[0049] In this embodiment, the connectivity index of the time-varying network is defined as the ratio of the number of node pairs with interconnected paths in the time-varying network to the total number of node pairs within a certain time interval. Specifically, the interconnected paths in the time-varying network can be cross-time transmission paths realized by means of the node storage capacity.

[0050] Step 2: Select a time interval and divide it into continuous time periods.

[0051] (2.1) Select a time interval T = [t0, t2):

[0052] In this embodiment, the start time t0 of the time interval is selected as 0, and the end time t2 of the time interval is 60 seconds. Then the time interval is: T = [0, 60);

[0053] (2.2) Divide the time interval into continuous time periods according to the moments when the network topology changes:

[0054] In this embodiment, the topology of the time-varying network changes at t1 = 20 seconds. Then the time interval T is divided into a time period τ1 = [0, 20) and a time period τ2 = [20, 60), and the number of time periods H = 2.

[0055] Step 3: Characterize the time-varying network.

[0056] The time-varying network can usually be represented by snapshot graphs, time-expanded graphs, and time-aggregated graphs. In this embodiment, but not limited to, the time-expanded graph G = (V, E, W) is used to depict the time-period connectivity relationship and node storage capacity of the time-varying network, as Figure 2 shown, where:

[0057] is the set of nodes, and the total number of physical nodes N = 4;

[0058] E = E t ∪E s is the set of edges,

[0059] is the set of transmission edges,

[0060] is the set of storage edges;

[0061]

[0062] is the set of weight values;

[0063] All weight values are set to 1 to represent the number of hops.

[0064] Step 4: Add N virtual nodes and 2·N·H virtual edges to the time-expanded graph G to form a time-expanded graph G' = {V', E', W'} containing virtual nodes.

[0065] In this embodiment, the number of virtual nodes N to be added is 4, and the number of virtual edges 2·N·H is 16. Among them, the virtual nodes are The virtual edges are And the weight values of all virtual edges are all set to 0, and the resulting time-expanded graph G' = {V', E', W'} is as Figure 3 shown, where:

[0066] is the set of expanded nodes;

[0067]

[0068] is the set of expanded edges;

[0069]

[0070] is the set of expanded weight values.

[0071] Step five, initialize the number C of connected virtual node pairs T .

[0072] C T is used to record the number of virtual node pairs for which it has been determined that there is a connected path between them, that is, the number of physical node pairs in the time-varying network that are connected. In this embodiment, initially, it has not been determined whether there is a connected path between any pair of virtual nodes, so let C T = 0.

[0073] Step six, initialize the number m of virtual node pairs for which the minimum-hop end-to-end path calculation has been completed.

[0074] m is used to record the number of virtual node pairs for which the minimum-hop end-to-end path calculation between them has been completed. In this embodiment, initially, the minimum-hop end-to-end path has not been calculated for any pair of virtual nodes, so m = 0.

[0075] Step seven, calculate the total number M of virtual node pairs.

[0076] In this embodiment, there are a total of N = 4 virtual nodes in the time-expanded graph G'. According to the formula the total number of virtual node pairs can be obtained as M = 6.

[0077] Step eight, arbitrarily select the i-th virtual node and the j-th virtual node in the time-expanded graph G', and use the shortest path algorithm to calculate the paths from to The minimum-hop end-to-end path P i,j , and the minimum-hop end-to-end path P from to j,i .

[0078] In this embodiment, the virtual node pair and are selected.Common shortest path algorithms include Dijkstra algorithm, Bellman-Ford algorithm, Floyd algorithm, etc. In this embodiment, the Dijkstra algorithm with the optimal time complexity is selected but not limited to calculate the minimum-hop end-to-end path.

