Patent Publication Number: US-11657311-B2

Title: Link prediction based on 3-step connectivity

Description:
RELATED APPLICATIONS 
     This application claims the benefit of U.S. Provisional Application No. 62/513,926, filed on Jun. 1, 2017, and U.S. Provisional Application No. 62/548,454, filed on Aug. 22, 2017. The entire teachings of the above applications are incorporated herein by reference. 
    
    
     GOVERNMENT SUPPORT 
     This invention was made with government support under P50HG004233 from National Institutes of Health, National Human Genome Research Institute Center of Excellence in Genome Science. The government has certain rights in the invention. 
    
    
     BACKGROUND 
     Network-based link prediction has a long history in social networks, providing a large arsenal of graph-based methods, routinely applied for social, biological, and technological networks as well. Most of these tools are either explicitly or implicitly based on the triadic closure principle (TCP), stating that similar nodes (sharing many of their neighbors) are likely to be connected. The TCP, however, has only limited applicability, unable to find important missing interactions, while predicting a large number of connections where there is no interaction in reality. 
     SUMMARY 
     Described herein are methods and systems that identify missing interactions in incompletely known datasets represented as complex networks. One example embodiment is a method of identifying missing connections in a complex network. The example method includes accessing an electronic representation of the network. The network includes nodes and links, the nodes represent entities, and the links represent interactions between the entities. The method further includes, for each pair of nodes in the network not directly connected by a link, (i) determining a number of paths connecting the pair of nodes, where each path is a length of at least three connections spanning between the pair of nodes in the network, and (ii) calculating a prediction score for the pair of nodes based on the number of paths connecting the pair of nodes. The method further includes ranking the pairs of nodes based on the prediction scores, resulting in an ordered list of node pairs, and selecting at least a subset of the pairs of nodes based on the ordered list of node pairs. The selected pairs of nodes represent missing connections in the network. 
     Another example embodiment is a system for identifying missing connections in a complex network. The system includes memory and a processor in communication with the memory. The memory stores an electronic representation of a network. The network includes nodes representing entities and links between nodes representing interactions between corresponding entities. The processor is configured to, for each pair of nodes in the network not directly connected, (i) determine a number of paths connecting the pair of nodes, where each path is a length of at least three connections spanning between the pair of nodes in the network, and (ii) calculate a prediction score for the pair of nodes based on the number of paths connecting the pair of nodes. The processor is further configured to rank the pairs of nodes based on the prediction scores, resulting in an ordered list of node pairs, and select at least a subset of the pairs of nodes based on the ordered list of node pairs. The selected pairs of nodes represent missing connections in the network. 
     The missing connections can be communicated to at least a portion of entities represented by the selected pairs of nodes. Connections can be created in the network between each of the selected pairs of nodes. In some embodiments, data can be collected regarding interactions among a plurality of entities, and the electronic representation of the network can be created from the collected data, where nodes in the network represent the entities, and connections between nodes in the network represent interactions between corresponding entities. 
     In some embodiments, a statistical significance can be calculated for each selected pair of nodes by comparing a connection between each pair of nodes to a pool of randomized networks with the same node degrees. In some embodiments calculating a prediction score for a pair of nodes can include calculating the prediction score based on the number of paths connecting the pair of nodes and a geometric mean of degrees of intermediate nodes between the pair of nodes. 
     In some embodiments, probabilities of existence of connections between the pairs of nodes can be estimated by, for each pair of nodes (i) performing a leave-one-out analysis, wherein a direct connection between the pair of nodes is left out of the network and each of the pair of nodes is scored in a context of a representation of the remaining network, (ii) adding the leave-one-out score of each of the pair of nodes to the ordered list of node pairs, (iii) estimating a probability of existence of a connection between the pairs of nodes based on the ordered list of node pairs, and (iv) assigning the estimated probability to the pair of nodes. 
     The network can be a social network, where the nodes of the network represent real-life individuals, and the connections between nodes in the network represent relationships between corresponding individuals. In such embodiments, the selected pairs of nodes can represent relationship recommendations between pairs of corresponding real-life individuals, and the relationship recommendations can be communicated to the individuals represented by the selected pairs of nodes. A connection between a pair of nodes can be created if at least one of the individuals represented by the nodes verifies a relationship between the individuals. 
     The network can be a protein network, where the nodes of the network represent real-life human proteins, and the connections between nodes in the network represent functional associations between corresponding proteins. In such embodiments, the selected pairs of nodes can represent new functional relationships between pairs of corresponding proteins. In such embodiments a protein sequence can be obtained from a patient and applied to the network to determine at least one of a disease afflicting the patient, a drug to use for treating the patient, and a potential reaction by the patient to a drug. 
     The network can be a professional network, where the nodes of the network represent real-life individuals and businesses, and the connections between nodes in the network represent employment relationships between corresponding individuals and companies. In such embodiments, the selected pairs of nodes can represent employment recommendations between corresponding individuals and companies, and the employment recommendations can be communicated to the individuals or companies represented by the selected pairs of nodes. 
     The network can represent real-life individuals and products, and the connections between nodes in the network can represent purchases of corresponding products by corresponding individuals. In such embodiments, the selected pairs of nodes can represent purchase recommendations between corresponding individuals and products. The purchase recommendations can be communicated to individuals represented by the selected pairs of nodes, and a connection can be created between a pair of nodes if the individual represented by the pair of nodes purchases a product represented by the pair of nodes. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The foregoing will be apparent from the following more particular description of example embodiments, as illustrated in the accompanying drawings in which like reference characters refer to the same parts throughout the different views. The drawings are not necessarily to scale, emphasis instead being placed upon illustrating embodiments. 
         FIG.  1    is a flow diagram illustrating a process of identifying missing connections in a complex network, according to an example embodiment. 
         FIGS.  2 A- 2 I  are graphs illustrating example paths of various lengths and connectivity. 
         FIGS.  3 A- 3 F  are graphs illustrating homomers versus heteromers. 
         FIGS.  4 A- 4 C  are schematic diagrams illustrating examples of protein binding. 
         FIGS.  4 D- 4 K  are graphs illustrating examples of protein connectivity. 
         FIGS.  5 A- 5 L  are graphs illustrating example computational and experimental validation of an example embodiment involving the human interactome. 
         FIGS.  6 A- 6 D  are graphs illustrating a simplified example of biological link prediction. 
         FIGS.  7 A- 7 G  are graphs illustrating a simplified example of gene duplication and link prediction. 
         FIG.  8    illustrates a computer network or similar digital processing environment in which embodiments may be implemented. 
         FIG.  9    is a diagram of an example internal structure of a computer in the computer system of  FIG.  8   . 
     
