Tree-like Graphs Research Articles

Analyze on the Learning Behavior of E-Learning

In e-learning, students must initiatively according to their own progress via computer and Internet. That is to say, students often handle contents transmitted from far side in front of computer by themselves. In the meanwhile students have to determine what to learn and where to go in each learning unit (or learning node), so, it relatively consumes mental and physical efforts during learning that results in cognitive overload, therefore, students are easy to feel anxious about e-learning content due to said cognitive overload. Purpose of this paper is to collect the learning behavior trace during e-learning activity, and then use decision tree ID 3 to discover strenuous place in the hyperlink of elearning in order to avoid anxiety of e-learning. Based on the analysis of ID 3, the curriculum designers of e-learning will be easy to visually understand students’ behavior on e-learning by treelike graph, and then compose a well-organized and adaptive e-learning curriculum.

Thermodynamics of the Lévy spin glass

We investigate the Lévy glass, a mean-field spin-glass model with power-law distributed couplings characterized by a divergent second moment. By combining extensively many small couplings with a spare random backbone of strong bonds the model is intermediate between the Sherrington-Kirkpatrick and the Viana-Bray models. A truncated version where couplings smaller than some threshold ε are neglected can be studied within the cavity method developed for spin glasses on locally treelike random graphs. By performing the limit ε→0 in a well-defined way we calculate the thermodynamic functions within replica symmetry and determine the de Almeida-Thouless line in the presence of an external magnetic field. Contrary to previous findings we show that there is no replica-symmetric spin-glass phase. Moreover we determine the leading corrections to the ground-state energy within one-step replica symmetry breaking. The effects due to the breaking of replica symmetry appear to be small in accordance with the intuitive picture that a few strong bonds per spin reduce the degree of frustration in the system.

Branch-and-Bound Algorithms for Enumerating Treelike Chemical Graphs with Given Path Frequency Using Detachment-Cut

Computational methods of enumerating chemical graphs have attained great importance in chemoinformatics since they lead to a variety of useful applications including structure determination of novel chemical compounds. Recently, Fujiwara et al. have presented an efficient branch-and-bound algorithm for enumerating treelike chemical graphs with given path frequency. In this paper, we augment Fujiwara et al.'s algorithm by introducing a new bounding operation called detachment-cut to reduce further the search space in the branch-and-bound framework. Experimental results on much chemical compound data show that our proposed algorithm achieves better performance than Fujiwara et al.'s algorithm in computation time. A program that implements our algorithm can be used freely via Web server.

Hamiltonian approach for explosive percolation

We present a cluster growth process that provides a clear connection between equilibrium statistical mechanics and an explosive percolation model similar to the one recently proposed by D. Achlioptas [Science 323, 1453 (2009)]. We show that the following two ingredients are sufficient for obtaining an abrupt (first-order) transition in the fraction of the system occupied by the largest cluster: (i) the size of all growing clusters should be kept approximately the same, and (ii) the inclusion of merging bonds (i.e., bonds connecting vertices in different clusters) should dominate with respect to the redundant bonds (i.e., bonds connecting vertices in the same cluster). Moreover, in the extreme limit where only merging bonds are present, a complete enumeration scheme based on treelike graphs can be used to obtain an exact solution of our model that displays a first-order transition. Finally, the presented mechanism can be viewed as a generalization of standard percolation that discloses a family of models with potential application in growth and fragmentation processes of real network systems.

Ising models on locally tree-like graphs

We consider ferromagnetic Ising models on graphs that converge locally to trees. Examples include random regular graphs with bounded degree and uniformly random graphs with bounded average degree. We prove that the "cavity" prediction for the limiting free energy per spin is correct for any positive temperature and external field. Further, local marginals can be approximated by iterating a set of mean field (cavity) equations. Both results are achieved by proving the local convergence of the Boltzmann distribution on the original graph to the Boltzmann distribution on the appropriate infinite random tree.

