K-core Decomposition Research Articles

Minimum spanning tree and k-core decomposition as measure of subject-specific EEG traits

The identification of subject-specific traits extracted from patterns of brain activity still represents an important challenge. The need to detect distinctive brain features, which is relevant for biometric and brain computer interface systems, has been also emphasized in monitoring the effect of clinical treatments and in evaluating the progression of brain disorders. In this study we propose an approach which aims to investigate the existence of a distinctive functional core (sub-network) using an unbiased reconstruction of network topology. Brain signals from a public and freely available EEG dataset were analysed using a phase synchronization based measure, minimum spanning tree and k-core decomposition. The analysis was performed for each classical brain rhythm separately. Highest classification rates from k-core decomposition were obtained in the gamma (EER = 0.130, AUC = 0.943) and high beta (EER = 0.172, AUC = 0.905) frequency bands. These results confirm that EEG analysis may represent an effective tool to identify subject-specific characteristics that may be of great impact for several bioengineering applications. However, despite the widespread use of these techniques, critical aspects should be considered when dealing with results from high-frequency scalp EEG.

Distinct types of eigenvector localization in networks.

The spectral properties of the adjacency matrix provide a trove of information about the structure and function of complex networks. In particular, the largest eigenvalue and its associated principal eigenvector are crucial in the understanding of nodes’ centrality and the unfolding of dynamical processes. Here we show that two distinct types of localization of the principal eigenvector may occur in heterogeneous networks. For synthetic networks with degree distribution P(q) ~ q−γ, localization occurs on the largest hub if γ > 5/2; for γ < 5/2 a new type of localization arises on a mesoscopic subgraph associated with the shell with the largest index in the K-core decomposition. Similar evidence for the existence of distinct localization modes is found in the analysis of real-world networks. Our results open a new perspective on dynamical processes on networks and on a recently proposed alternative measure of node centrality based on the non-backtracking matrix.

Investigating the relationship between [formula omitted]-core and [formula omitted]-core network decompositions

Network decomposition methods, such as the much used k-core analysis, are able to identify globally central regions of networks. The decomposition approaches are hierarchical and identify nested sets of nodes with increasing centrality properties. While most studies have been concerned with unweighted networks, i.e. k-core analysis, recent works have introduced network decomposition methods that apply to weighted networks. Here, we investigate the relationship between k-core decomposition for unweighted networks and s-core decomposition for weighted networks by systematically employing a link-weight scheme that gradually discretizes the link weights. We applied this approach to the Erdős–Rényi model and the scale-free configuration model for five different weight distributions, and two empirical networks, the US air traffic network and a Facebook network. We find that (1) both uniformly random and positively correlated link-weight distributions give rise to highly stable s-core decompositions with respect to discretization levels. (2) For negatively correlated link-weight distributions, the resulting s-core decomposition has no similarity to the k-cores. Since several combinations of network topology and link-weight distributions give rise to a core-structure that is highly similar to the full s-core for a large range of link-discretization levels, it is possible to significantly speed up the numerical s-core analysis for these situations.

Automated Identification of Core Regulatory Genes in Human Gene Regulatory Networks

Human gene regulatory networks (GRN) can be difficult to interpret due to a tangle of edges interconnecting thousands of genes. We constructed a general human GRN from extensive transcription factor and microRNA target data obtained from public databases. In a subnetwork of this GRN that is active during estrogen stimulation of MCF-7 breast cancer cells, we benchmarked automated algorithms for identifying core regulatory genes (transcription factors and microRNAs). Among these algorithms, we identified K-core decomposition, pagerank and betweenness centrality algorithms as the most effective for discovering core regulatory genes in the network evaluated based on previously known roles of these genes in MCF-7 biology as well as in their ability to explain the up or down expression status of up to 70% of the remaining genes. Finally, we validated the use of K-core algorithm for organizing the GRN in an easier to interpret layered hierarchy where more influential regulatory genes percolate towards the inner layers. The integrated human gene and miRNA network and software used in this study are provided as supplementary materials (S1 Data) accompanying this manuscript.