[0079] (8.1) Calculate the minimum-hop end-to-end path P from to i,j :

[0080] (8.1.1) Initialize P i,j as an empty set;

[0081] In this embodiment, the selected virtual node pair is and Therefore the minimum-hop end-to-end path from to 1,2 is denoted as P

[0082] (8.1.2) Initialize the parent node parameter of all nodes in G' to -1;

[0083] In this embodiment, that is, let

[0084] (8.1.3) Divide the nodes in G' into two parts, namely the selected node set S and the remaining node set D. Initially, S only contains That is

[0085] In this embodiment, the virtual node is selected. Therefore, initially

[0086] (8.1.4) Use the nodes outside S to form the remaining node set Suppose is a node in D, is the kth physical node u h in time period τ k , and determine whether there is an edge from node to node

[0087] If That is, there exists an edge Then to The hop count is Update The parent node parameter of is:

[0088] Otherwise, to The hop count is +∞;

[0089] In this embodiment, the remaining node set contains all nodes in V' except Then In G', there exists an edge Then to The hop count is Update The parent node parameter of At the same time, there also exists an edge Then to The hop count is Update The parent node parameter of Since there is no to any node in D except and The hop count

[0090] (8.1.5) Select a node with the minimum hop count from the remaining node set D and Add to the selected node set S, and delete from D to obtain to The minimum hop count

[0091] In this embodiment, to and The hop counts are both 0, while the hop counts to the remaining nodes in D are all +∞. To select a node with the minimum hop count, you can choose any one from and For example Add to S, that is, let Delete from D, that is, let Determine to The minimum number of hops is

[0092] (8.1.6) Using as the new intermediate point, compare the number of hops from using as the intermediate point to any node in D with the number of hops without passing through . If the number of hops without passing through is the r-th physical node u in time period τ l : r :

[0093] If then the number of hops without passing through is greater than the number of hops passing through , modify the hop value from to as: and update the parent node parameter of as: : where is the number of hops from to , is the number of hops from to ;

[0094] Otherwise, do not modify the hop value from to ;

[0095] In this embodiment, using as the intermediate point, modify the hop values from to each node in D:

[0096] For node Because So the hop value from to is modified to and the parent node parameter of

[0097] For the hop values from to the remaining nodes in D, do not modify them and they remain +∞;

[0098] (8.1.7) Determine whether the selected node set S contains all nodes in G':

[0099] If S = V', then S already contains all nodes in G', indicating that the construction of path P i,j is completed, and execute step (8.1.8);

[0100] Otherwise, return to step (8.1.5);

[0101] In this embodiment, the current set is not equal to That is, it does not contain all the nodes in G'. It is necessary to return to step (8.1.5) until S = V';

[0102] (8.1.8) Introduce an auxiliary variable U to record the nodes in the iteration process, and initialize and add U to P i,j That is, P i,j = P i,j ∪{U};

[0103] In this embodiment, initially, the auxiliary variable Add U to P 1,2 In it, we can get

[0104] (8.1.9) Determine whether U is

[0105] If Then the path search has been completed, and output P i,j ;

[0106] Otherwise, the path search is not completed, and execute step (8.1.10);

[0107] In this embodiment, because So continue to execute step (8.1.10).

[0108] (8.1.10) Obtain the parent node parameter p(U) of U, and determine whether p(U) exists:

[0109] If p(U) = -1, the parent node of U does not exist, then P i,j also does not exist;

[0110] Otherwise, add p(U) to P i,j That is, P i,j = P i,j ∪{p(U)}, update U = p(U), and return to step (8.1.9);

[0111] In this embodiment, the parent node parameter of Because it is not -1, so is added to P 1,2 In it, we can get It is necessary to return to step (8.1.9) until the path search is completed, and output the calculated minimum-hop end-to-end path The result is as Figure 4 shown.

