CN115333945B - Local topology inference method of online social network - Google Patents

Local topology inference method of online social network Download PDF

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CN115333945B
CN115333945B CN202210776030.6A CN202210776030A CN115333945B CN 115333945 B CN115333945 B CN 115333945B CN 202210776030 A CN202210776030 A CN 202210776030A CN 115333945 B CN115333945 B CN 115333945B
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季宏宇
李聪
郝旭
李翔
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Abstract

The invention belongs to the technical field of large-scale network data analysis, and particularly relates to a local topology inference method of an online social network. The invention adopts an independent cascade model as an information dynamics propagation model, and deduces partial continuous edges of a network with N nodes through the observed cascade result; i.e., when specifying a set of key nodes, the goal is to infer the edges around all nodes in the set, including the edges pointed to from those nodes and all edges pointed to those nodes. The invention is based on an online social network, fully grasps the characteristics of large scale, sparse connecting edges and strong heterogeneity of the online social network, considers the directionality of information propagation, and provides a network local inference algorithm for performing forward neighbor inference and backward neighbor inference on key nodes in the network to obtain a local network topology composed of forward neighbors and backward neighbors of the key nodes. The algorithm accuracy rate of local topology inference is significantly improved compared to global network topology inference algorithms.

Description

Local topology inference method of online social network
Technical Field
The invention belongs to the technical field of large-scale network data analysis, and particularly relates to a local topology inference method of an online social network.
Background
In recent years, there have been many excellent efforts to infer social networks that are not directly available. The online social platform gradually goes deep into the life of people and becomes an indispensable social way in the life of people, so that the relationship on the online social platform of the inferred user gradually becomes a research hotspot. However, most of the work at present is not inferred specifically for online social networks, and does not take full advantage of the unique properties and characteristics of online social networks. Under the background, the invention provides a local topology inference method which is more suitable for online social network inference aiming at the characteristics of large scale, strong sparsity and strong heterogeneity of the online social network.
The size of online social networks tends to be huge, and in such quantities, attempts to infer the edges of the entire network are almost impossible to accomplish. In fact, due to the strong heterogeneity of online social networks, many nodes are at very marginal locations in the network, and when trying to infer user edges, the real interests are often not all nodes in the network, but some of the more important nodes, which may be called key nodes. In other words, in most cases we only want to infer what the edges around the critical node are, not all nodes that are not active. However, in the conventional network inference algorithms, even if only a part of the edges around the nodes are to be inferred, the inference needs to be performed on the whole network, and in the context of the online social network, such large-scale network inference needs to be performed, so that a large amount of observation data is required, the calculation amount is large, and the completion is almost impossible. Therefore, if the method can realize targeted inference of the edge connection condition around the key node concerned by people, a great amount of redundant calculation can be effectively avoided, and only observation data related to the concerned node is needed, so that the method has great practical significance and application prospect. More importantly, as the local inference is carried out on the key nodes, the interference of other irrelevant nodes is avoided, and the accuracy of the inference is effectively improved.
Disclosure of Invention
In view of the above circumstances, the invention aims to provide a local topology inference method more suitable for online social network inference so as to improve inference accuracy based on the online social network, which has the characteristics of large online social network scale, sparse connecting edges and strong heterogeneity.
The invention adopts an independent cascade model as a model of information dynamics propagation, and the observed cascade result D= [ D ] 1 ，d 2 ，…，d C ] T Inferring a network with N nodes
Figure GDA0004232132880000011
Is connected with the part of the edge. Specifically, when the set of key nodes S is specified key When the goal is to infer the set S key Including both the edges pointed to from these nodes and all edges pointing to these nodes.
