The type of business relationships between the Internet autonomous systems (AS) determines the BGP inter-domain routing. Previous works on inferring AS relationships relied on the connectivity information between ASes. In this paper we infer AS relationships by analysing the routing polices of ASes encoded in the BGP attributes Communities and the Locpref. We accumulate BGP data from RouteViews, RIPE RIS and the public Route Servers in August 2010 and February 2011. Based on the routing policies extracted from data of the two BGP attributes, we obtain AS relationships for 39 links in our data, which include all links among the Tier-1 ASes and most links between Tier-1 and Tier-2 ASes. We also reveal a number of special AS relationships, namely the hybrid relationship, the partial-transit relationship, the indirect peering relationship and the backup links. These special relationships are relevant to a better understanding of the Internet routing. Our work provides a profound methodological progress for inferring the AS relationships.
In @cite_31 the Acyclic Type of Relationship (AToR) problem is defined. According to AToR when p2c relationships are assigned a directed edge, the resulting AS graph should be acyclic. In @cite_4 the authors validate the acyclicity of the AS graph and propose a heuristic to solve the maximal AToR problem.
@cite_31 @cite_4
2105742585 2121041505
The Internet connectivity in the autonomous system (AS) level reflects the commercial relationship between ASes. A connection between two ASes could be of type customer-provider when one AS is a provider of the other AS, or of type peer-peer, if they are peering ASes. This commercial relationship induces a global hierarchical structure which is a key ingredient in the ability to understand the topological structure of the AS connectivity graph. Unfortunately, it is very difficult to collect data regarding the actual type of the relationships between ASes, and in general this information is not part of the collected AS connectivity data. The Type of Relationship (ToR) problem attempts to address this shortcoming, by inferring the type of relationship between connected ASes based on their routing policies. However, the approaches presented so far are local in nature and do not capture the global hierarchical structure. In this work we define a novel way to infer this type of relationship from the collected data, taking into consideration both local policies and global hierarchy constrains. We define the Acyclic Type of Relationship AToR problem that captures this global hierarchy and present an efficient algorithm that allows determining if there is a hierarchical assignment without invalid paths. We then show that the related general optimization problem is NP-complete and present a 2 3 approximation algorithm where the objective function is to minimize the total number of local policy mismatches. We support our approach by extensive experiments and simulation results showing that our algorithms classify the type of relationship between ASes much better than all previous algorithms. An experimental study of the feasibility and accuracy of the acyclicity approach introduced in [14] for the inference of business relationships among autonomous systems (ASes) is provided. We investigate the maximum acyclic type-of-relationship problem: on a given set of AS paths, find a maximum-cardinality subset which allows an acyclic and valley-free orientation. Inapproximability and NP-hardness results for this problem are presented and a heuristic is designed. The heuristic is experimentally compared to most of the state-of-the-art algorithms on a reliable data set. It turns out that the proposed heuristic produces the least number of misclassified customer-to-provider relationships among the tested algorithms. Moreover, it is flexible in handling pre-knowledge in the sense that already a small amount of correct relationships is enough to produce a high-quality relationship classification. Furthermore, the reliable data set is used to validate the acyclicity assumptions. The findings demonstrate that acyclicity notions should be an integral part of models of AS relationships. Graph drawings are a basic component of user interfaces that display relationships between objects. Generating incrementally stable layouts is important for many applications. This paper describes DynaDAG, a new heuristic for incremental layout of directed acyclic graphs drawn as hierarchies, and its application in the DynaGraph system. In this paper we study the acyclic 3-colorability of some subclasses of planar graphs. First, we show that there exist infinite classes of cubic planar graphs that are not acyclically 3-colorable. Then, we show that every planar graph has a subdivision with one vertex per edge that is acyclically 3-colorable and provide a linear-time coloring algorithm. Finally, we characterize the series-parallel graphs for which every 3-coloring is