Inter-cluster Edges Research Articles

The Computational Complexity of the ChordLink Model

In order to visualize well-clustered graphs with many intra-cluster but few inter-cluster edges, hybrid approaches have been proposed. For example, ChordLink draws the clusters as chord diagrams and embeds these into a node-link diagram that represents the overall structure of the clustered graph. The ChordLink approach consists of four steps; node replication, node permutation, node merging, and chord insertion. In this paper, we focus on the optimization problems defined by two of these steps. We show that the decision version of the problem defined by node permutation is NP-complete and present an efficient algorithm for a special case. For chord insertion, we show that it is NP-complete to decide whether a crossing-free placement of the chords exists. Moreover, it is APX-hard to minimize the number of crossings among the chords. Our results answer an open question posed by Angori, Didimo, Montecchiani, Pagliuca, and Tappini, who introduced ChordLink [TVCG 2021].

Parameterized complexity of graph planarity with restricted cyclic orders

We study the complexity of testing whether a biconnected graph G=(V,E) is planar with the constraint that some cyclic orders of the edges incident to its vertices are allowed while some others are forbidden. The allowed cyclic orders are described by associating every vertex v of G with a set D(v) of FPQ-trees. Let tw be the treewidth of G and let Dmax be the maximum number of FPQ-trees per vertex. We show that the problem is FPT when parameterized by tw+Dmax, paraNP-hard when parameterized by Dmax, and W[1]-hard when parameterized by tw. We also consider NodeTrix planar representations of clustered graphs, where clusters are adjacency matrices and inter-cluster edges are non-intersecting simple curves. We prove that NodeTrix planarity with fixed sides is FPT when parameterized by the size of clusters plus the treewidth of the graph obtained by collapsing clusters to single vertices, provided that this graph is biconnected.

Hierarchical nearest neighbor descent, in-tree, and clustering

Recently, we have proposed a physically-inspired graph-theoretical method, called the Nearest Descent (ND), which is capable of organizing a dataset into an in-tree graph structure. Due to some beautiful and effective features, the constructed in-tree proves well-suited for data clustering. Although there exist some undesired edges (i.e., the inter-cluster edges) in this in-tree, those edges are usually very distinguishable, in sharp contrast to the cases in the famous Minimal Spanning Tree (MST). Here, we propose another graph-theoretical method, called the Hierarchical Nearest Neighbor Descent (HNND). Like ND, HNND also organizes a dataset into an in-tree, but in a more efficient way. Consequently, HNND-based clustering (HNND-C) is more efficient than ND-based clustering (ND-C) as well. This is well proved by the experimental results on five high-dimensional and large-size mass cytometry datasets. The experimental results also show that HNND-C achieves overall better performance than some state-of-the-art clustering methods.

Nearest Descent, In-Tree, and Clustering

Clustering aims at discovering the natural groupings in a dataset, prevalent in many disciplines that involve multivariate data analysis. In this paper, we propose a physically inspired graph-theoretical clustering method, which first makes the data points organized into an attractive graph, called In-Tree, via a physically inspired rule, called Nearest Descent (ND). The rule of ND works to select the nearest node in the descending direction of potential as the parent node of each node, which is fundamentally different from the classical Gradient Descent. The constructed In-Tree proves a very good candidate for clustering due to its particular features and properties. In the In-Tree, the original clustering problem is reduced to a problem of removing the inter-cluster edges from this graph. Pleasingly, those inter-cluster edges are usually so distinguishable that they can be easily determined by different automatic edge-cutting methods. We also propose a visualized strategy to validate the effectiveness of the automatic edge-cutting methods. The experimental results reveal that the proposed method is superior to the related clustering methods. The results also reveal the characteristics of different automatic cutting methods and the meaningfulness of the visualized strategy in increasing the reliability of the clustering results in practice.