The invention provides a local topology inference method of an online social network, which comprises the following specific steps:
step 1: specifying a set of key nodes S key For the key node set S key For one node u in the list, in the effective propagation time of the information, the node transmitting the information to the node u is called a forward neighbor of the node u; deducing the forward neighbor of the node u to obtain the local topology of the forward neighbor
Figure GDA0004232132880000021
Step 2: for a set of key nodes S key For one node u in the list, the node u transmits the information in the effective propagation time of the information, and the nodes receiving the information transmitted by the node u are called backward neighbors of the node u; deducing backward neighbors of the key node set node u to obtain a local topology of the backward neighbors of the node u
Figure GDA0004232132880000022
Step 3: local network topology inference, namely, local topology of forward neighbors inferred by node u
Figure GDA0004232132880000023
And backward neighbor-local topology->
Figure GDA0004232132880000024
Integrating to obtain the forward and backward neighbor topology of the node u>
Figure GDA0004232132880000025
Key node S key The corresponding local topology is->
Figure GDA0004232132880000026
In the invention, the specific flow of the step 1 is as follows:
step 1-1: finding a set of possible forward neighbors of node u based on timeliness of information propagation
Figure GDA0004232132880000027
Respectively using
Figure GDA0004232132880000028
To indicate the moment when node u and node v obtain information in the c-th propagation, in one propagation there must be +.>
Figure GDA0004232132880000029
Wherein t is max Is the age of the message propagation. Accordingly, all propagation cascades can be traversed, all possible forward neighbors of node u are found, forming the set +.>
Figure GDA00042321328800000210
Step 1-2: calculating the probability that node u gets a message from node v in one propagation c
Figure GDA00042321328800000211
Equation (1) gives the probability that node u gets the message from node v in the c-th propagation, where a v，u Representing the connected edge state of node pair (v, u), if there is a connected edge, a v，u =1, otherwise a v，u ＝0，μ v，u (τ) is the latency distribution of node pair (v, u).
Figure GDA00042321328800000212
Equation (2) gives that node u is in the c-th propagation, in
Figure GDA00042321328800000213
The probability of not being infected by other nodes than node v.
Figure GDA00042321328800000214
Step 1-3: computing a set of key nodes S key Likelihood function of forward neighbors and observed propagation results
Figure GDA00042321328800000215
Considering all possible forward neighbors in the c-th propagation, give node u +.>
Figure GDA0004232132880000031
The probability of obtaining information at the moment is shown in formula (3).
Figure GDA0004232132880000032
When all the cascade results are considered, the given node u produces a likelihood function of the propagation result D we observe, as shown in equation (4), where D u Is the set of cascades in which node u participates.
Figure GDA0004232132880000033
Step 1-4: the introduction of survival and risk functions simplifies the likelihood functions.
Introducing common functions, wherein the survival function and the risk function are respectively represented by the formulas (5) and (6), so as to obtain a simplified likelihood function, as shown by the formula (7):
Figure GDA0004232132880000034
Figure GDA0004232132880000035
Figure GDA0004232132880000036
step 1-5: computing edge gain of maximum likelihood function when adding edges (m, u)
Figure GDA0004232132880000037
Calculating edge gain when the state (m, u) of node pair is switched from 0 to 1 when there is a conjoined edge
Figure GDA0004232132880000038
As shown in equation (8). Considering all the cascade results, the edge gain of the node when the edge is increased is obtained>
Figure GDA0004232132880000039
I.e., equation (9).
Figure GDA00042321328800000310
Figure GDA00042321328800000311
Step 1-6: computing edge gain of maximum likelihood function when deleting edge (m, u)
Figure GDA00042321328800000312
The edge gain can be obtained in the same way by comparing the increment of the maximum likelihood function when the edge is added, as shown in the formula (10) and the formula (11).
Figure GDA0004232132880000041
Figure GDA0004232132880000042
Step 1-7: inferred node u forward neighbor local topology
Figure GDA0004232132880000043
Bringing edge gains into the Markov chain-Monte Carlo sampling framework for the node u and all its possible forward neighbors
Figure GDA00042321328800000414
Traversing the node pairs formed by the method to maximize likelihood function and obtain a local topology of deducing forward neighbors of the node u by +.>
Figure GDA0004232132880000045
And (3) representing.
In the invention, the specific flow of the step 2 is as follows:
step 2-1: finding a set of possible backward neighbors of node u based on timeliness of information propagation
Figure GDA0004232132880000046
The specific steps are the same as 1-1.