    
    
     DETAILED DESCRIPTION 
     A description of example embodiments follows. 
     Disclosed herein are 3-step link prediction (3SLP) methods and systems. As a case study of interest, 3SLP was validated for physical protein-protein interaction (PPI) networks. The approach is supported by both evolutionary arguments and structural evidences, predicting interacting neighbors based on their similarity to the already known neighbors. Intuitively, if two nodes share many of their neighbors then, as opposed to common belief, they might not be connected, but are likely to share some additional neighbors. Thus, instead of relying on indirect connectivity at distances of two steps (d=2) as in the TCP, interactions are predicted based on at least d=3 connectivity, leading to substantially improved predictions, validated both computationally and experimentally. The results can hold more generally, establishing a fundamental principle for link prediction in not only biological systems, but also in social and technological networks, including recommendation systems. 
       FIG.  1    is a flow diagram illustrating an example method  100  of identifying missing connections in a complex network. The example method  100  includes accessing  105  the network, which includes interactions between entities represented by links between nodes of the network. For each pair of nodes in the network not directly connected  110 , a number of paths connecting the pair of nodes is determined  115 . In one embodiment, each such path has a length of three connections in the network. A prediction score is calculated  120  for each pair of nodes based on the number of paths connecting the pair of nodes. After the prediction scores are calculated for the pairs of nodes not directly connected  125 , the pairs of nodes are ranked  130  based on the prediction scores, resulting in an ordered list of node pairs. At least a portion of the pairs of nodes is selected  135  based on the ordered list of node pairs. The resulting selected pairs of nodes represent missing connections in the network. 
     According to another example embodiment, a complex network can be created by collecting data regarding interactions among a plurality of entities and creating the network from the collected data. Nodes in the network can represent the entities, and connections between nodes in the network can represent interactions between corresponding entities. In a particular embodiment, this can include collecting experimental data about interacting human proteins. There exist about twenty thousand proteins in humans, and about 200 million pairs of such proteins. Collecting the data can be performed with a common assay version in a systematic way, without study biases. A protein interaction network can be created from the experimental data, where the nodes represent either individual protein isoforms or genes, summarizing the information of all related isoforms. The interactions can be represented as undirected unweighted links that can be extended to handle (linear) interaction weights or directional information. 
     For each protein pair in the network not directly connected, a score can be calculated using, for example, equation (1) described below. This can be accomplished by following and scoring each network path having a length of three and connecting the two nodes. One example score calculation of an individual path can be “1” divided the geometric mean of the degrees (number of interacting partners) of the intermediate nodes, along the path. Alternatively, with only a slight loss of performance (typically 0-20%), the score calculation for a path can be “1”, ignoring the degrees. The total probability score for a node pair can be the sum of individual scores for each path. The node pairs can be ranked according to the probability scores, where higher scores correspond to the top ranked predictions. 
     A probability of existence for each predicted interaction can be estimated in a self-consistent way. For each interacting pair of nodes in the original network, a leave-one-out analysis can be performed, where the interaction is left out of the network and its two end nodes are scored with equation (1) in the remaining network. The leave-one-out score of each original interaction is added to the ranked list of predicted interactions, and the ranked list is used to estimate the probability of existence for each predicted interaction as follows. The probability estimate can be the fraction of original links in a rank window of +/−n predictions around the novel prediction, where n=50 is found in practice to provide reasonable estimates. The estimated probability can be assigned to each predicted interaction, ranked previously using equation (1). 
     The top predictions for each node in the network can be selected by, for example, using a global or local (for the particular node) threshold, or based on an appropriate probability cut-off. The selected predictions can be published or communicated (e.g., provided to individuals associated with the nodes involved) along with their estimated probabilities. The selected predictions for each node can be integrated to the original network in a weighted manner, where the weight corresponds to the estimated probability of existence for the predicted interaction. 
     A statistical significance can be assigned to each predicted interaction by comparing the interaction to a pool of randomized networks with the same node degrees. A number of predicted interactions with the highest ranks, significance, or that are selected for some nodes can be subsequently tested experimentally. The sum of estimated probabilities for the selected predictions can serve as an estimate for the overall number of positive hits in the validation experiment. Experimentally validated interactions can be integrated into the network. 
     The 3-step link prediction (3SLP) platform has the following potential uses: 
     1. Improve existing recommendation systems, including but not limited to the following examples, such as friendship recommendations (FACEBOOK®) or media content recommendations, for example audio (SPOTIFY®) or video content (YOUTUBE®) recommendations, as well as recommendations for professional positions for job seeking candidates (LINKEDIN®). 
     2. Identify novel molecular interactions to enable a deeper understanding of molecular function, such as protein-protein, or protein-DNA interactions or drug-target interactions. 
     3. In annotated networks, such as tagged or labeled networks, the disclosed method can extend the existing annotations to previously unannotated nodes, performing automatic online tagging of media content or functional annotations in biological systems, such as Gene Ontology term or pathway annotations for unannotated proteins. 
     4. Identify novel disease related genes for complex diseases to help patient stratification and intervention strategies. 
     5. Identify missing relationships in relational data to improve decision support systems (DSS). 
     Embodiments can be used with social networks, such as FACEBOOK®, to identify and communicate friendship (or other) recommendations, or with online dating networks, such as OKCUPID®, to identify potential matches. Such embodiments can start with user-reported connection information, representing real-life relationships (e.g., friendships) between real-life people as a network. Profile information can be included in the network by assigning a node to each profile feature and a link connecting it to the relevant profiles. After determining predicted interactions in the network (as described above), the top predicted interactions (relationships) can be communicated to users as friendship (or other) recommendations. If a user confirms a recommendation, the predicted interaction can be added to the network. The embodiments, thus, improve the functionality of social networks. 
     Embodiments can also be used to identify functional associations, instead of physical connections. In such embodiments, the network can represent functions (e.g., a disease) as additional nodes. The function nodes can be connected to other nodes, representing known associations to the functions. When ranking the predicted interactions, a significance level of each prediction can be assessed by comparing it to randomized instances of the network, and ranking the prediction based on decreasing significance. 
     Embodiments can be used for functional annotation of proteins (e.g., disease, pathway, or Gene Ontology annotations). Starting with experimentally-validated functional information for a selected set of proteins, predicted interactions in the network can be determining as described above. The resulting predicted connections represent new functional associations for real-life proteins, which can be tested experimentally, thus improving the field of protein interactions. 
     Embodiments can also be used with professional networks, such as LINKEDIN®, to identify job (or other) recommendations. Such embodiments can start with user-reported job information representing real-life employment between real-life people and companies. Profile information can be included in the network by assigning a node to each profile feature and a link connecting it to the relevant profiles. After determining predicted interactions in the network (as described above), they can be filtered for predicted interactions between users and companies. The top predicted interactions (e.g., job recommendations) can be communicated to users (individuals and/or companies). The embodiments, thus, improve the functionality of professional networks. 
     With professional networks, such as LINKEDIN®, it is possible to endorse other users with various skills. The embodiments disclosed herein can be used to predict these skills based on available connection, profile, and skill information, in an automatic manner without user intervention. In such embodiments, the skills can be integrated into the network and the predictions can be filtered for user-skill predictions. Top predictions can be communicated to users, who can confirm the recommendations by accepting them. The confirmed recommendations can be added to the network. 
     Embodiments can also be used for personalized advertising or purchase recommendations. Purchasable items, such as real-life products from AMAZON® or media content (movies or music), can be represented in a network in place of functions (described above). After determining predicted interactions in the network (as described above), the predicted interactions can be communicated to potential buyers in the form of recommended purchases. If a potential buyer purchases the product(s), the corresponding predicted interaction(s) can be added to the network. 
