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12
A Geometric Preferential Attachment Model of Networks
 In Algorithms and Models for the WebGraph: Third International Workshop, WAW 2004
, 2004
"... We study a random graph Gn that combines certain aspects of geometric random graphs and preferential attachment graphs. This model yields a graph with powerlaw degree distribution where the expansion property depends on a tunable parameter of the model. The vertices of Gn are n sequentially generat ..."
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Cited by 32 (2 self)
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We study a random graph Gn that combines certain aspects of geometric random graphs and preferential attachment graphs. This model yields a graph with powerlaw degree distribution where the expansion property depends on a tunable parameter of the model. The vertices of Gn are n sequentially generated points x1, x2,..., xn chosen uniformly at random from the unit sphere in R 3. After generating xt, we randomly connect it to m points from those points in x1, x2,..., xt−1. 1
Random Deletion In A Scale Free Random Graph Process
, 2004
"... We study a dynamically evolving random graph which adds vertices and edges using preferential attachment and deletes vertices randomly. At time t, with probability #1 > 0 we add a new vertex u t and m random edges incident with u t . The neighbours of u t are chosen with probability proportional to ..."
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Cited by 16 (3 self)
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We study a dynamically evolving random graph which adds vertices and edges using preferential attachment and deletes vertices randomly. At time t, with probability #1 > 0 we add a new vertex u t and m random edges incident with u t . The neighbours of u t are chosen with probability proportional to degree. With probability # 0 we add m random edges to existing vertices where the endpoints are chosen with probability proportional to degree. With probability 1 #0 we delete a random vertex, if there are vertices left to delete. and with probability #0 we delete m random edges. Assuming that #+#1 +#0 > 1 and #0 is su#cently small, we show that for large k, t, the expected number of vertices of degree k is approximately dk t where as k ##, dk Ck 1# where # = 2(## 0 ) 3#1# 1 and C > 0 is a constant. Note that # can take any value greater than 1. 1
Competitioninduced preferential attachment
 Proceedings of the 31st International Colloquium on Automata, Languages and Programming (ICALP), 208221, Lecture Notes in Computer Science 3142
, 2004
"... Foundation Fellowship. Abstract. Models based on preferential attachment have had much success in reproducing the power law degree distributions which seem ubiquitous in both natural and engineered systems. Here, rather than assuming preferential attachment, we give an explanation of how it can aris ..."
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Cited by 9 (0 self)
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Foundation Fellowship. Abstract. Models based on preferential attachment have had much success in reproducing the power law degree distributions which seem ubiquitous in both natural and engineered systems. Here, rather than assuming preferential attachment, we give an explanation of how it can arise from a more basic underlying mechanism of competition between opposing forces. We introduce a family of onedimensional geometric growth models, constructed iteratively by locally optimizing the tradeoffs between two competing metrics. This family admits an equivalent description as a graph process with no reference to the underlying geometry. Moreover, the resulting graph process is shown to be preferential attachment with an upper cutoff. We rigorously determine the degree distribution for the family of random graph models, showing that it obeys a power law up to a finite threshold and decays exponentially above this threshold. We also introduce and rigorously analyze a generalized version of our graph process, with two natural parameters, one corresponding to the cutoff and the other a “fertility ” parameter. Limiting cases of this process include the standard BarabásiAlbert preferential attachment model and the uniform attachment model. In the general case, we prove that the process has a power law degree distribution up to a cutoff, and establish monotonicity of the power as a function of the two parameters.
Catching the ‘Network Science’ Bug: Insight and Opportunities for the Operations Researchers
 Operations Research
, 2009
"... Accepted for publication by ..."
An internet graph model based on tradeoff optimization
 European Physics Journal B
, 2004
"... be inserted by the editor) ..."
Degree Distribution of CompetitionInduced Preferential Attachment Graphs
 Combin. Probab. Comput
, 2005
"... We introduce a family of onedimensional geometric growth models, constructed iteratively by locally optimizing the tradeoffs between two competing metrics, and show that this family is equivalent to a family of preferential attachment random graph models with upper cutoffs. This is first explanatio ..."
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Cited by 6 (1 self)
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We introduce a family of onedimensional geometric growth models, constructed iteratively by locally optimizing the tradeoffs between two competing metrics, and show that this family is equivalent to a family of preferential attachment random graph models with upper cutoffs. This is first explanation of how preferential attachment can arise from a more basic underlying mechanism of local competition. We rigorously determine the degree distribution for the family of random graph models, showing that it obeys a power law up to a finite threshold and decays exponentially above this threshold. We also rigorously analyze a generalized version of our graph process, with two natural parameters, one corresponding to the cutoff and the other a “fertility ” parameter. We prove that the general model has a powerlaw degree distribution up to a cutoff, and establish monotonicity of the power as a function of the two parameters. Limiting cases of the general model include the standard preferential attachment model without cutoff and the uniform attachment model.
Diameters in preferential attachment models
, 2009
"... In this paper, we investigate the diameter in preferential attachment (PA) models, thus quantifying the statement that these models are small worlds. The models studied here are such that edges are attached to older vertices proportional to the degree plus a constant, i.e., we consider affine PAmo ..."
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Cited by 4 (0 self)
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In this paper, we investigate the diameter in preferential attachment (PA) models, thus quantifying the statement that these models are small worlds. The models studied here are such that edges are attached to older vertices proportional to the degree plus a constant, i.e., we consider affine PAmodels. There is a substantial amount of literature proving that, quite generally, PAgraphs possess powerlaw degree sequences with a powerlaw exponent τ> 2. We prove that the diameter of the PAmodel is bounded above by a constant times log t, where t is the size of the graph. When the powerlaw exponent τ exceeds 3, then we prove that log t is the right order for the diameter, by proving a lower bound of this order, both for the diameter as well as for the typical distance. This shows that, for τ> 3, distances are of the order log t. For τ ∈ (2, 3), we improve the upper bound to a constant times log log t, and prove a lower bound of the same order for the diameter. Unfortunately, this proof does not extend to typical distances. These results do show that the diameter is of order log log t. These bounds partially prove predictions by physicists that the typical distance in PAgraphs are similar to the ones in other scalefree random graphs, such as the configuration model and various inhomogeneous random graph models, where typical distances have been shown to be of order log log t when τ ∈ (2, 3), and of order log t when τ > 3.
Not All Scale Free Networks are Born Equal the Role of the Seed Graph in PPI Network Emulation
"... The (asymptotic) degree distributions of the best known “scale free ” network models are all similar and are independent of the seed graph used. Hence it has been tempting to assume that networks generated by these models are similar in general. In this paper we observe that several key topological ..."
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Cited by 1 (1 self)
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The (asymptotic) degree distributions of the best known “scale free ” network models are all similar and are independent of the seed graph used. Hence it has been tempting to assume that networks generated by these models are similar in general. In this paper we observe that several key topological features of such networks depend heavily on the specific model and the seed graph used. Furthermore, we show that starting with the “right” seed graph, the duplication model captures many topological features of publicly available PPI networks very well.
unknown title
, 2008
"... Asymptotic theory for the multidimensional random online nearestneighbour graph ..."
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Asymptotic theory for the multidimensional random online nearestneighbour graph