Showing papers on "Connectivity published in 2003"
TL;DR: The critical transmitting range for connectivity in wireless ad hoc networks is analyzed and insight into how mobility affects connectivity is yielded and useful trade offs between communication capability and energy consumption are revealed.
Abstract: We analyze the critical transmitting range for connectivity in wireless ad hoc networks. More specifically, we consider the following problem: assume n nodes, each capable of communicating with nodes within a radius of r, are randomly and uniformly distributed in a d-dimensional region with a side of length l; how large must the transmitting range r be to ensure that the resulting network is connected with high probability? First, we consider this problem for stationary networks, and we provide tight upper and lower bounds on the critical transmitting range for one-dimensional networks and nontight bounds for two and three-dimensional networks. Due to the presence of the geometric parameter l in the model, our results can be applied to dense as well as sparse ad hoc networks, contrary to existing theoretical results that apply only to dense networks. We also investigate several related questions through extensive simulations. First, we evaluate the relationship between the critical transmitting range and the minimum transmitting range that ensures formation of a connected component containing a large fraction (e.g., 90 percent) of the nodes. Then, we consider the mobile version of the problem, in which nodes are allowed to move during a time interval and the value of r ensuring connectedness for a given fraction of the interval must be determined. These results yield insight into how mobility affects connectivity and they also reveal useful trade offs between communication capability and energy consumption.
452 citations
24 Aug 2003
TL;DR: A novel local algorithm that returns all variables with direct edges to and from a target variable T as well as a local algorithms that returns the Markov Blanket of T, which are promising not only for discovery of local causal structure, and variable selection for classification, but also for the induction of complete BNs.
Abstract: Data Mining with Bayesian Network learning has two important characteristics: under conditions learned edges between variables correspond to casual influences, and second, for every variable T in the network a special subset (Markov Blanket) identifiable by the network is the minimal variable set required to predict T. However, all known algorithms learning a complete BN do not scale up beyond a few hundred variables. On the other hand, all known sound algorithms learning a local region of the network require an exponential number of training instances to the size of the learned region.The contribution of this paper is two-fold. We introduce a novel local algorithm that returns all variables with direct edges to and from a target variable T as well as a local algorithm that returns the Markov Blanket of T. Both algorithms (i) are sound, (ii) can be run efficiently in datasets with thousands of variables, and (iii) significantly outperform in terms of approximating the true neighborhood previous state-of-the-art algorithms using only a fraction of the training size required by the existing methods. A fundamental difference between our approach and existing ones is that the required sample depends on the generating graph connectivity and not the size of the local region; this yields up to exponential savings in sample relative to previously known algorithms. The results presented here are promising not only for discovery of local causal structure, and variable selection for classification, but also for the induction of complete BNs.
327 citations
01 Jun 2003
TL;DR: In this paper, the authors propose an approach to topology control based on the principle of maintaining the number of neighbors of every node equal to or slightly below a specific value k. The approach enforces symmetry on the resulting communication graph, thereby easing the operation of higher layer protocols.
Abstract: We propose an approach to topology control based on the principle of maintaining the number of neighbors of every node equal to or slightly below a specific value k. The approach enforces symmetry on the resulting communication graph, thereby easing the operation of higher layer protocols. To evaluate the performance of our approach, we estimate the value of k that guarantees connectivity of the communication graph with high probability. We then define k-Neigh, a fully distributed, asynchronous, and localized protocol that follows the above approach and uses distance estimation. We prove that k-Neigh terminates at every node after a total of 2n messages have been exchanged (with n nodes in the network) and within strictly bounded time. Finally, we present simulations results which show that our approach is about 20% more energy-efficient than a widely-studied existing protocol.
260 citations
16 Sep 2003
TL;DR: Two mobile agents having distinct identifiers and located in nodes of an unknown anonymous connected graph, have to meet at some node of the graph and fast deterministic algorithms for this rendezvous problem are presented.
Abstract: Two mobile agents having distinct identifiers and located in nodes of an unknown anonymous connected graph, have to meet at some node of the graph. We present fast deterministic algorithms for this rendezvous problem.
225 citations
TL;DR: An arc of a fuzzy graph is called strong if its weight is at least as great as the strength of connectedness of its end nodes when it is deleted.
