Layer reconstruction and missing link prediction of multilayer network with Maximum A Posteriori estimation
LLayer reconstruction and missing link prediction of multilayer network with
Maximum A Posteriori estimation
Junyao Kuang ∗ and Caterina Scoglio Department of Electrical and Computer Engineering,Kansas State University, Manhattan, KS 66506, USA (Dated: January 11, 2021)A multilayer network is composed of multiple layers, where different layers have the same set ofvertices but represent different types of interactions. Nevertheless, some layers are interdependentor structurally similar in the multilayer network. In this paper, we present a maximum a posteriori estimation based model to reconstruct a specific layer in the multilayer network. The
SimHash algorithm is used to compute the similarities between various layers. And the layers with similarstructures are used to determine the parameters of the conjugate prior. With this model, we canalso predict missing links and direct experiments for finding potential links. We test the methodthrough two real multilayer networks, and the results show that the maximum a posteriori estimationis promising in reconstructing the layer of interest even with large amount of missing links.
I. INTRODUCTION
Network analysis has been widely used in differentareas, such as information diffusion, infectious diseasespreading, gene co-expression analysis, and transporta-tion systems. People construct networks to analyze therelations of nodes in the system. Through networkanalysis, one can improve the robustness of networksor reduce systemic failure rates [1, 2]. For example,epidemiologists can use network analysis to predict thenumber of people infected by COVID-19 and providepieces of advice to policymakers at an early stage tocurb the spreading of the disease [3]. Also, throughnetwork analysis, biologists can detect potential rela-tions between genes within co-expression networks. Ingene co-expression networks, it is ubiquitous that mul-tiple layers are constructed through testing various bi-ological activities or functions [4, 5]. However, someedge identification experiments are expensive and time-consuming. Researchers are unable to obtain the entirenetwork with restricted resources in a short period. Analternative way to solve the problem is to reconstructthe layer of interest through the interdependent lay-ers obtained from existing data or simple experiments[8, 12, 13]. After reconstructing the target layer, peoplecan devote restricted resources to the node pairs withhigh expected number of edges. Various algorithmshave been proposed to reconstruct networks and pre-dict missing links [6–11, 14–20], and most of them arebased on generative models [21, 22]. In this paper, wewill also focus on the generative models.A stochastic generative model is to model the networkstructure by fitting the network features to a stochasticblock model [8, 14, 15, 23]. To reconstruct the tar-get layer in the multilayer network, we can either use ∗ Correspondence email address: [email protected] the target layer solely or take advantage of the struc-turally similar layers. In [14, 15], the authors presenteda degree-correlated stochastic block model to model asingle layer network. The authors reconstructed the sin-gle layer network by maximizing the likelihood throughexpectation-maximization algorithm. The method canalso help to detect overlapping communities. Paper [8]extended the single layer stochastic block model to mul-tilayer networks and used it to predict missing links anddetect overlapping communities. The experiments showthat the method is valid when the layers are similar orinterdependent. However, when the layers are indepen-dent, the authors tried all the layer combinations to findthe maximized likelihood. The layers that can improvethe maximum likelihood are detected as the interdepen-dent layers. Therefore, when there are numerous layers,it is impossible to try all the layer combinations to findthe maximum likelihood. The authors validated the al-gorithm through two multilayer networks by hiding 20%of links and non-links. In most cases, the number of lostedges is unknown to the people, and the results will besignificantly affected by the missing links. The goals ofthis work are to solve these issues.To avoid the brute force combination method men-tioned above, we will use network comparing methodsto detect structurally similar layers or the so-called in-terdependent layers. We compare the target layer withthe other layers when the target layer is partially knownto us. The authors [24, 25] compared and discussed the
