Set Of Cliques Research Articles

On-the-Fly Output Compression for Join-Based Graph Mining Algorithms

Many join-based graph mining (JGM) algorithms, such as triangle listing and clique enumeration, typically output data of such a large size that it often dominates the mining cost. Despite the significance of the I/O cost, only a few research focused on reducing the size of the output data. However, those techniques have limitation that they are highly specific to their corresponding graph mining algorithms. In this paper, through careful observations of output patterns, we propose a general compression solution that can be applied to an arbitrary JGM algorithm. The proposed technique first categorizes the nodes of a set of cliques into two lists, which contain overlapping and non-overlapping nodes, respectively. Then, we remove the redundant nodes from multiple sets of cliques by creating a Clique Constructor Tree using the overlapping and non-overlapping nodes. Finally, we compress the clique constructor trees by applying a novel flag-aligned word encoding technique. Our proposed technique performs compression on-the-fly and can easily be adopted by various JGM algorithms. Experiments on real datasets show that our proposed technique, which is adopted in a triangle listing algorithm, reduces the size of the output data and running time by seven times and about four times, respectively. Experimental results also show that a maximal clique listing algorithm can reduce the size of the output data by a factor of three.

On cliques and bicliques

Basic definitions are given in the next paragraph. We study second clique graphs of suspensions of graphs, K2(S(G)), and characterize them, in terms of an auxiliary biclique operator B which transforms a graph G into its biclique graph B(G). The characterization is then: K2(S(G))≅B(K(G)). We found a characterization of the graphs, G, that maximize |B(G)| for any given order n=|G|. This particular version of a biclique operator is new in the literature. The main motivation to study B(G) is an attempt to characterize the graphs G that maximize |K2(G)|, thus mimicking a result of Moon and Moser [12] that characterizes the graphs maximizing |K(G)|.The clique graph K(G) of a a graph G is the intersection graph of the set of all (maximal) cliques of G (and K2(G)=K(K(G))). The suspension S(G) of a graph G is the graph obtained from G by adding two new vertices which are adjacent to all other vertices, but not to each other. Here, a biclique (X, Y ) is an ordered pair of not necessarily disjoint subsets of vertices of G such that each x∈X is adjacent or equal to every y∈Y and such that (X, Y ) is maximal under component-wise inclusion. Finally B(G) is the graph whose vertices are the bicliques of G with adjacencies given by (X,Y)≃(X′,Y′) if and only if X∩X′≠∅ or Y∩Y′≠∅.

Junction trees constructions in Bayesian networks

Junction trees are used as graphical structures over which propagation will be carried out through a very important property called the ruining intersection property. This paper examines an alternative method for constructing junction trees that are essential for the efficient computations of probabilities in Bayesian networks. The new proposed method converts a sequence of subsets of a Bayesian network into a junction tree, in other words, into a set of cliques that has the running intersection property. The obtained set of cliques and separators coincide with the junction trees obtained by the moralization and triangulation process, but it has the advantage of adapting to any computational task by adding links to the Bayesian network graph.

Inference of time-delayed gene regulatory networks based on dynamic Bayesian network hybrid learning method.

Gene regulatory networks (GRNs) research reveals complex life phenomena from the perspective of gene interaction, which is an important research field in systems biology. Traditional Bayesian networks have a high computational complexity, and the network structure scoring model has a single feature. Information-based approaches cannot identify the direction of regulation. In order to make up for the shortcomings of the above methods, this paper presents a novel hybrid learning method (DBNCS) based on dynamic Bayesian network (DBN) to construct the multiple time-delayed GRNs for the first time, combining the comprehensive score (CS) with the DBN model. DBNCS algorithm first uses CMI2NI (conditional mutual inclusive information-based network inference) algorithm for network structure profiles learning, namely the construction of search space. Then the redundant regulations are removed by using the recursive optimization algorithm (RO), thereby reduce the false positive rate. Secondly, the network structure profiles are decomposed into a set of cliques without loss, which can significantly reduce the computational complexity. Finally, DBN model is used to identify the direction of gene regulation within the cliques and search for the optimal network structure. The performance of DBNCS algorithm is evaluated by the benchmark GRN datasets from DREAM challenge as well as the SOS DNA repair network in Escherichia coli, and compared with other state-of-the-art methods. The experimental results show the rationality of the algorithm design and the outstanding performance of the GRNs.