218 citations
01 Jan 2003
TL;DR: Methods characterizing average measures of connectivity, similarity of connection patterns, connectedness and components, paths, walks and cycles, distances, cluster indices, ranges and shortcuts, and node and edge cut sets are introduced and discussed in a neurobiological context.
Abstract: This paper summarizes a set of graph theory methods that are of special relevance to the computational analysis of neural connectivity patterns. Methods characterizing average measures of connectivity, similarity of connection patterns, connectedness and components, paths, walks and cycles, distances, cluster indices, ranges and shortcuts, and node and edge cut sets are introduced and discussed in a neurobiological context. A set of Matlab functions implementing these methods is available for download at http://php.indiana.edu/~osporns/graphmeasures.htm.
211 citations
16 Sep 2003
TL;DR: In this paper, asymptotically optimal O(log n)-approximation algorithms for three power assignment problems are presented for strong connectivity, symmetric connectivity and broadcast.
Abstract: Used for topology control in ad-hoc wireless networks, Power Assignment is a family of problems, each defined by a certain connectivity constraint (such as strong connectivity) The input consists of a directed complete weighted graph G=(V,c). The power of a vertex u in a directed spanning subgraph H is given by PH (u) = max uv ∈ E(H) c(uv). The power of H is given by \(p(H) = \sum_{u \in v}p{\sc H}(u)\), Power Assignment seeks to minimize p(H) while H satisfies the given connectivity constraint. We present asymptotically optimal O(log n)-approximation algorithms for three Power Assignment problems: Min-Power Strong Connectivity, Min-Power Symmetric Connectivity (the undirected graph having an edge uv iff H has both uv and vu must be connected) and Min-Power Broadcast (the input also has r ∈ V , and H must be a r-rooted outgoing spanning arborescence).
160 citations
TL;DR: The Graph k-Cut problem, that of finding a set of edges of minimum total weight, in an edge-weighted graph, such that their removal from the graph results in a graph having at least k connected components, is shown to be hard for the parameterized complexity class W.
147 citations
21 Sep 2003
TL;DR: An energy model whose minimum energy drawings reveal the clusters of the drawn graph is introduced, and this characterization shows in what sense the drawings separate clusters, and how the distance of separated clusters to the other nodes can be interpreted.
Abstract: We introduce an energy model whose minimum energy drawings reveal the clusters of the drawn graph. Here a cluster is a set of nodes with many internal edges and few edges to nodes outside the set. The drawings of the best-known force and energy models do not clearly show clusters for graphs whose diameter is small relative to the number of nodes. We formally characterize the minimum energy drawings of our energy model. This characterization shows in what sense the drawings separate clusters, and how the distance of separated clusters to the other nodes can be interpreted.
131 citations
01 Mar 2003
TL;DR: This approach emphasizes reachability via an algorithm within the implicit graph structure underlying a recommender dataset and allows us to consider questions relating algorithmic parameters to properties of the datasets.
Abstract: We present a novel framework for studying recommendation algorithms in terms of the ‘jumps’ that they make to connect people to artifacts. This approach emphasizes reachability via an algorithm within the implicit graph structure underlying a recommender dataset and allows us to consider questions relating algorithmic parameters to properties of the datasets. For instance, given a particular algorithm ‘jump,’ what is the average path length from a person to an artifact? Or, what choices of minimum ratings and jumps maintain a connected graph? We illustrate the approach with a common jump called the ‘hammock’ using movie recommender datasets.
130 citations
TL;DR: The problem of finding a most vital node of a given shortest path has been studied because of its implications in network management, where it is important to know in advance which component failure will affect network efficiency the most.
TL;DR: In this paper, it was shown that every fullerene graph G is cyclically 5-edge connected, i.e., G cannot be separated into two components, each containing a cycle, by deletion of fewer than five edges.
Abstract: It is shown that every fullerene graph G is cyclically 5-edge-connected, i.e., that G cannot be separated into two components, each containing a cycle, by deletion of fewer than five edges. The result is then generalized to the case of (k,6)-cages, i.e., polyhedral cubic graphs whose faces are only k-gons and hexagons. Certain linear and exponential lower bounds on the number of perfect matchings in such graphs are also established.