     Embodiments can also be used for personalized edgotyping using a protein-protein interaction (PPI) network. The predicted interactions provide better coverage and statistical power when used to map network perturbations of disease related mutations for individuals. Embodiments can also be used for personalized medicine, or personalized drug efficacy. Based on individually perturbed PPI networks, it is possible to stratify patients into disease subcategories and predict individual drug efficacy. A protein sequence can be obtained from a patient and applied to the network to determine a disease afflicting the patient, a drug to use for treating the patient, or a potential reaction by the patient to a drug. Such embodiments can be used as a diagnostic test prerequisite for being prescribed a drug. 
     Example features of 3SLP can include (1) in strong contrast to existing technologies based on the triangle closure principle, 3SLP goes one step beyond d=2 step connectivity and builds upon d=3 step connectivity, identifying the candidate interacting partners from a largely different and much larger pool, (2) a novel way to predict interacting neighbors for a studied node based on their similarity to the already known neighbors, instead of similarity to the studied node itself, (3) suitability to work with network datasets, where some or all the nodes are interacting based on heterophily and not on homophily as traditionally assumed in the field, and (4) besides providing a prioritized list of candidate interactions, 3SLP is able to estimate the probability of existence for each predicted interaction, using a Bayesian calibration based on a leave-one-out cross-validation scheme. 
     Prior link prediction methodologies rely on the triangle closure principle (TCP), which is proven to be completely wrong for most proteins. Even in the cases when the TCP works, 3SLP provides superior results as indicated by both computational and experimental validations. Example features of 3SLP include (1) ability to identify candidate interacting partners from an order of magnitude larger search pool compared to earlier methods relying on TCP, providing not only higher quality but also more candidates, (2) for protein-protein interaction networks 3SLP performs 2-10 times better than existing methodologies, (3) 3SLP can time and cost efficiently replace additional high-throughput network mapping screens; namely, after the result of the first (incomplete) screen the novel predicted links can be pairwise tested based on this screen instead of performing the same large-scale screen multiple times, recovering predominantly previously observed interaction only, and (4) 3SLP has been validated computationally on a wide range of additional complex networks (for example on social and technological datasets) and found an improved performance in all studied cases, when self-interacting nodes were assumed, indicating that 3SLP is a fundamental and more generally applicable tool than existing methodologies. 
     Example commercial applications of 3SLP can include (1) improvement of high-throughput network mapping methodologies by replacing additional screens with computational predictions, (2) application to recommendation systems to predict potential interactions between users or products and users, such as personal relationships or media content, (3) identification of novel drug targets, (4) finding of novel disease related genes, and (5) identification of missing relationships for decision support systems. 
     The following is a detailed example of 3-step link prediction (3SLP) methods and systems. 
     Biological function emerges from the complex interplay between molecules in cells. While protein-protein interaction (PPI) networks are probably the most studied instances of molecular interactions, current maps are still missing the majority of them. Network based link prediction has a long history in social networks, providing a large arsenal of graph-based algorithms, routinely applied for biological networks as well. Most of these tools are either explicitly or implicitly based on the triadic closure principle, stating that similar nodes (sharing many of their neighbors) are likely to be connected. This principle has only a limited applicability, mostly around proteins forming homomers while it fails for the rest of the interactome. Supported by both evolutionary arguments and structural evidences, disclosed herein is a new way to predict interacting neighbors, namely based on their similarity to the already known neighbors. Intuitively, if two nodes share many of their neighbors then, as opposed to common belief, they might not be connected, but are likely to share some additional neighbors. Thus, instead of relying on indirect connectivity at d=2 steps, interactions can be predicted based on d=3 connectivity, leading to substantially improved predictions, validated both computationally and experimentally. Results are expected to hold more generally, establishing a fundamental principle for link prediction in biological systems, helping to better understand the emergence of biological function and diseases. 
     Link prediction, aiming to identify missing links in a network using a wide range of information, from sequences to the 3D structure of analogous proteins, is a long-standing problem in the journey to map out the full interactome. Indeed, despite spectacular advances in the experimental mapping of the human interactome, the estimated size of the human protein-protein interaction (PPI) network of 10 5 -10 6  interactions, is far beyond the number of experimentally documented interactions so far. The gap is driven by inherent experimental limitations, such as saturation effects and systematic biases, which make it increasingly difficult to map the missing interactions. To obtain a more complete map in a reasonable time-scale, ongoing experimental mapping efforts must be guided by computational predictions, allowing them to focus and prioritize in-depth screening. 
     With the increasing coverage of the interactome, particular attention has been placed on network-based link prediction, which exploits the patterns of the already mapped part of the interactome to predict missing interactions. Virtually all network-based link prediction methods rely on the triadic closure principle, the expectation that the more common neighbors two nodes have, the more likely that they have a direct link between them (see  FIG.  2 A ). In other words, topologically similar nodes, i.e. those that share many of their neighbors, are expected to establish a direct connection. The hypothesis behind TCP is rooted in social network analysis, originating from archaic times as reflected in “similis simile gaudet” (like rejoices in like), quantifying the observation that the more common friends two individuals have, the higher the chance that they would know each other (see  FIGS.  2 A and  2 B ). In the case of protein interactions, the TCP hypothesis states that proteins that share a large number of common neighbors (CN) are likely to participate in the same functional mechanisms, increasing the likelihood that they directly interact with each other. Given its simple interpretation and even simpler implementations, TCP is explicitly or implicitly part of all network-based link prediction mechanisms in the biological space. 
       FIGS.  2 A- 2 I  are graphs illustrating paths of various lengths and connectivity.  FIG.  2 A  shows the connection probability based on the number of d=2 and d=3 paths as the function of the Jaccard similarity between the two nodes. The triadic closure principle of social networks predicts links based on d=2 connectivity, i.e. by closing possible triangles in the network by connecting the nodes that share the most neighbors. When aiming to find the dotted link, the TCP is the right principle, assuming that the more neighbors two nodes share, the larger the chance that they connect to each other. In this setting, the d=3 connectivity principle is unable to detect the dotted link.  FIG.  2 B  shows TCP predicted links between topologically similar nodes, sharing many of their neighbors.  FIG.  2 C  shows connection probability versus Jaccard similarity in direct systematic interactomes for three species. As opposed to the TCP hypothesis, based on a strong affinity of similar nodes to connect to each other, the opposite tendency is observed, where node similarity prevents direct connectivity, invalidating the TCP for all three studied organisms.  FIG.  2 D  shows the opposite tendency when studying the dotted link between two nodes without any shared neighbors. While this case falls behind the range of TCP, d=3 connectivity can successfully detect such a link, by counting the number of paths at distance d=3 between the two nodes.  FIG.  2 E  shows a representation of the interactome being organized fundamentally differently. Protein-protein interactions are realized by complementary interfaces, where two proteins with similar interfaces (X and Y) might share many of their neighbors while not being able to connect to each other. When looking for missing links in this biological setting, nodes sharing a large number of their neighbors are likely to share some additional neighbors as well, leading to a d=3 connectivity principle. Since natural networks are small worlds, the missing neighbors are relatively far away in the network.  FIG.  2 F  shows a connection probability plotted against d=3 connectivity, finding a positive tendency, indicating a strong connectivity pattern in the studied interactomes at three steps.  FIGS.  2 G- 2 I  show connection probability in the top 1000 node pairs ranked by different powers of the adjacency matrix, d, representing connectivity at distance d=2, . . . , 8. d=3 connectivity is found to be the most related to direct connectivity in each interactome. Note, that odd distances contain the d=3 paths as special cases. 
     Checking the validity of the fundamental assumption behind TCP (that the more common neighbors two proteins have, the more likely they will link to each other) can be accomplished by using the Jaccard similarity J=|N i  ∩N j |/|N i  ∪N j |, where N i  and N j  are the neighbors of nodes i and j, to measure the joined neighborhood of proteins i and j (see Supplemental Information, below, for an alternative similarity measure). 
     TCP predicts a positive correlation between Jaccard similarity and the probability that two nodes are connected (see  FIG.  2 A ). Interestingly, the opposite trend is observed: the larger the Jaccard similarity, the lower the chance of two proteins directly interacting with each other (see  FIG.  2 C ). This TCP paradox, representing a linking pattern that is the complete opposite of the TCP assumption, is observed in all high-quality direct physical interactomes explored, including human, yeast, and  arabidopsis.    
