Method for determining connectivity control set of social network

Abstract

The invention discloses a method for determining a connectivity control set of a social network, which comprises the following steps: acquiring a first support tree T1 in a connectivity graph G of the social network based on a breadth-first algorithm; wherein the node x1 of the connected graph G is the root of the first support tree; acquiring a node subset Z of the general graph G; obtaining a second support tree T in the connected graph G, so that the degree of each node in the node subset Z in the second support tree T is the same as that of the node in the connected graph G; leaves in the second support tree T are deleted, a connected graph H is obtained, and the node number of the connected graph H is an upper bound of the node number of the minimum connected control set of the connected graph G. According to the method and the device, a better path can be provided for message transmission by constructing the communication control set, so that the energy consumption is reduced, and the network life cycle is prolonged.

Description

Technical Field

[0001] The present application belongs to the technical field of social network analysis, and specifically relates to a method for determining a connected control set of a social network. Background Art

[0002] A dominating set D of a graph G is a subset of nodes in G such that any node in the graph is either in the set or adjacent to a node in the set. Dominating sets of graphs have applications in wireless network planning, network design, social network analysis, and other fields.

[0003] For example, a social network can be viewed as a graph, and a key feature of social networks is the ability to quickly communicate within the network. For example, in an emergency, all network users may need to be notified, but only a few specific members can be directly contacted. However, as long as every user is connected to at least one directly contactable member, the emergency information can be quickly disseminated to all network participants. This scenario can be modeled by constructing a dominating set of a graph.

[0004] Let D be a dominating set of G. If the subgraph derived from D is a connected graph, then D is called a connected dominating set. Connected dominating sets have the characteristic of connectivity, which facilitates information transmission. For large-scale wireless sensor networks, how to build a virtual backbone network is a hot research topic. A wireless sensor network is represented by an undirected graph G = (V, E), where the virtual backbone network is a subset U of the node set V. The following two conditions must be met: the subgraph derived from U is connected, and every node not in U is adjacent to at least one node in U. Therefore, the virtual backbone network is a connected dominating set of the graph G. In addition, connected dominating sets can also be applied to the selection of routers in the network, broadcast communications, and other network management.

[0005] The minimum dominating set of a graph G is the dominating set with the least number of nodes among all the dominating sets of G. Existing network designs have high costs and energy consumption. In order to reduce costs, finding the minimum connected dominating set is a goal. Summary of the Invention

[0006] The present application provides a method for determining a connectivity control set of a social network to solve the technical problems of high cost and energy consumption in existing network design.

[0007] To solve the above technical problems, the present application adopts a technical solution: a method for determining a connected control set of a social network, comprising:

[0008] S1. Obtain a first spanning tree T1 in a connected graph G of a social network based on a breadth-first algorithm; wherein node x1 of the connected graph G is the root of the first spanning tree; the number of nodes in the connected graph G is n; and the degree of a node in the connected graph G is d;

[0009] S2. Based on the first spanning tree T1, obtain the node subset Z of the general graph G = {x1, x2, ..., x k};

[0010] S3. Obtain a second spanning tree T in the connected graph G so that each node in the node subset Z has the same degree in the second spanning tree T as in the connected graph set G;

[0011] S4. Delete the leaves in the second spanning tree T to obtain a connected graph set H, where the number of nodes in the connected graph set H is an upper bound of the number of nodes in the minimum connected control set of the connected graph G; wherein, in the second spanning tree T, the leaves are nodes with a degree of 1.

[0012] Furthermore, in step S1, the degree of node x1 in the first spanning tree T1 is the same as that in G, and the distance from each node in the first spanning tree T1 to the node x1 is the same as the distance from each node in the connected graph G to the node x1.

[0013] Furthermore, the method of step S2 includes:

[0014] S21. Initialize the number of iterations, set the number of iterations i = 1, and select the node x in the connected graph G. i , let Z i ={x i}, Z′ i =V(G);

[0015] S22. Increase the iteration number i by 1 and determine whether Z' is an empty set;

[0016] S23. If yes, go to step S26;

[0017] S24. If not, select Z' i+1 Node x in i+1 , so that node x i+1 With Z i+1 The distance is the smallest, and |N G (x i )∩N G (Z)|≤1;

[0018] S25. At this time Z i+1 =Z i ∪{x1},Z' i+1 =Z' i -N G (x i ), repeat the above steps;

[0019] S26. Output Z = Z k ,Z={x1,x2,...,x k},and

[0020] Furthermore, the method of step S3 includes:

[0021] S31. Let Z = Z k , T = T1, iteration number i = 0, set Q is an empty set;

[0022] S32. Let i = i + 1, s = 0;