Step 2-2: in the calculation of the c-th propagation, backward neighbor set
Figure GDA0004232132880000047
Likelihood function of intermediate node i and likelihood function taking into account cascading +.>
Figure GDA0004232132880000048
Equation (12) gives the node's backward neighbor set in the c-th propagation
Figure GDA0004232132880000049
Likelihood function of intermediate node i. Equation (13) considers all cascaded likelihood functions.
Figure GDA00042321328800000410
Figure GDA00042321328800000411
Step 2-3: computing edge gain of maximum likelihood function when adding edges (j, i)
Figure GDA00042321328800000412
Equation (14) gives the edge gain for adding one edge (j, i) to node i in the c-th pass, and equation (15) gives the edge gain when all cascades are considered.
Figure GDA00042321328800000413
Figure GDA0004232132880000051
Figure GDA0004232132880000052
Step 2-4: computing edge gain of maximum likelihood function when deleting edge (j, i)
Figure GDA0004232132880000053
So equation (16) gives the edge gains for deleting one edge (j, i) for node i in the c-th propagation, respectively, and equation (17) gives the edge gains when all cascades are considered.
Figure GDA0004232132880000054
Figure GDA0004232132880000055
Step 2-5: inferred node u backward neighbor local topology
Figure GDA0004232132880000056
Bringing edge gains into the Markov chain-Monte Carlo sampling framework for the node u and all its possible forward neighbors
Figure GDA0004232132880000057
Traversing the node pairs formed by the method to maximize likelihood function and obtain a local topology of deducing forward neighbors of the node u by +.>
Figure GDA0004232132880000058
And (3) representing.
The innovation point of the invention is that:
(1) Aiming at the characteristic of large scale of the online social network, the idea of carrying out local network inference around key nodes is provided, so that a large amount of redundant calculation and a large amount of requirements on input data are avoided, and the inference of the connecting edges in the large-scale online social network is possible;
(2) The invention provides a network local topology inference algorithm, and carries out simulation verification in a real data set, and results show that the network local topology inference algorithm can accurately infer local edges of a network and has higher inference accuracy.
The invention is based on an online social network, fully grasps the characteristics of large scale, sparse connecting edges and strong heterogeneity of the online social network, considers the directionality of information propagation, and provides a network local inference algorithm for performing forward neighbor inference and backward neighbor inference on key nodes in the network aiming at the key nodes, thereby obtaining the local network topology consisting of the forward neighbors and the backward neighbors of the key nodes. The algorithm accuracy of the local topology inference is significantly improved compared to the global network topology inference algorithm.
Drawings
FIG. 1 is an independent cascading model in the network inference problem of the present invention.
FIG. 2 is a schematic diagram of a partial inference as discussed in the present invention.
FIG. 3 is a flow chart of a Markov chain-Monte Carlo sampling method in the network inference problem of the present invention.
FIG. 4 is a plot of local inferred accuracy versus number of key nodes for BA scaleless network (a) and Twitter sub-social network (b).
Detailed Description
In order that the above-recited objects and novel features of the present invention can be readily understood, a more particular description of the invention will be rendered by reference to specific embodiments thereof which are illustrated in the appended drawings.
An independent cascade model as shown in fig. 1 was used as a model for the propagation of information dynamics by the observed cascade result d= [ D ] 1 ，d 2 ，…，d C ] T Inferring a network with N nodes
Figure GDA0004232132880000061
Is connected with the part of the edge. Specifically, when the set of key nodes S is specified key When the goal is to infer the set S key Including both the edges pointed to from these nodes and all edges pointing to these nodes. Since the adjacency matrix is a common and visual representation, the method for storing the network edge condition is shown in FIG. 2, which shows an adjacency matrix diagram of the whole network, wherein the shaded portion is shown as S key ＝{key 1 ，key 2 We infer the target range at the time of }. Because of the directionality of information propagation, it is considered that the information propagates along the direction of the continuous edge, so that the neighbors of any node u are divided into forward neighbor nodes, that is, neighbors that can transfer information to u, and backward neighbor nodes, that is, nodes that can receive information sent by u. In the inference process, we will infer their forward neighbors and backward neighbors separately for each nodeTo the neighbors.