Convergence Results on Pulse Coupled Oscillator Protocols in Locally Connected Networks

This work provides new insights on the convergence of a locally connected network of pulse coupled oscillator (PCOs) (i.e., a bio-inspired model for communication networks) to synchronous and desynchronous states, and their implication in terms of the decentralized synchronization and scheduling in communication networks. Bio-inspired techniques have been advocated by many as fault-tolerant and scalable alternatives to produce self-organization in communication networks. The PCO dynamics in particular have been the source of inspiration for many network synchronization and scheduling protocols. However, their convergence properties, especially in locally connected networks, have not been fully understood, prohibiting the migration into mainstream standards. This work provides further results on the convergence of PCOs in locally connected networks and the achievable convergence accuracy under propagation delays. For synchronization, almost sure convergence is proved for $3$ nodes and accuracy results are obtained for general locally connected networks whereas, for scheduling (or desynchronization), results are derived for locally connected networks with mild conditions on the overlapping set of maximal cliques. These issues have not been fully addressed before in the literature.

Determination of the prime bound of a graph

Given a graph $G$, a subset $M$ of $V(G)$ is a module of $G$ if for each $v\in V(G)\setminus M$, $v$ is adjacent to all the elements of $M$ or adjacent to none of them. For instance, $V(G)$, $\emptyset$ and $\{v\}$ ($v\in V(G)$) are modules of $G$ called trivial. Given a graph $G$, $\omega_M(G)$ (respectively $\alpha_M(G)$) denotes the largest integer $m$ such that there is a module $M$ of $G$ which is a clique (respectively a stable) set in $G$ with $|M|=m$. A graph $G$ is prime if $|V(G)|\geq 4$ and if all its modules are trivial. The prime bound of $G$ is the smallest integer $p(G)$ such that there is a prime graph $H$ with $V(H)\supseteq V(G)$, $H[V(G)]=G$ and $|V(H)\setminus V(G)|=p(G)$. We establish the following. For every graph $G$ such that $\max(\alpha_M(G),\omega_M(G))\geq 2$ and $\log_2(\max(\alpha_M(G),\omega_M(G)))$ is not an integer, $p(G)=\lceil\log_2(\max(\alpha_M(G),\omega_M(G)))\rceil$. Then, we prove that for every graph $G$ such that $\max(\alpha_M(G),\omega_M(G))=2^k$ where $k\geq 1$, $p(G)=k$ or $k+1$. Moreover $p(G)=k+1$ if and only if $G$ or its complement admits exactly $2^k$ isolated vertices. Lastly, we show that $p(G)=1$ for every non prime graph $G$ such that $|V(G)|\geq 4$ and $\alpha_M(G)=\omega_M(G)=1$.

Distributed Maximal Clique Computation and Management

Maximal cliques are elementary substructures in a graph and instrumental in graph analysis such as the structural analysis of many complex networks, graph clustering and community detection, network hierarchy detection, emerging pattern mining, vertex importance measures, etc. However, the number of maximal cliques is also notoriously large even for many small real world graphs. This size problem gives rise to challenges in both computing and managing the set of maximal cliques. Many algorithms for computing maximal cliques have been proposed in the literature; however, most of them are sequential algorithms that cannot scale due to the high complexity of the problem, while existing parallel algorithms for computing maximal cliques are mostly immature and especially suffer from skewed workload. As for managing the set of maximal cliques, which is essential due to its large size, there is barely any efficient method for querying or updating the set of maximal cliques. In this paper, we first propose a distributed algorithm built on a share-nothing architecture for computing the set of maximal cliques. We effectively address the problem of skewed workload distribution due to high-degree vertices, which also leads to drastically reduced worst-case time complexity for computing maximal cliques in common real-world graphs. Then, we propose a set of fundamental query operations and efficient algorithms to process the queries, to aid more efficient and effective analysis of the set of maximal cliques. Finally, we also devise algorithms to support efficient update maintenance of the set of maximal cliques when the underlying graph is updated. We verify the efficiency of our algorithms for computing, querying, and updating the set of maximal cliques with a range of real-world graphs from different application domains.