[0112] (8.2) Calculate the minimum-hop end-to-end path P from to : j,i :

[0113] (8.2.1) Initialize P j,i as an empty set;

[0114] In this embodiment, the selected virtual node pair is and Therefore, the minimum-hop end-to-end path from is denoted as P 2,1 , and let

[0115] (8.2.2) Initialize the parent node parameter of all nodes in G' to -1;

[0116] In this embodiment, that is, let

[0117] (8.2.3) Divide the nodes in G' into two parts, namely the selected node set S and the remaining node set D. Initially, S only contains That is,

[0118] In this embodiment, the virtual node is selected. Therefore, initially

[0119] (8.2.4) Use the nodes outside S to form the remaining node set Suppose is a node in D, is the kth physical node u h in time period τ k , and determine whether there is an edge from node to node

[0120] If , that is, there is an edge , then the hop count from to is . Update the parent node parameter of

[0121] Otherwise, the hop count from to

[0122] is +∞; In this embodiment, the remaining node set contains all nodes in V' except u2. Then In G', since there exists an edge then the hop count from u2 to is Update the parent node parameter of At the same time, since there also exists an edge then the hop count from u2 to is Update the parent node parameter of Since there is no edge from u2 to any node in D other than and then the hop count

[0123] (8.2.5) Select a node with the minimum hop count from the remaining node set D Add to the selected node set S, and delete from D to obtain to the minimum hop count

[0124] In this embodiment, to and the hop counts are both 0, while the hop counts to the remaining nodes in D are all +∞. To select the node with the minimum hop count, one can be arbitrarily selected from and For example Add to S, that is, let Delete from D, that is, let Determine to the minimum hop count is

[0125] (8.2.6) Taking as the new intermediate point, compare the hop count from taking as the intermediate point to any node in D with the hop count without passing through The size of the hop count, is the r-th physical node u l in time period τ r :

[0126] If then the hop count without passing through is greater than the hop count passing through If the hop count is large, modify to The hop count value is: and update the parent node parameter of as: where is to the hop count, is to the hop count;

[0127] Otherwise, do not modify to the hop count value;

[0128] In this embodiment, taking as the midpoint, modify the hop counts of each node from

[0129] For node Because Therefore to the hop count value is modified to and the parent node parameter of

[0130] For to the remaining nodes in D, the hop count values are not modified and remain +∞;

[0131] (8.2.7) Determine whether the selected node set S contains all nodes in G':

[0132] If S = V', then S already contains all nodes in G', indicating that the construction of path P j,i is completed, and step (8.2.8) is executed;

[0133] Otherwise, return to step (8.2.5);

[0134] In this embodiment, the current set is not equal to that is, it does not contain all nodes in G', and it is necessary to return to execute step (8.2.5) until S = V';

[0135] (8.2.8) Introduce an auxiliary variable U to record the nodes in the iterative process, initialize and add U to P j,i in, that is, P j,i = P j,i ∪{U};

[0136] In this embodiment, initially, the auxiliary variable Add U to P 2,1 In, we can obtain

[0137] (8.2.9) Judge whether U is

[0138] If The path search has been completed, output P j,i ;

[0139] Otherwise, the path search is not completed, execute step (8.2.10);

[0140] In this embodiment, because So continue to execute step (8.2.10);

[0141] (8.2.10) Obtain the parent node parameter p(U) of U, and judge whether p(U) exists:

[0142] If p(U) = -1 and the U parent node does not exist, then P j,i Does not exist either;

[0143] Otherwise, add p(U) to, that is, P j,i = P j,i ∪{p(U)}, update U = p(U), return to step (8.2.9);

[0144] In this embodiment, The parent node parameter of is Because it is not -1, so Add to P 2,1 In, we can obtain Need to return to execute step (8.2.9) until the path search is completed, and output the calculated minimum-hop end-to-end path The result is as Figure 4 Shown

[0145] Step nine, update the number C of connected virtual node pairs T .

[0146] Increase the number m of virtual node pairs for which the minimum-hop end-to-end path calculation has been completed by 1, and judge the connection status of the virtual node pairs And :

[0147] If P i,j And P j,i Both exist, then Is connected to And increase the number C of connected virtual node pairs T By 1;

[0148] Otherwise, Is connected to Disconnected, C T The value is not updated.

[0149] In this embodiment, the number m of virtual node pairs for which the minimum-hop end-to-end path calculation has been completed after update is: m = 0 + 1 = 1. Since from to the minimum-hop end-to-end path P 1,2 , and from to the minimum-hop end-to-end path P 2,1 both exist, so and are connected, and the number C of connected virtual node pairs is updated T to: C T = 0 + 1 = 1.

