Edge Deletion Research Articles

Deviation estimates for Eulerian edit numbers of random graphs

Consider the random graph G(n,p) obtained by allowing each edge in the complete graph on n vertices to be present with probability p independent of the other edges. In this paper, we study the minimum number of edge edit operations needed to convert G(n,p) into an Eulerian graph. We obtain deviation estimates for three types Eulerian edit numbers based on whether we perform only edge additions or only edge deletions or a combination of both and show that with high probability, roughly n4 operations suffice in all three cases.

Robustness paradox of attack behavior induced by cascading failures: The larger the attack, the more vulnerable the network?

The robustness of complex networks responding to attacks has long been the focus of network science researching. Nonetheless, the precious studies mostly focus on network performance when facing malicious attacks and random failures while rarely pay attention to the influences of scales of attacking. It is wondering if it is an actual fact that the network is more fragile when attacking scale is exacerbated. In this paper, we are committed to exploring the influences related to the very factor of attacking scale from the perspective of cascading failure problem of dynamic network theory. We construct the model with a regular ranking edge deletion method by simulating attacking scale with [Formula: see text] and [Formula: see text] is denoted as attacked edge number. To be specific, we rank the edges according to initial distributed loads and delete edges in the ranked list, and subsequently observe the changes of robustness in the networks, including BA scale-free network, WS small-world network and several real traffic networks. During the process, an unusual counterintuitive phenomenon captures our attention that the network damages caused by attacks do not always grow with the increase of attacked edges number. We specifically demonstrate and analyze this abnormal cascading propagation phenomenon, ascribing this paradox to the dynamics of the load and the connections of the network structure. Our work may offer a new angle on better controlling the spread of cascading failure and remind the importance of effectively protecting networks from underlying dangers.

Controllability and Optimization of Complex Networks Based on Bridges

In a complex network, each edge has different functions on controllability of the whole network. A network may be out of control due to failure or attack of some specific edges. Bridges are a kind of key edges whose removal will disconnect a network and increase connected components. Here, we investigate the effects of removing bridges on controllability of network. Various strategies, including random deletion of edges, deletion based on betweenness centrality, and deletion based on degree of source or target nodes, are used to compare with the effect of removing bridges. It is found that the removing bridges strategy is more efficient on reducing controllability than the other strategies of removing edges for ER networks and scale-free networks. In addition, we also found the controllability robustness under edge attack is related to the average degree of complex networks. Therefore, we propose two optimization strategies based on bridges to improve the controllability robustness of complex networks against attacks. The effectiveness of the proposed strategies is demonstrated by simulation results of some model networks. These results are helpful for people to understand and control spreading processes of epidemic across different paths.

Strong Structural Controllability of Networks under Time-Invariant and Time-Varying Topological Perturbations

This paper investigates the robustness of strong structural controllability for linear time-invariant and linear time-varying directed networks with respect to structural perturbations, including edge deletions and additions. In this direction, we introduce a new construct referred to as a perfect graph associated with a network with a given set of control nodes. The tight upper bounds on the number of edges that can be added to, or removed from a network, while ensuring strong structural controllability, are then derived. Moreover, we obtain a characterization of critical edge-sets, the maximal sets of edges whose any subset can be respectively added to, or removed from a network, while preserving strong structural controllability. In addition, procedures for combining networks to obtain strongly structurally controllable network-of-networks are proposed. Finally, controllability conditions are proposed for networks whose edge weights, as well as their structures, can vary over time.

Effect of edge and vertex addition on Albertson and Bell indices

Topological graph indices have been of great interest in the research of several properties of chemical substances as it is possible to obtain these properties only by using mathematical calculations. The irregularity indices are the ones to determine the degree of irregularity of a graph. Albertson and Bell indices are two of them. Edge and vertex deletion and addition are important and useful methods in calculating several properties of a given graph. In this paper, the effects of adding a new edge or a new vertex to a graph on the Albertson and Bell indices are determined.

Dynamical algorithms for data mining and machine learning over dynamic graphs

AbstractIn many modern applications, the generated data is a dynamic network. These networks are graphs that change over time by a sequence of update operations (node addition, node deletion, edge addition, edge deletion, and edge weight change). In such networks, it is inefficient to compute from scratch the solution of a data mining/machine learning task, after any update operation. Therefore in recent years, several so‐called dynamical algorithms have been proposed that update the solution, instead of computing it from scratch. In this paper, first we formulate this emerging setting and discuss its high‐level algorithmic aspects. Then, we review state of the art dynamical algorithms proposed for several data mining and machine learning tasks, including frequent pattern discovery, betweenness/closeness/PageRank centralities, clustering, classification, and regression.This article is categorized under: Technologies > Structure Discovery and Clustering Technologies > Machine Learning Fundamental Concepts of Data and Knowledge > Big Data Mining