acyclic and provide a linear-time recognition algorithm for such graphs. The difference Laplacian on a square lattice in Rn has been stud- ied by many authors. In this paper an analogous difference operator is studied for an arbitrary graph. It is shown that many properties of the Laplacian in the continuous setting (e.g. the maximum principle, the Harnack inequality, and Cheeger's bound for the lowest eigenvalue) hold for this difference oper- ator. The difference Laplacian governs the random walk on a graph, just as the Laplace operator governs the Brownian motion. As an application of the theory of the difference Laplacian, it is shown that the random walk on a class of graphs is transient. The random walks we consider are defined as follows. Let K be a connected graph (i.e. a one dimensional simplicial complex). For a vertex x E K, let m(x) denote the number of edges emanating from x. The probability that a particle moves from x to another vertex y E K is l m(x) if x and y are connected by an edge and it is zero otherwise. As observed by Courant, Friedrichs and Lewy (CFL) for the case of a square lattice in the plane this random walk is intimately related to the difference analog of the Laplacian This paper introduces a new problem of inferring strings from graphs, and inferring strings from arrays. Given a graph G or an array A, we infer a string that suits the graph, or the array, under some condition. Firstly, we solve the problem of finding a string w such that the directed acyclic subsequence graph (DASG )o fw is isomorphic to a given graph G. Secondly, we consider directed acyclic word graphs (DAWGs) in terms of string inference. Finally, we consider the problem of finding a string w of a minimal size alphabet, such that the suffix array (SA )o f w is identical to a given permutation p = p1 ,...,p n of integers 1 ,...,n . Each of our three algorithms solving the above problems runs in linear time with respect to the input size. A heterogeneous information network (HIN) is a graph model in which objects and edges are annotated with types. Large and complex databases, such as YAGO and DBLP, can be modeled as HINs. A fundamental problem in HINs is the computation of closeness, or relevance, between two HIN objects. Relevance measures can be used in various applications, including entity resolution, recommendation, and information retrieval. Several studies have investigated the use of HIN information for relevance computation, however, most of them only utilize simple structure, such as path, to measure the similarity between objects. In this paper, we propose to use meta structure, which is a directed acyclic graph of object types with edge types connecting in between, to measure the proximity between objects. The strength of meta structure is that it can describe complex relationship between two HIN objects (e.g., two papers in DBLP share the same authors and topics). We develop three relevance measures based on meta structure. Due to the computational complexity of these measures, we further design an algorithm with data structures proposed to support their evaluation. Our extensive experiments on YAGO and DBLP show that meta structure-based relevance is more effective than state-of-the-art approaches, and can be efficiently computed. We consider the problem of learning causal directed acyclic graphs from an observational joint distribution. One can use these graphs to predict the outcome of interventional experiments, from which data are often not available. We show that if the observational distribution follows a structural equation model with an additive noise structure, the directed acyclic graph becomes identifiable from the distribution under mild conditions. This constitutes an interesting alternative to traditional methods that assume faithfulness and identify only the Markov equivalence class of the graph, thus leaving some edges undirected. We provide practical algorithms for finitely many samples, RESIT (regression with subsequent independence test) and two methods based on an independence score. We prove that RESIT is correct in the population setting and provide an empirical evaluation. A structured organization of information is typically required by symbolic processing. On the other hand, most connectionist models assume that data are organized according to relatively poor structures, like arrays or sequences. The framework described in this paper is an attempt to unify adaptive models like artificial neural nets and belief nets for the problem of processing structured information. In particular, relations between data variables are expressed by directed acyclic graphs, where both numerical and categorical values coexist. The general framework proposed in this paper can be regarded as an extension of both recurrent neural networks and hidden Markov models to the case of acyclic graphs. In particular we study the supervised learning problem as the problem of learning