[0150] Step ten, calculate the connectivity index value of the time-varying network.

[0151] Compare the number m of virtual node pairs for which the minimum-hop end-to-end path calculation has been completed in the current time-expanded graph G' with the total number M of virtual node pairs:

[0152] If m < M, the connectivity judgment of all virtual node pairs has not been completed. Return to step eight and re-select the virtual node pairs whose connectivity has not been judged.

[0153] Otherwise, the connectivity judgment of all virtual node pairs has been completed. Calculate and output the connectivity index value of this time-varying network:

[0154]

[0155] In this embodiment, the current m = 1 < M = 6, so the connectivity judgment of all virtual node pairs has not been completed. It is necessary to return to step eight and re-select the virtual node pairs whose connectivity has not been judged.

[0156] Assume that the next selected virtual node pair is and Execute the Dijkstra algorithm to calculate the minimum-hop end-to-end path from to as From to the minimum-hop end-to-end path is As Figure 5 shown, m = 1 + 1 = 2.

[0157] Since both paths P 1,3 and P 3,1 exist, so and are connected, and C is updated T= 2. Also, since m = 2 < M = 6, it is still necessary to return to step eight and repeat the virtual node pair selection operation until the virtual node pairs and and and and have all completed the connectivity judgment, and the following are obtained respectively:

[0158] Figure 6 The minimum-hop end-to-end path from to in and the minimum-hop end-to-end path from to in

[0159] Figure 7 The minimum-hop end-to-end path from to in and the minimum-hop end-to-end path from to in

[0160] Figure 8 The minimum-hop end-to-end path from to in and the minimum-hop end-to-end path from to in

[0161] Figure 9 The minimum-hop end-to-end path from to in and the minimum-hop end-to-end path from to in

[0162] Update the number m of virtual node pairs for which the minimum-hop end-to-end path calculation has been completed to m = 6 = M = 6. Since Figure 6 both P 1,4 and P 4,1 exist in Figure 7 both P 2,3 and P 3,2 exist in Figure 8 both P 2,4 and P 4,2 exist in Figure 9 both P 3,4 and P 4,3 exist, it is judged that and and and and are both connected, then finally update C T = 6.

[0163] According to the formula it can be calculated that the connectivity index value of this time-varying network is

[0164] Step Eleven, use the connectivity index value α of the time-varying network G,T to characterize the quality of the network connectivity.

[0165] If α G,T approaches 100%, then the connectivity of this time-varying network is good;

[0166] If α G,T approaches 0, then the connectivity of this time-varying network is poor.

[0167] In this embodiment, the connectivity index α G,T = 100%, indicating that the connectivity of this time-varying network is good, that is, within the selected time interval T = [0, 60), all physical nodes u1, u2, u3, and u4 can transmit data to each other pairwise.

[0168] The above description is only a specific example of the present invention and does not constitute any limitation to the present invention. Obviously, for professionals in the field, after understanding the content and principle of the present invention, various modifications and changes in form and details may be made without departing from the principle and structure of the present invention. However, these modifications and changes based on the idea of the present invention are still within the protection scope of the claims of the present invention.

