Finding missing edges and communities in incomplete networks

The number of common neighbours that two vertices have is a basic idea that suggests a mutual relationship between them. For example, it may be more likely that two people know each other if they have one or more acquaintances in common in a social network [40]. The function is defined as [8]

$$\begin{equation} {\rm score}(u,v) = | {\Gamma (u) \cap \Gamma (v)} |,\end{equation} \tag{ 1 }$$

where Γ(u) and Γ(v) represent the set of neighbours of vertices u and v, respectively.

These three coefficients have a similar definition, related to the probability of triangles in all connected edges of any two vertices [14, 16–18]. They are defined as [14]

$$\begin{equation} {\rm score}(u,v) = | {\Gamma (u) \cap \Gamma (v)} |/| {\Gamma (u) \cup \Gamma (v)} |, \end{equation} \tag{ 2 }$$

$$\begin{equation} {\rm score}(u,v) = \left| {\Gamma (u) \cap \Gamma (v)} \right|/\min (\left| {\Gamma (u)} \right|,\left| {\Gamma (v)} \right|), \end{equation} \tag{ 3 }$$

$$\begin{equation} {\rm score}(u,v) = \left| {\Gamma (u) \cap \Gamma (v)} \right|^2 /(\left| {\Gamma (u)} \right| \cdot \left| {\Gamma (v)} \right|). \end{equation} \tag{ 4 }$$

Adamic and Adar (AA) [13] proposed a similarity measure to define the similarity between two vertices in terms of the neighbours of the common neighbours of the two vertices. It is translated into [8]

$$\begin{equation} {\rm score}(u,v) = \sum\limits_{s \in \Gamma (u) \cap \Gamma (v)} {\frac{1}{{\log \left| {\Gamma (s)} \right|}}.} \end{equation} \tag{ 5 }$$

Resource allocation (RA) is a variant of the method of AA, which assumes that the common neighbours could transmit resources from one vertex to the other one, and is defined as [15]

$$\begin{equation} {\rm score}(u,v) = \sum\limits_{s \in \Gamma (u) \cap \Gamma (v)} {\frac{1}{{\left| {\Gamma (s)} \right|}}} . \end{equation} \tag{ 6 }$$

Preferential attachment (PA) shows that the probability of a connection between two arbitrary vertices is related to the number of neighbours of each vertex. The assumption of this method is that vertices prefer to connect with other vertices with a large number of neighbours. It is described as [40, 41]

$$\begin{equation} score(u,v) = \left| {\Gamma (u)} \right| \cdot \left| {\Gamma (v)} \right|. \end{equation} \tag{ 7 }$$

This method combines a maximum-likelihood method with a Markov chain Monte Carlo method to sample the hierarchical structure models with probability proportional to their likelihood from the given network. This model is a binary tree with n leaves (the vertices from the given network) and n−1 internal nodes, and a probability p_r is associated with each internal node r. The probability, p_r, of the deepest common ancestor of two vertices represents the probability of a connection between them.

This method uses the stochastic block model, in which vertices are placed into communities and a matrix Q includes the probabilities of connections between communities. Q_αβ is the probability of a connection between two vertices that are in community α and community β. Then, the probability of an edge can be calculated as explained in [3].

Because the hierarchical structure method (HRG) and block model (BM) methods have a higher computational complexity than other methods, we can only evaluate these on some small networks in our experiments reported below.

To determine the accuracy of edge prediction methods, a common measure is the AUC: the area under the ROC (receiver-operating characteristic) curve [42–44]. The interpretation of AUC is the probability that the score of a randomly-chosen missing edge is higher than that of a randomly-chosen pair of unconnected vertices.

Figure 2 shows a comparison of edge prediction methods on three incomplete versions of ER networks. Since ER networks are random networks, which do not have any vertex-to-vertex similarity, it is impossible to find missing edges by using edge prediction methods based on common neighbours. In figure 2, these methods have no effect with any type of missing edges. In contrast, the PA method relies on the number of neighbours of each vertex to find the relation between them. For example, PA performs worse in crawled networks, because the vertices at the boundary of crawled networks always have a low degree; in limited-degree networks, we limit vertices which have a high degree, so PA performs better than other methods on limited-degree networks.

