Showing papers on "Path graph published in 2014"

Fast iterative graph computation: a path centric approach

Abstract: Large scale graph processing represents an interesting challenge due to the lack of locality. This paper presents Path Graph for improving iterative graph computation on graphs with billions of edges. Our system design has three unique features: First, we model a large graph using a collection of tree-based partitions and use an path-centric computation rather than vertex-centric or edge-centric computation. Our parallel computation model significantly improves the memory and disk locality for performing iterative computation algorithms. Second, we design a compact storage that further maximize sequential access and minimize random access on storage media. Third, we implement the path-centric computation model by using a scatter/gather programming model, which parallels the iterative computation at partition tree level and performs sequential updates for vertices in each partition tree. The experimental results show that the path-centric approach outperforms vertex centric and edge-centric systems on a number of graph algorithms for both in-memory and out-of-core graphs.

Bootstrap Percolation in Power-Law Random Graphs

Abstract: A bootstrap percolation process on a graph is an "infection" process which evolves in rounds. Initially, there is a subset of infected nodes and in each subsequent round each uninfected node which has at least infected neighbours becomes infected and remains so forever. The parameter is fixed. Such processes have been used as models for the spread of ideas or trends within a network of individuals. We analyse this process in the case where the underlying graph is an inhomogeneous random graph, which exhibits a power-law degree distribution, and initially there are randomly infected nodes. The main focus of this paper is the number of vertices that will have been infected by the end of the process. The main result of this work is that if the degree sequence of the random graph follows a power law with exponent , where , then a sublinear number of initially infected vertices is enough to spread the infection over a linear fraction of the nodes of the random graph, with high probability. More specifically, we determine explicitly a critical function such that with the following property. Assuming that is the number of vertices of the underlying random graph, if , then the process does not evolve at all, with high probability as grows, whereas if , then there is a constant such that, with high probability, the final set of infected vertices has size at least . This behaviour is in sharp contrast with the case where the underlying graph is a random graph with . It follows from an observation of Balogh and Bollobas that in this case if the number of initially infected vertices is sublinear, then there is lack of evolution of the process. It turns out that when the maximum degree is , then depends also on . But when the maximum degree is , then .

Detection of Core-Periphery Structure in Networks Using Spectral Methods and Geodesic Paths

Abstract: We introduce several novel and computationally efficient methods for detecting "core--periphery structure" in networks. Core--periphery structure is a type of mesoscale structure that includes densely-connected core vertices and sparsely-connected peripheral vertices. Core vertices tend to be well-connected both among themselves and to peripheral vertices, which tend not to be well-connected to other vertices. Our first method, which is based on transportation in networks, aggregates information from many geodesic paths in a network and yields a score for each vertex that reflects the likelihood that a vertex is a core vertex. Our second method is based on a low-rank approximation of a network's adjacency matrix, which can often be expressed as a tensor-product matrix. Our third approach uses the bottom eigenvector of the random-walk Laplacian to infer a coreness score and a classification into core and peripheral vertices. We also design an objective function to (1) help classify vertices into core or peripheral vertices and (2) provide a goodness-of-fit criterion for classifications into core versus peripheral vertices. To examine the performance of our methods, we apply our algorithms to both synthetically-generated networks and a variety of networks constructed from real-world data sets.

The (revised) Szeged index and the Wiener index of a nonbipartite graph

Abstract: Hansen et al. used the computer program AutoGraphiX to study the differences between the Szeged index Sz(G) and the Wiener index W(G), and between the revised Szeged index Sz^*(G) and the Wiener index for a connected graph G. They conjectured that for a connected nonbipartite graph G with n>=5 vertices and girth g>=5, Sz(G)-W(G)>=2n-5, and moreover, the bound is best possible when the graph is composed of a cycle C"5 on 5 vertices and a tree T on n-4 vertices sharing a single vertex. They also conjectured that for a connected nonbipartite graph G with n>=4 vertices, Sz^*(G)-W(G)>=n^2+4n-64, and moreover, the bound is best possible when the graph is composed of a cycle C"3 on 3 vertices and a tree T on n-2 vertices sharing a single vertex. In this paper, we not only give confirmative proofs to these two conjectures but also characterize those graphs that achieve the two lower bounds.