A Maximal Clique Based Multiobjective Evolutionary Algorithm for Overlapping Community Detection

Detecting community structure has become one important technique for studying complex networks. Although many community detection algorithms have been proposed, most of them focus on separated communities, where each node can belong to only one community. However, in many real-world networks, communities are often overlapped with each other. Developing overlapping community detection algorithms thus becomes necessary. Along this avenue, this paper proposes a maximal clique based multiobjective evolutionary algorithm (MOEA) for overlapping community detection. In this algorithm, a new representation scheme based on the introduced maximal-clique graph is presented. Since the maximal-clique graph is defined by using a set of maximal cliques of original graph as nodes and two maximal cliques are allowed to share the same nodes of the original graph, overlap is an intrinsic property of the maximal-clique graph. Attributing to this property, the new representation scheme allows MOEAs to handle the overlapping community detection problem in a way similar to that of the separated community detection, such that the optimization problems are simplified. As a result, the proposed algorithm could detect overlapping community structure with higher partition accuracy and lower computational cost when compared with the existing ones. The experiments on both synthetic and real-world networks validate the effectiveness and efficiency of the proposed algorithm.

Community detection with the weighted parsimony criterion

Community detection in networks has been studied extensively in the last decade. Many criteria, expressing the quality of the partitions obtained, as well as a few exact algorithms and a large number of heuristics have been proposed. The parsimony criterion consists in minimizing the number of edges added or removed from the given network in order to transform it into a set of disjoint cliques. Recently Zhang, Qiu and Zhang have proposed a weighted parsimony model in which a weight coefficient is introduced to balance the numbers of inserted and deleted edges. These authors propose rules to select a good value of the coefficient, use simulated annealing to find optimal or near-optimal solutions and solve a series of real and artificial instances. In the present paper, an algorithm is proposed for solving exactly the weighted parsimony problem for all values of the parameter. This algorithm is based on iteratively solving the problem for a set of given values of the parameter using a row generation algorithm. This procedure is combined with a search procedure to find all lowest breakpoints of the value curve (i.e., the weighted sum of inserted and deleted edges). Computational results on a series of artificial and real world networks from the literature are reported. It appears that several partitions for the same network may be informative and that the set of solutions usually contains at least one intuitively appealing partition.

A Localized Implementation of the Iterative Proportional Scaling Procedure for Gaussian Graphical Models

In this article, we propose localized implementations of the iterative proportional scaling (IPS) procedure by the strategy of partitioning cliques for computing maximum likelihood estimations in large Gaussian graphical models. We first divide the set of cliques into several nonoverlapping and nonempty blocks, and then adjust clique marginals in each block locally. Thus, high-order matrix operations can be avoided and the IPS procedure is accelerated. We modify the Swendsen–Wang Algorithm and apply the simulated annealing algorithm to find an approximation to the optimal partition which leads to the least complexity. This strategy of partitioning cliques can also speed up the existing IIPS and IHT procedures. Numerical experiments are presented to demonstrate the competitive performance of our new implementations and strategies.

Clique-perfectness and balancedness of some graph classes

A graph is clique-perfect if the maximum size of a clique-independent set (a set of pairwise disjoint maximal cliques) and the minimum size of a clique-transversal set (a set of vertices meeting every maximal clique) coincide for each induced subgraph. A graph is balanced if its clique-matrix contains no square submatrix of odd size with exactly two ones per row and column. In this work, we give linear-time recognition algorithms and minimal forbidden induced subgraph characterizations of clique-perfectness and balancedness of P4-tidy graphs and a linear-time algorithm for computing a maximum clique-independent set and a minimum clique-transversal set for any P4-tidy graph. We also give a minimal forbidden induced subgraph characterization and a linear-time recognition algorithm for balancedness of paw-free graphs. Finally, we show that clique-perfectness of diamond-free graphs can be decided in polynomial time by showing that a diamond-free graph is clique-perfect if and only if it is balanced.