Near-optimal Distributed Triangle Enumeration via Expander Decompositions

We present improved distributed algorithms for variants of the triangle finding problem in the model. We show that triangle detection, counting, and enumeration can be solved in rounds using expander decompositions . This matches the triangle enumeration lower bound of by Izumi and Le Gall [PODC’17] and Pandurangan, Robinson, and Scquizzato [SPAA’18], which holds even in the model. The previous upper bounds for triangle detection and enumeration in were and , respectively, due to Izumi and Le Gall [PODC’17]. An -expander decomposition of a graph is a clustering of the vertices such that (i) each cluster induces a subgraph with conductance at least and (ii) the number of inter-cluster edges is at most . We show that an -expander decomposition with can be constructed in rounds for any and positive integer . For example, a -expander decomposition only requires rounds to compute, which is optimal up to subpolynomial factors, and a -expander decomposition can be computed in rounds, for any arbitrarily small constant . Our triangle finding algorithms are based on the following generic framework using expander decompositions, which is of independent interest. We first construct an expander decomposition. For each cluster, we simulate algorithms with small overhead by applying the expander routing algorithm due to Ghaffari, Kuhn, and Su [PODC’17] Finally, we deal with inter-cluster edges using recursive calls.

Inter-Cluster Transmission Control Using Graph Modal Barriers

In this paper we consider the problem of transmission across a graph and develop a method for effectively controlling/restricting it with limited resources. Transmission can represent information transfer across a social network, spread of a malicious virus across a computer network, or spread of an infectious disease across communities. The key insight is to appropriately reduce the capacity for transmission on inter-cluster edges of the graph. To that end, the key contribution of the paper is to develop algorithms for computing a real, positive-valued distribution over the edges that gives a measure of the role each edge plays in being an inter-cluster edge. We refer to this distribution as the “ resistance ” and its computation is based on the eigenvectors of the graph Laplacian. Selectively reducing the weights (implying reduced transmission rate) on the critical edges based on this computed distribution helps establish barriers between the clusters ( modal barriers ) and thus limit the transmission from one cluster to another. Our proposed method for edge weight reduction for inter-cluster transmission control depends only on the graph topology and not on the exact model of the transmission dynamics. Unlike other work on graph partitioning and clustering, we completely circumvent the associated computational complexities by assigning real values to edges instead of performing discrete graph cuts. This allows us to provide strong theoretical results on our proposed methods. We also develop approximations that allow low complexity distributed computation of the barrier weights using only neighborhood communication.

An MBO scheme for clustering and semi-supervised clustering of signed networks

We introduce a principled method for the signed clustering problem, where the goal is to partition a weighted undirected graph whose edge weights take both positive and negative values, such that edges within the same cluster are mostly positive, while edges spanning across clusters are mostly negative. Our method relies on a graph-based diffuse interface model formulation utilizing the Ginzburg-Landau functional, based on an adaptation of the classic numerical Merriman-Bence-Osher (MBO) scheme for minimizing such graph-based functionals. The proposed object ive function aims to minimize the total weight of inter-cluster positively-weighted edges, while maximizing the total weight of the inter-cluster negatively-weighted edges. Our method scales to large sparse networks, and can be easily adjusted to incorporate labelled data information, as is often the case in the context of semisupervised learning. We tested our method on a number of both synthetic stochastic block models and real-world data sets (including financial correlation matrices), and obtained promising results that compare favourably against a number of state-of-the-art approaches from the recent literature.

Multi-label symbolic value partitioning through random walks

Feature selection and symbolic value partitioning are effective knowledge reduction techniques in the field of data mining. A large body of feature selection methods has been proposed for multi-label data. By contrast, symbolic value partitioning for such data has not been studied. In this paper, we propose the multi-label symbolic value partitioning through random walks algorithm with two stages. In the first stage, an undirected weighted graph is constructed for each attribute. Each node corresponds to an attribute value and the weight of each edge corresponds to the similarity between two nodes. Similarity is defined based on the attribute value distribution for each label. In the second stage, a random walk algorithm is used to cluster attribute values. The average weight serves as the separation operator to sharpen the inter-cluster edges. We tested the new algorithm and seven popular feature selection algorithms on 13 datasets. The experimental results demonstrate the effectiveness of the proposed algorithm in reducing the data size and improving classification accuracy.