     Shown herein is that the TCP paradox is the result of taking intuitive experience from social networks into the biological space, while ignoring the structural and evolutionary mechanisms that drive protein interactions. Also shown is that by overcoming the TCP paradox, conceptually simple and biologically accurate prediction tools can be designed that not only avoid the pitfalls of TCP, but also significantly outperform all currently-available link prediction tools. 
     By connecting nodes with common neighbors, TCP relies on network paths of length d=2 (see  FIG.  2 E ). This has, however, some well-known limitations: no method based on TCP can recover a missing link connecting two nodes that lack shared neighbors (see  FIG.  2 D ). As disclosed herein, however, one can go beyond this d=2 connectivity by considering longer paths. 
     For example d=3 connectivity (see  FIGS.  2 D and  2 E ) considers the likelihood that node pairs linked by paths of length d=3 are connected. This d=3 connectivity overcomes the most elementary limitation of TCP, that it cannot predict links for nodes that lack shared neighbors (see  FIG.  2 B ). The more paths of length d=3 that exist between a node pair, the higher the chance that they will also have a direct link between them. In other words, the number of paths of length d=3 correlates positively with the likelihood of two nodes being connected (see  FIG.  2 F ). For completeness, the probability that two proteins, linked by paths of length d=2, . . . , 8 are connected in three organisms was measured, finding that in each the d=3 offers the highest probabilities of being directly connected (see  FIGS.  2 G- 2 I ). Equally important, the d=3 connection probability is two to three times higher than the d=2 (TCP) connection probability. 
     To better understand the roots of the TCP paradox and the unusual predicting power of d=3 connectivity, all homomers in the network were identified, capturing all proteins with self-interactions, and heteromers, i.e. the rest of the proteins. Currently, about 12% of the proteins in the human interactome are known to be homomers, hence only about 1% of the protein pairs consist of two homomers. Consequently, the vast majority of protein pairs (87%) in the human interactome capture interactions between two heteromers, representing proteins unable to interact with an identical copy of themselves. 
       FIGS.  3 A- 3 F  are graphs illustrating homomers versus heteromers.  FIGS.  3 A- 3 C  show connection probability versus Jaccard similarity in the three node pair classes, for the three example interactomes. There is a positive tendency between node similarity and connectivity between two homomers (top, circles) for several orders of magnitudes. While there is no strong tendency between heteromer and homomer nodes (squares), a negative tendency is observed for the vast majority of the node pairs of two heteromer nodes (triangles). Two similar nodes (with Jaccard similarity ˜1) are more than a thousand times more likely to be connected if both of them are homomers than none of them. Consequently, the applicability of the TCP is expected to be restricted to the relatively small fraction of homomer node pairs. The data is logarithmically binned based on the Jaccard similarity values.  FIGS.  3 D- 3 F  show connection probability versus d=3 connectivity in the three node pair classes, for three species. In strong contrast to Jaccard similarity shown in  FIGS.  3 A- 3 C , d=3 connectivity expresses an almost identical, strong positive tendency with the connection probability between all proteins, promising a unified way of predicting missing links even without making any distinction between homomers or heteromers. 
     Interestingly, TCP is valid for homomer pairs: the higher the Jaccard similarity between two homomers, the more likely that there is a direct link between them (see  FIGS.  3 A- 3 C ). Heteromer pairs follow the opposite pattern: high Jaccard similarity prevents the existence of a direct connection between them. For example, two proteins with a Jaccard similarity close to 1 are more than a thousand times more likely to directly interact with each other if both proteins are homomers than if they are both heteromers (see  FIGS.  3 A- 3 C ). In conclusion, TCP is valid for homomer pairs, but fails for heteromer pairs. As the number of heteromer pairs vastly dominates over the homomer pairs (87% vs. 1% of the pairs), when this distinction is not made, the dominance of heteromer pairs naturally leads to an overall negative correlation (see  FIG.  2 C ). 
     The differences between homomer and heteromer pairs vanish when focusing on d=3 connectivity, counting the number of paths of length three between two nodes. Indeed,  FIGS.  3 D- 3 F  shows the probability that two nodes are connected in the three node pair classes against d=3 connectivity, showing a clear positive correlation. In contrast to  FIGS.  3 A- 3 C , where there is qualitatively different behavior for homomers and heteromers, here the data points fall roughly on the same curve for all protein pairs. This indicates that the d=3 connectivity principle accurately captures not only the connectivity patterns of heteromers, but the connectivity pattern of the entire interactome. Two mechanisms can explain the observed patterns. 
       FIGS.  4 A- 4 C  are schematic diagrams illustrating protein binding, and  FIGS.  4 D- 4 K  are graphs illustrating protein connectivity.  FIGS.  4 A- 4 C  illustrate the structural reasons of the observed connectivity patterns, different binding interfaces being illustrated by different colors on the proteins. In order to form a self-interaction, homomer proteins require either a self-interacting interface ( FIG.  4 A ) or two complementary interfaces ( FIG.  4 B ). Heteromer proteins with similar interfaces are typically not connected, while share many of their neighbors.  FIGS.  4 D and  4 H  illustrate that, starting from an original protein (V), gene duplication leads to fundamentally different network structures.  FIGS.  4 E and  4 I  illustrate that homomers typically lead to locally dense cliques, while heteromers lead to much sparser, locally bipartite structures. Finding a missing (dotted) link appears to be a very different problem in the two cases.  FIGS.  4 F- 4 K  illustrate that in a dense graph of homomers, both the d=2 (TCP) and d=3 principles are able to detect the missing link. However, TCP fails to correctly identify the missing link between two heteromers, while the d=3 connectivity principle ( FIG.  4 G ) is expected to succeed in all classes. 
     As protein interactions primarily require complementary interfaces, a homomer protein must already possess both complementary interfaces ( FIG.  4 A ), or an interface that is able to bind to itself ( FIG.  4 B ). Consequently, if two homomer proteins interact, they are likely to share their neighbors as well, connecting to the same interface or interfaces that establish the self-interaction. Moreover, the more neighbors that two homomers share, the more likely that they interact, similarly to social networks. 
     In contrast, the interaction partners of a heteromer protein must have complementary interfaces (see  FIG.  4 C ). Two heteromers X and Y that have the same interface will have common neighbors (for example V), as they will interact with all proteins that have complementary interfaces. Since X and Y are heteromers, the shared interface is unable to connect to itself, hence X and Y will probably not directly interact. On the other hand, if two heteromer proteins interact (e.g., X and V) then they could only share common interaction partners, that have both complementary interfaces, which is again unlikely. To conclude, the structure based reasoning predicts that two proteins that share multiple neighbors (X and Y in  FIG.  2 D ) are likely to share some additional neighbors (e.g., D), as indicated by the blue links. This is the essence of the d=3 connectivity principle, connecting a neighbor (D) of protein X to protein Y in  FIG.  2 D , in three steps on the graph. 
     A similar conclusion is reached looking at gene duplication, a key evolutionary mechanism responsible for the emergence of new proteins. If protein V duplicates, the duplicated node (W) will maintain the links of the original protein (see  FIGS.  4 E and  4 I ). As homomer proteins can interact with themselves, the new protein (W) will have a direct link to the original protein (V) as well (see  FIG.  4 E ). In contrast, a duplicate of a heteromer protein will interact with the neighbors of the original heteromer, but will lack a direct link to it ( FIG.  4 I ). Starting from an original interaction between a pair of proteins, gene multiplication leads to fundamentally different network patterns, ranging from a fully connected clique to bipartite graphs. While mutations and diversification perturb these connectivity patterns, the overall topological differences in the neighborhood of homomer and heteromer proteins will prevail after duplication. 
     Next, the predictive power of d=3 connectivity was checked against the predictions of TCP. It was found that, to the contrary of TCP, the d=3 connectivity is able to find a missing link ( FIG.  4 K ), even for heteromers. The reason is that gene duplication induces the pattern that proteins sharing a large number of their neighbors are likely to share with some further neighbors, even if they are not directly connected. Hence, a missing partner is one step further away than expected by TCP, where nodes at distance d=2 would be connected. 
     Both the structural and evolutionary arguments suggest that similar proteins, i.e. those sharing either an interface or common evolutionary roots, are not necessarily connected, but are likely to connect to the same neighbors. Consequently, a potential interaction partner of a protein is not always similar to this protein, but is likely to be similar to the existing partners of the protein, underlying the d=3 connectivity principle. 
     Link prediction: Taken together, the d=3 connectivity principle describes the observed connectivity patterns of both heteromer and homomer proteins, in strong contrast to the d=2 connectivity (TCP), which correlates with the direct connectivity between homomer pairs only. 
     To turn this observation into a predictive framework, d=3 connectivity is quantified in PPI networks, in a way to account for the broad range of observed degrees. In  FIGS.  2 A- 2 I  and  FIGS.  3 A- 3 F , the number of paths at d=3 were measured, equivalent to the third power of the adjacency matrix, A 3 . Since high degree hubs induce a tremendous number of shortcuts in the network, improved predictions are expected by properly normalizing with the degrees of the intermediate nodes in the paths, similarly to various implementations of TCP. 
     To reduce the overwhelming effect of the hubs, the degree-normalized adjacency matrix Ã=D −1/2 AD −1/2  can be used, where D is the diagonal degree matrix, measuring the d=3 connectivity as AÃA. In other words, for each pair of nodes i and j, a degree normalized d=3 connectivity score is assigned: 
     