The invention provides a local topology inference method of an online social network, which comprises the following specific steps:
step 1: specifying a set of key nodes S key For the key node set S key For one node u, the node that transmits information to node u during the effective propagation time of the information is called the forward neighbor of node u. Deducing the forward neighbor of the node u to obtain the local topology of the forward neighbor
Figure GDA0004232132880000062
Step 2: for a set of key nodes S key For one node u in the list, the node u transmits the information during the effective propagation time of the information, and the nodes receiving the information transmitted by the node u are called backward neighbors of the node u. Deducing backward neighbors of the key node set node u to obtain a local topology of the backward neighbors of the node u
Figure GDA0004232132880000063
Step 3: local network topology inference, namely, local topology of forward neighbors inferred by node u
Figure GDA0004232132880000064
And backward neighbor-local topology->
Figure GDA0004232132880000065
Integrating to obtain the forward and backward neighbor topology of the node u>
Figure GDA0004232132880000066
Key node S key The corresponding local topology is->
Figure GDA0004232132880000071
In the invention, the specific flow of the step 1 is as follows:
step 1-1: finding possible front of node u based on timeliness of information propagationTo neighbor set
Figure GDA0004232132880000072
Node u, which obtains information from node v, must be t after node v obtains the message max Information is obtained during the time period. Respectively using
Figure GDA0004232132880000073
To indicate the moment when node u and node v obtain information in the c-th propagation, in one propagation there must be +.>
Figure GDA0004232132880000074
I.e. not all nodes infected before node u may be its forward neighbors, but also have to meet +.>
Figure GDA0004232132880000075
Accordingly, all propagation cascades can be traversed, all possible forward neighbors of node u are found, forming the set +.>
Figure GDA0004232132880000076
Step 1-2: calculating the probability that node u gets a message from node v in one propagation c
Figure GDA0004232132880000077
Equation (35) gives the probability that node u gets the message from node v in the c-th propagation, where a v，u Representing the connected edge state of node pair (v, u), if there is a connected edge, a v，u =1, otherwise a v，u ＝0，ρ v，u (τ) is the latency distribution of node pair (v, u).
Figure GDA0004232132880000078
Equation (36) gives that node u is in the c-th propagation, in
Figure GDA0004232132880000079
The probability of not being infected by other nodes than node v. Since we only consider that the first time node u gets information to be valid, if we want to ensure that node u gets information from node v, we need to ensure that +.>
Figure GDA00042321328800000715
Not previously infected by other possible forward neighbors.
Figure GDA00042321328800000710
Step 1-3: computing a set of key nodes S key Likelihood function of forward neighbors and observed propagation results
Figure GDA00042321328800000711
Node u is in c, possibly obtaining information from other nodes, so taking into account all possible forward neighbors in this propagation, giving node u +.>
Figure GDA00042321328800000712
The probability of obtaining information at the moment is shown in formula (20).
Figure GDA00042321328800000713
When all the cascade results are considered, the given node u produces a likelihood function of the propagation result D we observe, as shown in equation (38), where D u Is the set of cascades in which node u participates.
Figure GDA00042321328800000714
Step 1-4: the introduction of survival and risk functions simplifies the likelihood functions.
The usual functions are introduced, the survival function and the risk function are represented by formulas (22) and (23), respectively, and a simplified likelihood function is obtained as shown by formula (24).
Figure GDA0004232132880000081
Figure GDA0004232132880000082
Figure GDA0004232132880000083
Step 1-5: computing edge gain of maximum likelihood function when adding edges (m, u)
Figure GDA0004232132880000084
For either propagation c, we calculate for node u the increment of the maximum likelihood function when adding one conjoined edge (m, u), i.e., the edge gain when the state (m, u) of the node pair switches from 0 to 1 with a conjoined edge
Figure GDA0004232132880000085
As shown in equation (42). Although the formula appears to be complex, the main calculation amount of the (m, u) edge gain is only +.>
Figure GDA0004232132880000086
And->
Figure GDA0004232132880000087
Considering all the cascade results, the edge gain u of the node at the time of edge connection is increased, i.e., formula (43).