DeltaCon method, which compares the similarities ofnode pairs between two graphs. However, the similarityobtained through this method is highly affected by thenumber of nodes and links. And there are no universalcriteria and threshold to evaluate the similarity betweenthe two networks. Paper [26, 27] compared several net-work comparing methods. Including vertex/edge over-lapping, vertex/edge vector similarity, and signaturesimilarity. Among these comparing methods, the sig- a r X i v : . [ c s . S I] J a n nature similarity is the most promising method. Be-cause people can compare two networks by comparingthe network features, such as PageRank [28], eigenvec-tor centrality, betweenness centrality, degree centrality,etc. In this work, we will use the SimHash algorithm[26, 27] to compare the layers, which is in the signaturesimilarity family.In this paper, we present an alternative method forlayer reconstruction and missing link prediction. Thecontributions of this work can be summarized as fol-lows. Inspired by paper [8, 14, 15], we use the maximuma posteriori algorithm to reconstruct a target layer ina multilayer network. Similarly, we will use Poissondistribution to derive the posterior probabilities for thelayer of interest. And the Gamma distribution is usedas the conjugate prior [29]. The adjacency matrices ofthe interdependent layers are used to determine the pa-rameters of conjugate prior, and the contributions ofthe layers are weighted by the similarities. In experi-ments, we show that high similarity layers are criticalin improving the robustness of link prediction.The paper is organized as follows. First, we will de-rive the maximum a posteriori estimation for the layerof interest. Second, we review the
SimHash algorithmfor finding layers with similar features, and we will usethe eigenvector centrality for the comparison. Then, wepresent the method for identifying parameters of con-jugate Gamma prior under different circumstances. Fi-nally, we validate the proposed method through two realmultilayer networks, and compare the results with themaximum likelihood estimation algorithm.
II. LAYER RECONSTRUCTION INMULTILAYER NETWORKA.
Maximum a posteriori based stochastic blockmodel
In this section, we define the stochastic block modelfor both directed and undirected multilayer networks.We assume that θ is the parameter set that can recon-struct a layer in the multilayer network. The parameter D is used to denote the adjacency matrix of the targetlayer. Based on Bayes theorem, we can get the posteriorexpression for θ as P ( θ | D ) = P ( D | θ ) P ( θ ) P ( D ) , (1)where P ( θ | D ) , P ( θ | D ) , P ( θ ) and P ( D ) are the poste-rior probability of the parameter set θ , the likelihood ofthe network under the parameter set θ , the prior proba-bility of the parameter set θ and the marginal likelihoodwhich contains all the information of the network, re-spectively. Since P ( D ) is a constant, so P ( θ | D ) is proportional to the product of P ( D | θ ) and P ( D ) , andthe relation can be derived as P ( θ | D ) ∝ P ( D | θ ) P ( θ ) . (2)For the nodes in the network, we use E ij to denotethe expected number of links (which could be fractional)between node i and node j . So then we have θ = { E ij } ,for all of the node pairs. We use A ij to denote theadjacency matrix of the target layer. In the undirectedand unweighted network, the entries of the adjacencymatrix are either 0 or 1.Before we substitute any parameters into expression(2), we make the following assumptions. The links inthe target layer are independent and identically dis-tributed. In other words, the number of edges betweennode i and node j does not affect the relation betweennode i and node k . Further, we assume the expectednumber of links between two nodes is a Poisson distribu-tion. So we can rewrite expression (2) after substituting E ij and A ij as P ( { E ij } | D ) ∝ (cid:89) i,j e − E ij ( E ij ) A ij A ij ! P ( { E ij } ) ∝ (cid:89) i,j e − E