[0023] S33. Determine whether i is greater than k; if so, proceed to S39;

[0024] S34. If not, then determine d T (x i ) is less than d G (x i ); If yes, then go to S32; if d T (x i )=d G (x i ), then proceed to step S35;

[0025] S35. Order F i is the set of edges that are not associated with node x2 in the first spanning tree T1, in which case s = 1;

[0026] S36. Make the set Q = T + f i,s , get the circle C in the set Q, and the circle C passes through f i,s , then delete an edge e from the set Q that is not associated with any node in Z;

[0027] S37. At this time, the supporting tree T2 = Qe;

[0028] S38. Let s = s + 1, determine whether s is less than or equal to j i If yes, then go to S36; if no, then go to S32;

[0029] S39. Output the second spanning tree T, wherein the second spanning tree T has at least nodes, and the degree of each node in the second spanning tree T is equal to the degree of the node in the connected graph G.

[0030] Furthermore, the second spanning tree T has at least leaves.

[0031] Furthermore, in step S4, the number of nodes in the connected graph set H is not greater than

[0032] The beneficial effect of this application is that by constructing a connected dominating set, this application can provide a more optimal path for message transmission, thereby reducing energy consumption and extending the network lifecycle. In network design, the minimum connected dominating number can be used to determine the minimum number of nodes required to maintain reliable connections between all nodes in the network. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] Figure 1 This is a flow chart of an embodiment of a method for determining a connectivity control set of a social network of the present application;

[0034] Figure 2 It is a structural diagram of Example 1 of the present application;

[0035] Figure 3 This is a schematic structural diagram of Example 2 of the present application;

[0036] Figure 4 It is a structural diagram of Example 3 of the present application. DETAILED DESCRIPTION

[0037] In order to make the objectives, technical solutions and advantages of the present invention more clear, the present invention is further described in detail below with reference to specific embodiments.

[0038] In the following description, many specific details are set forth to facilitate a full understanding of the present invention. However, the present invention may also be implemented in other ways different from the description. Therefore, the present invention is not limited to the specific embodiments disclosed in the following specification.

[0039] The basic idea of ​​this application is to first find a spanning tree with as many leaves as possible, and then delete all the leaves to get a connected dominating set. In order to get a spanning tree with as many leaves as possible, we will first find a spanning tree with as many nodes as possible with consistent degrees.

[0040] Let the node set of G be V(G)={x1,x2,...,x n}. When d = 1, G has only one edge and its dominating set is easy to find. Therefore, we assume that d ≥ 2. Let F be a subgraph of G. If v is a node of F and d F (v) = d G (v), then v is called a degree-consistent node in F. For a node x1 in G, the set of nodes adjacent to x1 is denoted as N G (x i ), let N G (x i )=N G (x i )∪{x1}. And N G (x i ) is called x iThe degree, denoted as d G (x i ). If X is a node subset of G, then N G (X)=U x∈X N G (x). In a tree, nodes with degree 1 are called leaves.

[0041] See Figure 1 , Figure 1 1 is a flow chart of an embodiment of a method for determining a connectivity control set of a social network of the present application, the method comprising the following steps:

[0042] S1. Based on the breadth algorithm, obtain the first spanning tree T1 in the connected graph G of the social network; wherein the node x1 of the connected graph G is the root of the first spanning tree; the number of nodes in the connected graph G is n; and the degree of the node of the connected graph set G is d.

[0043] Specifically, we use a breadth-first algorithm to find a spanning tree T1 for the connected graph G of the social network, such that node x1 is the root of the tree, and thus x1 is a degree-consistent node. During the execution of the breadth-first algorithm, the distance from each node in the connected graph G to x1 can be determined.

[0044] S2. Based on the first spanning tree T1, obtain the node subset Z of the general graph G = {x1, x2, ..., x k}.

[0045] Specifically, step S2 includes:

[0046] S21. Initialize the number of iterations, set the number of iterations i = 1, and select the node x in the connected graph G. i , let Z i ={x i}, Z′ i =V(G);

[0047] S22. Increase the iteration number i by 1 and determine whether Z' is an empty set;

[0048] S23. If yes, go to step S26;

[0049] S24. If not, select Z' i+1 Node x in i+1 , so that node x i+1 With Z i+1 The distance is the smallest, and |N G (x i )∩N G (Z)|≤1;

[0050] S25. At this time Z i+1 =Z i ∪{x1},Z'i+1 =Z' i -N G (x i ), repeat the above steps;

[0051] S26. Output Z = Z k ,Z={x1,x2,...,x k},and

[0052] S3. Obtain a second spanning tree T in the connected graph G, such that the degree of each node in the node subset Z in the second spanning tree T is the same as the degree of each node in the node subset Z in the connected graph G.