Figure GDA0004232132880000088
Figure GDA0004232132880000089
Step 1-6: computing edge gain of maximum likelihood function when deleting edge (m, u)
Figure GDA00042321328800000810
The edge gain can be obtained in the same way by comparing the increment of the maximum likelihood function when the edge is added, as shown in the formula (44) and the formula (45).
Figure GDA00042321328800000811
Figure GDA0004232132880000091
Figure GDA0004232132880000092
Step 1-7: inferred node u forward neighbor local topology
Figure GDA0004232132880000093
Bringing the edge gains into the Markov chain-Monte Carlo sampling framework shown in FIG. 4 for the node u and all its possible forward neighbors
Figure GDA0004232132880000094
Traversing the node pairs formed by the method to maximize likelihood function and obtain a local topology of deducing forward neighbors of the node u by +.>
Figure GDA0004232132880000095
And (3) representing.
In the invention, the specific flow of the step 2 is as follows:
step 2-1: finding a set of possible backward neighbors of node u based on timeliness of information propagation
Figure GDA0004232132880000096
The specific steps are the same as 1-1.
Step 2-2: computing a set of key nodes S key Likelihood function of backward neighbor node i and observed propagation result
Figure GDA0004232132880000097
Equation (46) gives the node's backward neighbor set in the c-th propagation
Figure GDA0004232132880000098
The likelihood function of the intermediate node i, notably, is concerned only with the node i at S due to the local inference key In (c), the condition u e S is limited in the formula key And j E S key . Equation (47) considers all cascaded likelihood functions.
Figure GDA0004232132880000099
Figure GDA00042321328800000910
Step 2-3: computing edge gain of maximum likelihood function when adding edges (j, i)
Figure GDA00042321328800000911
Equation (48) gives the edge gain for adding one edge (j, i) to node i in the c-th pass, and equation (49) gives the edge gain when all cascades are considered.
Figure GDA00042321328800000912
Figure GDA0004232132880000101
Step 2-4: computing edge gain of maximum likelihood function when deleting edge (j, i)
Figure GDA0004232132880000102
So equation (50) gives the edge gains for deleting one edge (j, i) for node i in the c-th propagation, and equation (51) gives the edge gains when all cascades are considered.
Figure GDA0004232132880000103
Figure GDA0004232132880000104
Step 2-5: inferred node u backward neighbor local topology
Figure GDA0004232132880000105
Bringing the edge gains into the Markov chain-Monte Carlo sampling framework shown in FIG. 3 for the node u and all its possible backward neighbors
Figure GDA0004232132880000109
Traversing the node pairs formed by the method to maximize likelihood function and obtain a local topology of deducing forward neighbors of the node u by +.>
Figure GDA0004232132880000107
And (3) representing.
Taking a BA scaleless network comprising 1000 nodes and a Twitter sub-social network comprising 1973 nodes as examples, the algorithm of the present invention was used to make local network topology inferences. The parameters selected in this experiment were m=10，max l ag＝10，burn i n=10, and according to the number of network nodes, key node sets with different sizes are selected. The experimental results obtained are shown in table 1 and table 2. Wherein Table 1 and Table 2 are the accuracy of backward inference, forward inference, and local inference for BA scaleless networks and Twitter sub-social networks, respectively. FIG. 4 is a graph of BA scaleless networks and Twitter sub-social networks, local inference accuracy as a function of number of key nodes. It can be found that the accuracy of forward inference and local inference is less affected by the key nodes, while the accuracy of backward inference is more affected by the key nodes. In addition, the local inference accuracy on the BA scaleless network is generally higher than that on the Twitter sub-social network, which indicates that the type of network has a certain influence on the inference accuracy.
Table 1, local inferred accuracy of ba scaleless network.
Figure GDA0004232132880000108
Figure GDA0004232132880000111
Table 2, local inference accuracy for twitter sub-social networks.