       
         
           
             
               
                 
                   
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     where k m  is the degree of node m. The hypothesis is that node pairs with the highest scores are most likely to be connected by missing interactions. 
     To computationally verify the validity of equation (1), 50% of the links were removed and the recovery rate for the most highly ranked 2000 pairs was measured. The results were compared with the predictions of TCP and those of the preferential attachment (PA) principle, a link prediction method that is not based on the TCP. Indeed, PA assumes that the probability of a link between two nodes is proportional to the product of their degrees. TCP is represented by its simplest implementation, CN, counting the number of common neighbors. It was found that d=3 connectivity outperforms both the TCP and PA on the human interactome, independently of whether the node pairs are homomers or heteromers. 
       FIGS.  5 A- 5 L  are graphs illustrating computational and experimental validation on the human interactome.  FIGS.  5 A- 5 C  show results of a computational validation on the filtered human interactome, containing no keratins and restricted to one ORF per gene. 50% of the links were chosen randomly as the input for link prediction, averaged over 10 realizations. In all cases, the shaded range indicates the standard deviation of the results over 10 realizations. CRA stands for the community resource allocation index, performing best in the high-throughput validations ( FIGS.  5 D to  5 F ).  FIG.  5 C  shows the validation rate as a histogram for the top 100 and 500 predictions, respectively.  FIGS.  5 D- 5 F  show high-throughput experimental validation. The predicted links were tested against the human interactome, HI-III-123, generated with the same experimental assay as the input interactome (for details, see Supplemental Information). Besides the previously shown methods, 23 different methods were tested this way, highlighting CRA as the best available method for the top predictions. For a proper calibration, in F a positive set of 100 known interactions (“Known”) and random set (“RND”) of 100 node pairs were selected, connected to the same nodes as the top 500 predictions of the d=3 connectivity method.  FIGS.  5 G- 5 I  show direct validation with pairwise testing of the top 500 predicted links of CRA and the disclosed method. Again, the d=3 connectivity principle (blue) significantly outperformed CRA, providing 2 to 3 times better results. The pairs where the experiments were inconclusive were left out of the analyses. Remarkably, according to both experimental tests, the top 100 predictions validate about as well as the selected set of known interactions.  FIG.  5 J  illustrates experimentally validated predicted links with d=3 connectivity for retinitis pigmentosa (RP). The gray links represent all known interactions in HI-filtered for the two RP proteins, FAM161A and PRPF31 (yellow nodes). Eight of the novel, experimentally-validated links (blue) connect to at least one of these RP proteins, enabling a more complete understanding of the RP disease mechanism, including the top predicted link FAM161A-GOLGA2. As a manifestation of the link prediction principle, it is shown that if two nodes (yellow) share many of their neighbors then they might share even more, even if they are not directly connected as would be expected by the TCP.  FIGS.  5 K and  5 L  show a robustness analyses of the predictions with high-throughput validation against link removal ( FIG.  5 K ) or additional random links ( FIG.  5 L ) at the to 100, 500 and 2000 predictions, respectively. The method is robust even when less than 50% of the links are kept or are original. The standard deviation of the results is indicated by shaded ranges over the 10 realizations. 
     In the following, the combined performance of equation (1) is explored, not distinguishing between homomers or heteromers. It was found that for all studied organisms the d=3 connectivity principle significantly outperforms the d=2 connectivity and PA, having several times higher validation rates. Evidence was obtained that that results hold more generally, independently of the organism (fruit fly, mouse,  C. Elegans, S. Pombe ) and network origin (literature derived, Lit-BM-13, Biogrid), structure-based (Interactome3D) or mass-spectrometry based co-complex membership datasets (Bioplex, Qubic). 
     Experimental validation: To experimentally test the predictive power of equation (1), the study was restricted to a filtered human interactome (HI-filtered) of N=3727 proteins and E=9433 interactions between them, containing only one open reading frame (ORF) per gene and no keratins (see Supplemental Information) for HI-II-14. For a high-throughput validation, it is ideally needed to repeat the entire experimental screening independently, for the same proteins under identical experimental conditions. Such a dataset, consisting of three independent screens, has been recently generated (HI-III-123, REF). Note, that as any individual dataset, HI-III-123 is still incomplete, containing only 42.4% of the interactions in HI-filtered. Still, HI-III-123 allows a systematic evaluation of all computational methods, in the full range of predicted scores. It was, again, found that equation (1) significantly outperforms the other methods. For comparison, 23 link prediction methods were tested, finding that there is one method, called community resource allocation index (CRA) that performs best for the top predictions. CRA relies on a strong connectivity at both d=2 and d=3, hence its predictions represent a subset of the predictions disclosed herein. To control the performance of the experiment, a selection of known links in the interactome (“Known”) and random protein pairs (“RND”) were used as benchmarks (for details, see Supplemental Information). The obtained validation rate of the top 100 predictions, about 35%, is indistinguishable from the validation rate of the selected set of known links (35%) (see  FIG.  5 F ), indicating that the top predictions are as reliable in reproducibility as the already known protein-protein interactions. In the random set (RND) no validated hits were found, even though ˜1-5% would still indicate a high quality experiment. 
     To obtain a more complete validation that is not limited by the incompleteness of the existing interactome maps, direct pairwise testing experiments for the top 500 predicted links were also performed ( FIG.  5 G- 5 I ). It was found again that the top predicted links validate with the same rate as the already known interactions ( FIG.  5 I ). In particular, d=3 connectivity performs about three times better at the top 500 predictions than CRA, the best performing method in the literature. 
     Given the unavoidable presence of false negative and false positive interactions in the datasets, it was tested whether the results are robust against data incompleteness and noise. For this, a fraction of the links were randomly removed. In the high-throughput setting, it was found that the validation rate of the predicted interactions was fairly stable up to removal of even 60-70% of the interactome (see  FIG.  5 K ). While the false positive rate in the human interactome is expected to be very low (based on the low RRS discovery rate), it is typically not quantitatively known. To test the robustness of the technique against such a noise factor, a number of random links were included in the network. Again, the validation rates were found to be robust, even when the number of random links exceeds the number of original links (see  FIG.  5 L ). 