Critical percolation of virtually free groups and other tree-like graphs

This article presents a method for finding the critical probability $p_c$ for the Bernoulli bond percolation on graphs with the so-called tree-like structure. Such a graph can be decomposed into a tree of pieces, each of which has finitely many isomorphism classes. This class of graphs includes the Cayley graphs of amalgamated products, HNN extensions or general groups acting on trees. It also includes all transitive graphs with more than one end. The idea of the method is to find a multi-type Galton--Watson branching process (with a parameter $p$) which has finite expected population size if and only if the expected percolation cluster size is finite. This provides sufficient information about $p_c$. In particular, if the pairwise intersections of pieces are finite, then $p_c$ is the smallest positive $p$ such that $\operatorname {det}(M-1)=0$, where $M$ is the first-moment matrix of the branching process. If the pieces of the tree-like structure are finite, then $p_c$ is an algebraic number and we give an algorithm computing $p_c$ as a root of some algebraic function. We show that any Cayley graph of a virtually free group (i.e., a group acting on a tree with finite vertex stabilizers) with respect to any finite generating set has a tree-like structure with finite pieces. In particular, we show how to compute $p_c$ for the Cayley graph of a free group with respect to any finite generating set.

Analytical results for bond percolation andk-core sizes on clustered networks

An analytical approach to calculating bond percolation thresholds, sizes of k-cores, and sizes of giant connected components on structured random networks with nonzero clustering is presented. The networks are generated using a generalization of Trapman's [P. Trapman, Theor. Popul. Biol. 71, 160 (2007)] model of cliques embedded in treelike random graphs. The resulting networks have arbitrary degree distributions and tunable degree-dependent clustering. The effect of clustering on the bond percolation thresholds for networks of this type is examined and contrasted with some recent results in the literature. For very high levels of clustering the percolation threshold in these generalized Trapman networks is increased above the value it takes in a randomly wired (unclustered) network of the same degree distribution. In assortative scale-free networks, where the variance of the degree distribution is infinite, this clustering effect can lead to a nonzero percolation (epidemic) threshold.

A Separator Theorem for String Graphs and its Applications

A string graph is the intersection graph of a collection of continuous arcs in the plane. We show that any string graph with m edges can be separated into two parts of roughly equal size by the removal of $O(m^{3/4}\sqrt{\log m})$ vertices. This result is then used to deduce that every string graph with n vertices and no complete bipartite subgraph Kt,t has at most ctn edges, where ct is a constant depending only on t. Another application shows that locally tree-like string graphs are globally tree-like: for any ε > 0, there is an integer g(ε) such that every string graph with n vertices and girth at least g(ε) has at most (1 + ε)n edges. Furthermore, the number of such labelled graphs is at most (1 + ε)nT(n), where T(n) = nn−2 is the number of labelled trees on n vertices.

Feedback topology and XOR-dynamics in Boolean networks with varying input structure

We analyze a model of fixed in-degree random Boolean networks in which the fraction of input-receiving nodes is controlled by the parameter gamma. We investigate analytically and numerically the dynamics of graphs under a parallel XOR updating scheme. This scheme is interesting because it is accessible analytically and its phenomenology is at the same time under control and as rich as the one of general Boolean networks. We give analytical formulas for the dynamics on general graphs, showing that with a XOR-type evolution rule, dynamic features are direct consequences of the topological feedback structure, in analogy with the role of relevant components in Kauffman networks. Considering graphs with fixed in-degree, we characterize analytically and numerically the feedback regions using graph decimation algorithms (Leaf Removal). With varying gamma , this graph ensemble shows a phase transition that separates a treelike graph region from one in which feedback components emerge. Networks near the transition point have feedback components made of disjoint loops, in which each node has exactly one incoming and one outgoing link. Using this fact, we provide analytical estimates of the maximum period starting from topological considerations.

Anderson localization for radial tree-like random quantum graphs

We prove that certain random models associated with radial, tree-like, rooted quantum graphs exhibit Anderson localization at all energies. The two main examples are the random length model (RLM) and the random Kirchhoff model (RKM). In the RLM, the lengths of each generation of edges form a family of independent, identically distributed random variables (iid). For the RKM, the iid random variables are associated with each generation of vertices and moderate the current flow through the vertex. We consider extensions to various families of decorated graphs and prove stability of localization with respect to decoration. In particular, we prove Anderson localization for the random necklace model.

