GraphCrunch 2: Software tool for network modeling,alignment and clustering

Background  

Recent advancements in experimental biotechnology have produced large amounts of protein-protein interaction (PPI) data. The topology of PPI networks is believed to have a strong link to their function. Hence, the abundance of PPI data for many organisms stimulates the development of computational techniques for the modeling, comparison, alignment, and clustering of networks. In addition, finding representative models for PPI networks will improve our understanding of the cell just as a model of gravity has helped us understand planetary motion. To decide if a model is representative, we need quantitative comparisons of model networks to real ones. However, exact network comparison is computationally intractable and therefore several heuristics have been used instead. Some of these heuristics are easily computable "network properties," such as the degree distribution, or the clustering coefficient. An important special case of network comparison is the network alignment problem. Analogous to sequence alignment, this problem asks to find the "best" mapping between regions in two networks. It is expected that network alignment might have as strong an impact on our understanding of biology as sequence alignment has had. Topology-based clustering of nodes in PPI networks is another example of an important network analysis problem that can uncover relationships between interaction patterns and phenotype.     

Pairwise alignment of protein interaction networks.

With an ever-increasing amount of available data on protein-protein interaction (PPI) networks and research revealing that these networks evolve at a modular level, discovery of conserved patterns in these networks becomes an important problem. Although available data on protein-protein interactions is currently limited, recently developed algorithms have been shown to convey novel biological insights through employment of elegant mathematical models. The main challenge in aligning PPI networks is to define a graph theoretical measure of similarity between graph structures that captures underlying biological phenomena accurately. In this respect, modeling of conservation and divergence of interactions, as well as the interpretation of resulting alignments, are important design parameters. In this paper, we develop a framework for comprehensive alignment of PPI networks, which is inspired by duplication/divergence models that focus on understanding the evolution of protein interactions. We propose a mathematical model that extends the concepts of match, mismatch, and gap in sequence alignment to that of match, mismatch, and duplication in network alignment and evaluates similarity between graph structures through a scoring function that accounts for evolutionary events. By relying on evolutionary models, the proposed framework facilitates interpretation of resulting alignments in terms of not only conservation but also divergence of modularity in PPI networks. Furthermore, as in the case of sequence alignment, our model allows flexibility in adjusting parameters to quantify underlying evolutionary relationships. Based on the proposed model, we formulate PPI network alignment as an optimization problem and present fast algorithms to solve this problem. Detailed experimental results from an implementation of the proposed framework show that our algorithm is able to discover conserved interaction patterns very effectively, in terms of both accuracies and computational cost.     

Metabolic network alignment in large scale by network compression

Metabolic network alignment is a system scale comparative analysis that discovers important similarities and differences across different metabolisms and organisms. Although the problem of aligning metabolic networks has been considered in the past, the computational complexity of the existing solutions has so far limited their use to moderately sized networks. In this paper, we address the problem of aligning two metabolic networks, particularly when both of them are too large to be dealt with using existing methods. We develop a generic framework that can significantly improve the scale of the networks that can be aligned in practical time. Our framework has three major phases, namely the compression phase, the alignment phase and the refinement phase. For the first phase, we develop an algorithm which transforms the given networks to a compressed domain where they are summarized using fewer nodes, termed supernodes, and interactions. In the second phase, we carry out the alignment in the compressed domain using an existing network alignment method as our base algorithm. This alignment results in supernode mappings in the compressed domain, each of which are smaller instances of network alignment problem. In the third phase, we solve each of the instances using the base alignment algorithm to refine the alignment results. We provide a user defined parameter to control the number of compression levels which generally determines the tradeoff between the quality of the alignment versus how fast the algorithm runs. Our experiments on the networks from KEGG pathway database demonstrate that the compression method we propose reduces the sizes of metabolic networks by almost half at each compression level which provides an expected speedup of more than an order of magnitude. We also observe that the alignments obtained by only one level of compression capture the original alignment results with high accuracy. Together, these suggest that our framework results in alignments that are comparable to existing algorithms and can do this with practical resource utilization for large scale networks that existing algorithms could not handle. As an example of our method's performance in practice, the alignment of organism-wide metabolic networks of human (1615 reactions) and mouse (1600 reactions) was performed under three minutes by only using a single level of compression.     