     Case Study on Retinitis Pigmentosa: The ability to predict previously undetected parts of the interactome offers the possibility to gain insights about the mechanism of disease-related mutations. For this, the disclosed link predictions can be selectively tested for known disease genes, while follow-up studies can perform functional assays and edgotyping experiments in order to illuminate the function of these new interactions. To illustrate this potential, one can focus on the top predicted (and experimentally validated) link that connects FAM161A—involved in Retinitis Pigmentosa, (RP)—to GOLGA2 (see  FIG.  5 J ). RP is a genetic disorder leading to the loss of the retina&#39;s light sensitivity through progressive degeneration of the photoreceptor cells, being the leading cause of inherited blindness, lacking a cure. Remarkably, FAM161A and GOLGA2 share no neighbors in the HI-filtered interactome, hence a link between them could not be predicted by any TCP based method. In addition to the interaction of FAM161A with GOLGA2, five more novel interacting partners for FAM161A were identified, namely TRAF2, KIFC3, VPS52, REL, and TRIM23. TRIM23 is also identified as a new interacting partner for PRPF31 besides TRIM54. Three of the new interacting proteins (GOLGA2, TRIM23, and VPS52) play an important role in the Golgi apparatus, underlying the role of the Golgi apparatus in RP, as shown recently, where the interaction between FAM161A and KIFC3 has also been identified. As a summary, this case study serves as an example of the intuitive picture in  FIG.  2 I , where two proteins sharing many of their neighbors are likely to share some further neighbors. 
     Discussion: Protein-protein interactions (PPI) serve as a rich source of information, especially fruitful in predicting gene function, pathways and complexes, while playing a central role in a mechanistic understanding of cellular function under both normal and disease conditions. Besides providing an essential basis for functional proteomics and drug discovery, there is increasing evidence that disease related mutations are often manifested as perturbations of the physical interactions of the proteins in the human interactome. Moreover, gene mutations related to specific diseases organize into localized neighborhoods in the interactome, indicating the existence of disease modules. These observations establish the foundations of a systematic network framework to better understand human diseases. However, most of the human interactome is still uncharted, hindering further progress, such as a obtaining a detailed mechanistic understanding, e.g. by performing edgotyping analyses of disease mutations. In order to obtain a deeper understanding, the long-standing problem of network link prediction was revisited, and it was found that frequently-used methods are based on incorrect assumptions about the structure of PPI networks. 
     The computational analyses of biological networks is very challenging, affected heavily by study biases, incompleteness and false positive interactions. Even the most basic visualization of the interactome is prone to failure, known as the “hair ball” problem. Having no better options, the disclosed system-level intuition stems mostly from an understanding of social networks, where node (topological) similarity and connectivity are known to be correlated, according to the TCP. This hypothesis only works for homomer proteins, while fails for most of the proteins in the PPI network. As an alternative link prediction paradigm, a d=3 connectivity principle can be used, supported by both evolutionary and structural arguments. Both TCP and d=3 connectivity is based on the existence of node pairs sharing many of their neighbors. While the TCP intends to connect these two (topologically similar) nodes to each other, in the same situation the d=3 connectivity connects one of the not yet shared neighbors of them to the other node. Intuitively, a candidate neighbor should be (topologically) similar to the known neighbors of the studied protein and not to the protein itself. 
     After computationally checking the superior performance of d=3 connectivity on three species, the top predictions were validated experimentally for a restricted human interactome, HI-filtered. The top predictions validated with the same rate as already known interactions in both of the high-throughput and pairwise tests, indicating an optimal performance. This performance is found to be highly robust against incompleteness (link removal) or noise (randomly added links), qualifying the disclosed method to be used over a wide range of potential datasets. The disclosed method can be a great help to obtain more complete, high-quality interactomes, enabling a more detailed mechanical understanding of complex diseases. When put in a broader perspective, the results will likely lead to an improved set of biological network analyses methods that rely on more realistic assumptions than those borrowed from social sciences. 
     Supplemental Information 
     A. Analyzed Networks 
     In order to compensate for the incompleteness and strengths and weaknesses of each dataset, the results were tested on several interactomes over seven species. As a starting point its was intended to understand the structure of PPI networks including only direct (binary) physical interactions. For this purpose systematic, binary networks generated with yeast-two-hybrid experiments were used, such as the human (HI-II-14), yeast, and  arabidopsis  interactomes. Even though these networks were obtained in a systematic way to avoid study biases, they have a limited coverage and there might be certain biological limitations, leading to biases e.g. to a depletion in membrane proteins or potential overrepresentation of keratins. For a comparison, literature-curated PPI networks of direct physical interactions were included, such as Lit-BM-13, or the Biogrid datasets. In the case of the Biogrid datasets, only “direct interactions” and proteins assigned to the correct “taxid,” specific to the studied species were considered. These networks are prone to study biases, meaning that highly studied protein (with many publications) have a higher degree and a more complete coverage. 
     To show that the results hold more generally, beyond binary physical interactions, also considered were cocomplex proteomics datasets, such as the Bioplex and QUBIC networks, listing protein-protein associations (PPAs). Besides incompleteness, another issue is the (mostly unknown) number of false positives in these datasets. To test the results more precisely, also included was the Interactome3D network to see if the results hold also for the high quality but sparse set of available interactions with structural evidence. 
     B. Evaluation with Various Quality Measures AUC is Calculated as: 
     
