Computational approaches based on graph clustering algorithms are often used to complement the experimental approaches in the identification of protein complexes. Several studies (Li et al. 2010; Srihari and Leong 2013; Bhowmick and Seah 2016; Wu et al. 2019; Omranian et al. 2022) have categorized the existing computational approaches for protein complex prediction in multiple groups, such as (i) supervised (Qi et al. 2008; Shi et al. 2011) versus unsupervised (Spirin and Mirny 2003; Bader and Hogue 2003), (ii) using only the topological structure of PPI network (Enright 2002; Nepusz et al. 2012) versus integrating additional knowledge or data, such as gene expression (Feng et al. 2011), functional and evolutionary information (King et al. 2004; Sharan et al. 2005; Dost et al. 2008). Further, several protein complex gold standards of different species such as EcoCyc for Escherichia coli (Keseler et al. 2016), MIPS, SGD, and CYC2008 for Saccharomyces cerevisiae (Mewes 2004; Hong et al. 2007; Pu et al. 2008), and CORUM for Homo sapiens (Giurgiu et al. 2018), have been assembled to facilitate the comparison and evaluation of predicted complexes from different approaches.
min cut algorithm pdf download
Due to the incompleteness and noisiness of interactions data, a variety of computational approaches have been proposed as an alternative to experimental tools to predict protein interactions (Zeng 2016; Kovács et al. 2019; Wang et al. 2020). For instance, link prediction algorithms enable us to overcome some of the disadvantages of experimental approaches by identifying false-negative interactions in the PPI network. Therefore, the link prediction and graph clustering algorithms are jointly used to improve the performance of approaches for the prediction of protein complexes. One can employ a link prediction algorithm as a pre-processing step to tune the PPI network and then predict more accurate protein complexes. Alternatively, one can first employ a graph clustering algorithm to group the proteins that are more likely to interact together, and then apply a variety of local or global structure-based similarity measures to compute the possibility of protein interactions in the same cluster (Hu et al. 2021).
In contrast to the existing computational approaches, PC2P, GCC-v, and CUBCO (Omranian et al. 2021a; b; Omranian and Nikoloski 2022) represent parameter-free algorithms and compare the performance of their results with several state-of-the-art approaches across different species. These approaches detect a protein complex based on partitioning the network into biclique spanned subgraphs, which is also known as coherent network partition (CNP) (Angeleska and Nikoloski 2019; Angeleska et al. 2021). PC2P and GCC-v rely on local properties of the network by finding the minimum cut in complement of the second neighborhood of a node (Omranian et al. 2021a; b) and computing the clustering coefficient for each node to partition the network into biclique spanned subgraphs (Omranian et al. 2021a; b), respectively. Alternatively, CUBCO (Omranian and Nikoloski 2022) is based on the global properties of the network and utilizes global minimum cut to partition the network into biclique spanned subgraphs. Moreover, to overcome the incompleteness of PPI networks, CUBCO integrates link prediction (Kovács et al. 2019) as a pre-processing step to cluster more probable interacting proteins together. The three approaches show consistent performance across different species, in contrast to other approaches that obtain different ranking scores for different combinations of species and the corresponding gold standards.
The min-cut algorithm returns two node-subsets, \(S_1\) and \(S_2\), such that \(S_1\cup S_2=V(\overlineG )\) and an edge cut set (i.e. \(E_cut\)) that connects \(S_1\) to \(S_2\), \(E_cut = \left\ \left( u_i ,v_i \right)\left \right\\) where \(k=E_cut\). To make \(\overlineG \) disconnected and achieve the biclique spanned subgraph, \(C_i\), one of the node sets, \(u_i\) or \(v_i\), \(1\le i\le k\), must be removed from \(\overlineG \); therefore, the final biclique spanned subgraph is either \(C_i=\(S_1\cup S_2)/\bigcup _i=1^ku_i\\) or \(C_i=\(S_1\cup S_2)/\bigcup _i=1^kv_i\\). Finally, a score, which exhibits the cohesiveness of the two node-set in graph \(G\), is calculated as follows:
All the contending algorithms depend on multiple parameters, except PC2P, GCC-v, and CUBCO. To optimize the parameters and obtain the best performance for each contender based on different performance measures and combinations of PPI networks and gold standards is challenging. Finding the best parameters, by optimization of various performance measures, yields divergent predicted protein complexes. Therefore, it is impossible to do meaningful interpretation and combination of the findings. Hence, the default value of parameters is used as suggested in corresponding studies.