Stochastic Analysis

Stochastic Analysis

Treelike comparability graphs

An undirected graph is a treelike comparability graph if it admits a transitive orientation such that its transitive reduction is a tree. We show that treelike comparability graphs are distance hereditary. Utilizing this property, we give a linear time recognition algorithm. We then characterize permutation graphs that are treelike. Finally, we consider the Partitioning into Bounded Cliques problem on special subgraphs of treelike permutation graphs.

Double Scaling Limit of Semiclassical Orthogonal Polynomials in the Center of a Tree and the Painleve 1 Equation

The asymptotics of orthogonal polynomials with complex weight on complex contours was studied in [1] and [5]. It was shown that in the asymptotic limit, the zeros of these polynomials accumulate on tree-like graphs in and generically, the density of zeros vanishes like a square-root at each vertex of these tree graphs. In this article, we study the double scaling limit in which the density vanishes to an order higher than in an interior vertex. This is a new critical phenomenon that has no parallel in the case of orthogonal polynomials on the real line. Yet, surprisingly, the behavior of the orthogonal polynomial in the double scaling limit can be modeled by a solution of the Painleve 1 equation that was used previously [15] in the study of a different double scale limit.

Semi-nice tree-decompositions: The best of branchwidth, treewidth and pathwidth with one algorithm

Branchwidth and treewidth are connectivity parameters of graphs of high importance in algorithm design. By dynamic programming along the associated branch- or tree-decomposition one can solve most graph optimization problems in time linear in the graph size and exponential in the parameter. If one of these parameters is bounded on a class of graphs, then so is the other, but they can differ by a small constant factor and this difference can be crucial for the resulting runtime. In this paper we introduce semi-nice tree-decompositions and show that they combine the best of both branchwidth and treewidth. We first give simple algorithms to transform a given tree-decomposition or branch-decomposition into a semi-nice tree-decomposition. We then give two templates for dynamic programming along a semi-nice tree-decomposition, one for optimization problems over vertex subsets and another for optimization problems over edge subsets. We show that the resulting runtime will match or beat the runtimes achieved by doing dynamic programming directly on either a branch- or tree-decomposition. For example, given a graph G on n vertices with path-, tree- and branch-decompositions of width p w , t w and b w respectively, the Minimum Dominating Set problem on G is solved in time O ( n 2 min { 1.58 p w , 2 t w , 2.38 b w } ) by a single dynamic programming algorithm along a semi-nice tree-decomposition. On the applied side the immediate gain is that for each optimization problem one can achieve the benefits of both treewidth, branchwidth and pathwidth while developing and implementing only one dynamic programming algorithm. On the theoretical side the combination of the best properties of both branchwidth and treewidth in a single decomposition is a step towards a more general framework giving the fastest possible algorithms on tree-like graphs.

Enumerating Treelike Chemical Graphs with Given Path Frequency

The enumeration of chemical graphs satisfying given constraints is one of the fundamental problems in chemoinformatics. In this paper, we consider the problem of enumerating (i.e., listing) all treelike chemical graphs from a given path frequency. We propose an exact algorithm for enumerating all solutions to this problem on the basis of the branch-and-bound method. To further improve the efficiency of the enumeration, we introduce a new variant of the compound enumeration problem by adding a specification on the number of multiple bonds to the input and design another exact enumeration algorithm. The experimental results show that our algorithms can efficiently solve instances with larger sizes that are impossible to solve by the previous methods. In particular, we apply the latter algorithm to the enumeration problem of the special treelike chemical structures-alkane isomers. The theoretical and experimental results show that our algorithm works at least as fast as the state-of-the-art algorithms specially designed for generating alkane isomers, however using much less memory space.

Improved algorithms for enumerating tree-like chemical graphs with given path frequency.