       
         
           
             
               
                 
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     where by randomly selecting n pairs of a positive and negative link, larger score n′ times and an equal score n″ times is obtained for the positive link. The AUC value is between 0.5 and 1 and a higher value is assumed to indicate a better performance. 
     The Precision is defined as: 
     
       
         
           
             
               
                 
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     where L p  is the number of positive links in the top L number of predictions. The precision is between 0 and 1 and higher precision value means higher accuracy. 
     Being a frequently used performance measure for web search companies, the Normalized Discounted Cumulative Gain (NDCG) is known to be able to select the better ranking between any two, substantially different rankings. For binary classification the NDCG is given by: 
     
       
         
           
             
               
                 
                   
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     where the summation in the nominator runs over all positive instances, while the summation in the denominator quantifies the deal case, where the positive instances are the top ranked predictions. In practice, to deal with extreme degeneracies, for the candidates with a zero score, the last possible rank was assigned to each of them. 
     C. Calibration of the Predicted Score 
     A leave one out approach was used to assign predicted scores to the existing links. By having at hand all the scores for the existing and predicted links, a naive bayesian approach was used to estimate the probability that a predicted links is real. To be precise, the probability of a link to exist in a window of (at most) w=50 existing links was calculated before and after a link in the ranked list. 
     D. Details of the Pairwise Testing Experiment 
     In the experiments the top 500 predictions of the d=3 connectivity principle was tested against the top 500 predictions of CRA on the human network, HI-filtered. Additionally, for a proper benchmarking, included were literature curated, interactions with multiple evidence (Lit-BM-13), as well as a set of positive reference interactions (PRS) and 100 known interactions connected to at least one of the nodes in the top 500 predictions (Known). On the other side, to control the false positive rate, a set of random node pairs in the random reference set (RRS) and a similar set off 100 pairs centered around the top 500 predictions (RND) were selected. Summarizing all these pairs results in XXX non-redundant pairs. 
     In the experiments, each pair has been tested in both Y2H orientations (AD-DB vs. DB-AD) and a pair is considered to be validated if it was found in at least with one of the orientations. In the Y2H screening fresh overnight cultures of individual Y8930:DB-ORF yeast strains were mated against Y8800:AD-ORF strains. The growth phenotype of all pairs was tested in individual pair-wise tests. Previously developed liquid mating strategy was used with direct spotting of diploid yeast cells in 100 ul of YEPD using liquid handling robotics. After overnight incubation at 30° C., 2 ul of mated yeast culture were transferred to 100 ul of SC-Leu-Trp media to enrich for diploids. After overnight growth at 30° C., a 5 ul aliquot of the liquid culture was robotically spotted, onto both SC-Leu-Trp-His+3AT and SC-Leu-His+3AT+CHX solid media. After an incubation of three days, diploids that gave rise to growth on SC-Leu-Trp-His+3AT and failed to grow on SC-Leu-His+3AT+CHX were classified as pairwise positives. To confirm identity of the pairwise positive interactors, colonies were picked and processed for sequencing using the X×X system and aligned to the corresponding ORFs. Open-reading frame (ORF) clones encoding human proteins were obtained by PCR-based Gateway recombinational cloning following a protocol previously described. Prior to Y2H screening, diploid DB-ORF yeast strains were tested for auto-activation of the GAL1::HIS3 reporter gene in the absence of any AD-ORF plasmid as described. Individual DB-ORF yeast strains were mated with the Y8800 yeast strain transformed with empty pDEST-AD-CYH2 vector. Diploid cells were first selected on solid SC-Leu-Trp media and then transferred onto solid SC media lacking leucine, tryptophan and histidine and containing 1 mM 3AT (SC-Leu-Trp-His+3AT). Any DB-ORF yeast strains that grew on SC-Leu-Trp-His+3AT solid media were considered auto-activators and removed from the collection of DB-ORF yeast strains to be screened. 
     In principle, the nodes of the top predictions might have special characteristics which locally modify the recovery rate of their links. To be more specific, the selected positive set (“Known” links) contains 100 randomly selected known links (in HI-filtered) around the nodes involved in the top 500 predictions. Selecting a negative benchmark set is much more problematic due to the high false negative rate of the current experimental techniques. A frequently used alternative is the use of a random set, where node pairs are selected randomly and expected to be recovered with a rate given by the density of the network which is predicted to be less than 1% for the complete interactome (currently ˜0.14% in HI-filtered). Similarly to the positive set, 100 random links (“RND”) were selected in a way that at least one of the nodes is involved in the top 500 predictions. Note, that for a fraction of pairs the experiments turned out to be inconclusive, eventually left out from the analyses. Each pairwise tested positive pair was sequence confirmed by Sanger Sequencing with GeneWiz. 
     E. Functional Validation Based on GO-Term Analyses 
     While gene ontology (GO) terms are frequently used to evaluate the functional relevance of predicted interactions, proteins that are not connected but have a high degree of similarity might be more similar in their GO-annotations than interacting proteins. Indeed, the observations question the legitimacy of using GO-terms or functional annotations to evaluate the quality of the predicted physical links. 
     The annotations are obtained from the Gene Ontology Consortium (http://www.ebi.ac.uk/GOA). In order to avoid noise and circularity, the analyses were restricted to proteins only mapped to Entrez gene IDs, excluding the “NOT” qualifiers and considering only high confidence annotations associated with the evidence codes EXP, IDA, IMP, IGI, IEP, ISS, as listed below. 
     Experimental:
         Inferred from Experiment (EXP)   Inferred from Direct Assay (IDA)   Inferred from Mutant Phenotype (IMP)   Inferred from Genetic Interaction (IGI)   Inferred from Expression Pattern (IEP)       

     Computational:
         Inferred from Sequence or structural Similarity (ISS)       

