An efficient method for link prediction in weighted multiplex networks
 Shikhar Sharma†^{1} and
 Anurag Singh†^{2}Email author
Received: 21 March 2016
Accepted: 22 October 2016
Published: 5 November 2016
Abstract
Background
A great variety of artificial and natural systems can be abstracted into a set of entities interacting with each other. Such abstractions can very well represent the underlying dynamics of the system when modeled as the network of vertices coupled by edges. Prediction of dynamics in these structures based on topological attribute or dependency relations is an important task. Link Prediction in such complex networks is regarded useful in almost all types of networks as it can be used to extract missing information, identify spurious interactions, and evaluate network evolving mechanisms. Various similarity and likelihoodbased indices have been employed to infer different topological and relationbased information to form a link prediction algorithm. These algorithms, however, are too specific to the domain and do not encapsulate the generic nature of the realworld information. In most natural and engineered systems, the entities are linked with multiple types of associations and relations which play a factor in the dynamics of the network. It forms multiple subsystems or multiple layers of networked information. These networks are regarded as Multiplex Networks.
Methods
This work presents an approach for link prediction in Multiplex networks where the associations are learned from the multiple layers of networks for link prediction purposes. Most of the realworld networks are represented as weighted networks. Weight prediction coupled with Link Prediction can be of great use. Link scores are received using various similarity measures and used to predict weights. This work further proposes and testifies a strategy for weight prediction.
Results and Conclusions
This work successfully proposes an algorithm for Weight Prediction using Link similarity measures on multiplex networks. The predicted weights show very less deviation from their actual weights. In comparison to other indices, the proposed method has a far low error rate and outperforms them concerning the metric performance NRMSE.
Keywords
Link prediction Multiplex networks Complex networks Weighted networksBackground
Many systems and their interactions can be modeled as an abstraction which provides a suitably accurate representation of their dynamics. These dynamics can be simulated to discover new properties, interactions, and different representations. They can also be used to forecast or predict behavior. These entities can be modeled as nodes in a graph where the links/edges are representative of the interactions/relationships between the individuals. For instance, a simple friendship network can be represented with the nodes representing the individual under study and the edges representing if two individuals are friends. Alternatively, a normalized weight can be added to each link to denote the amount of friendship between two individuals derived from some other factors. Link prediction or forecasting the formation of a link in such systems with their abstraction as complex networks is useful in multiple ways. Link prediction attempts to formulate a likelihood of the existence of a link between two nodes. In many networks, whether a link exists or may exist in future is a matter of time and resource consuming experiments. Link prediction is an adequate substitute in such cases as the focus can be shifted to the links according to a higher likelihood obtained from a sophisticated model. In the age of growing network and demand for efficient network analysis mechanisms, link prediction becomes necessary in addressing the problem of missing link. Moreover, link prediction forms the basis of various recommended systems which are used in online marketing, ecommerce services, and many social networks. Link prediction in terrorist communication networks can help predict and intercept vital information about the issue of national security. Furthermore, link prediction also comes into play in various situations such as to strategize an efficient transport network and to study genetically transferred diseases. These networks can be studied and may contain many different aspects. They may be represented in the form of temporal networks [1] or information can also be visualized in the form of multilayer networks [2].
Various strategies have been adopted for link prediction. Some of the strategies are based on the local topological information from the network corresponding to an index of evaluation. Common neighbor (CN) [3] algorithm is one such strategy which relies on the fact that more the number of common neighbors of two unconnected nodes have at a given time, higher is the likelihood of them forming a link in future. Salton index [4] normalizes the common neighbor index by the degree of individual nodes. A new strategy has been proposed for predicting the missing links which also takes into account the number of links between two sets of uncommon neighbors of given nodes, in addition to their common neighbors [5]. Other local similaritybased indices like Jaccard index [6], preferential attachment index [7], hub depressed index, hub promoted index, [8] have also been used for link prediction. Indices are taking in account the global information from graphs (path ensemble and average commute time) such as Katz index [9] and cosinebased methods have also been successfully employed for link prediction. Strategies using the attributes of a node have been proposed to a good effect [10]. Furthermore, Bliss et al. [11] have successfully employed evolutionary algorithm to combine various indices to get an optimized result, whereas He et al. [12] have used the OWA operator for the purpose of combination. A brief description of some of these methods can be found in the next section. However, most realworld networks are better visualized in the form of multiple types of relationships rather than on a single one. For Instance, consider prediction of a social network attachment between two entities. The prediction would be more likely caused by a combination of factors such as mutual interests, spatial presence at certain times, common acquaintances. The target to be predicted is not just based on a single parameter but consists of a composite informational paradigm. For Instance, two persons being friends depend not just only on whether they share the same interests but also on other aspects such as time spent together, common acquaintances, and various other factors. It is not realistic to just establish and determine relationships based on a single parameter. A multiplex network is a network in which every pair of nodes can be connected by is a pair of nodes with different types of edges and can be visualized in the form of multiple layers. Such networks contain various types of information but are sparse in nature. A multiplex network stores each type of information as a separate layer and thus encapsulates a significant number of different relationships between the same entities. The current algorithms are highly network and context specific. They tend to rely on some underlying assumptions of the network rather than inferring those from the network itself. Their performance also relies on a large number of edges. Thus, these algorithms give mixed results on multiplex networks and are unable to use the information from different layers in an efficient manner. In this work, we propose an approach that can be utilized to predict links in multiplex networks where the extension of preexisting algorithms is not viable. Not much of work has been done in this regard, and recent works focus on a particular type of social network and not on a multiplex network in general. The importance of weights in networks is well known. The work proposes a weight prediction algorithm that depends on score assignment by the proposed likelihood algorithm to detect edge strength and similarity. These scores are used in a normalized manner to predict weights in a network. The normalization takes into account any significant differences between scores of edges under consideration. The methodology is compared to other indices based on normalized rootmeansquare error as the performance metric. It is the normalized square root of the squared sum of differences in weight prediction over some test cases. This work utilizes multiple sources of information, and combines them to make an optimal prediction strategy. It is context and network independent. It operates on any network where layers exhibit a positive corelation. The results depict the strength regarding a very little error. It is a scalable approach combining the idea of link and weight prediction and can be operated in parallel for networks with million of nodes.
The paper is organized as follows: In second section, various strategies for link prediction are discussed; in third section, a model for multiplex network is presented; in fourth section, a methodology with the help of algorithms for link prediction is proposed; in fifth section, results and analysis of the proposed algorithms are discussed; in sixth section, weights of the links are taken into consideration and a method for the weight prediction for the predicted link with the results and analysis is proposed; and in seventh section, a summary of the the analysis results of this research is provided.
Conceptual formalization of link prediction
Problem representation
Let a layer of network be represented by a graph G(V, E) where V is the set of nodes and E is the set of edges. Let . represent the cardinality of a set. Hence, we have V as the number of nodes and E as number of edges in the graph. Let n represent the number of nodes in the graph (V) and U represent the universal set containing all \(\frac{n.(n1)}{2}\) edges of the graph. The problem lies in assigning a likelihood to the existence of \(UE\) missing edges in the graph.
For a multiplex graph, let L represent the set of all layers. Between any two nodes, there can be L types of links. Some nodes may be totally disconnected in certain layers. The layer on which prediction has to be made is named as the target layer and represented by \(L^{T}\). Clearly, \(L=L^{T}UL^{P}\) and \(L L^{T}\) is \(L ^{P}\). The problem is to assign likelihood to a link in \(L^{T}\) based on information from the complete graph. It is important to note that \(L^{T}\) and \(L^{P}\) are used here as a set notation depicting the operations on corresponding set of vertices and edges. Similarly, the set of other layers is called as predictor layer set and is denoted by \(L^{P}\).
Let \(\mathrm {\Omega }\) be a subset of E which represents the set of edges which are used for testing the link prediction algorithm and are removed from the original graph. Let \(\mathrm {\kappa }\) denote the cardinality of \(\mathrm {\Omega }\) which is the number of edges in the test set.
It is important to note that the set of vertices, V is same for all the layers but each layer has a different subset of the set of edges E (all possible edges between nodes in set V). The target problem is missing link and weight prediction in multiplex networks.
Evaluation metrics
The link prediction algorithm assigns a likelihood score to each pair of nonobserved links (set \(UE\)) and the test set \(\mathrm {\Omega }\). Let S(i, j) denote the score for some node i, j \(\mathrm {\varepsilon }\) V. Following are the evaluation metrics that are most commonly used to test the prediction algorithm.
AUC
Precision
Link prediction methodologies
This section provides an overview to some of the indices used to assign a likelihood score between a pair of candidate nodes. These indices are often context specific and so is their performance across different systems. Some of the indices are described, and other are summarized in Table 1.
Common neighbors
The neighbors of a node N are all the nodes which share an edge directly with N. In this work, neighborhood is defined for all nodes within the same layer and not for nodes in a different layer.
Salton index
Jaccard’s coefficient
Katz index
Other indices
Some similarity measures for link prediction
Name of the index  Description 