Twelve metrics are commonly used to evaluate the predicted protein complexes from the contending algorithms, including maximum matching ratio, fraction match (Nepusz et al. 2012), sensitivity, positive predictive value, accuracy, and separation from (Brohée and van Helden 2006), precision, recall, and F-measure from (Liu et al. 2009), and precision+, recall+, and F-measure+ from (Maddi et al. 2019). Therefore, the predicted protein complexes are compared with complexes from gold standards across all organisms based on mentioned twelve metrics. Moreover, these metrics were selected since they have been employed in seminal studies (i.e. prediction of protein complexes) (Adamcsek et al. 2006; Liu et al. 2009; Nepusz et al. 2012; Wang et al. 2018). The twelve metrics are summarized into a composite score, which is the sum over MMR, FRM, ACC, and F-measure (Nepusz et al. 2012; Cao et al. 2018; Wang et al. 2018; Omranian et al. 2021a, b; Omranian and Nikoloski 2022). The definition and notations of evaluation metrics are comprehensively explained in the Additional file 1.
We present an algorithm for finding the minimum cut of an edge-weighted graph. It is simple in every respect. It has a short and compact description, is easy to implement and has a surprisingly simple proof of correctness. Its runtime matches that of the fastest algorithm known. The runtime analysis is straightforward. In contrast to nearly all approaches so far, the algorithm uses no flow techniques. Roughly speaking the algorithm consists of about V nearly identical phases each of which is formally similar to Prim's minimum spanning tree algorithm.
We give a deterministic approximation algorithm that computes $(2+\epsilon)$-approximate minimum $k$-cut in $O(m \log^3 n / \epsilon^2)$ time, via a $(1+\epsilon)$-approximation for an LP relaxation of $k$-cut.
Of course you can pick any cut you want (for example a cut just after the source, or just before the drain), if the algorithm is done correctly, then the sum of all these cuts is always 19. This is logical, since if the flow of one cut would be more (or less) then 19, then the existence of a cut that "advances" one node that has a flow of 19, means that in that node, flow has disappeared, or appeared.
It should be noted, that the Ford-Fulkerson method doesn't specify a method of finding the augmenting path.Possible approaches are using DFS or BFS which both work in $O(E)$.If all the capacities of the network are integers, then for each augmenting path the flow of the network increases by at least 1 (for more details see Integral flow theorem).Therefore, the complexity of Ford-Fulkerson is $O(E F)$, where $F$ is the maximal flow of the network.In the case of rational capacities, the algorithm will also terminate, but the complexity is not bounded.In the case of irrational capacities, the algorithm might never terminate, and might not even converge to the maximal flow.
Edmonds-Karp algorithm is just an implementation of the Ford-Fulkerson method that uses BFS for finding augmenting paths.The algorithm was first published by Yefim Dinitz in 1970, and later independently published by Jack Edmonds and Richard Karp in 1972.
The complexity can be given independently of the maximal flow.The algorithm runs in $O(V E^2)$ time, even for irrational capacities.The intuition is, that every time we find an augmenting path one of the edges becomes saturated, and the distance from the edge to $s$ will be longer if it appears later again in an augmenting path.The length of the simple paths is bounded by $V$.
The function maxflow will return the value of the maximal flow.During the algorithm, the matrix capacity will actually store the residual capacity of the network.The value of the flow in each edge will actually not be stored, but it is easy to extend the implementation - by using an additional matrix - to also store the flow and return it.
Karger's Algorithm is a randomized algorithm whose runtime is deterministic; that is, on every run, the time to execute will be bounded by a fixed time function in the size of the input but the algorithm may return a wrong answer with a small probability.
Karger's algorithm is a randomized algorithm (an algorithm which have some degree of randomness associated in its procedure) to compute a minimum cut of a connected, undirected, and unweighted graph G=(V,E)G=(V,E)G=(V,E). It is a "Monte Carlo" algorithm which means it may also produce a wrong output with a certain (usually low) probability.
The main idea of Karger's algorithm is based on the concept of edge contraction, where edge contraction means merging two nodes (say uuu and vvv) of the graph GGG into one node which is also termed as a supernode. All the edges connecting either to uuu or vvv are now attached to the merged node (supernode) which may result in a multigraph as shown in the image given below -
In Karger's algorithm, an edge is chosen randomly, and then the chosen edge is contracted which results in a supernode. The process continues until only two supernodes are remaining in the graph. Those two supernodes represent cut in the original graph GGG. 2ff7e9595c
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