TL;DR: The multiroot problem generalizes the min-cost $k$-connected subgraph problem and improves the approximation guarantee from $3 to $2+{\lfloor (k-1)/2 \rfloor}/{k} < 2.5".
Abstract: Given a graph (directed or undirected) with costs on the edges, and an integer $k$, we consider the problem of finding a $k$-node connected spanning subgraph of minimum cost. For the general instance of the problem (directed or undirected), there is a simple $2k$-approximation algorithm. Better algorithms are known for various ranges of $n,k$. For undirected graphs with metric costs Khuller and Raghavachari gave a $( 2+{2(k-1)}/{n})$-approximation algorithm. We obtain the following results: (i) For arbitrary costs, a $k$-approximation algorithm for undirected graphs and a $(k+1)$-approximation algorithm for directed graphs. (ii) For metric costs, a $(2+({k-1})/{n})$-approximation algorithm for undirected graphs and a $(2+{k}/{n})$-approximation algorithm for directed graphs. For undirected graphs and $k=6,7$, we further improve the approximation ratio from $k$ to $\lceil (k+1)/2 \rceil=4$; previously, $\lceil (k+1)/2 \rceil$-approximation algorithms were known only for $k \leq 5$. We also give a fast $3$-approximation algorithm for $k=4$. The multiroot problem generalizes the min-cost $k$-connected subgraph problem. In the multiroot problem, requirements $k_u$ for every node $u$ are given, and the aim is to find a minimum-cost subgraph that contains $\max\{k_u,k_v\}$ internally disjoint paths between every pair of nodes $u,v$. For the general instance of the problem, the best known algorithm has approximation ratio $2k$, where $k=\max k_u$. For metric costs there is a 3-approximation algorithm. We consider the case of metric costs, and, using our techniques, improve for $k \leq 7$ the approximation guarantee from $3$ to $2+{\lfloor (k-1)/2 \rfloor}/{k} < 2.5$.
TL;DR: The pebbling number of a connected graph is the smallest number f(G) such that any distribution of f (G) pebbles on G allows one pebble to be moved to any specified, but arbitrary vertex by a sequence of pebbled moves.
Abstract: Chung defined a pebbling move on a graph G to be the removal of two pebbles from one vertex and the addition of one pebble to an adjacent vertex. The pebbling number of a connected graph is the smallest number f(G) such that any distribution of f(G) pebbles on G allows one pebble to be moved to any specified, but arbitrary vertex by a sequence of pebbling moves. Graham conjectured that for any connected graphs G and H, f(G×H)≤ f(G)f(H). We prove Graham's conjecture when G is a cycle for a variety of graphs H, including all cycles. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 141–154, 2003
TL;DR: This note shows an application of MA ordering to the maximum flow problem with integral capacities to get a new polynomial-time algorithm.
TL;DR: If the maximum degree of a graph G with maximum degree can be colored properly in at most colors unless G = K+1 or G is an odd cycle, then there is a positive answer to Albertson's question when the pairwise distance between vertices of P is at least 8; moreover, this distance is optimal.
Abstract: Let G be a graph with maximum degree 3 not equal toK+1 and let P be a subset of vertices with pairwise distance, d(P ), between them at least 8. Let each vertex x be assigned a list of colors of size if x 2 V \P and 1 if x 2 P .W e prove that it is possible to color V (G) such that adjacent vertices receive dierent colors and each vertex has a color from its list. We show that d(P ) cannot be improved. This generalization of Brooks’ theorem answers the following question of Albertson positively: If G and P are objects described above, can any coloring of P in at most colors be extended to a proper coloring of G in at most colors? We say that a vertex-coloring of a graph G =( V,E )i sproper if the colors used on adjacent vertices are distinct. For an assignment of a color set (typically called a list) l(x) to each vertex x 2 V , we say that vertices are colored from their lists by a coloring c if c(x) 2 l(x) for each x 2 V ; c is called a list-coloring of G .A coloringc of V (G) extends a coloring c 0 of vertices in P if it is a proper coloring with c(x )= c 0 (x) for each x 2 P .W e denote by dG(x) the degree of x in a graph G and by G[X] the subgraph of G induced by a set of vertices X. The classic Brooks’ theorem states that any simple connected graph G with maximum degree can be colored properly in at most colors unless G = K+1 or G is an odd cycle. Recently, Albertson posed the following question. Take a graph described above, precolor a fixed set of vertices P in colors arbitrarily. Under what condition on P can we extend that coloring to a proper coloring of G in at most colors? He asks whether this condition is a large distance between the vertices in P . Albertson noticed though, that the maximum degree of a graph should be at least three. Indeed, it is easy to see that one cannot obtain a proper coloring of a path with an even number of vertices in two colors if the end-points are precolored in the same color. Here, we show that if the maximum degree is at least three, then there is a positive answer to Albertson’s question when the pairwise distance, d(P ), between vertices of P is at least 8; moreover, this distance is optimal. The color extension problem is closely related to the concept of a list-coloring
30 Jun 2003
TL;DR: It is shown that for any undirected graph with order n, the cover time and the mean hitting time with respect to P(1) are bounded by O(n2 log n) and O( n2), respectively.
