Edge Deletion Research Articles

An improved branching algorithm for the proper interval edge deletion problem

An improved branching algorithm for the proper interval edge deletion problem

On the edit distance function of the random graph

AbstractGiven a hereditary property of graphs $\mathcal{H}$ and a $p\in [0,1]$ , the edit distance function $\textrm{ed}_{\mathcal{H}}(p)$ is asymptotically the maximum proportion of edge additions plus edge deletions applied to a graph of edge density p sufficient to ensure that the resulting graph satisfies $\mathcal{H}$ . The edit distance function is directly related to other well-studied quantities such as the speed function for $\mathcal{H}$ and the $\mathcal{H}$ -chromatic number of a random graph.Let $\mathcal{H}$ be the property of forbidding an Erdős–Rényi random graph $F\sim \mathbb{G}(n_0,p_0)$ , and let $\varphi$ represent the golden ratio. In this paper, we show that if $p_0\in [1-1/\varphi,1/\varphi]$ , then a.a.s. as $n_0\to\infty$ , \begin{align*} {\textrm{ed}}_{\mathcal{H}}(p) = (1+o(1))\,\frac{2\log n_0}{n_0} \cdot\min\left\{ \frac{p}{-\log(1-p_0)}, \frac{1-p}{-\log p_0} \right\}. \end{align*} Moreover, this holds for $p\in [1/3,2/3]$ for any $p_0\in (0,1)$ .A primary tool in the proof is the categorization of p-core coloured regularity graphs in the range $p\in[1-1/\varphi,1/\varphi]$ . Such coloured regularity graphs must have the property that the non-grey edges form vertex-disjoint cliques.

Redefining the Graph Edit Distance

Graph edit distance has been used since 1983 to compare objects in machine learning when these objects are represented by attributed graphs instead of vectors. In these cases, the graph edit distance is usually applied to deduce a distance between attributed graphs. This distance is defined as the minimum amount of edit operations (deletion, insertion and substitution of nodes and edges) needed to transform a graph into another. Since now, it has been stated that the distance properties have to be applied [(1) non-negativity (2) symmetry (3) identity and (4) triangle inequality] to the involved edit operations in the process of computing the graph edit distance to make the graph edit distance a metric. In this paper, we show that there is no need to impose the triangle inequality in each edit operation. This is an important finding since in pattern recognition applications, the classification ratio usually maximizes in the edit operation combinations (deletion, insertion and substitution of nodes and edges) that the triangle inequality is not fulfilled.

Polynomial kernels for paw-free edge modification problems

Given a graph G and an integer k, the paw-free completion problem asks whether it is possible to add at most k edges to G to make it paw-free. The paw-free edge deletion problem is defined analogously. Sandeep and Sivadasan (IPEC 2015) asked whether these problems admit polynomial kernels. We answer both questions affirmatively by presenting, respectively, O(k)-vertex and O(k4)-vertex kernels for them. This is part of an ongoing program that aims at understanding the compressibility of H-free edge modification problems for general H.

AI-based network topology optimization system

Existing network topology planning does not fully consider the increasing network traffic and problem of uneven link capacity utilization, resulting in lower resource utilization and unnecessary investments in network construction. The AI-based network topology optimization system introduced in this paper builds a Long Short-Term Memory (LSTM) model for time series traffic forecasting, which uses NetworkX, a Python library, for graph analysis, dynamically optimizes the network topology by edge deletion or addition based on traffic over nodes, and ensures network load balancing when node traffic increases, mainly introducing the LSTM forecasting model building process, parameter optimization strategy, and network topology optimization in some detail. As it effectively enhances resource utilization, this system is vital to the optimization of complex network topology. The end of this paper looks forward to the future development of artificial intelligence, and suggests the possibility of how to cooperate with operator networks and how to establish cross-border ecological development.

Succinct dynamic de Bruijn graphs.

The de Bruijn graph is one of the fundamental data structures for analysis of high throughput sequencing data. In order to be applicable to population-scale studies, it is essential to build and store the graph in a space- and time-efficient manner. In addition, due to the ever-changing nature of population studies, it has become essential to update the graph after construction, e.g. add and remove nodes and edges. Although there has been substantial effort on making the construction and storage of the graph efficient, there is a limited amount of work in building the graph in an efficient and mutable manner. Hence, most space efficient data structures require complete reconstruction of the graph in order to add or remove edges or nodes. In this article, we present DynamicBOSS, a succinct representation of the de Bruijn graph that allows for an unlimited number of additions and deletions of nodes and edges. We compare our method with other competing methods and demonstrate that DynamicBOSS is the only method that supports both addition and deletion and is applicable to very large samples (e.g. greater than 15 billion k-mers). Competing dynamic methods, e.g. FDBG cannot be constructed on large scale datasets, or cannot support both addition and deletion, e.g. BiFrost. DynamicBOSS is publicly available at https://github.com/baharpan/dynboss. Supplementary data are available at Bioinformatics online.