The Network of Counterparty Risk: Analysing Correlations in OTC Derivatives.

Counterparty risk denotes the risk that a party defaults in a bilateral contract. This risk not only depends on the two parties involved, but also on the risk from various other contracts each of these parties holds. In rather informal markets, such as the OTC (over-the-counter) derivative market, institutions only report their aggregated quarterly risk exposure, but no details about their counterparties. Hence, little is known about the diversification of counterparty risk. In this paper, we reconstruct the weighted and time-dependent network of counterparty risk in the OTC derivatives market of the United States between 1998 and 2012. To proxy unknown bilateral exposures, we first study the co-occurrence patterns of institutions based on their quarterly activity and ranking in the official report. The network obtained this way is further analysed by a weighted k-core decomposition, to reveal a core-periphery structure. This allows us to compare the activity-based ranking with a topology-based ranking, to identify the most important institutions and their mutual dependencies. We also analyse correlations in these activities, to show strong similarities in the behavior of the core institutions. Our analysis clearly demonstrates the clustering of counterparty risk in a small set of about a dozen US banks. This not only increases the default risk of the central institutions, but also the default risk of peripheral institutions which have contracts with the central ones. Hence, all institutions indirectly have to bear (part of) the counterparty risk of all others, which needs to be better reflected in the price of OTC derivatives.

K-core decomposition of large networks on a single PC

Studying the topology of a network is critical to inferring underlying dynamics such as tolerance to failure, group behavior and spreading patterns. k -core decomposition is a well-established metric which partitions a graph into layers from external to more central vertices. In this paper we aim to explore whether k -core decomposition of large networks can be computed using a consumer-grade PC. We feature implementations of the "vertex-centric" distributed protocol introduced by Montresor, De Pellegrini and Miorandi on GraphChi and Webgraph. Also, we present an accurate implementation of the Batagelj and Zaversnik algorithm for k -core decomposition in Webgraph. With our implementations, we show that we can efficiently handle networks of billions of edges using a single consumer-level machine within reasonable time and can produce excellent approximations in only a fraction of the execution time. To the best of our knowledge, our biggest graphs are considerably larger than the graphs considered in the literature. Next, we present an optimized implementation of an external-memory algorithm (EMcore) by Cheng, Ke, Chu, and Özsu. We show that this algorithm also performs well for large datasets, however, it cannot predict whether a given memory budget is sufficient for a new dataset. We present a thorough analysis of all algorithms concluding that it is viable to compute k -core decomposition for large networks in a consumer-grade PC.

K-core decomposition of a protein domain co-occurrence network reveals lower cancer mutation rates for interior cores.

BackgroundNetwork biology currently focuses primarily on metabolic pathways, gene regulatory, and protein-protein interaction networks. While these approaches have yielded critical information, alternative methods to network analysis will offer new perspectives on biological information. A little explored area is the interactions between domains that can be captured using domain co-occurrence networks (DCN). A DCN can be used to study the function and interaction of proteins by representing protein domains and their co-existence in genes and by mapping cancer mutations to the individual protein domains to identify signals.ResultsThe domain co-occurrence network was constructed for the human proteome based on PFAM domains in proteins. Highly connected domains in the central cores were identified using the k-core decomposition technique. Here we show that these domains were found to be more evolutionarily conserved than the peripheral domains. The somatic mutations for ovarian, breast and prostate cancer diseases were obtained from the TCGA database. We mapped the somatic mutations to the individual protein domains and the local false discovery rate was used to identify significantly mutated domains in each cancer type. Significantly mutated domains were found to be enriched in cancer disease pathways. However, we found that the inner cores of the DCN did not contain any of the significantly mutated domains. We observed that the inner core protein domains are highly conserved and these domains co-exist in large numbers with other protein domains.ConclusionMutations and domain co-occurrence networks provide a framework for understanding hierarchal designs in protein function from a network perspective. This study provides evidence that a majority of protein domains in the inner core of the DCN have a lower mutation frequency and that protein domains present in the peripheral regions of the k-core contribute more heavily to the disease. These findings may contribute further to drug development.Electronic supplementary materialThe online version of this article (doi:10.1186/s13336-015-0016-6) contains supplementary material, which is available to authorized users.