Proof of a conjecture on distance energy change of complete multipartite graph due to edge deletion

Let Kn1,n2,…,nk be a complete k-partite graph with k≥2 and ni≥2 for i=1,2,…,k. The Turán graph T(n,k) is a complete k-partite graph of n vertices with sizes of partitions as equal as possible. The distance energy ED(G) of a graph G is defined as the sum of absolute values of distance eigenvalues of the graph G. Varghese et al. (2018) [11] conjectured thatED(Kn1,n2,…,nk)<ED(Kn1,n2,…,nk−e), where e is any edge of Kn1,n2,…,nk and proved that the above relation holds for k=2. Very recently, Tian et al. (2020) [10] confirmed that the above conjecture holds for T(n,k) with n≡0(modk) and T(n,3). They also mentioned a weaker conjecture as follows:ED(T(n,k))<ED(T(n,k)−e), where e is any edge of T(n,k) and k≥2, n≥2k. In this paper, we confirm that the former conjecture is true for k≥3 and then the latter conjecture follows immediately.

Motif discovery algorithms in static and temporal networks: A survey

AbstractMotifs are the fundamental components of complex systems. The topological structure of networks representing complex systems and the frequency and distribution of motifs in these networks are intertwined. The complexities associated with graph and subgraph isomorphism problems, as the core of frequent subgraph mining, directly impact the performance of motif discovery algorithms. Researchers have adopted different strategies for candidate generation and enumeration and frequency computation to cope with these complexities. Besides, in the past few years, there has been an increasing interest in the analysis and mining of temporal networks. In contrast to their static counterparts, these networks change over time in the form of insertion, deletion or substitution of edges or vertices or their attributes. In this article, we provide a survey of motif discovery algorithms proposed in the literature for mining static and temporal networks and review the corresponding algorithms based on their adopted strategies for candidate generation and frequency computation. As we witness the generation of a large amount of network data in social media platforms, bioinformatics applications and communication and transportation networks and the advance in distributed computing and big data technology, we also conduct a survey on the algorithms proposed to resolve the CPU-bound and I/O bound problems in mining static and temporal networks.

Efficient single-pair all-shortest-path query processing for massive dynamic networks

The single-pair all-shortest-path problem is to find all possible shortest paths, given a single source-destination pair in a graph. Due to the lack of efficient algorithms for single-pair all-shortest-path problem, many applications used diverse types of modifications to the existing shortest-path algorithms such as Dijkstra’s algorithm. Such approaches can facilitate the analysis of medium-sized static networks, but the heavy computational cost impedes their use for massive and dynamic real-world networks. In this paper, we present a novel single-pair all-shortest-path algorithm, which performs well on massive networks as well as dynamic networks. The efficiency of our algorithm stems from novel 2-hop label-based query processing on large-size networks. For dynamic networks, we also demonstrate how to incrementally maintain all shortest paths in 2-hop labels, which allows our algorithm to handle the topological changes of dynamic networks such as insertion or deletion of edges. We carried out experiments on real-world large datasets, and the results confirms the effectiveness of our algorithms for the single-pair all-shortest-path computation and the incremental maintenance of 2-hop labels.

Inheritance of convexity for the $$\mathcal {P}_{\min }$$-restricted game

We consider restricted games on weighted graphs associated with minimum partitions. We replace in the classical definition of Myerson restricted game the connected components of any subgraph by the sub-components corresponding to a minimum partition. This minimum partition $$\mathcal {P}_{\min }$$ is induced by the deletion of the minimum weight edges. We provide a characterization of the graphs satisfying inheritance of convexity from the underlying game to the restricted game associated with $$\mathcal {P}_{\min }$$ . Moreover, we prove that these graphs can be recognized in polynomial time.

Maximum reachability preserved graph cut

A class of reachability reduction problems were raised in the area of computer network security and software engineering. This paper studies such a reachability reduction problem on a vertex labeled graph. The reachability reduction here is modeled as maximum reachability preserved cut which is a variant of minimum multiway cut with a new objective function. Finding a maximum reachability preserved cut is to disconnect some labels specified in advance by edge deletion while preserving other reachable labels. It gives a way to model a large family of network problems based on graph model. We provide a comprehensive complexity analysis of this problem under different input settings. A landscape of the hardness hierarchy of this problem is shown in this paper, in which polynomial tractable and intractable cases are identified.