transductions from an input structured space to an output structured space, where transductions are assumed to admit a recursive hidden state-space representation. We introduce a graphical formalism for representing this class of adaptive transductions by means of recursive networks, i.e., cyclic graphs where nodes are labeled by variables and edges are labeled by generalized delay elements. This representation makes it possible to incorporate the symbolic and subsymbolic nature of data. Structures are processed by unfolding the recursive network into an acyclic graph called encoding network. In so doing, inference and learning algorithms can be easily inherited from the corresponding algorithms for artificial neural networks or probabilistic graphical model. Many underlying relationships among data in several areas of science and engineering, e.g., computer vision, molecular chemistry, molecular biology, pattern recognition, and data mining, can be represented in terms of graphs. In this paper, we propose a new neural network model, called graph neural network (GNN) model, that extends existing neural network methods for processing the data represented in graph domains. This GNN model, which can directly process most of the practically useful types of graphs, e.g., acyclic, cyclic, directed, and undirected, implements a function tau(G,n) isin IRm that maps a graph G and one of its nodes n into an m-dimensional Euclidean space. A supervised learning algorithm is derived to estimate the parameters of the proposed GNN model. The computational cost of the proposed algorithm is also considered. Some experimental results are shown to validate the proposed learning algorithm, and to demonstrate its generalization capabilities. Motivated by a problem of transmitting supplemental data over broadcast channels (Birk and Kol, INFOCOM 1998), we study the following coding problem: a sender communicates with n receivers R1,..., Rn. He holds an input x ∈ 0,01l n and wishes to broadcast a single message so that each receiver Ri can recover the bit xi. Each Ri has prior side information about x, induced by a directed graph Grain nodes; Ri knows the bits of a; in the positions j | (i,j) is an edge of G .G is known to the sender and to the receivers. We call encoding schemes that achieve this goal INDEXcodes for 0,1 n with side information graph G. In this paper we identify a measure on graphs, the minrank, which exactly characterizes the minimum length of linear and certain types of nonlinear INDEX codes. We show that for natural classes of side information graphs, including directed acyclic graphs, perfect graphs, odd holes, and odd anti-holes, minrank is the optimal length of arbitrary INDEX codes. For arbitrary INDEX codes and arbitrary graphs, we obtain a lower bound in terms of the size of the maximum acyclic induced subgraph. This bound holds even for randomized codes, but has been shown not to be tight. In this work we consider temporal networks, i.e. networks defined by a labeling ( ) assigning to each edge of an underlying graphG a set of discrete time-labels. The labels of an edge, which are natural numbers, indicate the discrete time moments at which the edge is available. We focus on path problems of temporal networks. In particular, we consider time-respecting paths, i.e. paths whose edges are assigned by ( ) a strictly increasing sequence of labels. We begin by giving two efficient algorithms for computing shortest time-respecting paths on a temporal network. We then prove that there is a natural analogue of Menger’s theorem holding for arbitrary temporal networks. Finally, we propose two cost minimization parameters for temporal network design. One is the temporality of G, in which the goal is to minimize the maximum number of labels of an edge, and the other is the temporal cost of G, in which the goal is to minimize the total number of labels used. Optimization of these parameters is performed subject to some connectivity constraint. We prove several lower and upper bounds for the temporality and the temporal cost of some very basic graph families such as rings, directed acyclic graphs, and trees. Examines the vertex deletion problem for weighted directed acyclic graphs (WDAGs). The objective is to delete the fewest number of vertices so that the resulting WDAG has no path of length > spl delta . Several simplified versions of this problem are shown to be NP-hard. However, the problem is solved in linear time when the WDAG is a rooted tree, and in quadratic time when the WDAG is a series-parallel graph. > Database schemes (which, intuitively, are collections of table skeletons) can be viewed as hypergraphs. (A hypergraph is a generalization of an ordinary undirected graph, such that an edge need not contain exactly two nodes, but can instead contain an arbitrary nonzero number of nodes.) Unlike the situation for ordinary undirected graphs, there are several natural, nonequivalent notions of