Seed selection strategy in global network alignment without destroying the entire structures of functional modules

Background

Network alignment is one of the most common biological network comparison methods. Aligning protein-protein interaction (PPI) networks of different species is of great important to detect evolutionary conserved pathways or protein complexes across species through the identification of conserved interactions, and to improve our insight into biological systems. Global network alignment (GNA) problem is NP-complete, for which only heuristic methods have been proposed so far. Generally, the current GNA methods fall into global heuristic seed-and-extend approaches. These methods can not get the best overall consistent alignment between networks for the opinionated local seed. Furthermore These methods are lost in maximizing the number of aligned edges between two networks without considering the original structures of functional modules.

Methods

We present a novel seed selection strategy for global network alignment by constructing the pairs of hub nodes of networks to be aligned into multiple seeds. Beginning from every hub seed and using the membership similarity of nodes to quantify to what extent the nodes can participate in functional modules associated with current seed topologically we align the networks by modules. By this way we can maintain the functional modules are not damaged during the heuristic alignment process. And our method is efficient in resolving the fatal problem of most conventional algorithms that the initialization selected seeds have a direct influence on the alignment result. The similarity measures between network nodes (e.g., proteins) include sequence similarity, centrality similarity, and dynamic membership similarity and our algorithm can be called Multiple Hubs-based Alignment (MHA).

Results

When applying our seed selection strategy to several pairs of real PPI networks, it is observed that our method is working to strike a balance, extending the conserved interactions while maintaining the functional modules unchanged. In the case study, we assess the effectiveness of MHA on the alignment of the yeast and fly PPI networks. Our method outperforms state-of-the-art algorithms at detecting conserved functional modules and retrieves in particular 86% more conserved interactions than IsoRank.

Conclusions

We believe that our seed selection strategy will lead us to obtain more topologically and biologically similar alignment result. And it can be used as the reference and complement of other heuristic methods to seek more meaningful alignment results.

     

Discovering large conserved functional components in global network alignment by graph matching

Background

Aligning protein-protein interaction (PPI) networks is very important to discover the functionally conserved sub-structures between different species. In recent years, the global PPI network alignment problem has been extensively studied aiming at finding the one-to-one alignment with the maximum matching score. However, finding large conserved components remains challenging due to its NP-hardness.

Results

We propose a new graph matching method GMAlign for global PPI network alignment. It first selects some pairs of important proteins as seeds, followed by a gradual expansion to obtain an initial matching, and then it refines the current result to obtain an optimal alignment result iteratively based on the vertex cover. We compare GMAlign with the state-of-the-art methods on the PPI network pairs obtained from the largest BioGRID dataset and validate its performance. The results show that our algorithm can produce larger size of alignment, and can find bigger and denser common connected subgraphs as well for the first time. Meanwhile, GMAlign can achieve high quality biological results, as measured by functional consistency and semantic similarity of the Gene Ontology terms. Moreover, we also show that GMAlign can achieve better results which are structurally and biologically meaningful in the detection of large conserved biological pathways between species.

Conclusions

GMAlign is a novel global network alignment tool to discover large conserved functional components between PPI networks. It also has many potential biological applications such as conserved pathway and protein complex discovery across species. The GMAlign software and datasets are avaialbile at https://github.com/yzlwhu/GMAlign.

     

Protein complexes discovery based on protein-protein interaction data via a regularized sparse generative network model

Detecting protein complexes from protein interaction networks is one major task in the postgenome era. Previous developed computational algorithms identifying complexes mainly focus on graph partition or dense region finding. Most of these traditional algorithms cannot discover overlapping complexes which really exist in the protein-protein interaction (PPI) networks. Even if some density-based methods have been developed to identify overlapping complexes, they are not able to discover complexes that include peripheral proteins. In this study, motivated by recent successful application of generative network model to describe the generation process of PPI networks and to detect communities from social networks, we develop a regularized sparse generative network model (RSGNM), by adding another process that generates propensities using exponential distribution and incorporating Laplacian regularizer into an existing generative network model, for protein complexes identification. By assuming that the propensities are generated using exponential distribution, the estimators of propensities will be sparse, which not only has good biological interpretation but also helps to control the overlapping rate among detected complexes. And the Laplacian regularizer will lead to the estimators of propensities more smooth on interaction networks. Experimental results on three yeast PPI networks show that RSGNM outperforms six previous competing algorithms in terms of the quality of detected complexes. In addition, RSGNM is able to detect overlapping complexes and complexes including peripheral proteins simultaneously. These results give new insights about the importance of generative network models in protein complexes identification.     