Parameterized Dynamic Cluster Editing

We introduce a dynamic version of the NP-hard graph modification problem Cluster Editing. The essential point here is to take into account dynamically evolving input graphs: having a cluster graph (that is, a disjoint union of cliques) constituting a solution for a first input graph, can we cost-efficiently transform it into a “similar” cluster graph that is a solution for a second (“subsequent”) input graph? This model is motivated by several application scenarios, including incremental clustering, the search for compromise clusterings, or also local search in graph-based data clustering. We thoroughly study six problem variants (three modification scenarios edge editing, edge deletion, edge insertion; each combined with two distance measures between cluster graphs). We obtain both fixed-parameter tractability as well as (parameterized) hardness results, thus (except for three open questions) providing a fairly complete picture of the parameterized computational complexity landscape under the two perhaps most natural parameterizations: the distances of the new “similar” cluster graph to (1) the second input graph and to (2) the input cluster graph.

Finding the root graph through minimum edge deletion

The line graph of a graph G has one node per each edge of G, two of them being adjacent only when the corresponding edges have a node of G in common. In this work, we consider the problem of finding the minimum number of edges to delete so that the resulting graph is a line graph, which presents an interesting application in haplotyping of diploid organisms. We propose an Integer Linear Programming formulation for this problem. We compare our approach with the only other existing formulation for the problem and explore the possibility of combining both of them. Finally, we present a computational study to compare the different approaches proposed.

Near-critical spanning forests and renormalization

We study random two-dimensional spanning forests in the plane that can be viewed both in the discrete case and in their appropriately taken scaling limits as a uniformly chosen spanning tree with some Poissonian deletion of edges or points. We show how to relate these scaling limits to a stationary distribution of a natural coalescent-type Markov process on a state space of abstract graphs with real-valued edge weights. This Markov process can be interpreted as a renormalization flow. This provides a model for which one can rigorously implement the formalism proposed by the third author in order to relate the law of the scaling limit of a critical model to a stationary distribution of such a renormalization/Markov process. When starting from any two-dimensional lattice with constant edge weights, the Markov process does indeed converge in law to this stationary distribution that corresponds to a scaling limit of UST with Poissonian deletions. The results of this paper heavily build on the convergence in distribution of branches of the UST to $\mathrm{SLE}_{2}$ (a result by Lawler, Schramm and Werner) as well as on the convergence of the suitably renormalized length of the loop-erased random walk to the “natural parametrization” of the $\mathrm{SLE}_{2}$ (a recent result by Lawler and Viklund).

ON THE SIGNED MATCHINGS OF GRAPHS

For a graph $G$ and any $v\in V(G)$, $E_{G}(v)$ is the set of all edges incident with $v$. A function $f:E(G)\rightarrow \{-1,1\}$ is called a signed matching of $G$ if $\sum_{e\in E(v)}f(e) \leq 1$ for every $ {v\in V(G)}$. For a signed matching $x$, set $x(E(G))=\sum_{e\in E(G))}x(e)$. The signed matching number of $G$, denoted by $\beta_1'(G)$, is the maximum $x(E(G))$ where the maximum is taken over all signed matching over $G$. In this paper we obtain the signed matching number of some families of graphs and study the signed matching number of subdivision and edge deletion of edges of graph.

Subexponential algorithm for d-cluster edge deletion: Exception or rule?

We study the question of finding a set of at most k edges, whose removal makes the input n-vertex graph a disjoint union of s-clubs (graphs of diameter s). Komusiewicz and Uhlmann [DAM 2012] showed that Cluster Edge Deletion (i.e., for the case of 1-clubs (cliques)), cannot be solved in time 2o(k)nO(1) unless the Exponential Time Hypothesis (ETH) fails. But, Fomin et al. [JCSS 2014] showed that if the number of cliques in the output graph is restricted to d, then the problem (d-Cluster Edge Deletion) can be solved in time O(2O(dk)+m+n). We show that assuming ETH, there is no algorithm solving 2-Club Cluster Edge Deletion in time 2o(k)nO(1). Further, we show that the same lower bound holds in the case of s-Clubd-Cluster Edge Deletion for any s≥2 and d≥2.

A non-linear reverse-engineering method for inferring genetic regulatory networks.

Hematopoiesis is a highly complex developmental process that produces various types of blood cells. This process is regulated by different genetic networks that control the proliferation, differentiation, and maturation of hematopoietic stem cells (HSCs). Although substantial progress has been made for understanding hematopoiesis, the detailed regulatory mechanisms for the fate determination of HSCs are still unraveled. In this study, we propose a novel approach to infer the detailed regulatory mechanisms. This work is designed to develop a mathematical framework that is able to realize nonlinear gene expression dynamics accurately. In particular, we intended to investigate the effect of possible protein heterodimers and/or synergistic effect in genetic regulation. This approach includes the Extended Forward Search Algorithm to infer network structure (top-down approach) and a non-linear mathematical model to infer dynamical property (bottom-up approach). Based on the published experimental data, we study two regulatory networks of 11 genes for regulating the erythrocyte differentiation pathway and the neutrophil differentiation pathway. The proposed algorithm is first applied to predict the network topologies among 11 genes and 55 non-linear terms which may be for heterodimers and/or synergistic effect. Then, the unknown model parameters are estimated by fitting simulations to the expression data of two different differentiation pathways. In addition, the edge deletion test is conducted to remove possible insignificant regulations from the inferred networks. Furthermore, the robustness property of the mathematical model is employed as an additional criterion to choose better network reconstruction results. Our simulation results successfully realized experimental data for two different differentiation pathways, which suggests that the proposed approach is an effective method to infer the topological structure and dynamic property of genetic regulations.