An ensemble based on a bi-objective evolutionary spectral algorithm for graph clustering

Graph clustering is a challenging pattern recognition problem whose goal is to identify vertex partitions with high intra-group connectivity. This paper investigates a bi-objective problem that maximizes the number of intra-cluster edges of a graph and minimizes the expected number of inter-cluster edges in a random graph with the same degree sequence as the original one. The difference between the two investigated objectives is the definition of the well-known measure of graph clustering quality: the modularity. We introduce a spectral decomposition hybridized with an evolutionary heuristic, called MOSpecG, to approach this bi-objective problem and an ensemble strategy to consolidate the solutions found by MOSpecG into a final robust partition. The results of computational experiments with real and artificial LFR networks demonstrated a significant improvement in the results and performance of the introduced method in regard to another bi-objective algorithm found in the literature. The crossover operator based on the geometric interpretation of the modularity maximization problem to match the communities of a pair of individuals was of utmost importance for the good performance of MOSpecG. Hybridizing spectral graph theory and intelligent systems allowed us to define significantly high-quality community structures.

The complexity of cluster-connectivity of wireless sensor networks

Wireless sensor networks consist of sensor devices with limited computational capabilities and memory operating in bounded energy resources; hence, network optimisation and algorithmic development in minimising the total energy or power while maintaining the connectivity of the underlying network are crucial for their design and maintenance. We consider a generalised system model of wireless sensor networks whose node set is decomposed into multiple clusters, and show that the decision and the associated minimisation problems of the connectivity of clustered wireless sensor networks appear to be computationally intractable - completeness and hardness, respectively, for the non-deterministic polynomial-time complexity class. An approximation algorithm is devised to minimise the number of end nodes of inter-cluster edges within a factor of 2 of the optimum for the cluster-connectivity.

Computing NodeTrix Representations of Clustered Graphs

NodeTrix representations are a popular way to visualize clustered graphs; they represent clusters as adjacency matrices and inter-cluster edges as curves connecting the matrix boundaries. We study the complexity of constructing NodeTrix representations focusing on planarity testing problems, and we show several $\mathbb{NP}$-completeness results and some polynomial-time algorithms. Building on such algorithms we develop a JavaScript library for NodeTrix representations aimed at reducing the crossings between edges incident to the same matrix.

The Complexity of Cluster-Connectivity of Wireless Sensor Networks

Wireless sensor networks consist of sensor devices with limited computational capabilities and memory operating in bounded energy resources; hence, network optimisation and algorithmic development in minimising the total energy or power while maintaining the connectivity of the underlying network are crucial for their design and maintenance. We consider a generalised system model of wireless sensor networks whose node set is decomposed into multiple clusters, and show that the decision and the associated minimisation problems of the connectivity of clustered wireless sensor networks appear to be computationally intractable - completeness and hardness, respectively, for the non-deterministic polynomial-time complexity class. An approximation algorithm is devised to minimise the number of end nodes of inter-cluster edges within a factor of 2 of the optimum for the cluster-connectivity.

Semi-supervised learning through adaptive Laplacian graph trimming

Graph-based semi-supervised learning (GSSL) attracts considerable attention in recent years. The performance of a general GSSL method relies on the quality of Laplacian weighted graph (LWR) composed of the similarity imposed on input examples. A key for constructing an effective LWR is on the proper selection of the neighborhood size K or ε on the construction of KNN graph or ε-neighbor graph on training samples, which constitutes the fundamental elements in LWR. Specifically, too large K or ε will result in “shortcut” phenomenon while too small ones cannot guarantee to represent a complete manifold structure underlying data. To this issue, this study attempts to propose a method, called adaptive Laplacian graph trimming (ALGT), to make an automatic tuning to cut improper inter-cluster shortcut edges while enhance the connection between intra-cluster samples, so as to adaptively fit a proper LWR from data. The superiority of the proposed method is substantiated by experimental results implemented on synthetic and UCI data sets.