Figure GDA0004232132880000112
/>

Claims (1)

1. A local topology inference method of an online social network is characterized in that an independent cascade model is adopted as an information dynamics propagation model, and an observed cascade result D= [ D ] is adopted 1 ，d 2 ，…，d C ] T Inferring a network with N nodes
Figure FDA00042321328700000115
Is connected with the edge of the part; in particular, when referring toSet of fixed key nodes S key When the goal is to infer the set S key Including the edges pointed out from these nodes, and all edges pointing to these nodes; the method comprises the following specific steps:
step 1: specifying a set of key nodes S key For the key node set S key For one node u in the list, in the effective propagation time of the information, the node transmitting the information to the node u is called a forward neighbor of the node u; deducing the forward neighbor of the node u to obtain the local topology of the forward neighbor
Figure FDA0004232132870000011
Step 2: for a set of key nodes S key For one node u in the list, the node u transmits the information in the effective propagation time of the information, and the nodes receiving the information transmitted by the node u are called backward neighbors of the node u; deducing backward neighbors of the node u in the key node set to obtain a local topology of the backward neighbors of the node u
Figure FDA0004232132870000012
Step 3: local network topology inference, namely, local topology of forward neighbors inferred by node u
Figure FDA0004232132870000013
And backward neighbor-local topology->
Figure FDA0004232132870000014
Integrating to obtain the forward and backward neighbor topology of the node u>
Figure FDA0004232132870000015
Set of key nodes S key The corresponding local topology is->
Figure FDA0004232132870000016
Deducing a key node set S described in step 1 key The forward neighbor of the intermediate node u is obtained to obtain the local topology of the forward neighbor of the node u
Figure FDA0004232132870000017
The specific flow is as follows:
step 1-1: finding a set of possible forward neighbors of node u from information propagation
Figure FDA0004232132870000018
Step 1-2: calculating the probability that node u gets a message from node v in one propagation c
Figure FDA0004232132870000019
Step 1-3: calculating likelihood functions of forward neighbors of key node u and observed propagation results
Figure FDA00042321328700000110
Step 1-4: leading in survival function and risk function to simplify likelihood function;
step 1-5: computing edge gain of maximum likelihood function when adding edges (m, u)
Figure FDA00042321328700000111
Step 1-6: computing edge gain of maximum likelihood function when deleting edge (m, u)
Figure FDA00042321328700000112
Step 1-7: inferred node u forward neighbor local topology
Figure FDA00042321328700000113
Deducing the key node set S in the step 2 key Backward neighbor of middle node u, concrete flowThe process is as follows:
step 2-1: finding a set of possible backward neighbors of node u based on timeliness of information propagation
Figure FDA00042321328700000114
Step 2-2: calculating likelihood functions of possible backward neighbor nodes i of key node u and observed propagation results
Figure FDA0004232132870000021
Step 2-3: computing edge gain of maximum likelihood function when adding edges (j, i)
Figure FDA0004232132870000022
Step 2-4: computing edge gain of maximum likelihood function when deleting edge (j, i)
Figure FDA0004232132870000023
Step 2-5: inferred node u backward neighbor local topology
Figure FDA0004232132870000024
In step 1:
step 1-1 of finding out a set of possible forward neighbors of the node u according to the timeliness of information propagation
Figure FDA0004232132870000025
The operation flow of (1) is as follows:
respectively using
Figure FDA0004232132870000026
To indicate the moment when node u and node v obtain information in the c-th propagation, in one propagation there must be +.>
Figure FDA0004232132870000027
Wherein t is max Is the age of message propagation; accordingly, all propagation cascades are traversed, all possible forward neighbors of the node u are found, and a set is formed
Figure FDA0004232132870000028
In the calculation of one propagation c described in step 1-2, the probability of obtaining a message from node v by node u +.>
Figure FDA0004232132870000029
The method of (1) is as follows:
in the c-th propagation, the probability that node u gets a message from node v is:
Figure FDA00042321328700000210
wherein a is v，u Representing the connected edge state of node pair (v, u), if there is a connected edge, a v，u =1, otherwise a v，u ＝0，ρ v，u (τ) is the latency distribution of node pair (v, u);