Adamic/Adar [14]  Counting of common features by weighting rarer features more heavily 
Preferential attachment  Product of degree of nodes 
Katz index  Ensemble of all paths with more weight to shorter paths 
Random walk with restart [15]  Steps in which a random walker reaches from one node to another 
Resource allocation [16]  Assigns scores according to resource distribution between candidate vertices with common neighbors as transmitters 
Various other similar indices based on topological and probabilistic models have been proposed whose discussion is out of the scope of this work.
Multiplex networks
Monoplex networks are a network which represents a single property and its related dynamics. There is a single layer consisting of nodes and edges. They are restricted to the representation of a single property. The majority of the realworld systems exhibit dynamics that are not just an outcome of an individual property but rather are dependent on a combination of many such properties. A pair of the node may have multiple relationships which govern the overall dynamics of a network. Most recently, there have been increasingly intense efforts to investigate networks with multiple types of connections [17]. Many realworld complex systems function through multiple layers of distinct interactions among constituents and the interplay between these interaction layers [17, 18]. It was recognized decades ago that it is crucial to study systems by constructing multiple social networks using different types of ties among the same set of individuals [19]. The different types of interactions are modeled using multiplex networks, and there exist a plethora of methods for the representation and behavior inference [2].
In Fig. 1, there are five nodes with three different types of relationships represented by the colors red, blue, and green. It can be considered as representative of a network of five places where each color represents whether a particular type of mode of transport [road (R), water (B), air (G)] is available to commute between these places. One may plan a journey on combinations of these factors according to different weights attributing to cost, distances, and time. Thus the optimum way of travel requires a combination of many different types of information. This information can be even more heterogeneous in nature than the above example as evident by the example dataset used for analysis in this work which models five different types of social relationships. Moreover, one can visualize such networks by preserving all nodes and keeping a particular color of edges at a certain time. This way we would have many graphs equal to the number of different colors or different types of relationships. Thereby, these can be visualized as different layers. The only interlayer connection that is present is between the same node in both layers. Information from all such layers can be used to predict a link in another layer. The link prediction methods at present focus on considering a single type of relationship from a particular graph and ignore other factors. These networks help in discovering new types of relationships and characteristics. Multiplex networks are realistic representation where they can take all types of information influencing a certain dynamic to predict the outcomes in future.