Sublinear update time randomized algorithms for dynamic graph regression

A well-known problem in data science and machine learning is linear regression, which is recently extended to dynamic graphs. Existing exact algorithms for updating the solution of dynamic graph regression require at least a linear time (in terms of n: the size of the graph). However, this time complexity might be intractable in practice.In the current paper, we utilize subsampled randomized Hadamard transformand CountSketchto propose the first sublinear update time randomized algorithms for regression of general dynamic graphs. Suppose that we are given a n×d matrix embedding M of the graph, where d≪n and M has certain properties. Let r be the number of samples required by subsampled randomized Hadamard transform for a 1±ϵ approximation, which is a sublinear of n. Our first algorithm supports edge insertion and edge deletion and updates the approximate solution in O(rd) time. Our second algorithm is based on CountSketchand supports edge insertion, edge deletion, node insertion and node deletion. It updates the approximate solution in O(qd) time, where q=O(d2ϵ2log6(d/ϵ)).

Complexity and algorithms for constant diameter augmentation problems

We study the following problem: for given integers d,k and graph G, can we obtain a graph with diameter d via at most k edge deletions? We determine the computational complexity of this and related problems for different values of d.

I/O efficient k-truss community search in massive graphs

Community detection that discovers all densely connected communities in a network has been studied a lot. In this paper, we study online community search for query-dependent communities, which is a different but practically useful task. Given a query vertex in a graph, the problem is to find meaningful communities that the vertex belongs to in an online manner. We propose a community model based on the k-truss concept, which brings nice structural and computational properties. We design a compact and elegant index structure which supports the efficient search of k-truss communities with a linear cost with respect to the community size. We also investigate the k-truss community search problem in a dynamic graph setting with frequent insertions and deletions of graph vertices and edges. In addition, to support k-truss community search over massive graphs which cannot entirely fit in main memory, we propose I/O-efficient algorithms for query processing under the semi-external model. Extensive experiments on massive real-world networks demonstrate the effectiveness of our k-truss community model, the efficiency, and the scalability of our in-memory and semi-external community search algorithms.

Editing to Prime Graphs

Given a graph G, a subset M of V(G) is a module of G if for each $$v\in V(G)\setminus M$$ and $$x,y\in M$$ , $$vx \in E(G)$$ if and only if $$vy \in E(G)$$ . The trivial modules of G are $$\varnothing ,\{u\}(u\in V(G))$$ and V(G). Let G be a graph with at least four vertices. The graph G is prime if all its modules are trivial. We are interested in the minimum number $$\delta (G)$$ of edge modifications (edge deletions and/or additions) that are needed to transform G into a prime graph. We prove that $$\delta (G) \le v(G)-1$$ , where $$v(G) = |V(G)|$$ , and that the upper bound is uniquely reached by the complete graphs and their complements. We then characterize the graphs G such that $$\delta (G) = v(G)-2$$ or $$v(G)-3$$ , respectively. Lastly, we look for the $$\delta $$ -critical graphs, i.e. the graphs G for which $$\delta (H) > \delta (G)$$ for every graph H obtained from G by deleting one vertex. We prove that the $$\delta $$ -critical graphs are precisely the half graphs and their complements.

Symmetric continuous subgraph matching with bidirectional dynamic programming

In many real datasets such as social media streams and cyber data sources, graphs change over time through a graph update stream of edge insertions and deletions. Detecting critical patterns in such dynamic graphs plays an important role in various application domains such as fraud detection, cyber security, and recommendation systems for social networks. Given a dynamic data graph and a query graph, the continuous subgraph matching problem is to find all positive matches for each edge insertion and all negative matches for each edge deletion. The state-of-the-art algorithm TurboFlux uses a spanning tree of a query graph for filtering. However, using the spanning tree may have a low pruning power because it does not take into account all edges of the query graph. In this paper, we present a symmetric and much faster algorithm SymBi which maintains an auxiliary data structure based on a directed acyclic graph instead of a spanning tree, which maintains the intermediate results of bidirectional dynamic programming between the query graph and the dynamic graph. Extensive experiments with real and synthetic datasets show that SymBi outperforms the state-of-the-art algorithm by up to three orders of magnitude in terms of the elapsed time.

The change of Seidel energy of tripartite Turán graph due to edge deletion

Let be the Seidel matrix of a graph G. The Seidel energy of G, denoted by , is defined to be the sum of absolute values of all eigenvalues of the Seidel matrix of G. In this paper, we consider the change of Seidel energy of graphs due to edge deletion. It is proved that the Seidel energy of the tripartite Turán graph of order is always increased when an edge is deleted.

Combining Clickstream Analyses and Graph-Modeled Data Clustering for Identifying Common Response Processes

Complex interactive test items are becoming more widely used in assessments. Being computer-administered, assessments using interactive items allow logging time-stamped action sequences. These sequences pose a rich source of information that may facilitate investigating how examinees approach an item and arrive at their given response. There is a rich body of research leveraging action sequence data for investigating examinees’ behavior. However, the associated timing data have been considered mainly on the item-level, if at all. Considering timing data on the action-level in addition to action sequences, however, has vast potential to support a more fine-grained assessment of examinees’ behavior. We provide an approach that jointly considers action sequences and action-level times for identifying common response processes. In doing so, we integrate tools from clickstream analyses and graph-modeled data clustering with psychometrics. In our approach, we (a) provide similarity measures that are based on both actions and the associated action-level timing data and (b) subsequently employ cluster edge deletion for identifying homogeneous, interpretable, well-separated groups of action patterns, each describing a common response process. Guidelines on how to apply the approach are provided. The approach and its utility are illustrated on a complex problem-solving item from PIAAC 2012.