The Algebraic Bayesian Network Tertiary Structure

Algebraic Bayesian network (ABN) tertiary structure is required to both a random minimal join graph and the minimal join graph set. In addition, it is required to find the best or an optimal secondary structure over the given ABN primary structure. The goal of the work is to create a clear definition of the ABN tertiary structure and associated objects on the basis of analysis of existing approaches and study of their properties. All existing approaches to the definition of "clique", "clique set" and "clique graph", and classification of the maximal join graph cliques are overviewed. A unified vocabulary for describing the associated objects satisfying the criteria of non-redundancy and systematization completeness is suggested. Tertiary polystructure is defined as a family of graphs constructed on subsets of the maximum join graph narrowings set whose edges are matched to specific relationships defined in the article. Tertiary structure is defined as a directed graph whose edges lead from the parent vertices to their sons, and whose vertices are the maximum join graph narrowing on weights of the edges and the vertices, as well as on the empty weight (parent graph over extended set of useful cliques).

Intelligent Sales Prediction for Pharmaceutical Distribution Companies: A Data Mining Based Approach

One of the problems of pharmaceutical distribution companies (PDCs) is how to control inventory levels in order to prevent costs of excessive inventory and to prevent losing customers due to drug shortage. Consequently, the purpose of this study is to propose a novel method to forecast sales of PDCs. The presented method is a combination of network analysis tools and time series forecasting methods. Due to lack of enough past sales records of each drug, an explorative network based analysis is conducted to find clique sets and group members and to use comembers’ sales data in their sales prediction. Afterwards, time series sales forecasting models were built with three different approaches including ARIMA methodology, neural network, and an advanced hybrid neural network approach. The offered hybrid method by applying each drug and its comembers past records facilitates capturing both linear and nonlinear patterns of sales accurately. The performance of the proposed method was evaluated by a real dataset provided by one of the leading PDCs in Iran. The results indicated that the proposed method is able to cope with low number of past records while it forecasts medicines sales accurately.

Probabilistic Joint Image Segmentation and Labeling by Figure-Ground Composition

We propose a layered statistical model for image segmentation and labeling obtained by combining independently extracted, possibly overlapping sets of figure-ground (FG) segmentations. The process of constructing consistent image segmentations, called tilings, is cast as optimization over sets of maximal cliques sampled from a graph connecting all non-overlapping figure-ground segment hypotheses. Potential functions over cliques combine unary, Gestalt-based figure qualities, and pairwise compatibilities among spatially neighboring segments, constrained by T-junctions and the boundary interface statistics of real scenes. Building on the segmentation layer, we further derive a joint image segmentation and labeling model (JSL) which, given a bag of FGs, constructs a joint probability distribution over both the compatible image interpretations (tilings) composed from those segments, and over their labeling into categories. The process of drawing samples from the joint distribution can be interpreted as first sampling tilings, followed by sampling labelings conditioned on the choice of a particular tiling. We learn the segmentation and labeling parameters jointly, based on maximum likelihood with a novel estimation procedure we refer to as incremental saddle-point approximation. The partition function over tilings and labelings is increasingly more accurately approximated by including incorrect configurations that are rated as probable by candidate models during learning. State of the art results are reported on the Berkeley, Stanford and Pascal VOC datasets, where an improvement of 28 % was achieved for the segmentation task only (tiling), and an accuracy of 47.8 % was obtained on the test set of VOC12 for semantic labeling (JSL).

The cluster deletion problem for cographs

The min-edge clique partition problem asks to find a partition of the vertices of a graph into a set of cliques with the fewest edges between cliques. This is a known NP-complete problem and has been studied extensively in the scope of fixed-parameter tractability (FPT) where it is commonly known as the Cluster Deletion problem. Many of the recently-developed FPT algorithms rely on being able to solve Cluster Deletion in polynomial time on restricted graph structures.We prove new structural properties of cographs which characterize how a largest clique interacts with the rest of the graph. These results imply a remarkably simple polynomial time algorithm for Cluster Deletion on cographs. In contrast, we observe that Cluster Deletion remains NP-hard on a hereditary graph class which is slightly larger than cographs.