Layer 1: Representative of two individuals having Lunch together.

Layer 2: Representative of two individuals having a social connection via Facebook.

Layer 3: Representative of two individuals coauthoring a publication.

Layer 4: Representative of two individuals having Leisure together.

Layer 5: Representative of two individuals working together.
Proposed methodology for link prediction in multiplex networks
The proposed method uses the information from all other layers of the network for the purpose of link prediction at an individual layer known as the target layer. Other layers are referred to as predictor layers. Each layer influences prediction on a target layer to a different extent. Some form a good representation of the target layer, whereas others may create a poor representation. This information is required for a good overall link prediction. Relationship and representational information between the target layer and each of the predictor layer are extracted from the network structure and links. This information is tested to get a measure of how well does a particular predictor layer represent the target layer. The final link prediction score is a weighted result of the above outcomes.
More specifically, the final score is assigned as a weighted combination of scores of each layer. The weights are extracted by checking the link correspondence between two layers using likelihood of a link being present in the target layer given the link is present in the predictor layer. This orders the predictor layers concerning relative importance for a specific target layer.
A pictorial representation of the proposed algorithm is given in Fig. 3. Each predictor layer provides the information about the likelihood of the presence of a link in a target layer. The likelihood is added to assign a score to each pair of nodes. These are then sorted and tested for precision. Similarly, a nonobserved link and an observed link are both picked randomly and compared by likelihood score assigned to obtain the AUC.
The first algorithm (Algorithm 1) is used to assign the likelihood of a link existing in a layer based on information obtained from the other layers. For each layer, the likelihood is individually calculated and used as a weight. The overall likelihood is a combination of the likelihoods obtained individually [22]. The likelihood is calculated by observing a previous snapshot (denotes a subset of the graph after removing the edges used for performance testing purposes) of the graph and getting an estimation of the dependency of the probability of the existence of a link in the target layer given a predictor layer. The same is carried out for each pair. The second algorithm (Algorithm 2) is used to find the AUC measure for the proposed method. It picks up two edges in an iterative manner, one from the training set and another edge which was not a part of the graph (nonobserved). It compares the likelihoods and gets incremented in case the likelihood of an edge that was predicted correctly is greater than a nonobserved edge. The algorithm to obtain precision (Algorithm 3) sorts all edges based on the assigned scores and checks the number of relevant (correctly predicted) edges which contribute to the top of sorted scores.
The algorithm inputs a network and returns it by attaching scores to each edge based on a likelihood of formation.
The procedure linkInPredictorLayer is used to obtain information if a link is found to be present in a predictor layer at some snapshot of the network. It assesses a predictor layer for the presence of an edge and uses a likelihood estimate to get a prediction for the presence of an edge in the target layer. It returns true (a value of 1) if an edge is present in the predictor layer else it returns a value of 0. The same is repeated for all pairs of layers, and the likelihood is combined to assign a final score. Network structure at an earlier snapshot is known. After obtaining the likelihood and combination of opinion from predictor layers, an idea about the existence of a link in the future can be made.
Results and analysis of proposed link prediction methodology
Observed AUC values
CN  JA  PA  Proposed  

