Weak ties phenomenon in PPI networks

A network consists of two basic elements: vertices and edges. Many measurements are developed to characterize the role of a node for structure and function including random walk-based indices [43], PageRank score [44]. In comparison, the study of the edge's role is less extensive.

Actually, edges in a network usually have two roles to play: some contribute to the global connectivity like the ones connecting two clusters while others enhance the locality like the ones inside a cluster. In social networks, the two roles are reflected as two important phenomena, being respectively the homophily [45] and weak ties effects [46]. Homophily demonstrates that connections are more likely to be formed among individuals with close background, common characteristics. On the other hand, the weak ties phenomenon shows that the less similar individuals are prone to be connected with weaker strength. These weak ties have important roles to play in maintaining the global connectivity. It has been proved that the weak ties phenomenon exists in the mobile communication [39] and document networks [40]. But, the weak ties effect for PPI networks remains to be tested.

where s is the size of a connected subgraph, N is the size of the whole network and the sum includes all connected components. An obvious gap occurs when the network disintegrates [47].

where (u, υ) is the edge with u, υ being the endpoints, C _u is the size of the maximal clique containing vertex u and C_{(u, υ)}is the size of the maximal clique containing (u, υ). It, however, can not distinguish the bridges and non-bridges because it fails to take into account the difference between a pair of vertices. The bridggness value for each edge in a clique is 1 according to Eq.(2). It is unreasonable because intuitively the larger the size of a clique is, the lower the probability for some edge in the clique being a bridge is. For example, edges in 3-clique are more prone to be bridges than ones in 8-clique.

where J(u, υ) is the Jaccard similarity, i.e., $J\left(u,\upsilon \right)=\frac{\left|N\left(u\right)\cap N\left(\upsilon \right)\right|}{\left|N\left(u\right)\cap N\left(\upsilon \right)\right|}$ with N(u) being the neighbors of vertex u, and C _u\υ is the size of the maximal clique containing u without υ. The 1- J(u, υ) measures the dissimilarity between the pair of endpoints while the latter component quantifies the relation between the neigbors of two endpoints. The physical interpretation of Eq.(3) is that only these edges whose endpoints are less similar in topological and maintain the global connectivity are the bridges. Compared with Eq.(2), the new index is more reasonable, for example, for an edge in a m-clique is $\frac{2\left(m-1\right)}{m^2}$, which decreases as the size of a clique increases.

Similar to Ref. [39], we quantify the weak ties phenomenon according to an edge percolation process. Generally speaking, if the weak ties phenomenon exists in terms of content similarity, the network will disintegrate much faster when we remove edges successively in ascending order of content similarity than in descending order. Figure 2 (a) shows R _GC decreases much faster when the less similar edges are removed firstly. As shown in Figure 2 (b), a sharp peak occurs when the edges removed from the weakest to the strongest one, demonstrating the disintegration of the networks involved. Careful comparison of Figure 2 (a)(b) further shows that no percolation phase transition appears since there is no clear peak. These strongly supports the weak ties phenomenon in the PPI networks. In addition to the existence of weak ties phenomenon, we also have great interest in quantifying the edges' role of maintaining global connectivity. How good the bridgeness characterizes the weak ties phenomenon has been investigated in Figure 2 (c)(d). Figure 2 (c) indicates that R _GC decreases much faster when the stronger bridges are removed firstly. As shown in Figure 2 (d), a sharp peak occurs when the edges removed from the strongest to the weakest one, demonstrating the disintegration of the networks involved. It is enough to assert that the bridgeness is an excellent alternative to describe the tie strength. To make a fair comparison between the index [40] in and ours, we also investigated how the networks changes in terms of bridgeness in Eq.(2) as shown in Figure 2 (e)(f). Compared Figure 2 (c)(d) with Figure 2 (e)(f), we can easily conclude that the network disintegrated more quickly (the bigger gaps in R _GC and $\tilde{S}$) when the novel bridgeness is adopted, indicating that the new index is more efficient in characterizing the bridges in networks.

where A is the adjacency matrix of the network involved, (A ^k ) _ij denotes the number of walks of length k connecting vertex υ _i and υ _j , and β is parameter controlling the relevant importance of each component. The long walks receive greater weights when β > 1 while the short ones get more attention if β < 1. Here, we set β = 0.618. The result is showed in Figure 3. It demonstrates that there is a negative correlation between bridgeness and topological similarity, i.e., the weaker the similarity between a pair of proteins is, the stronger its bridgeness is.