ij ( E ij ) A ij P ( { E ij } ) . (3)To reconstruct the probabilistic model of the tar-get layer, we need to specify the prior distribution of P ( { E ij } ) . Since the conjugated distribution for the ex-ponential family is Gamma distribution. i.e. P ( { E ij } ) = β α ij ij Γ( α ij ) E α ij − ij e − β ij E ij ∝ E α ij − ij e − β ij E ij , (4)where α ij , β ij and Γ( α ij ) are the shape parameter,the scale parameter, and The gamma function of thegamma distribution, respectively. Substituting theabove conjugate distribution into expression (3), we ob-tain P ( { E ij } | D ) ∝ (cid:89) i,j e − ( β +1) E ij ( E ij ) A ij + α − . (5)So the problem now has been simplified to the maxi-mum a posteriori estimation problem, i.e., Finding theparameters E ij that maximize the posterior. However,we still need to specify the expression for E ij . We follow[8] and use the tensors to factorize E ij .In the multilayer network, we assume there are N vertices. And the target layer has K communities. Amembership vector s iz , which represents the outgoingdegree of node i within community z , is assigned toeach node. Similarly, a membership vector t jz is used todenote the incoming degree of node j within community z . Moreover, a control vector w z is used to controlthe density of links within the community z . Based onthe above assumptions, the expected number of edgesbetween node i and j will be the summation of expectedlinks across all the K communities. Which is E ij = K (cid:88) z s iz t jz w z . (6)However, equation (5) is still intractable after we sub-stitute equation (6) into it. After taking the log likeli-hood of equation (5), we have L ( θ | D ) = (cid:88) i,j [( A ij + α ij −
1) log E ij − ( β ij + 1) E ij ]= (cid:88) i,j [( A ij + α ij −
1) log K (cid:88) z s iz t jz w z − ( β ij + 1) K (cid:88) z s iz t jz w z ] , (7)where L ( θ | D ) is the log posterior.To find the maximized posterior for the expression(7), we apply the Jensen’s inequality log x ≥ log x andwe can get log K (cid:88) z s iz t jz w z = log K (cid:88) z q ijz s iz t jz w z q ijz ≥ K (cid:88) z q ijz log s iz t jz w z q ijz = K (cid:88) z q ijz (log s iz t jz w z − log q ijz ) . (8)The equality is satisfied when q ijz = s iz t jz w z (cid:80) z s iz t jz w z . (9)After leaving out constant terms, equation (7) can besimplified to L ( θ | D ) ≥ (cid:88) i,j,k [( A ij + α ij − q ijz log s iz t jz w z − ( β ij + 1) s iz t jz w z ] . (10)Then if we take the derivation of equation (10), wecan get the expressions for s iz , t jz and w z as s iz = (cid:80) j ( A ij + α ij − q ijz (cid:80) j ( β ij + 1) t jz w z , (11) t jz = (cid:80) i ( A ij + α ij − q ijz (cid:80) i ( β ij + 1) s iz w z , (12) w z = (cid:80) i,j ( A ij + α ij − q ijz (cid:80) i,j ( β ij + 1) s iz t jz . (13)Assigning random initial values for s iz , t jz and w z ,then update equation (9), (11), (12) and (13) alterna-tively. The optimized values for s iz , t jz and w z canbe found until the expressions converged. The max-imum posterior is also achieved, when the algorithmconverges. B. Similarity and layer comparison
If the target layer is unknown to the people, this stepwill be skipped. And we will use the layers that arebelieved interdependent or similar to determine the pa-rameters of conjugate prior. However, if the adjacencymatrix is partially known, the similarity and layer com-parison is recommended.To compare the layers in the multilayer network, thefirst thing we need to do is extracting the features ofthe layers [26, 27]. In our experiments, we will use theeigenvector centrality as the feature to compare the lay-ers in the multilayer network. Because the eigenvectorcentralities measure the influences of nodes on a layer.A node has high eigenvector centrality if the node is con-nected by plenty of nodes with high eigenvector central-ity. Other types of centrality measures such as degreecentrality, edge betweenness centrality, or their combi-nations can also be used according to the purposes ofthe experiments or types of multilayer networks.We extract the eigenvector centrality for the nodeswithin layer r and denote it by an N × vector C r . The N entries of the vector are the eigenvector centralityvalues of all the nodes. Then we apply the SimHash algorithm to map the feature (eigenvector centrality inour experiments) of the network into binary bits. Theprocedure for implementing the