[0053] Specifically, the method of step S3 includes:

[0054] S31. Let Z = Z k , T = T1, iteration number i = 0, set Q is an empty set;

[0055] S32. Let i = i + 1, s = 0;

[0056] S33. Determine whether i is greater than k; if so, proceed to S39;

[0057] S34. If not, then determine d T (x i ) is less than d G (x i ); If yes, then go to S32; if d T (x i )=d G (x i ), then proceed to step S35;

[0058] S35. Order F i is the set of edges that are not associated with node x2 in the first spanning tree T1, in which case s = 1;

[0059] S36. Make the set Q = T + f i,s , get the circle C in the set Q, and the circle C passes through f i,s , then delete an edge e from the set Q that is not associated with any node in Z;

[0060] S37. At this time, the supporting tree T2 = Qe;

[0061] S38. Let s = s + 1, determine whether s is less than or equal to j i If yes, then go to S36; if no, then go to S32;

[0062] S39. Output the second spanning tree T, wherein the second spanning tree T has at least nodes, and the degree of each node in the second spanning tree T is equal to the degree of the node in the connected graph G.

[0063] When x k After processing, we get the second spanning tree T of the connected graph G, so that there is at least nodes, and each node has at least The degree of the node in T is equal to its degree in G. Therefore, there are at least leaves.

[0064] The specific process is as follows:

[0065] Let L be the set of all leaves of T. From the construction of Z, we know that for any two nodes x in Z i ,x j , there are |N G (x i )∩N G (Z)|≤1. Therefore,

[0066]

[0067] For any node x i , there is d T (x i )=d, and So we can get:

[0068]

[0069] Where S is a subset of nodes in a connected graph G. For any node u in G, if it is either in S or adjacent to a node in S, then S is called a dominating set of G. If S is a dominating set of G and the subgraph derived from the nodes of S is connected, then S is called a connected dominating set of G. The number of nodes contained in the connected dominating set with the least number of nodes in G is called the connected dominating number of G, denoted by γ C (G).

[0070] Let λ(G) be the number of leaves in the spanning tree with the least number of leaves in G. Then

[0071] γ C (G) = n - λ(G) (3);

[0072] S4. Delete the leaves in the second spanning tree T to obtain a connected graph set H, where the number of nodes in the connected graph set H is an upper bound of the number of nodes in the minimum connected control set of the connected graph G; in the spanning tree, a node with a degree of 1 is called a leaf.

[0073] Specifically, all leaves in the second spanning tree T obtained in step S3 are deleted to obtain a connected subgraph H of G. It can be seen that the number of nodes contained in the connected subgraph H is at most

[0074] For any node in G, if it is not a leaf of the second spanning tree T, then it is in the connected subgraph H. If it is a leaf of the second spanning tree T, then it is adjacent to a node in the connected subgraph H. Therefore, the set of nodes in the connected subgraph H is a connected dominating set of G.

[0075] Example 1

[0076] like Figure 2 As shown, the present application provides an embodiment, which is a 4-regular connected graph G with 10 nodes: x1, x2, ..., x 10 This example is a social network that contains 10 members, and each member can only contact 4 other members. Each member is represented by a node. If two members can contact each other, an edge is used to connect the nodes representing the two members. This forms a graph G. Figure 2 .

[0077] In an emergency, if x1 has a message to convey to every member, we can solve it by constructing a connected control set of graph G.

[0078] Example 2

[0079] like Figure 3 As shown, according to the method of the present application, a spanning tree T1 of G is first obtained by a breadth-first algorithm, and a spanning tree of the embodiment can be obtained by using the breadth-first algorithm.

[0080] For this embodiment, the following algorithm can be used to obtain a node subset Z = {x1, x6} of the embodiment.

[0081] The specific algorithm is as follows:

[0082] Let Z1={x1},Z'1=V(G)-N G (x 1i )={x6,x7,x8,x9,x 10}.

[0083] In Z'1, it is obvious from T1 that the nodes closest to x1 are x6, x7, x8, and x9.

[0084] At the above four nodes, the following conditions are met|N G (x6)∩N G (Z)|≤1,|N G (x7)∩N G(Z)|≤1,|N G (x8)∩N G (Z)|≤1,|N G (x9)∩N G (Z)|≤1;

[0085] Select node x6, set Z2 = {x1, x6}, and then set Z'2 = Z'1-N G (x6). Then Z'2 is an empty set.

[0086] Let Z = Z'2, then Z is the desired node set.

[0087] Example 3

[0088] See Figure 4 , a schematic diagram of a spanning tree with four nodes of consistent degree obtained by performing a spanning tree transformation on Z according to an embodiment of the present application, where {x1, x2, x4, x6} is a connected dominating set of G. If x1 has a message to convey to each member, the message first propagates among x1, x2, x4, and x6 before being transmitted to the remaining six members.

