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Inequalities for the Number of Walks in Graphs
"... We investigate the growth of the number wk of walks of length k in undirected graphs as well as related inequalities. In the first part, we derive the inequalities w2a+c · w2(a+b)+c ≤ w2a · w2(a+b+c) and w2a+c(v, v) · w2(a+b)+c(v, v) ≤ w2a(v, v) · w2(a+b+c)(v, v) for the number wk(v, v) of closed ..."
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Cited by 1 (1 self)
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We investigate the growth of the number wk of walks of length k in undirected graphs as well as related inequalities. In the first part, we derive the inequalities w2a+c · w2(a+b)+c ≤ w2a · w2(a+b+c) and w2a+c(v, v) · w2(a+b)+c(v, v) ≤ w2a(v, v) · w2(a+b+c)(v, v) for the number wk(v, v) of closed
Random walks for image segmentation
 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 2006
"... A novel method is proposed for performing multilabel, interactive image segmentation. Given a small number of pixels with userdefined (or predefined) labels, one can analytically and quickly determine the probability that a random walker starting at each unlabeled pixel will first reach one of the ..."
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Cited by 387 (21 self)
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A novel method is proposed for performing multilabel, interactive image segmentation. Given a small number of pixels with userdefined (or predefined) labels, one can analytically and quickly determine the probability that a random walker starting at each unlabeled pixel will first reach one
Evolving Virtual Creatures
 in SIGGRAPH 94 Conference Proceedings, ser. Annual Conference Series
, 1994
"... This paper describes a novel system for creating virtual creatures that move and behave in simulated threedimensional physical worlds. The morphologies of creatures and the neural systems for controlling their muscle forces are both generated automatically using genetic algorithms. Different fitnes ..."
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Cited by 386 (1 self)
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fitness evaluation functions are used to direct simulated evolutions towards specific behaviors such as swimming, walking, jumping, and following. A genetic language is presented that uses nodes and connections as its primitive elements to represent directed graphs, which are used to describe both
Cones of matrices and setfunctions and 01 optimization
 SIAM JOURNAL ON OPTIMIZATION
, 1991
"... It has been recognized recently that to represent a polyhedron as the projection of a higher dimensional, but simpler, polyhedron, is a powerful tool in polyhedral combinatorics. We develop a general method to construct higherdimensional polyhedra (or, in some cases, convex sets) whose projection a ..."
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Cited by 347 (7 self)
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of inequalities, such that already the first system includes clique, odd hole, odd antihole, wheel, and orthogonality constraints. In particular, for perfect (and many other) graphs, this first system gives the vertex packing polytope. For various classes of graphs, including tperfect graphs, it follows
Computing communities in large networks using random walks
 J. of Graph Alg. and App. bf
, 2004
"... Dense subgraphs of sparse graphs (communities), which appear in most realworld complex networks, play an important role in many contexts. Computing them however is generally expensive. We propose here a measure of similarities between vertices based on random walks which has several important advan ..."
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Cited by 226 (3 self)
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Dense subgraphs of sparse graphs (communities), which appear in most realworld complex networks, play an important role in many contexts. Computing them however is generally expensive. We propose here a measure of similarities between vertices based on random walks which has several important
Randomwalk computation of similarities between nodes of a graph, with application to collaborative recommendation
 IEEE Transactions on Knowledge and Data Engineering
"... ABSTRACT This work presents a new perspective on characterizing the similarity between elements of a database or, more generally, nodes of a weighted, undirected, graph. It is based on a Markovchain model of random walk through the database. More precisely, we compute quantities (the average commu ..."
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Cited by 194 (19 self)
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ABSTRACT This work presents a new perspective on characterizing the similarity between elements of a database or, more generally, nodes of a weighted, undirected, graph. It is based on a Markovchain model of random walk through the database. More precisely, we compute quantities (the average
Walks and the spectral radius of graphs
, 2008
"... Given a graph G, write µ (G) for the largest eigenvalue of its adjacency matrix, ω (G) for its clique number, and wk (G) for the number of its kwalks. We prove that the inequalities wq+r (G) wq (G) ≤ µr ω (G) − 1 (G) ≤ ω (G) wr (G) hold for all r> 0 and odd q> 0. We also generalize a number ..."
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Given a graph G, write µ (G) for the largest eigenvalue of its adjacency matrix, ω (G) for its clique number, and wk (G) for the number of its kwalks. We prove that the inequalities wq+r (G) wq (G) ≤ µr ω (G) − 1 (G) ≤ ω (G) wr (G) hold for all r> 0 and odd q> 0. We also generalize a
On graph kernels: Hardness results and efficient alternatives
 IN: CONFERENCE ON LEARNING THEORY
, 2003
"... As most ‘realworld’ data is structured, research in kernel methods has begun investigating kernels for various kinds of structured data. One of the most widely used tools for modeling structured data are graphs. An interesting and important challenge is thus to investigate kernels on instances tha ..."
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Cited by 184 (6 self)
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problem. It is also shown that computing an inner product in a feature space indexed by all possible graphs, where each feature counts the number of subgraphs isomorphic to that graph, is NPhard. On the other hand, inner products in an alternative feature space, based on walks in the graph, can
On the number of walks in a graph
 Appl. Math. Letters (submitted
, 2003
"... Abstract The aim of this note is to call attention to a simple regularity regarding the number of walks in a finite graph G. Let Wk denote the number of walks of length k ( ≥ 0) in G. Then W 2 a+b ≤ W2a W2b holds for all a, b ∈ N0 while equality holds exclusively either (I) for all a, b ∈ N0 (in cas ..."
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Cited by 8 (1 self)
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Abstract The aim of this note is to call attention to a simple regularity regarding the number of walks in a finite graph G. Let Wk denote the number of walks of length k ( ≥ 0) in G. Then W 2 a+b ≤ W2a W2b holds for all a, b ∈ N0 while equality holds exclusively either (I) for all a, b ∈ N0 (in
A Recursive Greedy Algorithm for Walks in Directed Graphs
 PROC. OF IEEE FOCS
, 2005
"... Given an arcweighted directed graph G = (V, A, ℓ) and a pair of nodes s, t, we seek to find an st walk of length at most B that maximizes some given function f of the set of nodes visited by the walk. The simplest case is when we seek to maximize the number of nodes visited: this is called the ori ..."
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Cited by 57 (3 self)
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Given an arcweighted directed graph G = (V, A, ℓ) and a pair of nodes s, t, we seek to find an st walk of length at most B that maximizes some given function f of the set of nodes visited by the walk. The simplest case is when we seek to maximize the number of nodes visited: this is called
Results 1  10
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1,091