SimHash algorithm isas follows:1. Generate N different binary numbers with φ bits,and store the N binary numbers to a matrix. So we canget an N × φ matrix H , where the entries in the matrixare either 0 or 1. Each row of matrix H corresponds toan entry in vector C r . Accordingly, the eigenvector cen-tralities of the nodes are projected to a φ -dimensionalspace;2. Obtaining the weighted feature matrix H r of layer r . We then set all the zeros of H to -1, and multiply eachrow of matrix H by its corresponding entry in vector C r .i.e. H r ( i, j ) = C r ( i ) H ( i, j ) . (14)Each row of matrix H r represents the weighted digestof a node in layer r . The digest H dr for the entire layer r can be obtained through summing up the entries of H r across the columns. i.e. H dr ( j ) = (cid:80) Ni H r ( i, j ) . Theentries of H dr are affected by the eigenvector centralitiesin vector C r . Finally, we set H dr ( j ) to 1 if H dr ( j ) greateror equal to 0, otherwise set H dr ( j ) to 0;3. The similarity between layer m and m (cid:48) is measuredby µ m,m (cid:48) = 1 − Hamming ( H dm , H dm (cid:48) ) φ (15), where Hamming ( H dm , H dm (cid:48) ) is the Hamming distance. C. Identify the parameters of the Gammadistribution prior
In the above analysis, we derived the expressions forreconstructing a target layer of a multilayer network.However, the parameters α ij and β ij are yet to be de-termined. In an L layers multilayer network, differentlayers have the same set of vertices, but we will only usethe layers with similar structures and features. To re-construct the layer of interest, we consider the problemin two scenarios.In the first scenario, we assume we have no informa-tion about the target layer, i.e., The entries of the adja-cency matrix are all zeros. The similarity between thetarget layer and the rest layers cannot be determined.We will assume the similarities are all one. And cal-culate the parameters of the Gamma distribution prioras (cid:40) α ij = (cid:80) L (cid:48) r A rij β ij = L (cid:48) , (16)where r , L (cid:48) and A rij are the r th layer, the layers thatwe believe similar to the target layer and the adjacencymatrix of layer r , respectively. If (cid:80) Lr (cid:54) = m A rij less than 1,we will set α ij to 1, and set β ij to to make equation(11), (12) and (13) converge.In this scenario, we do not have any information re-garding the layer of interest. i.e. A ij = 0 . If we takethe zeros into the calculation, this equals that we as-sume zero presences of all the edges in the target layer.To avoid this issue, we take the entries of A ij as theratio of α ij and β ij , i.e., A ij = α ij /β ij . The entries areassigned as the average presences of the layer, whichwe believe similar to the target layer. People can applythis condition to identify the probabilities of links be-fore doing expensive and time-consuming experiments[12, 13]. And researchers can prioritize their experi-ments according to the link posterior probabilities.In the second scenario, we assume the entries of A ij are partially known, and there are missing links in thetarget layer. In this case, we consider using the layerswith high similarities to reconstruct the target layer and predict missing links. More specifically, we use the sim-ilarity method introduced above to compare the layersand find the interdependent layers. Then we use thesimilarities to help us identify the parameters of theGamma distribution prior.The parameters of the Gamma distribution priorare obtained through the weighted similarities and ex-pressed as α ij = max ( (cid:80) L (cid:48) r (cid:54) = m µ m,r A rij , β ij = max ( (cid:80) L (cid:48) r (cid:54) = m µ m,r , (cid:80) L (cid:48) r (cid:54) = m µ m,r α ij ) , (17)where m , L (cid:48) and µ m,r are the target layer, the layersthat have high similarities with the target layer andthe similarity between the layer m and r , respectively.Similarly, if (cid:80) L (cid:48) r (cid:54) = m µ m,r A rij greater than 0 but less than1, we will set α ij to 1, and set β ij to (cid:80) L (cid:48) r (cid:54) = m µ m,r α ij tokeep the mean of Gamma distribution unchanged. If α ij equals to 0, we will set α ij to 1, and set β ij to to make the iterations converge.In this scenario, the layers’ contributions are weightedby the similarities. The layers with high similarities aremore likely to affect the reconstruction. III. EXPERIMENTAL VALIDATION
