- Open Access
An efficient method for link prediction in weighted multiplex networks
© The Author(s) 2016
- Received: 21 March 2016
- Accepted: 22 October 2016
- Published: 5 November 2016
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 likelihood-based indices have been employed to infer different topological and relation-based 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 real-world 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.
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 real-world 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.
- Link prediction
- Multiplex networks
- Complex networks
- Weighted networks
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, e-commerce 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  or information can also be visualized in the form of multilayer networks .
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)  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  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 . Other local similarity-based indices like Jaccard index , preferential attachment index , hub depressed index, hub promoted index,  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  and cosine-based methods have also been successfully employed for link prediction. Strategies using the attributes of a node have been proposed to a good effect . Furthermore, Bliss et al.  have successfully employed evolutionary algorithm to combine various indices to get an optimized result, whereas He et al.  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 real-world 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 pre-existing 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 root-mean-square 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 co-relation. 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.
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.
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.
Some similarity measures for link prediction
Name of the index
Counting of common features by weighting rarer features more heavily
Product of degree of nodes
Ensemble of all paths with more weight to shorter paths
Random walk with restart 
Steps in which a random walker reaches from one node to another
Resource allocation 
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.
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 real-world 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 . Many real-world 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 . The different types of interactions are modeled using multiplex networks, and there exist a plethora of methods for the representation and behavior inference .
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 inter-layer 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 co-authoring a publication.
Layer 4: Representative of two individuals having Leisure together.
Layer 5: Representative of two individuals working together.
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.  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 single-layered 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.
Description of dataset
Number of nodes
Number of edges
Number of layers
Neutrinos, detector, enhancements, anisotropy, point source
Proposed methodology for weight prediction in multiplex networks
Results and analysis for weight prediction
Observed average NRMSE values
Number of tests
The NRMSE is the square root of squared difference of predicted values normalized by number of tests and range of predicted weights .
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.
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.
The authors declare that they have no competing interests.
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