IDENTIFYING IMPORTANT CLASSES OF LARGE SOFTWARE SYSTEMS THROUGH K-CORE DECOMPOSITION

In a large software project, the number of classes, and the dependencies between them, generally increase as software evolves. The size and scale of the system often makes it difficult to easily identify the important components in a particular software product. To address this problem, we model software as a network, where the classes are the vertices in the network and the dependencies are the edges, and apply K-core decomposition to identify a core subset of vertices as potentially important classes. We study three open source Java projects over a 10-year period and demonstrate, using different metrics, that the K-core decomposition of the network can help us identify the key classes of the corresponding software. Specifically, we show that the vertices with the highest core number represent the important classes and demonstrate that the core-numbers of classes with similar functionalities evolve at similar trends.

Identifying Important Packages of Object-Oriented Software Using Weighted k-Core Decomposition

AbstractIdentifying important entities in software systems has many implications for effective resource allocation. Complex network research opens new opportunities for identifying important entities from software networks. However, the existing methods only focus on identifying important classes. Little work has been done on the identification of important packages. Moreover, the metrics they used to quantify the class importance are only designed for unweighted software networks and cannot fit in with the weighted software networks. To overcome these limitations, in this article, we introduce the weighted k-core decomposition method (Wk-core) to identify the important packages. First, we use a weighted software network to describe packages and their internal dependencies. Second, we use Wk-core to partition a software network into a layered structure. Then, the packages that are denoted by the nodes within the main core are the identified important packages. To evaluate our method, we use a variant of the susceptible–infectious–recovered model to examine the spreading influence of the nodes in six real weighted software networks. The results show that our method can well identify influential nodes, better than other four methods (i.e., original k-core decomposition, degree centrality, closeness centrality, and betweenness centrality methods). Furthermore, we demonstrate our method on two software networks and show that the important packages identified by our method are more meaningful from a software engineering perspective when compared with the other methods.

Analyzing the structure of earthquake network by [formula omitted]-core decomposition

Earthquake network is a novel methodology to analyze the relationships among the earthquake events. k-core decomposition is one of the common measures to identify the most important nodes in complex networks. In this paper, we first verify the hierarchical structure of the earthquake network, which is constructed by the method based on space–time influence domain, and use the k-core decomposition to visualize the topology in order to figure out the structure and the central part of earthquake networks. On observation of the evolution of the maximum coreness, it is found that it has some sudden changes with the occurrence of major shocks. The highest core, which has high clustering characteristic, is tend to be located in the area that can directly or indirectly cause the major shocks. By comparing the energy distribution in each shell, we find that the energy proportion of the highest core is high, especially before the major shocks.

Efficient Core Maintenance in Large Dynamic Graphs

The k-core decomposition in a graph is a fundamental problem for social network analysis. The problem of k-core decomposition is to calculate the core number for every node in a graph. Previous studies mainly focus on k-core decomposition in a static graph. There exists a linear time algorithm for k-core decomposition in a static graph. However, in many real-world applications such as online social networks and the Internet, the graph typically evolves over time. Under such applications, a key issue is to maintain the core numbers of nodes given the graph changes over time. In this paper, we propose a new efficient algorithm to maintain the core number for every node in a dynamic graph. Our main result is that only certain nodes need to update their core numbers given the graph is changed by inserting/deleting an edge. We devise an efficient algorithm to identify and recompute the core numbers of such nodes. The complexity of our algorithm is independent of the graph size. In addition, to further accelerate the algorithm, we develop two pruning strategies by exploiting the lower and upper bounds of the core number. Finally, we conduct extensive experiments over both real-world and synthetic datasets, and the results demonstrate the efficiency of the proposed algorithm.

K-core percolation on multiplex networks.