Constructing a reliable network

Gavin et al [8] have pointed out that the core of a complex has relatively more interactions while the attachments bind to the core proteins to form a biological complex, implying that the connectivity of a core is better than the whole complex.

To assess the topological proximity of a core, the measure of proximity of a pair of vertices should be handled beforehand. The most commonly used one is the graph distance, that is, the length of the shortest path connecting the pair of vertices. This quantity, however, is not appropriate for the biological networks largely because of two drawbacks: first, it does not take into account the local structural feature of the networks; second, it is very susceptible to the noises, e.g., a single missing edge effects the proximity, significantly. Thus, vertices connected by paths of various lengthes are likely to be functionality closer than vertices connected via a single path. In detail, give an edge, say (u,υ), it is reasonable to consider that the information transferred from u to υ through the right channels. The more the channels are, the better the connectivity is. Actually, in biological network, the genetic information is transferred by the pathways. From the aspect of graph theory, it is natural to consider the channels as various walks connecting u, υ. Likewise, we also take into consideration the strength of paths: the strength of the effect via longer paths with more intermediate vertices is very likely to be lower than those via shorter ones with fewer intermediaries. Given a walk of length k, say υ₁→υ₂ → ... υ _{k+ 1}, its strength is defined as the product of the weights on each edge in the walk, i.e., ${\prod }_{i=1}^k{w}_{i,i+1}$ where w _{i, j} is the weight on the edge (υ _i , υ _{i +1}).

The larger the bridgeness of an interaction is, the less weight it is.

where W is a matrix with element (W) _ij = D(i, j).

For any protein pairs, if the similarity between them is large enough, we have enough reason to believe they should be connected, otherwise, un-connected. Therefore, the proteins among a core should connect each other. To construct a virtual and reliable network for the original PPI network, similar to [25], a definition is proposed as

There are two good physic interpretations for Φ(G, τ): first of all, if the similarity of a pair of proteins is considered as the reliable score on the corresponding edge, Φ(G) can be considered as a reliable network of the original one; second, it can be understood as a perturbation of the original network by adding edges between vertices if there are enough short walks connecting them and deleting edges between vertex pairs if there are fewer short walks connecting them.

In this way, the core of a protein complex corresponds to a maximal clique in the virtual network. In the follows, we design algorithm to discover complexes by extracting cores and attachments, respectively.

A core-attachment algorithm

The first task is to extract all the maximal cliques in the virtual network, known as the classic all cliques problem-an NP-hard problem [48]. Therefore, the exact algorithms are prohibited largely due to the complexity. The heuristic algorithms are selected in order to avoid the time issue. The Coach algorithm detects dense subgraphs very quickly and accurately from each vertex's neighborhood graphs [24]. We adopt the Protein-complex core mining algorithm in the Coach to identify approximately all cliques in the communicability graph Φ(G). Of course, others can be used to identify the cliques, for example, the greedy algorithm, the tabu search and so on.

What we would like to point out is that, although we adopt the same strategy to detect the cores, our algorithm differ greatly from Coach algorithm for two reasons: first, our algorithm detects core in a virtual network based on the weak ties phenomenon, while the Coach on the original network; second, the strategies for the attachment vary greatly.

which quantifies the average closeness of υ to U from the aspect of connectivity. The larger cl(υ, U) is, the more walks connecting υ and the core. Thus, a vertex υ ∊ CS(U) is selected as an attachment when the $cl\left(\upsilon ,U\right)\ge acl\left(U\cup N\left(U\right)\right)=\frac{{\sum }_{\upsilon \in N\left(U\right)}cl\left(\upsilon ,U\right)}{\left|N\left(U\right)\right|+\left|U\right|}$, indicating that the selected attachment has more connection ways with U than the average connectivity in N(U).