Estimating network parameters using random walks

Abstract: Sampling from large graphs is an area of great interest, especially since the emergence of huge structures such as Online Social Networks and the World Wide Web (WWW). The large scale properties of a network can be summarized in terms of parameters of the underlying graph, such as the total number of vertices, edges and triangles. However, the large si ze of these networks makes it computationally expensive to obtain such structural properties of the underlying graph by exhaustive search. If we can estimate these properties by taking small but representative samples from the network, then size is no longer such a problem. In this paper we present a general framework to estimate network properties using random walks. These methods work under the assumption we are able to obtain local characteristics of a vertex during each step of the random walk, for example the number of its neighbours, and their labels. As examples of this approach, we present practical methods to estimate the total number of edges/links m, number of vertices/nodes n and number of connected triads of vertices (triangles) t in graphs. We also give a general method to count any type of small connected subgraph, of which vertices, edges and triangles are specific examples. Additionally we present experimental estimates for n, m, t we obtained using our methods on real or synthetic networks. The synthetic networks were random graphs with power-law degree distributions and designed to have a large number of triangles. We used these graphs as they tend to correspond to the structure of large online networks. The real networks were samples of the WWW and social networks obtained from the SNAP database. In order to test that the methods are indeed practical, the total number of steps made by the walk was limited to at most the size n of the network. In fact the estimates appear to converge to the correct value at a lower number of steps, indicating that our proposed methods are feasible in practice.

Locally Estimating Core Numbers

Abstract: Graphs are a powerful way to model interactions and relationships in data from a wide variety of application domains. In this setting, entities represented by vertices at the 'center' of the graph are often more important than those associated with vertices on the 'fringes'. For example, central nodes tend to be more critical in the spread of information or disease and play an important role in clustering/community formation. Identifying such 'core' vertices has recently received additional attention in the context of network experiments, which analyze the response when a random subset of vertices are exposed to a treatment (e.g. Inoculation, free product samples, etc). Specifically, the likelihood of having many central vertices in any exposure subset can have a significant impact on the experiment. We focus on using k-cores and core numbers to measure the extent to which a vertex is central in a graph. Existing algorithms for computing the core number of a vertex require the entire graph as input, an unrealistic scenario in many real world applications. Moreover, in the context of network experiments, the sub graph induced by the treated vertices is only known in a probabilistic sense. We introduce a new method for estimating the core number based only on the properties of the graph within a region of radius a#x03B4; around the vertex, and prove an asymptotic error bound of our estimator on random graphs. Further, we empirically validate the accuracy of our estimator for small values of a#x03B4; on a representative corpus of real data sets. Finally, we evaluate the impact of improved local estimation on an open problem in network experimentation posed by Ugander et al.

Mining statistically significant connected subgraphs in vertex labeled graphs

Abstract: The steady growth of graph data in various applications has resulted in wide-spread research in finding significant sub-structures in a graph. In this paper, we address the problem of finding statistically significant connected subgraphs where the nodes of the graph are labeled. The labels may be either discrete where they assume values from a pre-defined set, or continuous where they assume values from a real domain and can be multi-dimensional. We motivate the problem citing applications in spatial co-location rule mining and outlier detection. We use the chi-square statistic as a measure for quantifying the statistical significance. Since the number of connected subgraphs in a general graph is exponential, the naive algorithm is impractical. We introduce the notion of contracting edges that merge vertices together to form a super-graph. We show that if the graph is dense enough to start with, the number of super-vertices is quite low, and therefore, running the naive algorithm on the super-graph is feasible. If the graph is not dense, we provide an algorithm to reduce the number of super-vertices further, thereby providing a trade-off between accuracy and time. Empirically, the chi-square value obtained by this reduction is always within 96% of the optimal value, while the time spent is only a fraction of that for the optimal. In addition, we also show that our algorithm is scalable and it significantly enhances the ability to analyze real datasets.