We generalize the theory of k-core percolation on complex networks to k-core percolation on multiplex networks, where k≡(k(1),k(2),...,k(M)). Multiplex networks can be defined as networks with vertices of one kind but M different types of edges, representing different types of interactions. For such networks, the k-core is defined as the largest subgraph in which each vertex has at least k(i) edges of each type, i=1,2,...,M. We derive self-consistency equations to obtain the birth points of the k-cores and their relative sizes for uncorrelated multiplex networks with an arbitrary degree distribution. To clarify our general results, we consider in detail multiplex networks with edges of two types and solve the equations in the particular case of Erdős-Rényi and scale-free multiplex networks. We find hybrid phase transitions at the emergence points of k-cores except the (1,1)-core for which the transition is continuous. We apply the k-core decomposition algorithm to air-transportation multiplex networks, composed of two layers, and obtain the size of (k(1),k(2))-cores.

Evolution characteristics of the network core in the Facebook.

Statistical properties of the static networks have been extensively studied. However, online social networks are evolving dynamically, understanding the evolving characteristics of the core is one of major concerns in online social networks. In this paper, we empirically investigate the evolving characteristics of the Facebook core. Firstly, we separate the Facebook-link(FL) and Facebook-wall(FW) datasets into 28 snapshots in terms of timestamps. By employing the k-core decomposition method to identify the core of each snapshot, we find that the core sizes of the FL and FW networks approximately contain about 672 and 373 nodes regardless of the exponential growth of the network sizes. Secondly, we analyze evolving topological properties of the core, including the k-core value, assortative coefficient, clustering coefficient and the average shortest path length. Empirical results show that nodes in the core are getting more interconnected in the evolving process. Thirdly, we investigate the life span of nodes belonging to the core. More than 50% nodes stay in the core for more than one year, and 19% nodes always stay in the core from the first snapshot. Finally, we analyze the connections between the core and the whole network, and find that nodes belonging to the core prefer to connect nodes with high k-core values, rather than the high degrees ones. This work could provide new insights into the online social network analysis.

Gene interaction map: a paradigm for identifying significant pathways responsible for rheumatoid arthritis

Rheumatoid arthritis (RA) is a chronic autoimmune disease that causes inflammation of the joints and may cause inflammation of other tissues in the body. It is mainly caused by combination of factors including abnormal autoimmune response, genetic susceptibility and some environmental or biological triggers. A number of pathways have been shown to be affected wherein numerous molecules are known to be involved. Uncovering the molecular pathways in this disease becomes more difficult as it is complicated with genetic, environmental and more over inflammatory and autoimmune parameters. Hence, network biology approach is used to identify the important pathways and molecules which play significant role in RA through Gene Interaction Map (GIP) of 1,200 nodes and 5,286 edges. Our studies elucidate the relationship between topological properties of GIP and the role played by molecules in cellular system, which helps in defining the organizational mechanism used in cellular system. K-core decomposition method identified novel genes and revealed the correlation between toll-like receptor, MAPK signaling, apoptosis, t cell receptor signaling and epithelial cell signaling pathways.

Evolution of air transport network of China 1930–2012

This paper analyzes the evolution process of the air transport network of China (ATNC) since 1930. Based on the network analysis results, the ATNC has significantly improved in connectivity based on (1) rising alpha, beta and gamma indices, (2) declining diameter and centre index and (3) decreasing average path length and increasing clustering coefficient. The network centralization index reveals a fluctuation phase before 1952, a pre-1980 centralization phase before the economic reform era, a centralization phase after the mid-1990s deregulation, and a decentralization phase between. The k-core decomposition method helps identify the evolution of core network and hierarchy of the ATNC over time. The spatial development model characterizes its structure change in six stages: (1) scattered development, (2) trunk line connection, (3) circular linkage, (4) hub formation, (5) a complex network structure, and (6) emerging multi-airport systems.