In this section, we validate the method introduced inPart.2. First, the testing layer and the remaining layersare compared through the eigenvector centrality basedsignature similarity. Second, the effectiveness of the re-construction process is validated on two real multilayernetworks.the first multilayer network we use to demonstrateour method has nine layers [30]. Each layer representsa highly variable region (HVR). Each node is a malariaparasite gene, and a link is detected if two nodes sharean exact match of significant length. The multilayernetwork has 307 nodes with heterogeneous degree dis-tributions and structures across the nine layers, and thenine layers have a different number of communities interms of Louvain algorithm[32]. We show the first threelayers in Fig. 1(a) through the KiNG software [31].The FAO (Food and Agriculture Organization) tradenetwork [33] is used to validate the similarity compar-ison and network reconstruction. The FAO multilayernetwork is composed of 364 layers, and each layer repre-sents a product. A link is detected if there are tradingsof the product between the two countries. We show thefirst three layers of the FAO network as in Fig. 1(b). (a) The first three layers ofHVR network. (b) The first three layers ofFAO network.
Figure 1: Layer 1, 2, and 3 of the HVR and FAOmultilayer network. Same nodes share the same planecoordinates.
A. Validation of the eigenvector centrality basedlayer comparison
We performed the
SimHash
Algorithm to both theHVR and FAO multilayer networks. The similarities areshown in Fig. 2. Each layer is alternatively set as thetarget layer, and we compare the target layer with theremaining layers. Therefore, we can get L − similarityvalues for each layer.The similarity values for each target layer are shownin the interquartile ranges. In figure 2(a), we can seethat almost all of the nine layers of the HVR networkhave similarities less than 0.8, although layer 5, 6, 7, 8,and 9 are closer than layer 1, 2, 3, and 4. Fig. 2(b)showsthe similarities of the FAO network. To make it clear,we only showed the results of the first 20 layers. Itcan be seen that the interquartile ranges of the FAOnetwork are wider since there are more layers. Also,we can observe that the FAO network has more similarlayers than the HVR network.Given a target layer with missing links, people do notknow how many links are lost. When we utilize the restlayers to reconstruct the layer of interest, we hope thatthe similarities vary insignificantly with respect to themissing links. Fig. 3 shows the similarities when thevarious percent of links removed. We randomly choosethree layers to show the results. In the figure, we cansee that the similarities do not change much even af-ter we remove 60% and 80% links. This is because thestructures of the layers are maintained, and the relativevalues of the eigenvector centralities do not change sig-nificantly. Therefore, the digests of the layers do notchange much since they are binary bits determined bythe eigenvector centralities. B. Validation of layer reconstruction and linkprediction
The method we presented in this work is mainly forlayer reconstruction and missing link estimation. We as- s i m il a r i t y (a) similarities of the HVR network. layer0.50.60.70.80.9 s i m il a r i t y (b) similarities of the FAO network. Figure 2: The layer comparison of the HVR and FAOnetworks. Results are averaged over 10 Monte Carlocross-validations. Panel a) shows the similaritiesbetween each target layer and the remaining layers inthe HVR network. Panel b) shows the similaritiesbetween the first 20 layers and all the rest layers in theFAO networks.sume that the relation between the target layer and therest layers can be pre-assigned or determined throughthe structural comparison as we introduced above.Besides, the number of communities K also needs tobe specified. The community is not similar to the intu-itive communities as detected through traditional meth-ods [34, 35] like the Louvain algorithm [32] or Girvan-Newman algorithm [36]. Since the nodes can belong tomultiple communities, and the nodes in