Information Hiding for Medical Privacy Protection based on GHM and Canny Edge Detection

In order to protect the personal and diagnosis information of patient, GHM and Canny edge detection are employed in this algorithm to process the original CT digital image. The original image will be firstly processed GHM for a global analysis . According to different importance degrees, CT images can be distributed into LL2, LH2, HL2 and HH2 by GHM multi-wavelet. After that, the distributed sub-images will be processed by Canny edge detection. For example, the edges of LL2 (with the highest important degree) and HH2 (with the lowest important degree) are detected by strong Canny method. Then many thin edges can be obtained, which are useless for medical diagnosis, privacy and diagnosis information can be hid by deletion and comparison processing of invalid edges. The experimental results indicate that the robustness and capacity of watermarked CT images can be improved after this algorithm.

Quantifying the total effect of edge interventions in discrete multistate networks

Developing efficient computational methods to assess the impact of external interventions on the dynamics of a network model is an important problem in systems biology. This paper focuses on quantifying the global changes that result from the application of an intervention to produce a desired effect, which we define as the total effect of the intervention. The type of mathematical models that we will consider are discrete dynamical systems which include the widely used Boolean networks and their generalizations. The potential interventions can be represented by a set of nodes and edges that can be manipulated to produce a desired effect on the system. We use a class of regulatory rules called nested canalizing functions that frequently appear in published models and were inspired by the concept of canalization in evolutionary biology. In this paper, we provide a polynomial normal form based on the canalizing properties of regulatory functions. Using this polynomial normal form, we give a set of formulas for counting the maximum number of transitions that will change in the state space upon an edge deletion in the wiring diagram. These formulas rely on the canalizing structure of the target function since the number of changed transitions depends on the canalizing layer that includes the input to be deleted. We also present computations on random networks to compare the exact number of changes with the upper bounds provided by our formulas. Finally, we provide statistics on the sharpness of these upper bounds in random networks.

On the Change of Distance Energy of Complete Bipartite Graph due to Edge Deletion

The distance energy of a graph is defined as the sum of absolute values of distance eigenvalues of the graph. The distance energy of a graph plays an important role in many fields. By constructing a new polynomial, we transform a problem on the sum of the absolute values of the roots of a quadratic polynomial into a problem on the largest root of a cubic polynomial. Hence, we give a new and shorter proof on the change of distance energy of a complete bipartite graph due to edge deletion, which was given by Varghese et al.

The partial duplication random graph with edge deletion

The partial duplication random graph with edge deletion

On 2-Clubs in Graph-Based Data Clustering: Theory and Algorithm Engineering

Editing a graph into a disjoint union of clusters is a standard optimization task in graph-based data clustering. Here, complementing classical work where the clusters shall be cliques, we focus on clusters that shall be 2-clubs, that is, subgraphs of diameter at most two. This naturally leads to the two NP-hard problems $\rm{2-C \small{LUB}}$ $\rm{C{\small LUSTER}}$ $\rm{E{\small DITING}}$ (the editing operations are edge insertions and edge deletions) and $\rm{2-C \small{LUB}}$ $\rm{C{\small LUSTER}}$ $\rm{V{\small ERTEX}}$ $\rm{D{\small ELETION}}$ (the editing operations are vertex deletions). Answering an open question from the literature, we show that $\rm{2-C \small{LUB}}$ $\rm{C{\small LUSTER}}$ $\rm{E{\small DITING}}$ is W[2]-hard with respect to the number of edge modifications, thus contrasting the fixed-parameter tractability result for the classical $\rm{C{\small LUSTER}}$ $\rm{E{\small DITING}}$ problem (considering cliques instead of 2-clubs). Then, focusing on $\rm{2-C \small{LUB}}$ $\rm{C{\small LUSTER}}$ $\rm{V{\small ERTEX}}$ $\rm{D{\small ELETION}}$, which is easily seen to be fixed-parameter tractable, we show that under standard complexity-theoretic assumptions it does not have a polynomial-size problem kernel when parameterized by the number of vertex deletions. Nevertheless, we develop several effective data reduction and pruning rules, resulting in a competitive solver, outperforming a standard ILP formulation in most instances of an established biological test data set.

Linear Kernels for Edge Deletion Problems to Immersion-Closed Graph Classes

Suppose ${\mathcal{F}}$ is a finite family of graphs. We consider the following meta-problem, called $\mathcal{F}$-Immersion Deletion: given a graph $G$ and integer $k$, decide whether the deletion...

Kernel for Kt-free Edge Deletion

In the Kt-free Edge Deletion problem the input is a graph G and an integer k, and the goal is to decide whether there is a set of at most k edges of G whose removal results in a graph with no clique of size t. In this paper we give a kernel to this problem with O(kt−1) vertices and edges.