The procedure can be described as following:

Step 1: Compute the bridgeness for each interaction in PPI network G according to Eq.(3);

Step 2: Compute similarity matrix S based on Eqs.(5)(6);

Step 3: Construct the virtual network Φ(G) with a predefined threshold τ;

Step 4: Extract the cores using Protein-complex core mining algorithm [24];

Step 5: Detect the attachments for each core.

Performance measures

The biological significance of the numerically computed modules can be validated by comparing the experimentally determined complexes (will be introduced in result section).

Data

The Database of Interaction Proteins [31] (DIP)(http://dip.doe-mbi.ucla.edu/[version yeast20071104]) data is adopted, which consists of 4,928 proteins and 17,201 interactions. To evaluate the protein complexes predicted by our algorithm, a benchmark set was constructed from the the MIPS [52], Aloy et al. [53] and the SGD database [54] based on the Gene Ontology (GO) notations, which consists of 428 protein complexes [50].

F-measure and coverage rate

Figure 4 shows the overall comparison in terms of F-measure and coverage rate on the DIP data. Although it is 2.9% lower than Coach algorithm, the F-measure of our algorithm Type II is 43.2%, which is 16.7%, 16.5% and 6.0% higher than MCL, DPClus and DECAFF, respectively. It demonstrates that our algorithm can predict protein complexes very accurately. From Figure 4, it is very easy to see that our method obtains the highest coverage rate of 42.8%, which is 7.9%, 9.6%, 11.4% and 16.2% higher than Coach,MCL, DPClus and DECAFF, respectively. It shows that the predicted complexes by our algorithm can cover the most proteins involved in the real complexes. From Figure 4, we can make a conclusion that our algorithm is obviously outperform the MCL, DPClus and DECAFF, and it makes a better balance between the F-measure and Coverage rate than the Coach. Compared Type I with Type II, we discovered that the Type II is much better than Type I, demonstrating that the efficiency of the proposed bridgeness. Such results further demonstrate that the critical phenomenon in the PPI can be used for enhancing the prediction accuracy.

P-value

To further investigate the biological significance of the predicted complexes, the P-value is adopted here. The functional homogeneity P-value is the probability that a given set of proteins is enriched by a given functional group merely by chance, following the hypergeometric distribution. It is the probability of cooccurrence of proteins with common functions. Accordingly, a low P-value of a predicted complex indicates that the collective occurrence of these proteins in the complex does not merely combine by chance and thus achieves high statistical significance. The values are calculated by the GO::TermFinder [55].

We discarded all clusters with P-value above a cutoff threshold. In the experiments, we chose a cutoff of 1 × 10^-2 for each protein complex because it offers a compromise between complex-cluster matching rate and a clustering passing rate.

Size and density distributions

Because the above experiments are sufficient to prove that the superiority of the proposed bridgeness, we only focused on the Type II method in the forthcoming experiment.

The P- values of predicted complexes by our algorithm support that the roles of interactions in PPI networks is promising on enhancing the accuracy of prediction. The module size distribution of predicted protein complexes for each compared methods on the DIP network has been shown in Figure 5. From it we can conclude that the major trend generated by our algorithm is very similar to that of the complexes in the benchmark set, which suggest that the definition of protein complex based on the weak tie effect is reasonable. However, the Coach can identify much less modules than these in the benchmark set, and its trend is different from that of the benchmark set. What we would like to point out is that the size distributions of the DPClus and MCL algorithms are very different from the previous ones.

Notice that our algorithm is quite different from those based on discovering the dense subgraphs because it makes use of the weak ties effect. To verify the difference on the densities of the predicted complexes, we compared the Coach algorithm with our method in terms of the graph densities of the predicted complexes, shown in the Figure 6. It is easy to figure out that more than 50% complexes predicted by the Coach algorithm whose densities are more than 0.9, while only 40% complexes predicted by our method whose densities are larger than 0.9. Furthermore, our algorithm can discover more protein complexes whose densities in range [0.6 0.9], which suggests that the density is not the only manner to characterize the protein complex and others are necessary and reasonable.