Abstract: It is just amazing that both of the mean hitting time and the cover time of a random walk on a finite graph, in which the vertex visited next is selected from the adjacent vertices at random with the same probability, are bounded by O(n3) for any undirected graph with order n, despite of the lack of global topological information. Thus a natural guess is that a better transition matrix is designable if more topological information is available. For any undirected connected graph G = (V,E), let P(β) = (puvβ)u,v∈V be a transition matrix defined by puvβ = exp [-βU(u, v)]/Σw∈N(u) exp [-βU(u, w)] for u∈V, v∈N(u), where β is a real number, N(u) is the set of vertices adjacent to a vertex u, deg(u) = |N(u)|, and U(., .) is a potential function defined as U(u, v) = log (max {deg(u), deg(v)}) for u∈V, v∈N(u).
In this paper, we show that for any undirected graph with order n, the cover time and the mean hitting time with respect to P(1) are bounded by O(n2 log n) and O(n2), respectively. We further show that P(1) is best possible with respect to the mean hitting time, in the sense that the mean hitting time of a path graph of order n, with respect to any transition matrix, is Ω(n2).
TL;DR: In this article, the maximal number of sign graphs of an eigenvalue of a generalized Laplacian of a tree was characterized and an O(n 2 )-time algorithm was given to find an Eigenvector with maximum number of signs.
TL;DR: This paper finds the graphs on which the zeroth-order connectivity index, equal to the sum of degrees of vertices of G(n,m) raised to the power -½, attains maximum.
TL;DR: In this article, the source location problem with flow requirements in directed networks is addressed. But the flow requirements are not defined. And the source locations are not specified. But flow requirements do not change.
Abstract: (2003). Source location problem with flow requirements in directed networks. Optimization Methods and Software: Vol. 18, No. 4, pp. 427-435.
TL;DR: In this article, a multiset analog of Lovasz's version of the Kruskal-Katona Theorem and an analog of the Bollobas-Thomason threshold result were shown.
30 Jun 2003
TL;DR: This paper gives a complete complexity classification of the role assignment problem for connected graphs and shows further corollaries for disconnected graphs and related problems.
Abstract: A graph G is R-role assignable if there is a locally surjective homomorphism from G to R, i.e. a vertex mapping r: VG → VR, such that the neighborhood relation is preserved: r(NG(u)) = NR(r(u)). Kristiansen and Telle conjectured that the decision problem whether such a mapping exists is an NP-complete problem for any connected graph R on at least three vertices. In this paper we prove this conjecture, i.e. we give a complete complexity classification of the role assignment problem for connected graphs. We show further corollaries for disconnected graphs and related problems.
Journal Article•
TL;DR: The problem of finding a planar embedding of a (planar) graph with a prescribed Euclidean length on every edge has been shown to be tractable in linear time as discussed by the authors.
Abstract: We consider the problem of finding a planar embedding of a (planar) graph with a prescribed Euclidean length on every edge. There has been substantial previous work on the problem without the planarity restrictions, which has close connections to rigidity theory, and where it is easy to see that the problem is NP-hard. In contrast, we show that the problem is tractable-indeed, solvable in linear time on a real RAM-for planar embeddings of planar 3-connected triangulations, even if the outer face is not a triangle. This result is essentially tight: the problem becomes NP-hard if we consider instead planar embeddings of planar 3-connected infinitesimally rigid graphs, a natural relaxation of triangulations in this context.