Finding maximal homogeneous clique sets

Many datasets can be encoded as graphs with sets of labels associated with the vertices. We consider this kind of graphs and we propose to look for patterns called maximal homogeneous clique sets, where such a pattern is a subgraph that is structured in several large cliques and where all vertices share enough labels. We present an algorithm based on graph enumeration to compute all patterns satisfying user-defined constraints on the number of separated cliques, on the size of these cliques, and on the number of labels shared by all the vertices. Our approach is tested on real datasets based on a social network of scientific collaborations and on a biological network of protein---protein interactions. The experiments show that the patterns are useful to exhibit subgraphs organized in several core modules of interactions. Performances are reported on real data and also on synthetic ones, showing that the approach can be applied on different kinds of large datasets.

Finding maximal cliques in massive networks

Maximal clique enumeration is a fundamental problem in graph theory and has important applications in many areas such as social network analysis and bioinformatics. The problem is extensively studied; however, the best existing algorithms require memory space linear in the size of the input graph. This has become a serious concern in view of the massive volume of today's fast-growing networks. We propose a general framework for designing external-memory algorithms for maximal clique enumeration in large graphs. The general framework enables maximal clique enumeration to be processed recursively in small subgraphs of the input graph, thus allowing in-memory computation of maximal cliques without the costly random disk access. We prove that the set of cliques obtained by the recursive local computation is both correct (i.e., globally maximal) and complete. The subgraph to be processed each time is defined based on a set of base vertices that can be flexibly chosen to achieve different purposes. We discuss the selection of the base vertices to fully utilize the available memory in order to minimize I/O cost in static graphs, and for update maintenance in dynamic graphs. We also apply our framework to design an external-memory algorithm for maximum clique computation in a large graph.

Enumeration of the perfect sequences of a chordal graph

A graph is chordal if and only if it has no chordless cycle of length more than three. The set of maximal cliques in a chordal graph admits special tree structures called clique trees. A perfect sequence is a sequence of maximal cliques obtained by using the reverse order of repeatedly removing the leaves of a clique tree. This paper addresses the problem of enumerating all the perfect sequences. Although this problem has statistical applications, no efficient algorithm has been proposed. There are two difficulties with developing this type of algorithm. First, a chordal graph does not generally have a unique clique tree. Second, a perfect sequence can normally be generated by two or more distinct clique trees. Thus it is hard using a straightforward algorithm to generate perfect sequences from each possible clique tree. In this paper, we propose a method to enumerate perfect sequences without constructing clique trees. As a result, we have developed the first polynomial delay algorithm for dealing with this problem. In particular, the time complexity of the algorithm on average is O ( 1 ) for each perfect sequence.

Understanding the scalability of Bayesian network inference using clique tree growth curves

One of the main approaches to performing computation in Bayesian networks (BNs) is clique tree clustering and propagation. The clique tree approach consists of propagation in a clique tree compiled from a BN, and while it was introduced in the 1980s, there is still a lack of understanding of how clique tree computation time depends on variations in BN size and structure. In this article, we improve this understanding by developing an approach to characterizing clique tree growth as a function of parameters that can be computed in polynomial time from BNs, specifically: (i) the ratio of the number of a BN's non-root nodes to the number of root nodes, and (ii) the expected number of moral edges in their moral graphs. Analytically, we partition the set of cliques in a clique tree into different sets, and introduce a growth curve for the total size of each set. For the special case of bipartite BNs, there are two sets and two growth curves, a mixed clique growth curve and a root clique growth curve. In experiments, where random bipartite BNs generated using the BPART algorithm are studied, we systematically increase the out-degree of the root nodes in bipartite Bayesian networks, by increasing the number of leaf nodes. Surprisingly, root clique growth is well-approximated by Gompertz growth curves, an S-shaped family of curves that has previously been used to describe growth processes in biology, medicine, and neuroscience. We believe that this research improves the understanding of the scaling behavior of clique tree clustering for a certain class of Bayesian networks; presents an aid for trade-off studies of clique tree clustering using growth curves; and ultimately provides a foundation for benchmarking and developing improved BN inference and machine learning algorithms.

On distance-regular graphs with smallest eigenvalue at least − m

A non-complete geometric distance-regular graph is the point graph of a partial linear space in which the set of lines is a set of Delsarte cliques. In this paper, we prove that for a fixed integer m ⩾ 2 , there are only finitely many non-geometric distance-regular graphs with smallest eigenvalue at least − m, diameter at least three and intersection number c 2 ⩾ 2 .