Global and Local Information in Clustering Labeled Block Models

The stochastic block model is a classical cluster-exhibiting random graph model that has been widely studied in statistics, physics, and computer science. In its simplest form, the model is a random graph with two equal-sized clusters, with intracluster edge probability $p$ , and intercluster edge probability $q$ . We focus on the sparse case, i.e., $p, q = O(1/n)$ , which is practically more relevant and also mathematically more challenging. A conjecture of Decelle, Krzakala, Moore, and Zdeborova, based on ideas from statistical physics, predicted a specific threshold for clustering. The negative direction of the conjecture was proved by Mossel, Neeman, and Sly (2012), and more recently, the positive direction was independently proved by Massoulie and Mossel, Neeman, and Sly. In many real network clustering problems, nodes contain information as well. We study the interplay between node and network information in clustering by studying a labeled block model, where in addition to the edge information, the true cluster labels of a small fraction of the nodes are revealed. In the case of two clusters, we show that below the threshold, a small amount of node information does not affect recovery. On the other hand, we show that for any small amount of information, efficient local clustering is achievable as long as the number of clusters is sufficiently large (as a function of the amount of revealed information).

Erratum—Exact Algorithms for the Clustered Vehicle Routing Problem

The clustered vehicle routing problem (CluVRP) is a generalization of the capacitated vehicle routing problem (CVRP) were customers are partitioned into disjoint clusters. As in the CVRP, all the customers must be visited exactly once, but a vehicle visiting one customer in a cluster must visit all the remaining customers therein before leaving it. Hence, when the first and last customer to be visited in a cluster C are p and q, finding the optimal sequence to visit the remaining customers in the cluster is equal to finding the minimum cost Hamiltonian path with p and q as endpoints. Battarra et al. (2014) presented a new compact integer programming formulation for the CluVRP. This formulation exploits the special substructure of the clusters such that only inter-cluster connections have to be determined. The authors show that this formulation provides a stronger linear relaxation than the CluVRP formulation based on the tradional CVRP two-index formulation. In the sequence of propositions that show the equivalence of the new formulation with the traditional CluVRP formulation, an erroneous notation was introduced in Battarra et al. (2014). In particular, this appears in the proof of Lemma 2, which is rewritten here in correct form. We emphasize that the incorrect notation in the proof does not affect the overall validity of Lemma 2, hence has no effect on the correctness of the paper and on the results reported therein. Let „4S5 denote the set of edges connecting the vertices in a vertex set S ⊆ V to those outside the set, and the set of edges inside a vertex set S by E4S5. The decision variables x ij and x ∗∗ ij are equal to the number of times a vehicle traverses edge 4i1 j5 for a solution of the traditional and the new formulation, respectively. The values of x ij are determined through the following transformation from x ij . For each inter-cluster edge 4i1 j5 ∈ Ē, set x ij = x ∗∗ ij . Initialize x ij = 0 for all intra-cluster edges 4i1 j5 ∈ E. For all 4p1 q5 ∈ E and for all 4i1 j5 ∈ P4p1q5, increment x ij by x ∗∗ pq . For the details of the CluVRP formulations, we refer the reader to Battarra et al. (2014). Lemma 2. Given a set of customer vertices S ⊂ V \809, with C being the minimal set of clusters covering S, i.e., ∀i ∈ S, ∃C ∈ C: i ∈C, the inequality ∑

Density-based graph model summarization: Attaining better performance and efficiency

Several algorithms based on PageRank algorithm have been proposed to rank the document sentences in the multidocument summarization field and LexRank and T-LexRank algorithms are well known examples. In literature different concepts such as weighted inter-cluster edge, cluster-sensitive graph model and document-sensitive graph model have been proposed to improve LexRank and T-LexRank algorithms (e.g. DsR-G, DsR-Q) for multi-document summarization. In this paper, a density-based graph model for multi-document summarization is proposed by adding the concept of density to LexRank and T-LexRank algorithms. The resulting generic multi-document summarization systems, DensGS and DensGSD were evaluated on DUC 2004 while the query-based variants, DensQS, DensQSD were evaluated on DUC 2006, DUC 2007 and TAC 2010 task A. ROUGE measure was used in the evaluation. Experimental results show that density concept improves LexRank and T-LexRank algorithms and outperforms previous graph-based models (DsR-G and DsR-Q) in generic and query-based multi-document summarization tasks. Furthermore, the comparison of the number of iterations indicates that the density-based algorithm is faster than the other algorithms based on PageRank.