[0089] 1. Take a spanning tree as shown in Example 2, denoted as T1. In T1, x1 is a degree-consistent node.

[0090] 2. Add the edge x6x7 to T1 to obtain a unique cycle x1x2x6x7x3x1. Delete the edge x3x7 to obtain another spanning tree T2 in the embodiment.

[0091] 3. Add the edge x6x9 to T2 to obtain a unique cycle x1x2x6x9x5x1. Delete the edge x5x9 to obtain a spanning tree T3 in this embodiment.

[0092] 4. In T3, there are 6 leaves x3,x5,x7,x8,x9,x 10 .

[0093] 5. Let L = {x3,x5,x7,x8,x9,x 10}Finish.

[0094] like Figure 4 As shown, the present application provides an algorithm for a connectivity control set of Example 3.

[0095] 1. Place the leaves in T3 x3, x5, x7, x8, x9, x 10 Delete them all and get a connected graph H.

[0096] 2. The node set V(H) = {x1, x2, x4, x6} of H is a connected control set of G.

[0097] Finally, it should be noted that if x1 has a message to convey to each member, the message will first be transmitted between x1, x2, x4, and x6, and then conveyed to the remaining 6 members.

[0098] The above description is merely an embodiment of the present application and does not limit the patent scope of the present application. Any equivalent structure or equivalent process transformation made using the contents of the present application specification and drawings, or directly or indirectly applied in other related technical fields, are also included in the patent protection scope of the present application.

Claims

1. A method for determining a connected control set of a social network, characterized in that: The following steps are involved: S1. Obtain a first spanning tree T1 in a connected graph G of a social network based on a breadth-first algorithm; wherein a node x1 of the connected graph G is the root of the first spanning tree; the number of nodes in the connected graph G is n; and the degree of each node in the connected graph G is d; S2. Based on the first spanning tree T1, obtain the node subset Z of the general graph G = {x1, x2, ..., x k }; S3. Obtain a second spanning tree T in the connected graph G, such that the degree of each node in the node subset Z in the second spanning tree T is the same as that in the connected graph G; S4. Delete leaves from the second spanning tree T to obtain a connected graph H, where the number of nodes in the connected graph H is an upper bound on the number of nodes in the minimum connected dominating set of the connected graph G. In the second spanning tree T, the leaves are nodes with degree 1.

2. The method according to claim 1, characterized in that In step S1, the degree of the node x1 in the first spanning tree T1 is the same as that in G, and the distance from each node in the first spanning tree T1 to the node x1 is the same as the distance from each node in the connected graph G to the node x1.

3. The method according to claim 1, characterized in that The method of step S2 comprises: S21. Initialize the number of iterations, set the number of iterations i = 1, select the node x in the connected graph G i , let Z i ={x i }, Z' i =V(G); S22. Increase the iteration number i by 1 and determine whether Z' is an empty set; S23. If yes, go to step S26; S24. If not, select Z' i+1 Node x in i+1 , so that the node x i+1 With Z i+1 The distance is the smallest, and |N G (x i )∩N G (Z)|≤1; S25. At this time Z i+1 =Z i ∪{x1},Z' i+1 =Z' i -N G (x i ), repeat the above steps; S26. Output Z = Z k ,Z={x1,x2,...,x k },and 4. The method according to claim 3, characterized in that The method of step S3 comprises: S31. Let Z = Z k , T = T1, iteration number i = 0, set Q is an empty set; S32. Let i = i + 1, s = 0; S33. Determine whether i is greater than k; if so, proceed to S39; S34. If not, then determine d T (x i ) is less than d G (x i ); If yes, then go to S32; if d T (x i )=d G (x i ), then proceed to step S35; S35. Order F i is the set of edges not associated with node x2 in the first spanning tree T1, in which case s=1; S36. Make the set Q = T + f i,s , get the circle C in the set Q, and the circle C passes through f i,s , then delete an edge e from the set Q that is not associated with any node in Z; S37. At this time, the supporting tree T2 = Qe; S38. Let s = s + 1, determine whether s is less than or equal to j i If yes, then go to S36; if no, then go to S32; S39. Output the second spanning tree T, wherein the second spanning tree T has at least nodes, and the degree of each of the nodes in the second spanning tree T is equal to the degree of the node in the connected graph G.

5. The method according to claim 4, characterized in that The second spanning tree T has at least leaves.

6. The method according to claim 5, characterized in that In step S4, the number of nodes in the connected graph set H is not greater than Therefore, the number of nodes in the minimum connected control set of the connected graph G is not greater than

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