Edge Deletion Research Articles

Synchrony and Elementary Operations on Coupled Cell Networks

Given a finite graph (network), let every node (cell) represent an individual dynamics given by a system of ordinary differential equations, and every arrow (edge) encode the dynamical influence of the tail node on the head node. We have then defined a coupled cell system that is associated with the given network structure. Subspaces that are defined by equalities of cell coordinates and left invariant under every coupled cell system respecting the network structure are called synchrony subspaces. They are completely determined by the network structure and form a complete lattice under set inclusions. We analyze the transition of the lattice of synchrony subspaces of a network that is caused by structural changes in the network topology, such as deletion and addition of cells or edges, and rewirings of edges. We give sufficient, and in some cases both sufficient and necessary, conditions under which lattice elements persist or disappear.

Sufficient conditions for edit-optimal clusters

Cluster Editing is the problem of turning a graph into a cluster graph, that is, a disjoint union of cliques, by a minimum number of edge edits. Cluster Deletion is similarly defined, with edge deletions only. We propose a local notion of edit-optimality: Informally, we say that a crown (a certain type of labeled graph) is edit-optimal if it yields a cluster in some optimal solution to Cluster Editing, in every graph containing this crown as a subgraph. Then we give sufficient conditions for edit-optimality, e.g., in terms of vertex degrees. A condition for Cluster Deletion applies a theorem of Landau (1953) [7] on degree sequences of tournaments. The conditions are particularly suited for planted models of clusterings, and for networks that only partially exhibit a clear cluster structure.

Simple Deterministic Algorithms for Fully Dynamic Maximal Matching

A maximal matching can be maintained in fully dynamic (supporting both addition and deletion of edges) n -vertex graphs using a trivial deterministic algorithm with a worst-case update time of O ( n ). No deterministic algorithm that outperforms the naïve O ( n ) one was reported up to this date. The only progress in this direction is due to Ivković and Lloyd, who in 1993 devised a deterministic algorithm with an amortized update time of O (( n + m ) √2/2 ), where m is the number of edges. In this article, we show the first deterministic fully dynamic algorithm that outperforms the trivial one. Specifically, we provide a deterministic worst-case update time of O (√ m ). Moreover, our algorithm maintains a matching, which in fact is a 3/2-approximate maximum cardinality matching (MCM). We remark that no fully dynamic algorithm for maintaining (2 − ϵ)-approximate MCM improving upon the naïve O ( n ) was known prior to this work, even allowing amortized time bounds and randomization. For low arboricity graphs (e.g., planar graphs and graphs excluding fixed minors), we devise another simple deterministic algorithm with sublogarithmic update time. Specifically, it maintains a fully dynamic maximal matching with amortized update time of O (log n /log log n ). This result addresses an open question of Onak and Rubinfeld [2010]. We also show a deterministic algorithm with optimal space usage, which for arbitrary graphs maintains a maximal matching in amortized O (√ m ) time and uses only O ( n + m ) space.

On Hilbert bases of cuts

A Hilbert basis is a set of vectors X⊆Rd such that the integer cone (semigroup) generated by X is the intersection of the lattice generated by X with the cone generated by X. Let ℋ be the class of graphs whose set of cuts is a Hilbert basis in RE (regarded as {0,1}-characteristic vectors indexed by edges). We show that ℋ is not closed under edge deletions, subdivisions, nor 2-sums. Furthermore, no graph having K6∖e as a minor belongs to ℋ. This corrects an error in Laurent (1996).For positive results, we give conditions under which the 2-sum of two graphs produces a member of ℋ. Using these conditions we show that all K5⊥-minor-free graphs are in ℋ, where K5⊥ is the unique 3-connected graph obtained by uncontracting an edge of K5. We also establish a relationship between edge deletion and subdivision. Namely, if G′ is obtained from G∈ℋ by subdividing e two or more times, then G∖e∈ℋ if and only if G′∈ℋ.

Editing to Eulerian graphs

The Eulerian Editing problem asks, given a graph G and an integer k, whether G can be modified into an Eulerian graph using at most k edge additions and edge deletions. We show that this problem is polynomial-time solvable for both undirected and directed graphs. We generalize these results for problems with degree parity constraints and degree balance constraints, respectively. We also consider the variants where vertex deletions are permitted. Combined with known results, this leads to full complexity classifications for both undirected and directed graphs and for every subset of the three graph operations.