Robustness of cluster synchronous patterns in small-world networks with inter-cluster co-competition balance.

All edges in the classical Watts and Strogatz's small-world network model are unweighted and cooperative (positive). By introducing competitive (negative) inter-cluster edges and assigning edge weights to mimic more realistic networks, this paper develops a modified model which possesses co-competitive weighted couplings and cluster structures while maintaining the common small-world network properties of small average shortest path lengths and large clustering coefficients. Based on theoretical analysis, it is proved that the new model with inter-cluster co-competition balance has an important dynamical property of robust cluster synchronous pattern formation. More precisely, clusters will neither merge nor split regardless of adding or deleting nodes and edges, under the condition of inter-cluster co-competition balance. Numerical simulations demonstrate the robustness of the model against the increase of the coupling strength and several topological variations.

CiSE: A Circular Spring Embedder Layout Algorithm

We present a new algorithm for automatic layout of clustered graphs using a circular style. The algorithm tries to determine optimal location and orientation of individual clusters intrinsically within a modified spring embedder. Heuristics such as reversal of the order of nodes in a cluster and swap of neighboring node pairs in the same cluster are employed intermittently to further relax the spring embedder system, resulting in reduced inter-cluster edge crossings. Unlike other algorithms generating circular drawings, our algorithm does not require the quotient graph to be acyclic, nor does it sacrifice the edge crossing number of individual clusters to improve respective positioning of the clusters. Moreover, it reduces the total area required by a cluster by using the space inside the associated circle. Experimental results show that the execution time and quality of the produced drawings with respect to commonly accepted layout criteria are quite satisfactory, surpassing previous algorithms. The algorithm has also been successfully implemented and made publicly available as part of a compound and clustered graph editing and layout tool named CHISIO.

Sublinear Fully Distributed Partition with Applications

We present new efficient deterministic and randomized distributed algorithms for decomposing a graph with n nodes into a disjoint set of connected clusters with radius at most k−1 and having O(n 1+1/k ) intercluster edges. We show how to implement our algorithms in the distributed $\mathcal{CONGEST}$model of computation, i.e., limited message size, which improves the time complexity of previous algorithms (Moran and Snir in Theor. Comput. Sci. 243(1–2):217–241, 2000; Awerbuch in J. ACM 32:804–823, 1985; Peleg in Distributed Computing: A Locality-Sensitive Approach, 2000) from O(n) to O(n 1−1/k ). We apply our algorithms for constructing low stretch graph spanners and network synchronizers in sublinear deterministic time in the $\mathcal{CONGEST}$model.

Isoperimetric properties of weighted graphs related to the Laplacian spectrum and canonical correlations

The relation between isoperimetric properties and Laplacian spectra of weighted graphs is investigated. The vertices are classified into k clusters with „few" inter-cluster edges of „small" weights (area) and „similar" cluster sizes (volumes). For k=2 the Cheeger constant represents the minimum requirement for the area/volume ratio and it is estimated from above by v?1(2-?1), where ?1 is the smallest positive eigenvalue of the weighted Laplacian. For k?2 we define the k-density of a weighted graph that is a generalization of the Cheeger constant and estimated from below by Si=1k-1?i and from above by c2 Si=1k-1 ?i, where 0<?1=…=Sk-1 are the smallest Laplacian eigenvalues and the constant c?1 depends on the metric classification properties of the corresponding eigenvectors. Laplacian spectra are also related to canonical correlations in a probabilistic setup.