acyclicity for hypergraphs (and hence for database schemes). A large number of desirable properties of database schemes fall into a small number of equivalence classes, each completely characterized by the degree of acyclicity of the scheme. This paper is intended to be an informal introduction, in which the focus is mainly on the originally studied (and least restrictive) degree of acyclicity. We consider fair allocation of indivisible items under an additional constraint: there is an undirected graph describing the relationship between the items, and each agent's share must form a connected subgraph of this graph. This framework captures, e.g., fair allocation of land plots, where the graph describes the accessibility relation among the plots. We focus on agents that have additive utilities for the items, and consider several common fair division solution concepts, such as proportionality, envy-freeness and maximin share guarantee. While finding good allocations according to these solution concepts is computationally hard in general, we design efficient algorithms for special cases where the underlying graph has simple structure, and or the number of agents -or, less restrictively, the number of agent types- is small. In particular, despite non-existence results in the general case, we prove that for acyclic graphs a maximin share allocation always exists and can be found efficiently. We present a definition of cause and effect in terms of decision-theoretic primitives and thereby provide a principled foundation for causal reasoning. Our definition departs from the traditional view of causation in that causal assertions may vary with the set of decisions available. We argue that this approach provides added clarity to the notion of cause. Also in this paper, we examine the encoding of causal relationships in directed acyclic graphs. We describe a special class of influence diagrams, those in canonical form, and show its relationship to Pearl's representation of cause and effect. Finally, we show how canonical form facilitates counterfactual reasoning. Several authors have investigated the correspondence between graph rewriting and term rewriting. Almost invariably they have considered only acyclic graphs. Yet cyclic graphs naturally arise from certain optimizations in implementing functional languages. They correspond to infinite terms, and their reductions correspond to transfinite term-reduction sequences, which have recently received detailed attention. We formalize the close correspondence between finitary cyclic graph rewriting and a restricted form of infinitary term rewriting, called rational term rewriting. This subsumes the known relation between finitary acyclic graph rewriting and finitary term rewriting. Surprisingly, the correspondence breaks down for general infinitary rewriting Abstract We consider the problem of orienting the edges of a planar graph in such a way that the out-degree of each vertex is minimized. If, for each vertex v , the out-degree is at most d , then we say that such an orientation is d -bounded. We prove the following results: • Each planar graph has a 5-bounded acyclic orientation, which can be constructed in linear time. • Each planar graph has a 3-bounded orientation, which can be constructed in linear time. • A 6-bounded acyclic orientation, and a 3-bounded orientation, of each planar graph can each be constructed in parallel time O(log n log ∗ n ) on an EREW PRAM, using O( n log n log ∗ n ) processors. As an application of these results, we present a data structure such that each entry in the adjacency matrix of a planar graph can be looked up in constant time. The data structure uses linear storage, and can be constructed in linear time. The algorithm for Lovasz Local Lemma by Moser and Tardos gives a constructive way to prove the existence of combinatorial objects that satisfy a system of constraints. We present an alternative probabilistic analysis of the algorithm that does not involve reconstructing the history of the algorithm from the witness tree. We apply our technique to improve the best known upper bound to acyclic chromatic index. Specifically we show that a graph with maximum degree Δ has an acyclic proper edge coloring with at most ⌈3.74(Δ − 1)⌉ + 1 colors, whereas the previously known best bound was 4(Δ − 1). The same technique is also applied to improve corresponding bounds for graphs with bounded girth. An interesting aspect of this application is that the probability of the "undesirable" events do not have a uniform upper bound, i.e. it constitutes a case of the asymmetric Lovasz Local Lemma. We consider the following class of problems: given a graph, find the maximum number of nodes inducing a subgraph that satisfies a desired property π, such as planar, acyclic, bipartite, etc. We show that this problem is hard to approximate for any property π on directed or undirected graphs that is nontrivial and hereditary on induced subgraphs.