Efficient Coverage Hole Detection Algorithm Based on the Simplified Rips Complex in Wireless Sensor Networks

The appearance of coverage holes in the network leads to transmission links being disconnected, thereby resulting in decreasing the accuracy of data. Timely detection of the coverage holes can effectively improve the quality of network service. Compared with other coverage hole detection algorithms, the algorithms based on the Rips complex have advantages of high detection accuracy without node location information, but with high complexity. This paper proposes an efficient coverage hole detection algorithm based on the simplified Rips complex to solve the problem of high complexity. First, Turan’s theorem is combined with the concept of the degree and clustering coefficient in a complex network to classify the nodes; furthermore, redundant node determination rules are designed to sleep redundant nodes. Second, according to the concept of the complete graph, redundant edge deletion rules are designed to delete redundant edges. On the basis of the above two steps, the Rips complex is simplified efficiently. Finally, from the perspective of the loop, boundary loop filtering and reduction rules are designed to achieve coverage hole detection in wireless sensor networks. Compared with the HBA and tree-based coverage hole detection algorithm, simulation results show that the proposed hole detection algorithm has lower complexity and higher accuracy and the detection accuracy of the hole area is up to 99.03%.

Fast, Accurate and Provable Triangle Counting in Fully Dynamic Graph Streams

Given a stream of edge additions and deletions, how can we estimate the count of triangles in it? If we can store only a subset of the edges, how can we obtain unbiased estimates with small variances? Counting triangles (i.e., cliques of size three) in a graph is a classical problem with applications in a wide range of research areas, including social network analysis, data mining, and databases. Recently, streaming algorithms for triangle counting have been extensively studied since they can naturally be used for large dynamic graphs. However, existing algorithms cannot handle edge deletions or suffer from low accuracy. Can we handle edge deletions while achieving high accuracy? We propose T hink D, which accurately estimates the counts of global triangles (i.e., all triangles) and local triangles associated with each node in a fully dynamic graph stream with additions and deletions of edges. Compared to its best competitors, T hink D is (a) Accurate: up to 4.3 × more accurate within the same memory budget, (b) Fast: up to 2.2 × faster for the same accuracy requirements, and (c) Theoretically sound: always maintaining estimates with zero bias (i.e., the difference between the true triangle count and the expected value of its estimate) and small variance. As an application, we use T hink D to detect suddenly emerging dense subgraphs, and we show its advantages over state-of-the-art methods.

Parameterized Aspects of Strong Subgraph Closure

Motivated by the role of triadic closures in social networks, and the importance of finding a maximum subgraph avoiding a fixed pattern, we introduce and initiate the parameterized study of the StrongF-closure problem, where F is a fixed graph. This is a generalization of Strong Triadic Closure, whereas it is a relaxation of F-free Edge Deletion. In StrongF-closure, we want to select a maximum number of edges of the input graph G, and mark them as strong edges, in the following way: whenever a subset of the strong edges forms a subgraph isomorphic to F, then the corresponding induced subgraph of G is not isomorphic to F. Hence, the subgraph of G defined by the strong edges is not necessarily F-free, but whenever it contains a copy of F, there are additional edges in G to forbid that strong copy of F in G. We study StrongF-closure from a parameterized perspective with various natural parameterizations. Our main focus is on the number k of strong edges as the parameter. We show that the problem is FPT with this parameterization for every fixed graph F, whereas it does not admit a polynomial kernel even when $$F =P_3$$. In fact, this latter case is equivalent to the Strong Triadic Closure problem, which motivates us to study this problem on input graphs belonging to well known graph classes. We show that Strong Triadic Closure does not admit a polynomial kernel even when the input graph is a split graph, whereas it admits a polynomial kernel when the input graph is planar, and even d-degenerate. Furthermore, on graphs of maximum degree at most 4, we show that Strong Triadic Closure is FPT with the above guarantee parameterization $$k - \mu (G)$$, where $$\mu (G)$$ is the maximum matching size of G. We conclude with some results on the parameterization of StrongF-closure by the number of edges of G that are not selected as strong.