Spanning Trees with a Bounded Number of Branch Vertices in a Claw-Free Graph

Abstract: Let k be a non-negative integer. A branch vertex of a tree is a vertex of degree at least three. We show two sufficient conditions for a connected claw-free graph to have a spanning tree with a bounded number of branch vertices: (i) A connected claw-free graph has a spanning tree with at most k branch vertices if its independence number is at most 2k + 2. (ii) A connected claw-free graph of order n has a spanning tree with at most one branch vertex if the degree sum of any five independent vertices is at least n ? 2. These conditions are best possible. A related conjecture also is proposed.

Graph Homomorphisms between Trees

Abstract: In this paper we study several problems concerning the number of homomorphisms of trees We begin with an algorithm for the number of homomorphisms from a tree to any graph By using this algorithm and some transformations on trees, we study various extremal problems about the number of homomorphisms of trees These applications include a far reaching generalization and a dual of Bollobas and Tyomkyn's result concerning the number of walks in trees Some other main results of the paper are the following Denote by $\hom(H,G)$ the number of homomorphisms from a graph $H$ to a graph $G$ For any tree $T_m$ on $m$ vertices we give a general lower bound for $\hom(T_m,G)$ by certain entropies of Markov chains defined on the graph $G$ As a particular case, we show that for any graph $G$, $$\exp(H_{\lambda}(G))\lambda^{m-1}\leq\hom(T_m,G),$$ where $\lambda$ is the largest eigenvalue of the adjacency matrix of $G$ and $H_{\lambda}(G)$ is a certain constant depending only on $G$ which we call the spectral entropy of $G$ We also show that if $T_m$ is any fixed tree and $$\hom(T_m,P_n)>\hom(T_m,T_n),$$for some tree $T_n$ on $n$ vertices, then $T_n$ must be the tree obtained from a path $P_{n-1}$ by attaching a pendant vertex to the second vertex of $P_{n-1}$ All the results together enable us to show that among all trees with fixed number of vertices, the path graph has the fewest number of endomorphisms while the star graph has the most

Locating sets and numbers of graphs associated to commutative rings

Abstract: For a graph G(V, E) with order n ≥ 2, the locating code of a vertex v is a finite vector representing distances of v with respect to vertices of some ordered subset W of V(G). The set W is a locating set of G(V, E) if distinct vertices have distinct codes. A locating set containing a minimum number of vertices is a minimum locating set for G(V, E). The locating number denoted by loc(G) is the number of vertices in the minimum locating set. Let R be a commutative ring with identity 1 ≠ 0, the zero-divisor graph denoted by Γ(R), is the (undirected) graph whose vertices are the nonzero zero-divisors of R with two distinct vertices joined by an edge when the product of vertices is zero. We introduce and investigate locating numbers in zero-divisor graphs of a commutative ring R. We then extend our definition to study and characterize the locating numbers of an ideal based zero-divisor graph of a commutative ring R.

Graph travelsal operator and extensible framework inside a column store

Abstract: A system, computer-implemented method, and a computer-readable storage medium for a traversal of a property graph, are provided. The edge table of the property graph is divided into a plurality of fragments. A first fragment is selected for traversal using a set of selected vertices, where the traversal identifies a set of edges. Based on the set of edges, a set of adjacent vertices is determined during the traversal. A set of discovered vertices in the property graph is determined based on the set of selected vertices and the set of adjacent vertices.

The approximate Loebl-Koml\'os-S\'os Conjecture I: The sparse decomposition

Abstract: In a series of four papers we prove the following relaxation of the Loebl-Komlos-Sos Conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\frac12+\alpha)n$ vertices of degree at least $(1+\alpha)k$ contains each tree $T$ of order $k$ as a subgraph. The method to prove our result follows a strategy similar to approaches that employ the Szemeredi regularity lemma: we decompose the graph $G$, find a suitable combinatorial structure inside the decomposition, and then embed the tree $T$ into $G$ using this structure. Since for sparse graphs $G$, the decomposition given by the regularity lemma is not helpful, we use a more general decomposition technique. We show that each graph can be decomposed into vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties. In this paper, we introduce this novel decomposition technique. In the three follow-up papers, we find a combinatorial structure suitable inside the decomposition, which we then use for embedding the tree.