Claims

1. A method for characterizing the connectivity of a time-varying network based on a time-expanded graph, characterized in that The following are included: (1) Define the connectivity index of the time-varying network as the ratio of the number of node pairs with mutually connected paths in the network within a certain time range to the total number of node pairs; (2) Select a time interval and divide it into continuous time periods; (2a) Selected time interval T = [t0, t H ), where t0 ≥ 0 represents the start time, and t H > t0 ≥ 0 represents the end time; (2b) According to the topological change times t1, t2,..., t h ,..., t H-1 , the time interval T is divided into a series of consecutive time periods τ1, τ2,..., τ h ,..., τ H , where τ h = [t h-1 , t h represents the h-th time period, t h-1 is the start time of τ h , and t h is the end time of τ h , and H is the total number of time periods; (3) Characterize the time-varying network: (3a) Define the set of nodes Indicates the i-th physical node u h within the time period τ i , N is the total number of physical nodes; (3b) Define the edge set $E = E t \cup E s , where represents the transmission edge set, represents to transmission link, is the $j$-th physical node $u$ h within the time period $\tau$ j ; represents the storage edge set, represents to storage capacity, is the $i$-th physical node $u$ h+1 within the time period $\tau$ i ; (3c) Define the set of edge weight values Denote the edge with the weight value on it, for the j-th physical node u r within the time period τ j , set to represent the hop count; (3d) According to the above definitions of each set, use the original time-expanded graph G=(V, E, W) to characterize the time-varying network with N physical nodes, including time-period connectivity relationships and node storage capabilities; (4) Add N virtual nodes {u i} and 2·N·H virtual edges to obtain a time-expanded graph G' = {V', E', W'} with virtual nodes, where V' is the set of expanded nodes, E' is the set of expanded edges, and W' is the set of expanded weight values, and set the weight values of each virtual edge to 0; (5) Define the number C of connected virtual node pairs T , and initialize C T = 0, which is used to record the number of physical node pairs with interconnected paths in the network within a given time interval T; (6) Define the number m of virtual node pairs for which the minimum-hop end-to-end path calculation has been completed, and initialize m = 0; (7) Calculate the total number of virtual node pairs (8) Arbitrarily select the $i$-th virtual node $u$ in the time-expanded graph $G'$ i and the $j$-th virtual node $u$ j . Use the shortest path algorithm to calculate the minimum-hop end-to-end path $P$ i from $u$ j to $u$ i,j in $G'$, and the minimum-hop end-to-end path $P$ j from $u$ i to $u$ j,i ; (9)Increment the number of virtual node pairs for which the minimum-hop end-to-end path calculation has been completed by 1, i.e., let m = m + 1, and determine the connectivity of the virtual node pair u i and u j : If P i,j and P j,i both exist, then u i is connected to u j , and the number C T of connected virtual node pairs is incremented by 1, i.e., let C T = C T + 1; Otherwise, u i is not connected to u j and the value of C T is not updated; (10) Compare the number m of virtual node pairs for which the minimum-hop end-to-end path calculation has been completed in the current time-expanded graph G' with the total number M of virtual node pairs: If m < M, the connectivity judgment for all virtual node pairs has not been completed. Return to step (8) and re-select a virtual node pair whose connectivity has not been judged; Otherwise, all virtual node pairs have completed the connectivity judgment, and the connectivity index value of the time-varying network is calculated and output Execute step (11); (11) The connectivity index value 0 ≤ α of the time-varying network G,T ≤ 100% characterizes the quality of the network connectivity: If α G,T approaches 100%, the connectivity of the time-varying network is good at that time; If α G,T approaches 0, the connectivity of the time-varying network is poor at that time.

2. The method according to claim 1, characterized in that, The time-expanded graph G' = {V', E', W'} containing virtual nodes obtained in (4) is implemented as follows: (4a)Construct the extended node set is the total set of all nodes in the original time-expanded graph G and all added virtual nodes; (4b) Construct the extended edge set is the total set of all transmission edges, storage edges, and all added virtual edges in the original time-expanded graph G; (4c)Construct an extended weight value set It is the total set of the weight values on all transmission edges, the weight values on all storage edges, and the weight values on all added virtual edges in the time-expanded graph G. Among them, is the weight value on any virtual edge , is the weight value on any virtual edge .