     Note, that there were no instances of ISO, ISA or ISM annotations. The input dataset was generated on 2016-11-28 09:32. The following version was used: http://purl.obolibrary.org/obo/go/releases/2016-11-26/go.owl goa_human.gaf. 
     F. Topological Similarity Vs. Connection Probability 
     There are several established ways to quantify topological similarity, most of them differing only in the (degree) normalization scheme of the number of shared neighbors, frequently used for similarity-based link prediction in favor of the TCP. To further support the findings shown in  FIGS.  2 J- 2 I , shown in  FIGS.  6 A- 6 D  is the same dataset when the similarity is quantified by the hypergeometric p-value. The results suggest, that in datasets focusing on complexes or homomer dominated regions of the interactome, the TCP might provide reasonably good predictions, however, on systematic interactomes, dominated by heteromers, there is a need to go beyond the TCP and, as indicated before, the natural candidate is to use a d=3 connectivity principle. The universality of this observation is reflected in the fact that it replicates for all studied networks, as shown in  FIGS.  7 A- 7 G  with the Jaccard-index as a similarity measure. 
       FIGS.  6 A- 6 D  are graphs illustrating a simplification of biological link prediction.  FIG.  6 A  represents that redundancy is a key element of biological networks. Typically there are alternative pathways, meaning that two (functionally similar) nodes (e.g., nodes  605  and  610 ) are connected through multiple shared neighbors (e.g., nodes  615  and  620 ).  FIG.  6 B  represents that current maps of the network are incomplete, missing many links (e.g., missing link  625 ).  FIG.  6 C  represents the celebrated triadic closure principle that predicts links based on closing possible triangles in the network, while often missing the true opportunities.  FIG.  6 D  represents a missing link  625  in a redundant network that can be conveniently uncovered by the d=3 connectivity principle, by closing paths of length  3 . 
       FIGS.  7 A- 7 G  are graphs illustrating a simplification of gene duplication and link prediction.  FIGS.  7 A- 7 C  illustrates, starting from an original link, gene duplication leading to fundamentally different network structures in the three classes, after duplicating both the first ( FIG.  7 B ) and the second node ( FIG.  7 C ). While a link between two heteromers yields a bipartite structure, a link between a homomer and a heteromer leads to a core-periphery structure. Dense clusters or cliques emerge only when starting from a link between two homomers.  FIG.  7 D  illustrates, at the current level of network incompleteness, that not all the resulted links are seen.  FIGS.  7 E- 7 G  illustrate the top predictions provided by three network based link prediction methods. While both the preferential attachment principle (PA) ( FIG.  7 E ) and the common neighbors method (CN) ( FIG.  7 F ) fail to correctly identify the missing link between two heteromers, the d=3 connectivity principle ( FIG.  7 G ) is expected to work well in all three classes. 
     G. Degree Difference Between Homomers and Heteromers 
     As a consequence of both the structural and evolutionary arguments, homomers are expected to have a higher degree compared to heteromers, due to a surplus of links towards similar homomer nodes in dense clusters (local cores and cliques), while heteromer nodes mainly connect to dissimilar nodes only. Indeed, as a group, in all three studied organisms, homomers have a higher degree compared to the rest of the proteins. Due to the degree difference, there is a strong bias towards homomers, leading to ˜9% of the links connecting two homomers and 50% of the links connecting two heteromers in the human intercom. 
     H. Interpretation as a Feature Selection Approach 
     This can be understood by realizing that the d=3 connectivity can detect all paths at 2-steps as well if at least one of the nodes involved is a homomer, providing an additional step by the self-interaction. d=3 connectivity relies on the same network patterns in the data, while predicting links differently. The key ingredient for both methods is the existence of topologically similar nodes, sharing some of their neighbors. Traditionally, in the TCP it is assumed that such topologically similar nodes should be also connected. On the contrary, a different hypothesis is formulated, stating that a potential neighbor should be topologically similar to the already known neighbors of the node. While this interpretation might hold more generally, even besides biological networks, it is noted, that due to the elegant simplicity of the d=3 connectivity principle, there are other possible alternative interpretations as well 
     Digital Processing Environment 
       FIG.  8    illustrates a computer network or similar digital processing environment in which embodiments of the disclosed systems and methods may be implemented. Client computer(s)/devices  50  and server computer(s)  60  provide processing, storage, and input/output devices executing application programs and the like. The client computer(s)/devices  50  can also be linked through communications network  70  to other computing devices, including other client devices/processes  50  and server computer(s)  60 , via communication links  75  (e.g., wired or wireless network connections). The communications network  70  can be part of a remote access network, a global network (e.g., the Internet), a worldwide collection of computers, local area or wide area networks, and gateways that currently use respective protocols (TCP/IP, Bluetooth®, etc.) to communicate with one another. Other electronic device/computer network architectures are suitable. 
       FIG.  9    is a diagram of an example internal structure of a computer (e.g., client processor/device  50  or server computers  60 ) in the computer system of  FIG.  8   . Each computer  50 ,  60  contains a system bus  79 , where a bus is a set of hardware lines used for data transfer among the components of a computer or processing system. The system bus  79  is essentially a shared conduit that connects different elements of a computer system (e.g., processor, disk storage, memory, input/output ports, network ports) that enables the transfer of information between the elements. Attached to the system bus  79  is an I/O device interface  82  for connecting various input and output devices (e.g., keyboard, mouse, displays, printers, speakers) to the computer  50 ,  60 . A network interface  86  allows the computer to connect to various other devices attached to a network (e.g., network  70  of  FIG.  8   ). Memory  90  provides volatile storage for computer software instructions  92  and data  94  used to implement an embodiment of the present invention (e.g., process  100   FIG.  1   ). Disk storage  95  provides non-volatile storage for computer software instructions  92  and data  94  used to implement an embodiment of the present invention. A central processor unit  84  is also attached to the system bus  79  and provides for the execution of computer instructions. The disk storage  95  or memory  90  can provide storage for a database. Embodiments of a database can include a SQL database, text file, or other organized collection of data. 
     In one embodiment, the processor routines  92  and data  94  are a computer program product (generally referenced  92 ), including a non-transitory computer-readable medium (e.g., a removable storage medium such as one or more DVD-ROM&#39;s, CD-ROM&#39;s, diskettes, tapes) that provides at least a portion of the software instructions for the invention system. The computer program product  92  can be installed by any suitable software installation procedure, as is well known in the art. In another embodiment, at least a portion of the software instructions may also be downloaded over a cable communication and/or wireless connection. 
     It should be understood that the example embodiments described herein may be implemented in many different ways. In some instances, the various methods, systems, and devices described herein may each be implemented by a physical, virtual, or hybrid general purpose computer. The computer systems  50 ,  60  may be transformed into machines that execute methods described herein, for example, by loading software instructions into either memory  90  or non-volatile storage  95  for execution by the CPU  84 . 
     Embodiments or aspects thereof may be implemented in the form of hardware, firmware, or software. If implemented in software, the software may be stored on any non-transient computer readable medium that is configured to enable a processor to load the software or subsets of instructions thereof. The processor then executes the instructions and is configured to operate or cause an apparatus to operate in a manner as described herein. 
     Further, firmware, software, routines, or instructions may be described herein as performing certain actions and/or functions of the data processors. However, it should be appreciated that such descriptions contained herein are merely for convenience and that such actions in fact result from computing devices, processors, controllers, or other devices executing the firmware, software, routines, instructions, etc. 
     It should be understood that the flow diagrams, block diagrams, and network diagrams may include more or fewer elements, be arranged differently, or be represented differently. But it further should be understood that certain implementations may dictate the block and network diagrams and the number of block and network diagrams illustrating the execution of the embodiments be implemented in a particular way. 
     Accordingly, further embodiments may also be implemented in a variety of computer architectures, physical, virtual, cloud computers, or some combination thereof, and, thus, the data processors described herein are intended for purposes of illustration only and not as a limitation of the embodiments. 
     While example embodiments have been particularly shown and described, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the scope of the embodiments encompassed by the appended claims.