these communi-ties have various expected degrees. The parameter w k is used to control the density of links within differentcommunities. Moreover, there are no sound ways to de-tect such communities. In this method, we evaluate the O D \ H U V L P L O D U L W \ O D \ H U O D \ H U O D \ H U (a) Similarity comparison with edges removed of HVRnetwork. O D \ H U V L P L O D U L W \ O D \ H U O D \ H U O D \ H U (b) Similarity comparison with edges removed of FAOnetwork. Figure 3: Similarity comparison with edges removed.The blue, red, and green represent layers 1, 6, and 9.The three layers are removed with 20%, 40%, 60%,and 80% links, and then are compared with the layersin the original network. The horizontal axis is thelayers in the original network. Lighter color with morelinks removed. We average the results over 10 MonteCarlo cross-validations.number of communities by balancing the optimized pos-terior and computational complexity. Since the vector s ik and t jk are of dimension K , so there are K + 2 N K parameters. In each iteration, the computational com-plexity is O ( N K ) . So the time consumed in each it-eration is proportional to the number of communities.In the experiments, we input the initial number of com-munities determined by the Louvain algorithm and findthe optimal K by varying the number until the poste-rior does not improve significantly. Also, we limit thenumber of communities to less than 20 in consideringrunning time. In practice, we find that the optimized K locates in a wide range, from 10 to 15, the posteriors are quite close. The results shown in our experimentsare all based on ten communities. O D \ H U $ 8 & O D \ H U V O D \ H U V O D \ H U V O D \ H U V Figure 4: AUCs obtained with different number ofsimilar layers. Results are averaged over 10 MonteCarlo cross-validations.To evaluate the results, we follow [8, 9, 37] and use thereceiver-operator curve (ROC) and the area under thecurve (AUC) to validate the effectiveness of our method.The AUC in this method works as follows. Take a valuefrom the set { E ij , i < j } as the threshold and we de-note it as E thres , we can get the True Positive Rateby comparing the percent of true links with E ij greaterthan threshold E thres . Similarly, we can discover theFalse Positive Rate by comparing the percent of non-links with E ij greater than E thres . Repeat the processfor all of the values in the set { E ij , i < j } , and plot theTPR versus FPR. The area under the curve is the AUC.The model is perfect when the AUC approaching 1, and0.5 means the model is similar to a fair coin tossing. O D \ H U $ 8 & O R Z V L P L O D U L W \ P L G G O H V L P L O D U L W \ K L J K V L P L O D U L W \ Figure 5: The AUCs were obtained from differentsimilarity ranges. Results are averaged over 10 MonteCarlo cross-validations.When there are more similar layers, we can get moreaccurate results. Namely, we can still obtain quite closeresults for E ij in the target layer when there are manysimilar layers, even if we have no data of the targetlayer. In this experiment, we choose the first nine lay-ers of the FAO network to show the results as there isthe same number of layers in the HVR network. Andthe similarity ranges in the FAO multilayer network arewider. In Fig. 2(b), we know that the similarities be-tween the first nine layers and the rest layers of theFAO network range from 0.5 to 1. To compare how thenumber of layers affects the results, we choose 5, 10,20, and 30 layers with highest similarities, respectively.And assume we have no information for the layers of in-terest as in the first scenario of Part.2(c). The AUCs ofthe nine layers are shown in Fig. 4. The augmentationof numbers of similar layers enhances the AUCs, butthe improvements attenuated as the number of layers isgreater than 10. So in the undirected network, ten highsimilarity layers are enough to help to reconstruct thelayer. O D \ H U $ 8 & U H P R Y H G U H P R Y H G U H P R Y H G U H P R Y H G U H P R Y H G (a) The MLE based AUCs of HVR network. O D \ H U $ 8 & U H P R Y H G U H P R Y H G U H P R Y H G U H P R Y H G U H P R Y H G (b) The MAP based AUCs of HVR network. Figure 6: Comparison of the MLE and MAP methodsfor the HVR network. Results are averaged over 10Monte Carlo cross-validations.Then, we will