Gallai's Conjecture For Graphs of Girth at Least Four

Abstract: In 1966, Gallai conjectured that for any simple, connected graph G having n vertices, there is a path-decomposition of G having at most paths. In this article, we show that for any simple graph G having girth , there is a path-decomposition of G having at most paths, where is the number of vertices of odd degree in G and is the number of nonisolated vertices of even degree in G.

The k-metric dimension of a graph: Complexity and algorithms

Abstract: Given a connected graph G = (V,E), a set S ⊆ V is said to be a k-metric generator for G if the elements of any pair of vertices of G are distinguished by at least k elements of S, i.e., for any two different vertices u,v ∈ V , there exist at least k vertices w1,w2,...,wk ∈ S such that dG(u,wi) 6 dG(v,wi) for every i ∈ {1,...,k}. A metric generator of minimum cardinality is called a k-metric basis and its cardinality the k-metric dimension of G. We show that the problem of computing the k-metric dimension of graphs is NP-Complete. However, the problem is solved in linear time for the particular case of trees. A connected graph G is k-metric dimensional if k is the largest integer such that there exists a k-metric basis for G. We also show that the problem of finding the integer k such that a graph G is k-metric dimensional can be solved in polynomial time.

A lower bound of the surviving rate of a planar graph with girth at least seven

Abstract: Let G be a connected graph with n?2 vertices. Suppose that a fire breaks out at a vertex v of G. A firefighter starts to protect vertices. At each time interval, the firefighter protects one vertex not yet on fire. At the end of each time interval, the fire spreads to all the unprotected vertices that have a neighbor on fire. Let sn(v) denote the maximum number of vertices in G that the firefighter can save when a fire breaks out at vertex v. The surviving rate ?(G) of G is defined to be ? v?V(G)sn(v)/n 2, which is the average proportion of saved vertices. In this paper, we show that if G is a planar graph with n?2 vertices and having girth at least 7, then $\rho(G)>\frac{1}{301}$ .

The Annihilating-Ideal Graph of a Commutative Ring with Respect to an Ideal

Abstract: For a commutative ring R with identity, the annihilating-ideal graph of R, denoted 𝔸𝔾(R), is the graph whose vertices are the nonzero annihilating ideals of R with two distinct vertices joined by an edge when the product of the vertices is the zero ideal We will generalize this notion for an ideal I of R by replacing nonzero ideals whose product is zero with ideals that are not contained in I and their product lies in I and call it the annihilating-ideal graph of R with respect to I, denoted 𝔸𝔾 I (R) We discuss when 𝔸𝔾 I (R) is bipartite We also give some results on the subgraphs and the parameters of 𝔸𝔾 I (R)

Learning a hidden graph

Abstract: We study the problem of learning a hidden graph by edge-detecting queries, each of which tells whether a set of vertices induces an edge of the hidden graph or not. We provide a new information-theoretic lower bound and give a more efficient adaptive algorithm to learn a general graph with $$n$$ vertices and $$m$$ edges in $$m\log n+10m+3n$$ edge-detecting queries.

A Note on Large Rainbow Matchings in Edge-coloured Graphs

Abstract: A rainbow subgraph in an edge-coloured graph is a subgraph such that its edges have distinct colours. The minimum colour degree of a graph is the smallest number of distinct colours on the edges incident with a vertex over all vertices. Kostochka, Pfender, and Yancey showed that every edge-coloured graph on n vertices with minimum colour degree at least k contains a rainbow matching of size at least k, provided \({n\geq \frac{17}{4}k^2}\) . In this paper, we show that n ≥ 4k − 4 is sufficient for k ≥ 4.

Interval deletion is fixed-parameter tractable

Abstract: We study the minimum interval deletion problem, which asks for the removal of a set of at most k vertices to make a graph on n vertices into an interval graph. We present a parameterized algorithm of runtime 10k · nO(1) for this problem, thereby showing its fixed-parameter tractability.