This paper considers the problem of enumerating all non-isomorphic tree-like chemical graphs with given path frequency, where "tree-like" means that the graph can be viewed as a tree if multiple edges (i.e., edges with the same end points) and a benzene ring are treated as one edge and one vertex, respectively, and "path frequency" is a vector of the numbers of specified vertex-labeled paths that must be realized in every output. This and related problems have several potential applications such as classification of chemical compounds, structure determination using mass-spectrum and/or NMR and design of novel chemical compounds. For this problem, several studies have been done. Recently, Fujiwara et al. (2008) showed two formulations and for each of them, they gave a branch-and-bound algorithm, which combined efficient enumeration of non-isomorphic trees with bounding operations based on the path frequency and the atom-atom bonds to avoid the generation of invalid trees. In this paper, based on their work and a result of Nagamochi (2006), we introduce two new bounding operations, the detachment-cut and the H-cut, to further reduce the size of the search space. We performed computational experiments to compare our proposed algorithms with those of Fujiwara et al. (2008) using some chemical compound data obtained from the KEGG LIGAND database (http://www.genome.jp/kegg/ligand.html). The results show that our proposed algorithms are much faster than their algorithms.

ON COMPUTING LONGEST PATHS IN SMALL GRAPH CLASSES

The longest path problem is the one that finds a longest path in a given graph. While the graph classes in which the Hamiltonian path problem can be solved efficiently are widely investigated, few graph classes are known to be solved efficiently for the longest path problem. Among those, for trees, a simple linear time algorithm for the longest path problem is known. We first generalize the algorithm, and show that the longest path problem can be solved efficiently for some tree-like graph classes by this approach. We next propose two new graph classes that have natural interval representations, and show that the longest path problem can be solved efficiently on these classes.

Decomposition of Smith graphs in maximal reflexive cacti

The spectrum of a graph is the family of eigenvalues of its ( 0 , 1 ) adjacency matrix. A simple graph is reflexive if its second largest eigenvalue λ 2 does not exceed 2. The graphic property λ 2 ⩽ 2 is a hereditary one, i.e. every induced subgraph of a reflexive graph preserves this property and that is why reflexive graphs are usually represented through maximal graphs. Cacti, or treelike graphs, are graphs whose all cycles are mutually edge-disjoint. The set of simple connected graphs characterized by the property λ 1 = 2 , where λ 1 is the largest eigenvalue, is known as the set of Smith graphs. It consists of cycles of all possible lengths and some trees. If two trees T 1 and T 2 have such vertices u 1 ∈ T 1 and u 2 ∈ T 2 which, after their identification u 1 = u 2 = u give a Smith tree, we say that that Smith tree can be split at its vertex u into T 1 and T 2 . It has turned out that several classes of maximal reflexive cacti can be described in the following way: we start from certain essential cyclic structure with two characteristic vertices c 1 and c 2 , and then form a family of maximal connected reflexive cacti by splitting Smith trees, and by attaching their parts to c 1 and c 2 . This way of decomposition of Smith trees leads to an interesting phenomenon of so-called pouring of Smith trees between two vertices.

Distinguishing Infinite Graphs

The distinguishing number $D(G)$ of a graph $G$ is the least cardinal number $\aleph$ such that $G$ has a labeling with $\aleph$ labels that is only preserved by the trivial automorphism. We show that the distinguishing number of the countable random graph is two, that tree-like graphs with not more than continuum many vertices have distinguishing number two, and determine the distinguishing number of many classes of infinite Cartesian products. For instance, $D(Q_{n}) = 2$, where $Q_{n}$ is the infinite hypercube of dimension ${n}$.

Multicyclic treelike reflexive graphs

A simple graph is reflexive if its second largest eigenvalue does not exceed 2. A graph is treelike (sometimes also called a cactus) if all its cycles (circuits) are mutually edge-disjoint. In a lot of cases one can establish whether a given graph is reflexive by identifying and removing a single cut-vertex (Theorem 1). In this paper we prove that, if this theorem cannot be applied to a connected treelike reflexive graph G and if all its cycles do not have a common vertex (do not form a bundle), such a graph has at most five cycles (Theorem 2). On the same conditions, in Theorem 3 we find all maximal treelike reflexive graphs with four and five cycles.