Effects of the parameters

This subsection is devoted to investigate how the parameters τ and β used effects the performance. The value of τ controls the size of a core, the total number of cores in the virtual graph, and the connectivity 'strength' of the network involved. Therefore, we studied its effect on the size of the virtual network. Figure 7 shows how the number of edges in the virtual network changes for various values of τ. From it, we can see that the size of the virtual graph decreases dramatically when the value of τ increases from 0 to 0.4. Specifically, the size is approximately 3 × 10⁴ if τ = 0. 02. The reason is that when the value of τ increases, only the edges whose connectivity is strong enough are maintained.

The parameter β controls the weights on the edges. Thus, we study its effect on the accuracy of prediction in terms of F-measure and coverage rate. Figure 8 demonstrates that the F-measure decreases, while the coverage rate increases when β increases. A possible reason is that the size of a maximal clique in the virtual network decreases when β increases, resulting in many small cores by dividing the large cores in the virtual graphs with small β. As β increases, more and more proteins in the PPI data are covered because the number of predicted protein complexes increases. For this reason, the coverage rate keeps increasing. To make a good balance between the F-measure and coverage rate, we set β = 0.618.

Robustness analysis

The robustness analysis on the proposed algorithm was discussed in this subsection. The benchmark networks adopted here originated from Ref. [51]. In detail, from the protein complexes annotated in the MIPS database [52], an interaction network named a test graph is constructed by regarding each protein as a vertex and connecting each pair of nodes in the same complexes. The test graph has a poor value for assessing the robustness of the algorithms because each protein complex corresponds to a clique in the test graph. To solve this problem, the altered graphs are constructed from the test graph by adding or deleting the edges in various proportions. For the sake of convenience, the altered graph is denoted by AG _{add, del} where add and del show the percentage of added and deleted edges, respectively.

In this experiment, only the MCL and Coach algorithms are selected for a comparison. The reason is that it is reported that the MCL is the most robust algorithms [51], and the Coach algorithm is the best core-attachment based method.

The Figure 9(A) shows how the geometric accuracy fluctuates as the number of edges increases. Increasing proportions of edges were randomly added to the test graph from 0% to 100%. Both the MCL and our algorithm are barely affected by the additions of up to 100% edges, while the performance of Coach is acceptable for low values of noise, they change dramatically when the percentage of added edges increases to 40%. A good reason is that as the percentage of added edges increases, the added edges connecting to the vertices in different cliques yield larger complexes(through merging the small complexes). In this case, the altered graph is not suitable for correctly extracting the complexes by the Coach algorithm. However, our algorithm can remove the noise dramatically because it extracts the protein complexes in a virtual network, where some of the added edges are filtered by increasing the value of the threshold τ.

Figure 9(B) displays the impact of edge addition on the separation. We can see that both the MCL and our algorithm have good performances when the percentage of the added edges increases to 80%, while the performance of the Coach algorithm decreases when the percentage of added edges increases to 20%. The impacts of edge removals on the geometric accuracy and separation are shown in Figure 9(C)(D), respectively. Figure 9(C) demonstrates that both the MCL and our algorithm outperform the Coach algorithm. A possible reason is that, as more and more edges are deleted, it becomes more and more difficult to re-obtain the deleted edges. When the percentage of removed edges is more than 20%, the virtual network constructed by our algorithm differs greatly from the original test graph. The general trends in Figure 9(D) are similar to those displayed in Figure 9(C).

Figure 9 (A-D) are the results on the networks being either added or removed edges, while Figure 9 (E-H) are the results on the networks involving both addition and removal. Figure 9 (E) demonstrates the effect of edge addition on the altered network from which 40% of the edges have been deleted previously. From it one can easily draw a conclusion that, when the addition less than 50%, the MCL outperforms the Coach and our algorithm, but when the the addition greater than 50%, both methods outperform the MCL. There is a good explanation: since the Coach and our algorithm are clique-based method, edge deletion destroys the structure of cliques, decreasing their performance; when more and more edges are added, some of the cliques destroyed previously are recovered, enhancing their performance. Furthermore, these two algorithms are barely affected by addition that is up to 100%, as the MCL decreases significantly the edges start to increase gradually. The values of separation on this type of altered network are shown in Figure 9 (F), where the MCL is at its the best performance. However, both the Coach and our algorithm are more stable than the MCL. The results on edge deletion on the altered network from which 40% of the edges have been added previously are shown in Figure 9 (G-H), which are similar to those in Figure 9 (E-F).