node u in the c-th propagation, in
Figure FDA00042321328700000211
The probability that a node other than node v has not been infected is:
Figure FDA00042321328700000212
step 1-3 calculating the Key node set S key Likelihood function of forward neighbors and observed propagation results
Figure FDA00042321328700000213
The operation flow of (1) is as follows:
giving the node taking into account all possible forward neighbors in the c-th propagationu in this propagation
Figure FDA00042321328700000214
The probability of obtaining information at the moment is shown in the formula (3):
Figure FDA00042321328700000215
when all the cascade results are considered, giving the likelihood function that node u produces the observed propagation result D, as shown in equation (4), where D u Is the set of cascades in which node u participates:
Figure FDA00042321328700000216
the operation flow of introducing the survival function and the risk function simplified likelihood function in the step 1-4 is as follows:
introducing common functions, wherein the survival function and the risk function are respectively represented by the formulas (5) and (6), and further obtaining a simplified likelihood function, wherein the simplified likelihood function is represented by the formula (7):
Figure FDA0004232132870000031
Figure FDA0004232132870000032
Figure FDA0004232132870000033
computing edge gain of maximum likelihood function when adding edges (m, u) as described in steps 1-5
Figure FDA0004232132870000034
The method of (1) is as follows:
calculating edge gain when the state (m, u) of node pair is switched from 0 to 1 when there is a conjoined edge
Figure FDA0004232132870000035
As shown in formula (8); considering all the cascade results, the edge gain of the node when the edge is increased is obtained>
Figure FDA0004232132870000036
Namely, formula (9):
Figure FDA0004232132870000037
Figure FDA0004232132870000038
computing edge gains of maximum likelihood functions when deleting edges (m, u) as described in steps 1-6
Figure FDA0004232132870000039
The specific method comprises the following steps:
comparing the increment of the maximum likelihood function when the edge is added, and obtaining edge gain according to the same processing mode, wherein the edge gain is shown as a formula (10) and a formula (11):
Figure FDA00042321328700000310
Figure FDA00042321328700000311
step 1-7 the inferred node u forward neighbor is a local topology
Figure FDA00042321328700000312
The specific method comprises the following steps:
bringing edge gains into the Markov chain-Monte Carlo sampling framework for the node u and all its possible forward neighbors
Figure FDA0004232132870000041
Traversing the node pairs formed by the method to maximize likelihood function and obtain a local topology of deducing forward neighbors of the node u by +.>
Figure FDA0004232132870000042
A representation;
in step 2:
step 2-1 of finding out the possible backward neighbor set of the node u according to the timeliness of information propagation
Figure FDA0004232132870000043
The specific operation is the same as the step 1-1;
step 2-2 calculating backward neighbor set in the c-th propagation
Figure FDA0004232132870000044
The likelihood function calculation formula of the middle node i is as follows:
Figure FDA0004232132870000045
the calculation taking into account all the concatenated likelihood functions is:
Figure FDA0004232132870000046
computing edge gain of maximum likelihood function when adding edges (j, i) as described in step 2-3
Figure FDA0004232132870000047
The calculation formula is the following formula (14), and the calculation formula of the edge gain when all cascading is considered is the following formula (15);
Figure FDA0004232132870000048
Figure FDA0004232132870000049
edge gain of maximum likelihood function when deleting the edge (j, i) in step 2-4
Figure FDA00042321328700000410
The expression is the following expression (16), and the expression of the edge gain when all cascading is considered is the following expression (17):
Figure FDA0004232132870000051
Figure FDA0004232132870000052
step 2-5 the inferred node u backward neighbor is a local topology
Figure FDA0004232132870000053
The specific way is to bring the edge gain into the Markov chain-Monte Carlo sampling framework for the node u and all its possible forward neighbors +.>
Figure FDA0004232132870000054
Traversing the node pairs formed by the method to maximize likelihood function and obtain a local topology of deducing forward neighbors of the node u by +.>
Figure FDA0004232132870000055
And (3) representing.
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