Layer 1  0.79  0.75  0.71  0.85 
Layer 2  0.83  0.84  0.79  0.88 
Layer 3  0.1  0.71  0.72  0.8 
Layer 4  0.81  0.8  0.79  0.93 
Layer 5  0.8  0.82  0.83  0.83 
Observed precision values
CN  JA  PA  Proposed  

Layer 1  0.11  0.43  0.21  0.95 
Layer 2  0.33  0.41  0.3  0.83 
Layer 3  0.8  0.1  0.2  0.98 
Layer 4  0.2  0.11  0.2727  0.61 
Layer 5  0.29  0.16  0.29  0.61 
In Algorithm 2, randEgde returns a random edge from the augmented Set and edgeScore returns the likelihood of the augmented edge.
Weight prediction
Weighted networks
Consider, for instance, Fig. 4, which represents the modes of traveling between different nodes/places as described earlier using colored edge representation: road (Red), air (Green), and water (Blue). Assuming each link is bidirectional, let the weights on the links represent the volume of flow from each node. Assuming addition of a new node, F is indicated by a link prediction algorithm; it is also important to have an indication of what would be the volume (weight) of the link that will be developed (u and x); and the type of network that will best suit the cause. For example, if there is a very low volume of commuters to F, we could only add just a single road link. This will depend on the weights/volume on other links and their dynamics.
Different methodologies have been used for the purpose of weight prediction in the network. Zhao et al. [24] identify some reliable routes to predict weights. Furthermore, common neighbors, Jaccard and preferential attachment and other methods can also be used for predicting weights utilizing the underlying property they rely on. For example, if two nodes share the high number of common neighbors, the weight of the link between them depends more on the weights of the edge they form with their common neighbors. Similarly, the Jaccard measure assumes higher values for pairs of nodes which share a higher proportion of common neighbors about the total number of neighbors they have biased the weight prediction accordingly. However, these methods take into only account prediction for a singlelayered network. We extend the idea from the proposed link prediction algorithm for multiplex networks for the purpose of weight prediction. Weight prediction in the multiplex network is based on the scoring of proposed multiplex likelihood assignment method.
Dataset
Description of dataset
Parameter  Description 

Number of nodes  514 
Number of edges  7153 
Number of layers  16 
Some keywords  Neutrinos, detector, enhancements, anisotropy, point source 
Proposed methodology for weight prediction in multiplex networks
Results and analysis for weight prediction
Observed average NRMSE values
Section  Number of tests  CN  JA  PA  Proposed 

1  50  0.0623  0.0801  0.1207  0.0212 
2  100  0.084  0.07488  0.10407  0.00955 
3  200  0.0455  0.00389  0..0823  0.001919 
4  500  0.0076  0.0056  0.0912  0.0017 
5  1000  0.01956  0.00790  0.1025  0.00162 
The NRMSE is the square root of squared difference of predicted values normalized by number of tests and range of predicted weights [26].
Conclusions
A network is an abstraction of various kinds of dynamics that govern a process. A model for prediction of these dynamics is of significant importance in many fields viz social networks, biological networks, and transportation networks. Existing link prediction algorithms focuses on a single type of information. A multiplex network facilitates link prediction using information of different types which is a more accurate representation of reality. This work successfully proposes an algorithm for link prediction on generalized multiplex networks. It interprets the relations, and their relative importance in the whole network visualized in the form of multiple layers. The algorithm establishes a likelihood of the presence of edges in one layer with respect to the other layer. This process is repeated for all layers to obtain a likelihood measure individually. The combination of the likelihoods is taken to pass final judgment. The method uses multilayer information rather than the intralayer information. The validation proves that the algorithm outperforms the common neighbor, Jaccard’s, preferential attachment and provides validation on some intuitive observations. Multiplex networks are closer to reality as they encapsulate various heterogeneous relations rather than just one. The algorithm successfully employs information from all of the layers and combines to give favorable link prediction results on a multiplex network. Furthermore, combined with the target layer information, the algorithm can prove to be an extension of single layer indices over multiplex networks. The proposed likelihood assignment algorithm for link prediction performs well over a multiplex dataset. The network assumes the positive correlation between relations depicted by each layer. It is used for scoring a network to obtain similar edges whose weights are further averaged in a normalized manner to assign a weight to an edge. The predicted weights have very less deviation from actual weights. Compared to other indices, the proposed method has a far low error rate and outperforms them concerning the metric performance NRMSE.
Notes
Declarations
Authors' contributions
Both the authors designed the research, data collection, and experiments and contributed to the proposed methods and to the writing of the manuscript. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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