Molecular network control through boolean canalization.

Boolean networks are an important class of computational models for molecular interaction networks. Boolean canalization, a type of hierarchical clustering of the inputs of a Boolean function, has been extensively studied in the context of network modeling where each layer of canalization adds a degree of stability in the dynamics of the network. Recently, dynamic network control approaches have been used for the design of new therapeutic interventions and for other applications such as stem cell reprogramming. This work studies the role of canalization in the control of Boolean molecular networks. It provides a method for identifying the potential edges to control in the wiring diagram of a network for avoiding undesirable state transitions. The method is based on identifying appropriate input-output combinations on undesirable transitions that can be modified using the edges in the wiring diagram of the network. Moreover, a method for estimating the number of changed transitions in the state space of the system as a result of an edge deletion in the wiring diagram is presented. The control methods of this paper were applied to a mutated cell-cycle model and to a p53-mdm2 model to identify potential control targets.

Quantifying loss of information in network-based dimensionality reduction techniques

To cope with the complexity of large networks, a number of dimensionality reduction techniques for graphs have been developed. However, the extent to which information is lost or preserved when these techniques are employed has not yet been clear. Here we develop a framework, based on algorithmic information theory, to quantify the extent to which information is preserved when network motif analysis, graph spectra and spectral sparsification methods are applied to over twenty different biological and artificial networks. We find that the spectral sparsification is highly sensitive to high number of edge deletion, leading to significant inconsistencies, and that graph spectral methods are the most irregular, capturing algebraic information in a condensed fashion but largely losing most of the information content of the original networks. However, the approach shows that network motif analysis excels at preserving the relative algorithmic information content of a network, hence validating and generalizing the remarkable fact that despite their inherent combinatorial possibilities, local regularities preserve information to such an extent that essential properties are fully recoverable across different networks to determine their family group to which they belong to (eg genetic vs social network). Our algorithmic information methodology thus provides a rigorous framework enabling a fundamental assessment and comparison between different data dimensionality reduction methods thereby facilitating the identification and evaluation of the capabilities of old and new methods.

Quantifying dynamical spillover in co-evolving multiplex networks.

Multiplex networks (a system of multiple networks that have different types of links but share a common set of nodes) arise naturally in a wide spectrum of fields. Theoretical studies show that in such multiplex networks, correlated edge dynamics between the layers can have a profound effect on dynamical processes. However, how to extract the correlations from real-world systems is an outstanding challenge. Here we introduce the Multiplex Markov chain to quantify correlations in edge dynamics found in longitudinal data of multiplex networks. By comparing the results obtained from the multiplex perspective to a null model which assumes layers in a network are independent, we can identify real correlations as distinct from simultaneous changes that occur due to random chance. We use this approach on two different data sets: the network of trade and alliances between nation states, and the email and co-commit networks between developers of open source software. We establish the existence of “dynamical spillover” showing the correlated formation (or deletion) of edges of different types as the system evolves. The details of the dynamics over time provide insight into potential causal pathways.

Graph energy change due to any single edge deletion

The energy of a graph is the sum of the absolute values of its eigenvalues. We propose a new problem on graph energy change due to any single edge deletion. Then we survey the literature for existing partial solution of the problem, and mention a conjecture based on numerical evidence. Moreover, we prove in three different ways that the energy of a cycle graph decreases when an arbitrary edge is deleted except for the order of 4.

Bootstrap percolation and the geometry of complex networks

On a geometric model for complex networks (introduced by Krioukov et al.) we investigate the bootstrap percolation process. This model consists of random geometric graphs on the hyperbolic plane having N vertices, a dependent version of the Chung–Lu model. The process starts with infection rate p=p(N). Each uninfected vertex with at least r≥1 infected neighbors becomes infected, remaining so forever. We identify a function pc(N)=o(1) such that a.a.s. when p≫pc(N) the infection spreads to a positive fraction of vertices, whereas when p≪pc(N) the process cannot evolve. Moreover, this behavior is “robust” under random deletions of edges.