show that the layers with high similar- O D \ H U $ 8 & U H P R Y H G U H P R Y H G U H P R Y H G U H P R Y H G U H P R Y H G (a) The MLE based AUCs of FAO network. O D \ H U $ 8 & U H P R Y H G U H P R Y H G U H P R Y H G U H P R Y H G U H P R Y H G (b) The MAP based AUCs of FAO network. Figure 7: Comparison of the MLE and MAP methodsfor the FAO network, where the 10 layers with highestsimilarities are used. Results are averaged over 10Monte Carlo cross-validations.ities can provide more accurate results. In this exper-iment, for each target layer, we choose ten layers withhighest similarities (greater than 0.8), 10 layers withsimilarities range from 0.7 to 0.8, and 10 layers withsimilarities range from 0.6 to 0.7, respectively. Simi-larly, we use these layers to reconstruct the target layersand show the results in figure 5. The ten layers withhigh similarities return the best performance. Similari-ties in the range of 0.7 to 0.8 provide fair good results.In contrast, the ten layers with similarities range from0.6 to 0.7 produce the worst results. In constructing atarget layer in a multilayer network, it is recommendedto use the layers with similarities greater than 0.7.In Fig. 6 and 7, we compared the results obtainedthrough Maximum Likelihood Estimation (MLE) andthe maximum a posteriori (MAP) estimation. For thelayer of interest, we remove 20%, 40%, 60%, 80%, and100% of links, respectively. We compare the removednetwork with other layers to obtain the similarities. Byapplying the MLE and MAP method, we can obtainthe E ij for all the node pairs in the target layer. Wethen compare the E ij with the original network, whichwe used as the ground truth. We apply the secondscenario as introduced in part two (c) to the 20%, 40%,60%, and 80% links removed conditions. And apply thefirst scenario to the 100% links removed condition.In the HVR network, most of the similarities are lessthan 0.8, which means the layers are more heteroge-neous than the layers in the FAO network. In both Fig.6(a) and 7(a), the AUCs obtained through the MLEvarying with respect to the percent of links removed.In contrast, the ranges of the AUCs obtained throughthe MAP method are narrower. Especially when thelayers are more similar as in Fig. 7(b). IV. CONCLUSION
To model the layer of interest, we take advantage ofthe high similarities layers. First, we derived the max-imum a posteriori estimation for the target layer re-construction, and used the Gamma distribution as theconjugate prior. Second, we introduced the similarityand layer comparison process. Then, we separated theproblem into two scenarios. In the first scenario, if thetarget layer is unknown to us, we use the layers thatare believed to have high similarities. In the secondscenario, if the target layer is partially known, we canuse the comparing method to find similar layers. Thesimilar layers are then used to determine the parametersof the conjugate prior. However, there are still some drawbacks to themethod we presented. It is required that layers usedto find the prior must be similar to the layer of interest.The similarities need to be pre-determined or detectedthrough feature comparisons. Besides, the number ofcommunities needs to be specified. Although variousmethods have been proposed to detect communities fornodes [32, 34–36]. To our knowledge, there are no soundand reliable ways to obtain multiple communities. Inthis method, we tested multiple numbers for K , andfind the optimal K by balancing the convergence timeand maximum posterior. Further, we did not quantifyhow the number of similar layers combining the targetlayer affects the posterior. We leave these questions forfuture works.The method we proposed shows promising resultswhen we can identify the similarities between the tar-get layer and the supporting layers. The results are lesslikely affected by the percent of missing links, whichis generally unknown to the researchers. On the con-trary, the results obtained through the MLE methodare significantly affected by the percent of missing links.We believe that the results obtained through our MAPmethod can help researchers prioritize their experi-ments, especially when there is no information aboutthe layer of interest. ACKNOWLEDGEMENTS
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