TL;DR: In this article, a d'Alembert type formula for the solution of the Cauchy problem for the wave equation on finite weighted networks is presented. But the authors do not discuss the periodicity in time of the solution in terms of the spectrum of the discrete graph associated with the network and present two significant examples to illustrate the general analysis.
TL;DR: A simplification of the algorithm of Salembier e.a.ved.
Posted Content•
TL;DR: It is conjectured that natural selection tuned the average connectivity in such a way that the network reaches a sparse graph of connections and the scaling laws and the essential ingredients for building a modular net emerge for free close to such transition.
Abstract: Background: Recent models of genome-proteome evolution have shown that some of the key traits displayed by the global structure of cellular networks might be a natural result of a duplication-diversification (DD) process One of the consequences of such evolution is the emergence of a small world architecture together with a scale-free distribution of interactions Here we show that the domain of parameter space were such structure emerges is related to a phase transition phenomenon At this transition point, modular architecture spontaneously emerges as a byproduct of the DD process
Results: Although the DD models lack any functionality and are thus free from meeting functional constraints, they show the observed features displayed by the real proteome maps when tuned close to a sharp transition point separating a highly connected graph from a disconnected system Close to such boundary, the maps are shown to display scale-free hierarchical organization, behave as small worlds and exhibit modularity
Conclusions: It is conjectured that natural selection tuned the average connectivity in such a way that the network reaches a sparse graph of connections One consequence of such scenario is that the scaling laws and the essential ingredients for building a modular net emerge for free close to such transition
TL;DR: In this paper, three exact and well-known algorithms for maximum common subgraph detection are described, a large database containing various categories of pairs of graphs is presented, and the performance of the three algorithms is evaluated.
Abstract: A graph g is called a maximum common subgraph of two graphs, g1 and g2, if there exists no other common subgraph of g1 and g2 that has more nodes than g. For the maximum common subgraph problem, exact and inexact algorithms are known from the literature. Nevertheless, until now no effort has been done for characterizing their performance, mainly for the lack of a large database of graphs. In this paper, three exact and well-known algorithms for maximum common subgraph detection are described. Moreover, a large database containing various categories of pairs of graphs (e.g. randomly connected graphs, meshes, bounded valence graphs...), having a maximum common subgraph of at least two nodes, is presented, and the performance of the three algorithms is evaluated on this database.
TL;DR: This paper provides an efficient algorithm for solving the problem of restoring, by means of a minimum weight set of replacement edges, the 2 -edge-connectivity of a 2-edge-connected communication network undergoing a link failure in polynomial time.
Abstract: Given a 2-edge-connected, real weighted graph G with n vertices and m edges, the 2-edge-connectivity augmentation problem is that of finding a minimum weight set of edges of G to be added to a spanning subgraph H of G to make it 2-edge-connected. While the general problem is NP-hard and 2 -approximable, in this paper we prove that it becomes polynomial time solvable if H is a depth-first search tree of G . More precisely, we provide an efficient algorithm for solving this special case which runs in O(M · ?(M,n)) time, where ? is the classic inverse of Ackermann's function and M=m · ?(m,n) . This algorithm has two main consequences: first, it provides a faster 2 -approximation algorithm for the general 2 -edge-connectivity augmentation problem; second, it solves in O(m · ?(m,n)) time the problem of restoring, by means of a minimum weight set of replacement edges, the 2 -edge-connectivity of a 2-edge-connected communication network undergoing a link failure.
21 Sep 2003
TL;DR: Barycentric placements exist for every connected periodic graph, are unique up to affine transformations, and provide a versatile tool not only in drawing, but also in computation.
Abstract: We study barycentric placement of vertices in periodic graphs of dimension 2 or higher. Barycentric placements exist for every connected periodic graph, are unique up to affine transformations, and provide a versatile tool not only in drawing, but also in computation. Example applications include symmetric convex drawing in dimension 2 as well as determining topological types of crystals and computing their ideal symmetry groups.
TL;DR: An attempt is made to link connectivity augmentation and connectivity orientation, by considering degree-specified and minimum cardinality augmentation of graphs so that the resulting graph admits an orientation satisfying a prescribed edge-connection requirement, such as (k,l)-edge-connectivity.