Dynamic Maintenance of a Shortest-Path Tree on Homogeneous Batches of Updates

A dynamic graph algorithm is called batch if it is able to update efficiently the solution of a given graph problem after multiple updates at a time (i.e., a batch) take place on the input graph. In this article, we study batch algorithms for maintaining a single-source shortest-path tree in graphs with positive real edge weights. In particular, we focus our attention on homogeneous batches, that is, either incremental (containing only edge insertion and weight decrease operations) or decremental (containing only edge deletion and weight increase operations) batches, which model realistic dynamic scenarios like transient vertex failures in communication networks and traffic congestion/decongestion phenomena in road networks. We propose two new algorithms to process either incremental or decremental batches, respectively, and a combination of these two algorithms that is able to process arbitrary sequences of incremental and decremental batches. All these algorithms are update sensitive ; namely, they are efficient with respect to the number of vertices in the shortest-path tree that change their parents and/or their distances from the source as a consequence of a batch. This makes unfeasible an effective comparison on a theoretical basis of our new algorithms with the solutions known in the literature, which in turn are analyzed with respect to others and different parameters. For this reason, in order to evaluate the quality of our approach, we provide also an extensive experimental study including our new algorithms and the most efficient previous batch algorithms. Our experimental results complement previous studies and show that the various solutions can be consistently ranked on the basis of the type of homogeneous batch and of the underlying network. As a result, our work can be helpful in selecting a proper solution depending on the specific application scenario.

Chordal Editing is Fixed-Parameter Tractable

Graph modification problems are typically asked as follows: is there a small set of operations that transforms a given graph to have a certain property. The most commonly considered operations include vertex deletion, edge deletion, and edge addition; for the same property, one can define significantly different versions by allowing different operations. We study a very general graph modification problem which allows all three types of operations: given a graph $G$ and integers $k_1$, $k_2$, and $k_3$, the \textsc{chordal editing} problem asks whether $G$ can be transformed into a chordal graph by at most $k_1$ vertex deletions, $k_2$ edge deletions, and $k_3$ edge additions. Clearly, this problem generalizes both \textsc{chordal vertex/edge deletion} and \textsc{chordal completion} (also known as \textsc{minimum fill-in}). Our main result is an algorithm for \textsc{chordal editing} in time $2^{O(k\log k)}\cdot n^{O(1)}$, where $k:=k_1+k_2+k_3$ and $n$ is the number of vertices of $G$. Therefore, the problem is fixed-parameter tractable parameterized by the total number of allowed operations. Our algorithm is both more efficient and conceptually simpler than the previously known algorithm for the special case \textsc{chordal deletion}.

Editing to a Graph of Given Degrees

We consider the Editing to a Graph of Given Degrees problem that asks for a graph G, non-negative integers d,k and a function δ:V(G)→{1,…,d}, whether it is possible to obtain a graph G′ from G such that the degree of v is δ(v) for any vertex v by at most k vertex or edge deletions or edge additions. We construct an FPT-algorithm for Editing to a Graph of Given Degrees parameterized by d+k. We complement this result by showing that the problem has no polynomial kernel unless NP⊆coNP/poly.

A Note on the Minimum H-Subgraph Edge Deletion

In this note we consider the following problem: Given a graph [Formula: see text] and a subgraph [Formula: see text], what is the smallest subset [Formula: see text] of edges in [Formula: see text] that needs to be deleted from the graph to make it [Formula: see text]-free? Several algorithmic results are presented. First, using the general framework of Courcelle [9], we show that, for a fixed subgraph [Formula: see text], the problem can be solved in linear time on graphs of bounded treewidth. It is known that the constant hidden in the big-O notation of Courcelle algorithm is big which makes the approach impractical. Thus, we present two explicit linear time dynamic programming algorithms on graphs of bounded treewidth for restricted settings of the problem with reasonable constants. Third, using the linear time algorithm for graphs of bounded treewidth, we design a Baker's type polynomial time approximation scheme for the problem on planar graphs.

Edge deletion problems: Branching facilitated by modular decomposition

Edge deletion problems ask for a minimum set of edges whose deletion makes a graph have a certain property. When this property can be characterized by a finite set of forbidden induced subgraphs, the problem can be solved in fixed-parameter time by a naive bounded search tree algorithm. Sometimes deleting an edge to break an erstwhile forbidden induced subgraph might introduce new ones, which may involve the neighbors of the original forbidden induced subgraph. Therefore, in considering possible ways to break a forbidden induced subgraph one naturally takes its neighborhood into consideration. This observation easily yields more efficient branching rules, but a naive implementation will require too many tedious case analyses. Here we take advantage of modular decomposition, which allows us to focus on far simpler quotient graphs instead of the original graphs. They together yield simple improved algorithms for the edge deletion problems to chain graphs and trivially perfect graphs.