Identification of functional modules from conserved ancestral protein-protein interactions

MOTIVATION: The increasing availability of large-scale protein-protein interaction (PPI) data has fueled the efforts to elucidate the building blocks and organization of cellular machinery. Previous studies have shown cross-species comparison to be an effective approach in uncovering functional modules in protein networks. This has in turn driven the research for new network alignment methods with a more solid grounding in network evolution models and better scalability, to allow multiple network comparison. RESULTS: We develop a new framework for protein network alignment, based on reconstruction of an ancestral PPI network. The reconstruction algorithm is built upon a proposed model of protein network evolution, which takes into account phylogenetic history of the proteins and the evolution of their interactions. The application of our methodology to the PPI networks of yeast, worm and fly reveals that the most probable conserved ancestral interactions are often related to known protein complexes. By projecting the conserved ancestral interactions back onto the input networks we are able to identify the corresponding conserved protein modules in the considered species. In contrast to most of the previous methods, our algorithm is able to compare many networks simultaneously. The performed experiments demonstrate the ability of our method to uncover many functional modules with high specificity. AVAILABILITY: Information for obtaining software and supplementary results are available at http://bioputer.mimuw.edu.pl/papers/cappi.     

Asymmetric comparison and querying of biological networks

Comparing and querying the protein-protein interaction (PPI) networks of different organisms is important to infer knowledge about conservation across species. Known methods that perform these tasks operate symmetrically, i.e., they do not assign a distinct role to the input PPI networks. However, in most cases, the input networks are indeed distinguishable on the basis of how the corresponding organism is biologically well characterized. In this paper a new idea is developed, that is, to exploit differences in the characterization of organisms at hand in order to devise methods for comparing their PPI networks. We use the PPI network (called Master) of the best characterized organism as a fingerprint to guide the alignment process to the second input network (called Slave), so that generated results preferably retain the structural characteristics of the Master network. Technically, this is obtained by generating from the Master a finite automaton, called alignment model, which is then fed with (a linearization of) the Slave for the purpose of extracting, via the Viterbi algorithm, matching subgraphs. We propose an approach able to perform global alignment and network querying, and we apply it on PPI networks. We tested our method showing that the results it returns are biologically relevant.     

Decomposing PPI networks for complex discovery

Background

Protein complexes are important for understanding principles of cellular organization and functions. With the availability of large amounts of high-throughput protein-protein interactions (PPI), many algorithms have been proposed to discover protein complexes from PPI networks. However, existing algorithms generally do not take into consideration the fact that not all the interactions in a PPI network take place at the same time. As a result, predicted complexes often contain many spuriously included proteins, precluding them from matching true complexes.

Results

We propose two methods to tackle this problem: (1) The localization GO term decomposition method: We utilize cellular component Gene Ontology (GO) terms to decompose PPI networks into several smaller networks such that the proteins in each decomposed network are annotated with the same cellular component GO term. (2) The hub removal method: This method is based on the observation that hub proteins are more likely to fuse clusters that correspond to different complexes. To avoid this, we remove hub proteins from PPI networks, and then apply a complex discovery algorithm on the remaining PPI network. The removed hub proteins are added back to the generated clusters afterwards. We tested the two methods on the yeast PPI network downloaded from BioGRID. Our results show that these methods can improve the performance of several complex discovery algorithms significantly. Further improvement in performance is achieved when we apply them in tandem.

Conclusions

The performance of complex discovery algorithms is hindered by the fact that not all the interactions in a PPI network take place at the same time. We tackle this problem by using localization GO terms or hubs to decompose a PPI network before complex discovery, which achieves considerable improvement.

     

Functional topology in a network of protein interactions

MOTIVATION: The building blocks of biological networks are individual protein-protein interactions (PPIs). The cumulative PPI data set in Saccharomyces cerevisiae now exceeds 78 000. Studying the network of these interactions will provide valuable insight into the inner workings of cells. RESULTS: We performed a systematic graph theory-based analysis of this PPI network to construct computational models for describing and predicting the properties of lethal mutations and proteins participating in genetic interactions, functional groups, protein complexes and signaling pathways. Our analysis suggests that lethal mutations are not only highly connected within the network, but they also satisfy an additional property: their removal causes a disruption in network structure. We also provide evidence for the existence of alternate paths that bypass viable proteins in PPI networks, while such paths do not exist for lethal mutations. In addition, we show that distinct functional classes of proteins have differing network properties. We also demonstrate a way to extract and iteratively predict protein complexes and signaling pathways. We evaluate the power of predictions by comparing them with a random model, and assess accuracy of predictions by analyzing their overlap with MIPS database. CONCLUSIONS: Our models provide a means for understanding the complex wiring underlying cellular function, and enable us to predict essentiality, genetic interaction, function, protein complexes and cellular pathways. This analysis uncovers structure-function relationships observable in a large PPI network.     