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Figures 3–5 show comparisons of edge prediction methods on three incomplete versions of LFR networks (named LFR1, LFR2, and LFR3, respectively), using the following parameters:

• 1  
  n = 100, 〈k〉 = 5, k_max = 15, τ₁ = 2, τ₂ = 1, µ = 0.3, c_min = 10, c_max = 20.
• 2  
  n = 1000, 〈k〉 = 10, k_max = 25, τ₁ = 2, τ₂ = 1, µ = 0.3, c_min = 10, c_max = 20.
• 3  
  n = 1000, 〈k〉 = 10, k_max = 25, τ₁ = 2, τ₂ = 1, µ = 0.3, c_min = 20, c_max = 40.

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The results of CN, AA and RA index are similar, because they are all based on common neighbours. They perform best on LFR2 networks because these have a higher clustering coefficient than LFR1 and LFR3 (see table 2). Jaccard, meet/min, and geometric also consider common neighbours, but are affected by other conditions. For example, Jaccard performs best in limited-degree networks because the degrees of many vertices are similar, especially when there are more missing edges. Jaccard is not appropriate when there is a missing edge between a high-degree vertex and a low-degree one.

PA performs well in limited-degree networks, and is not affected by the community size and the clustering coefficient, because PA is appropriate to the mechanism of limited-degree networks. On the contrary, vertices at the periphery usually have a low degree in crawled networks, and it is impossible to know whether the endvertices of missing edges in the crawled and random-deletion networks have high degree.

The HRG and BM methods are better than other edge prediction methods in random-deletion networks, but they do not perform well in crawled networks. In limited-degree networks, these two methods perform similar to other edge prediction methods.

Figure 6 shows comparisons of edge prediction methods on three incomplete versions of small real-world networks. Basically, BM and HRG work better than other methods in random-deletion and limited-degree networks. HRG performs well in all types of incomplete networks if the original network is a hierarchical structure (see figures 6(g)–(i)). PA performs well in limited-degree networks, and sometimes it is even close to or better than BM. CN, RA and AA perform better than BM and HRG in crawled networks if the original network has a high clustering coefficient (see 'terrorists' in table 2).

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Figure 7 shows comparisons of edge prediction methods on three incomplete versions of larger real-world networks; these are too large for BM and HRG to process in a reasonable time. The results of RA, AA and CN are similar, and they are better than other prediction methods in crawled and random-deletion networks. PA performs well on limited-degree networks.

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In general, BM and HRG perform better than other methods in random-deletion and limited-degree networks. However, they are too slow to apply to large networks. In addition to these two methods, CN, RA and AA perform well, while PA usually works well in limited-degree networks. All methods do better on limited-degree networks than on crawled and random-deletion networks.

We now investigate the effect of missing data on community detection algorithms: COPRA [45], CliqueMod [46], CNM [47], Wakita and Tsurumi [48], Walktrap [49], and the Louvain method [50]. In this part, we construct the incomplete networks on the induced subnetworks to keep them connected, as explained in section 3.3, and we compare all of them, including the induced subnetworks, with the original networks. For artificial networks, we use the normalized mutual information measure [51] to compare the known partition with the partition found by each algorithm. For real-world networks, since we do not know the real community structure, we use two community detection algorithms (COPRA and the Louvain method), which do not require to be told the number of communities. We run these two algorithms on the original network first, to obtain a partition, and choose the one with the maximum modularity as the 'real' partition. Then, we compare the partition obtained by each algorithm for all incomplete networks with that 'real' partition.

Figure 8 shows the results of different community detection algorithms on the induced subnetworks and on three incomplete versions of LFR networks (parameters n = 1000, 〈k〉 = 20, k_max = 50, τ₁ = 2, τ₂ = 1, µ = 0.1, c_min = 50, c_max = 100; the results are averaged over 100 random networks with the same parameters). All of the results of the induced subnetworks are better than those of the incomplete networks. We can also see that the results for crawled networks are usually worse than those for other incomplete networks. To understand the reason for this, we calculated the number of removed intercommunity edges, intracommunity edges and the remaining intercommunity edges in these incomplete networks (see table 3). This shows that random-deletion and limited-degree networks usually lose more intercommunity edges and fewer intracommunity edges than crawled networks. This explains why results are worst for crawled networks. For the other incomplete networks, communities can be found well even when there are many missing edges.

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Finally, we examine three real networks: email, scientometrics and C. elegans. Figure 9 shows the performance of COPRA and the Louvain method on induced networks and the three incomplete networks, compared with the original networks. The results are broadly similar to those of artificial networks: the results of random-deletion and limited-degree networks are better than those of crawled networks.

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