Discovering the structure of complex networks by minimizing cyclic bandwidth sum

Abstract: Getting a labeling of vertices close to the structure of the graph has been proved to be of interest in many applications e.g., to follow smooth signals indexed by the vertices of the network. This question can be related to a graph labeling problem known as the cyclic bandwidth sum problem. It consists in finding a labeling of the vertices of an undirected and unweighted graph with distinct integers such that the sum of (cyclic) difference of labels of adjacent vertices is minimized. Although theoretical results exist that give optimal value of cyclic bandwidth sum for standard graphs, there are neither results in the general case, nor explicit methods to reach this optimal result. In addition to this lack of theoretical knowledge, only a few methods have been proposed to approximately solve this problem. In this paper, we introduce a new heuristic to find an approximate solution for the cyclic bandwidth sum problem, by following the structure of the graph. The heuristic is a two-step algorithm: the first step consists of traversing the graph to find a set of paths which follow the structure of the graph, using a similarity criterion based on the Jaccard index to jump from one vertex to the next one. The second step is the merging of all obtained paths, based on a greedy approach that extends a partial solution by inserting a new path at the position that minimizes the cyclic bandwidth sum. The effectiveness of the proposed heuristic, both in terms of performance and time execution, is shown through experiments on graphs whose optimal value of CBS is known as well as on real-world networks, where the consistence between labeling and topology is highlighted. An extension to weighted graphs is also proposed.

Some Remarks on the Graph Γ I (R)

Abstract: Let R be a commutative ring with nonzero identity and Z(R) its set of zero-divisors. The zero-divisor graph of R is Γ(R), with vertices Z(R)∖{0} and distinct vertices x and y are adjacent if and only if xy = 0. For a proper ideal I of R, the ideal-based zero-divisor graph of R is Γ I (R), with vertices {x ∈ R∖I | xy ∈ I for some y ∈ R∖I} and distinct vertices x and y are adjacent if and only if xy ∈ I. In this article, we study the relationship between the two graphs Γ(R) and Γ I (R). We also determine when Γ I (R) is either a complete graph or a complete bipartite graph and investigate when Γ I (R) ≅ Γ(S) for some commutative ring S.

Vertex-Colored Encompassing Graphs

Abstract: It is shown that every disconnected vertex-colored plane straight line graph with no isolated vertices can be augmented (by adding edges) into a connected plane straight line graph such that the new edges respect the coloring and the degree of every vertex increases by at most two. The upper bound for the increase of vertex degrees is best possible: there are input graphs that require the addition of two new edges incident to a vertex. The exclusion of isolated vertices is necessary: there are input graphs with isolated vertices that cannot be augmented to a connected vertex-colored plane straight line graph.

On $r$-Simple $k$-Path

Abstract: An r-simple k-path is a path in the graph of length k that passes through each vertex at most r times. The r-SIMPLE k-PATH problem, given a graph G as input, asks whether there exists an r-simple k-path in G. We first show that this problem is NP-Complete. We then show that there is a graph G that contains an r-simple k-path and no simple path of length greater than 4logk/logr. So this, in a sense, motivates this problem especially when one’s goal is to find a short path that visits many vertices in the graph while bounding the number of visits at each vertex.

Forcing $k$-Repetitions in Degree Sequences

Abstract: One of the most basic results in graph theory states that every graph with at least two vertices has two vertices with the same degree. Since there are graphs without $3$ vertices of the same degree, it is natural to ask if for any fixed $k$, every graph $G$ is "close" to a graph $G'$ with $k$ vertices of the same degree. Our main result in this paper is that this is indeed the case. Specifically, we show that for any positive integer $k$, there is a constant $C=C(k)$, so that given any graph $G$, one can remove from $G$ at most $C$ vertices and thus obtain a new graph $G'$ that contains at least $\min\{k,|G|-C\}$ vertices of the same degree. Our main tool is a multidimensional zero-sum theorem for integer sequences, which we prove using an old geometric approach of Alon and Berman.