S-core network decomposition: A generalization ofk-core analysis to weighted networks

A broad range of systems spanning biology, technology, and social phenomena may be represented and analyzed as complex networks. Recent studies of such networks using k-core decomposition have uncovered groups of nodes that play important roles. Here, we present s-core analysis, a generalization of k-core (or k-shell) analysis to complex networks where the links have different strengths or weights. We demonstrate the s-core decomposition approach on two random networks (ER and configuration model with scale-free degree distribution) where the link weights are (i) random, (ii) correlated, and (iii) anticorrelated with the node degrees. Finally, we applythes-core decomposition approach to the protein-interaction network of the yeast Saccharomyces cerevisiaeinthe context of two gene-expression experiments: oxidative stress in response to cumene hydroperoxide (CHP), and fermentation stress response (FSR). We find that the innermost s-cores are (i) different from innermost k-cores, (ii) different for the two stress conditions CHP and FSR, and (iii) enriched with proteins whose biological functions give insight into how yeast manages these specific stresses.

Percolation on random networks with arbitraryk-core structure

The k-core decomposition of a network has thus far mainly served as a powerful tool for the empirical study of complex networks. We now propose its explicit integration in a theoretical model. We introduce a hard-core random network (HRN) model that generates maximally random networks with arbitrary degree distribution and arbitrary k-core structure. We then solve exactly the bond percolation problem on the HRN model and produce fast and precise analytical estimates for the corresponding real networks. Extensive comparison with real databases reveals that our approach performs better than existing models, while requiring less input information.

Identifying Influential Nodes in Large-Scale Directed Networks: The Role of Clustering

Identifying influential nodes in very large-scale directed networks is a big challenge relevant to disparate applications, such as accelerating information propagation, controlling rumors and diseases, designing search engines, and understanding hierarchical organization of social and biological networks. Known methods range from node centralities, such as degree, closeness and betweenness, to diffusion-based processes, like PageRank and LeaderRank. Some of these methods already take into account the influences of a node’s neighbors but do not directly make use of the interactions among it’s neighbors. Local clustering is known to have negative impacts on the information spreading. We further show empirically that it also plays a negative role in generating local connections. Inspired by these facts, we propose a local ranking algorithm named ClusterRank, which takes into account not only the number of neighbors and the neighbors’ influences, but also the clustering coefficient. Subject to the susceptible-infected-recovered (SIR) spreading model with constant infectivity, experimental results on two directed networks, a social network extracted from delicious.com and a large-scale short-message communication network, demonstrate that the ClusterRank outperforms some benchmark algorithms such as PageRank and LeaderRank. Furthermore, ClusterRank can also be applied to undirected networks where the superiority of ClusterRank is significant compared with degree centrality and k-core decomposition. In addition, ClusterRank, only making use of local information, is much more efficient than global methods: It takes only 191 seconds for a network with about nodes, more than 15 times faster than PageRank.

Streaming algorithms for k-core decomposition

A k -core of a graph is a maximal connected subgraph in which every vertex is connected to at least k vertices in the subgraph. k -core decomposition is often used in large-scale network analysis, such as community detection, protein function prediction, visualization, and solving NP-Hard problems on real networks efficiently, like maximal clique finding. In many real-world applications, networks change over time. As a result, it is essential to develop efficient incremental algorithms for streaming graph data. In this paper, we propose the first incremental k -core decomposition algorithms for streaming graph data. These algorithms locate a small subgraph that is guaranteed to contain the list of vertices whose maximum k -core values have to be updated, and efficiently process this subgraph to update the k -core decomposition. Our results show a significant reduction in run-time compared to non-incremental alternatives. We show the efficiency of our algorithms on different types of real and synthetic graphs, at different scales. For a graph of 16 million vertices, we observe speedups reaching a million times, relative to the non-incremental algorithms.

Distributed k-Core Decomposition

Several novel metrics have been proposed in recent literature in order to study the relative importance of nodes in complex networks. Among those, k-coreness has found a number of applications in areas as diverse as sociology, proteinomics, graph visualization, and distributed system analysis and design. This paper proposes new distributed algorithms for the computation of the k-coreness of a network, a process also known as k-core decomposition. This technique 1) allows the decomposition, over a set of connected machines, of very large graphs, when size does not allow storing and processing them on a single host, and 2) enables the runtime computation of k-cores in “live” distributed systems. Lower bounds on the algorithms complexity are given, and an exhaustive experimental analysis on real-world data sets is provided.