A multi-network clustering method for detecting protein complexes from multiple heterogeneous networks

Background

The accurate identification of protein complexes is important for the understanding of cellular organization. Up to now, computational methods for protein complex detection are mostly focus on mining clusters from protein-protein interaction (PPI) networks. However, PPI data collected by high-throughput experimental techniques are known to be quite noisy. It is hard to achieve reliable prediction results by simply applying computational methods on PPI data. Behind protein interactions, there are protein domains that interact with each other. Therefore, based on domain-protein associations, the joint analysis of PPIs and domain-domain interactions (DDI) has the potential to obtain better performance in protein complex detection. As traditional computational methods are designed to detect protein complexes from a single PPI network, it is necessary to design a new algorithm that could effectively utilize the information inherent in multiple heterogeneous networks.

Results

In this paper, we introduce a novel multi-network clustering algorithm to detect protein complexes from multiple heterogeneous networks. Unlike existing protein complex identification algorithms that focus on the analysis of a single PPI network, our model can jointly exploit the information inherent in PPI and DDI data to achieve more reliable prediction results. Extensive experiment results on real-world data sets demonstrate that our method can predict protein complexes more accurately than other state-of-the-art protein complex identification algorithms.

Conclusions

In this work, we demonstrate that the joint analysis of PPI network and DDI network can help to improve the accuracy of protein complex detection.

     

Fully automated protein complex prediction based on topological similarity and community structure

To understand the function of protein complexes and their association with biological processes, a lot of studies have been done towards analyzing the protein-protein interaction (PPI) networks. However, the advancement in high-throughput technology has resulted in a humongous amount of data for analysis. Moreover, high level of noise, sparseness, and skewness in degree distribution of PPI networks limits the performance of many clustering algorithms and further analysis of their interactions.In addressing and solving these problems we present a novel random walk based algorithm that converts the incomplete and binary PPI network into a protein-protein topological similarity matrix (PP-TS matrix). We believe that if two proteins share some high-order topological similarities they are likely to be interacting with each other. Using the obtained PP-TS matrix, we constructed and used weighted networks to further study and analyze the interaction among proteins. Specifically, we applied a fully automated community structure finding algorithm (Auto-HQcut) on the obtained weighted network to cluster protein complexes. We then analyzed the protein complexes for significance in biological processes. To help visualize and analyze these protein complexes we also developed an interface that displays the resulting complexes as well as the characteristics associated with each complex.Applying our approach to a yeast protein-protein interaction network, we found that the predicted protein-protein interaction pairs with high topological similarities have more significant biological relevance than the original protein-protein interactions pairs. When we compared our PPI network reconstruction algorithm with other existing algorithms using gene ontology and gene co-expression, our algorithm produced the highest similarity scores. Also, our predicted protein complexes showed higher accuracy measure compared to the other protein complex predictions.     

Diffusion Model Based Spectral Clustering for Protein-Protein Interaction Networks

Background

A goal of systems biology is to analyze large-scale molecular networks including gene expressions and protein-protein interactions, revealing the relationships between network structures and their biological functions. Dividing a protein-protein interaction (PPI) network into naturally grouped parts is an essential way to investigate the relationship between topology of networks and their functions. However, clear modular decomposition is often hard due to the heterogeneous or scale-free properties of PPI networks.

Methodology/Principal Findings

To address this problem, we propose a diffusion model-based spectral clustering algorithm, which analytically solves the cluster structure of PPI networks as a problem of random walks in the diffusion process in them. To cope with the heterogeneity of the networks, the power factor is introduced to adjust the diffusion matrix by weighting the transition (adjacency) matrix according to a node degree matrix. This algorithm is named adjustable diffusion matrix-based spectral clustering (ADMSC). To demonstrate the feasibility of ADMSC, we apply it to decomposition of a yeast PPI network, identifying biologically significant clusters with approximately equal size. Compared with other established algorithms, ADMSC facilitates clear and fast decomposition of PPI networks.