Fully Dynamic Maximal Matching in $O(\log n)$ Update Time

We present an algorithm for maintaining maximal matching in a graph under addition and deletion of edges. Our algorithm is randomized and it takes expected amortized $O(\log n)$ time for each edge update, where $n$ is the number of vertices in the graph. While there exists a trivial $O(n)$ time algorithm for each edge update, the previous best known result for this problem is due to Ivkovicź and Lloyd [Lecture Notes in Comput. Sci. 790, Springer-Verlag, London, 1994, pp. 99--111]. For a graph with $n$ vertices and $m$ edges, they gave an $O( {(n+ m)}^{0.7072})$ update time algorithm which is sublinear only for a sparse graph. For the related problem of maximum matching, Onak and Rubinfeld [Proceedings of STOC'10, Cambridge, MA, 2010, pp. 457--464] designed a randomized algorithm that achieves expected amortized $O(\log^2 n)$ time for each update for maintaining a $c$-approximate maximum matching for some unspecified large constant $c$. In contrast, we can maintain a factor 2 approximate maximum matching in expected amortized $O(\log n )$ time per update as a direct corollary of the maximal matching scheme. This in turn also implies a 2-approximate vertex cover maintenance scheme that takes expected amortized $O(\log n )$ time per update.

Directed triadic closure and edge deletion mechanism induce asymmetry in directed edge properties

Many directed real world networks, such as the WWW, genetic regulation networks and economic networks exhibit significant differences between the properties of the incoming and outgoing edges, while the differences exhibited by other networks, such as Social Netw. are far more limited. This phenomenon is most evident in the differences between the distributions of incoming and outgoing degrees and direct clustering coefficients. There is currently no generic network generation model that would reproduce and tune these observed dissimilarities. We propose and empirically validate a simple and realistic model that can explain the emergence of the dissimilarities between the incoming and outgoing network degrees and clustering coefficients by combining directed triadic closure, random edge addition and directed edge removal. Surprisingly, we find that the difference between in and out degree distributions is attributed to asymmetries in the edge removal, highlighting the neglected yet crucial importance of edge removal mechanisms to the static and dynamic properties of real world networks. The model is governed by only two parameters: the first tunes the randomness of the edge addition mechanism, while the second controls the difference between the in and out degrees. The combination of these parameters reproduces the observed variety of directed degree distributions and clustering coefficients. Further comparisons of the model’s microscopic dynamics against the empirically observed evolution of real world social network confirms that while quite simple, the model properly describes both the edge addition and deletion processes in directed networks.

The effect of vertex or edge deletion on the metric dimension of graphs

The effect of vertex or edge deletion on the metric dimension of graphs

Parameterized Algorithms for Finding Square Roots

We show that the following two problems are fixed-parameter tractable with parameter $$k$$k: testing whether a connected $$n$$n-vertex graph with $$m$$m edges has a square root with at most $$n-1+k$$n-1+k edges and testing whether such a graph has a square root with at least $$m-k$$m-k edges. Our first result implies that squares of graphs obtained from trees by adding at most $$k$$k edges can be recognized in polynomial time for every fixed $$k\ge 0$$k?0; previously this result was known only for $$k=0$$k=0. Our second result is equivalent to stating that deciding whether a graph can be modified into a square root of itself by at most $$k$$k edge deletions is fixed-parameter tractable with parameter $$k$$k.

Efficient local updates for undirected graphical models

We present a new Bayesian approach for undirected Gaussian graphical model determination. We provide some graph theory results for local updates that facilitate a fast exploration of the graph space. Specifically, we show how to locally update, after either edge deletion or inclusion, the perfect sequence of cliques and the perfect elimination order of the nodes associated to an oriented, directed acyclic version of a decomposable graph. Building upon the decomposable graphical models framework, we propose a more flexible methodology that extends to the class of nondecomposable graphs. Posterior probabilities of edge inclusion are interpreted as a natural measure of edge selection uncertainty. When applied to a protein expression data set, the model leads to fast estimation of the protein interaction network.