3. The method according to claim 1, wherein Calculate the minimum-hop end-to-end path P from the i-th virtual node u i to the j-th virtual node u j as follows: i,j ​ (8a) Initialize P i,j to be an empty set; (8b) For each node in G' = {V', E', W'}, define the parent node parameter p(·) to record the previous hop node of this node in the minimum-hop end-to-end path P i,j and initialize p(·) = -1; (8c) Divide the nodes in G' into two parts, namely the selected node set S and the remaining node set D. Initially, S only contains u i , that is, S = {u i}; (8d) Use the nodes other than S to form the remaining node set D = V' / {u i} and assume is a node in D, is the k-th physical node u h within the time period τ k . Determine whether there is an edge i from node u to node If That is, there exists an edge Then the hop count from u i to is Update the parent node parameter of to be Otherwise, u i to has a hop count of +∞; (8e) Select a node with the minimum number of hops from the remaining node set D to u i The node with the minimum number of hops Add to the selected node set S, and Delete from D to obtain the minimum number of hops from u i to The minimum number of hops (8f) With as the new midpoint, compare the hop count from u i With as the midpoint to any node in D and the hop count without passing through . is the r-th physical node u l within the time period τ r : If then it does not pass through The hop count is greater than that passing through Modify the hop count value of u i to as: And update The parent node parameter of as: Where is the hop count from u i to and is the hop count from to; Otherwise, do not modify u i to jump value; (8g) Judge whether the selected node set S contains all nodes in G'; If S = V', then S already contains all the nodes in G', indicating that the construction of path P has been completed, and step (8h) is executed; i,j is constructed, and step (8h) is executed; Otherwise, execute step (8e); (8h) Introduce an auxiliary variable U to record the nodes in the iterative process, and initialize U = u j , and add U to P i,j , that is, P i,j = P i,j ∪ {U}; (8i) Determine whether U is u i : If U = u i , the path search is completed, and P is output i,j ; Otherwise, the path search has not been completed. Execute step (8j); (8j) Obtain the parent node parameter p(U) of U and judge whether p(U) exists; If p(U) = -1 and the parent nodes of U do not exist, then P i,j also does not exist; Otherwise, add p(U) to P i,j in, that is, P i,j = P i,j ∪ {p(U)}, update U = p(U), and return to step (8i).

4. The method according to claim 1, characterized in that The calculation in (8) for the minimum-hop end-to-end path P j from the j-th virtual node u i to the i-th virtual node u j,i is implemented as follows: (8k) Initialize P j,i is an empty set; For each node in G' = {V', E', W'}, define the parent node parameter p(·) to record the previous hop node of this node in the minimum-hop end-to-end path P j,i and initialize p(·) = -1; (8m) Divide the nodes in G' into two parts, namely the selected node set S and the remaining node set D. Initially, S only contains u j , that is, S = {u j}; (8n) Use the nodes other than S to form the remaining node set D = V' / {u j}. Assume is a node in D, is the k-th physical node u h in time period τ k . Determine whether there is an edge j from node u to node If That is, there exists an edge Then the hop count from u j to is Update the parent node parameter of as: Otherwise, u j to has a hop count of +∞; (8o) Select a node with the minimum number of hops from the remaining node set D to u j with the minimum number of hops Add to the selected node set S, and delete it from D to obtain the minimum number of hops from u j to with the minimum number of hops (8p) Using as the new midpoint, compare the number of hops from u j Using as the midpoint to any node in D with the number of hops without passing through . is the r-th physical node u l within the time period τ r : If then it does not pass through The hop count is greater than the hop count passing through then modify u j to The hop count value is: and update The parent node parameter of is: where is the hop count from u j to and is the hop count from to; Otherwise, do not modify u j to jump value; (8q) Judge whether the selected node set S contains all nodes in G'; If S = V', then S already contains all the nodes in G', indicating that the construction of path P j,i has been completed, and step (8r) is executed; Otherwise, execute step (8o); (8r) Introduce an auxiliary variable U to record the nodes during the iteration process, and initialize U = u i , and add U to P j,i , that is, P j,i = P j,i ∪ {U}; (8s) Determine whether U is u j : If U = u j , then the path search is completed and P is output j,i ; Otherwise, the path search has not been completed. Execute step (8t); (8t) Obtain the parent node parameter p(U) of U and judge whether p(U) exists; If p(U) = -1 and there is no parent node of U, then P j,i also does not exist; Otherwise, add p(U) to P, i.e., P j,i = P j,i ∪ {p(U)}, update U = p(U), and return to step (8s).

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