Conclusions/Significance

ADMSC is proposed by introducing the power factor that adjusts the diffusion matrix to the heterogeneity of the PPI networks. ADMSC effectively partitions PPI networks into biologically significant clusters with almost equal sizes, while being very fast, robust and appealing simple.     

Biomolecular network motif counting and discovery by color coding

Protein-protein interaction (PPI) networks of many organisms share global topological features such as degree distribution, k-hop reachability, betweenness and closeness. Yet, some of these networks can differ significantly from the others in terms of local structures: e.g. the number of specific network motifs can vary significantly among PPI networks. Counting the number of network motifs provides a major challenge to compare biomolecular networks. Recently developed algorithms have been able to count the number of induced occurrences of subgraphs with k < or = 7 vertices. Yet no practical algorithm exists for counting non-induced occurrences, or counting subgraphs with k > or = 8 vertices. Counting non-induced occurrences of network motifs is not only challenging but also quite desirable as available PPI networks include several false interactions and miss many others. In this article, we show how to apply the 'color coding' technique for counting non-induced occurrences of subgraph topologies in the form of trees and bounded treewidth subgraphs. Our algorithm can count all occurrences of motif G' with k vertices in a network G with n vertices in time polynomial with n, provided k = O(log n). We use our algorithm to obtain 'treelet' distributions for k < or = 10 of available PPI networks of unicellular organisms (Saccharomyces cerevisiae Escherichia coli and Helicobacter Pyloris), which are all quite similar, and a multicellular organism (Caenorhabditis elegans) which is significantly different. Furthermore, the treelet distribution of the unicellular organisms are similar to that obtained by the 'duplication model' but are quite different from that of the 'preferential attachment model'. The treelet distribution is robust w.r.t. sparsification with bait/edge coverage of 70% but differences can be observed when bait/edge coverage drops to 50%.     

AlignNemo: a local network alignment method to integrate homology and topology

Local network alignment is an important component of the analysis of protein-protein interaction networks that may lead to the identification of evolutionary related complexes. We present AlignNemo, a new algorithm that, given the networks of two organisms, uncovers subnetworks of proteins that relate in biological function and topology of interactions. The discovered conserved subnetworks have a general topology and need not to correspond to specific interaction patterns, so that they more closely fit the models of functional complexes proposed in the literature. The algorithm is able to handle sparse interaction data with an expansion process that at each step explores the local topology of the networks beyond the proteins directly interacting with the current solution. To assess the performance of AlignNemo, we ran a series of benchmarks using statistical measures as well as biological knowledge. Based on reference datasets of protein complexes, AlignNemo shows better performance than other methods in terms of both precision and recall. We show our solutions to be biologically sound using the concept of semantic similarity applied to Gene Ontology vocabularies. The binaries of AlignNemo and supplementary details about the algorithms and the experiments are available at: sourceforge.net/p/alignnemo.     

Efficient estimation of graphlet frequency distributions in protein-protein interaction networks

MOTIVATION: Algorithmic and modeling advances in the area of protein-protein interaction (PPI) network analysis could contribute to the understanding of biological processes. Local structure of networks can be measured by the frequency distribution of graphlets, small connected non-isomorphic induced subgraphs. This measure of local structure has been used to show that high-confidence PPI networks have local structure of geometric random graphs. Finding graphlets exhaustively in a large network is computationally intensive. More complete PPI networks, as well as PPI networks of higher organisms, will thus require efficient heuristic approaches. RESULTS: We propose two efficient and scalable heuristics for finding graphlets in high-confidence PPI networks. We show that both PPI and their model geometric random networks, have defined boundaries that are sparser than the 'inner parts' of the networks. In addition, these networks exhibit 'uniformity' of local structure inside the networks. Our first heuristic exploits these two structural properties of PPI and geometric random networks to find good estimates of graphlet frequency distributions in these networks up to 690 times faster than the exhaustive searches. Our second heuristic is a variant of a more standard sampling technique and it produces accurate approximate results up to 377 times faster than the exhaustive searches. We indicate how the combination of these approaches may result in an even better heuristic. AVAILABILITY: Supplementary information is available at http://www.cs.toronto.edu/~natasha/BIOINF-2005-0946/Supplementary.pdf. Software implementing the algorithms is available at http://www.cs.toronto.edu/~natasha/BIOINF-2005-0946/estimate_grap-hlets.html. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.     

An iterative network partition algorithm for accurate identification of dense network modules

A key step in network analysis is to partition a complex network into dense modules. Currently, modularity is one of the most popular benefit functions used to partition network modules. However, recent studies suggested that it has an inherent limitation in detecting dense network modules. In this study, we observed that despite the limitation, modularity has the advantage of preserving the primary network structure of the undetected modules. Thus, we have developed a simple iterative Network Partition (iNP) algorithm to partition a network. The iNP algorithm provides a general framework in which any modularity-based algorithm can be implemented in the network partition step. Here, we tested iNP with three modularity-based algorithms: multi-step greedy (MSG), spectral clustering and Qcut. Compared with the original three methods, iNP achieved a significant improvement in the quality of network partition in a benchmark study with simulated networks, identified more modules with significantly better enrichment of functionally related genes in both yeast protein complex network and breast cancer gene co-expression network, and discovered more cancer-specific modules in the cancer gene co-expression network. As such, iNP should have a broad application as a general method to assist in the analysis of biological networks.     

Global geometric affinity for revealing high fidelity protein interaction network

Protein-protein interaction (PPI) network analysis presents an essential role in understanding the functional relationship among proteins in a living biological system. Despite the success of current approaches for understanding the PPI network, the large fraction of missing and spurious PPIs and a low coverage of complete PPI network are the sources of major concern. In this paper, based on the diffusion process, we propose a new concept of global geometric affinity and an accompanying computational scheme to filter the uncertain PPIs, namely, reduce the spurious PPIs and recover the missing PPIs in the network. The main concept defines a diffusion process in which all proteins simultaneously participate to define a similarity metric (global geometric affinity (GGA)) to robustly reflect the internal connectivity among proteins. The robustness of the GGA is attributed to propagating the local connectivity to a global representation of similarity among proteins in a diffusion process. The propagation process is extremely fast as only simple matrix products are required in this computation process and thus our method is geared toward applications in high-throughput PPI networks. Furthermore, we proposed two new approaches that determine the optimal geometric scale of the PPI network and the optimal threshold for assigning the PPI from the GGA matrix. Our approach is tested with three protein-protein interaction networks and performs well with significant random noises of deletions and insertions in true PPIs. Our approach has the potential to benefit biological experiments, to better characterize network data sets, and to drive new discoveries.     

Coherent activation of a synthetic mammalian gene network

A quantitative analysis of naturally-occurring regulatory networks, especially those present in mammalian cells, is difficult due to their high complexity. Much simpler gene networks can be engineered in model organisms and analyzed as isolated regulatory modules. Recently, several synthetic networks have been constructed in mammalian systems. However, most of these engineered mammalian networks have been characterized using steady-state population level measurements. Here, we use an integrated experimental-computational approach to analyze the dynamical response of a synthetic positive feedback network in individual mammalian cells. We observe a switch-like activation of the network with variable delay times in individual cells. In agreement with a stochastic model of the network, we find that increasing the strength of the positive feedback results in a decrease in the mean delay time and a more coherent activation of individual cells. Our results are important for gaining insight into biological processes which rely on positive feedback regulation.     

A binary matrix factorization algorithm for protein complex prediction

BACKGROUND: Identifying biologically relevant protein complexes from a large protein-protein interaction (PPI) network, is essential to understand the organization of biological systems. However, high-throughput experimental techniques that can produce a large amount of PPIs are known to yield non-negligible rates of false-positives and false-negatives, making the protein complexes difficult to be identified. RESULTS: We propose a binary matrix factorization (BMF) algorithm under the Bayesian Ying-Yang (BYY) harmony learning, to detect protein complexes by clustering the proteins which share similar interactions through factorizing the binary adjacent matrix of a PPI network. The proposed BYY-BMF algorithm automatically determines the cluster number while this number is pre-given for most existing BMF algorithms. Also, BYY-BMF's clustering results does not depend on any parameters or thresholds, unlike the Markov Cluster Algorithm (MCL) that relies on a so-called inflation parameter. On synthetic PPI networks, the predictions evaluated by the known annotated complexes indicate that BYY-BMF is more robust than MCL for most cases. On real PPI networks from the MIPS and DIP databases, BYY-BMF obtains a better balanced prediction accuracies than MCL and a spectral analysis method, while MCL has its own advantages, e.g., with good separation values.