Results 1 
8 of
8
Community detection in graphs
, 2009
"... The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of th ..."
Abstract

Cited by 801 (1 self)
 Add to MetaCart
The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Such
Statistics of lattice animals (polyominoes) and polygons
 Journal of Physics, Series A
"... We have developed an improved algorithm that allows us to enumerate the number of site animals (polyominoes) on the square lattice up to size 46. Analysis of the resulting series yields an improved estimate, τ = 4.062570(8), for the growth constant of lattice animals and confirms to a very high degr ..."
Abstract

Cited by 20 (2 self)
 Add to MetaCart
We have developed an improved algorithm that allows us to enumerate the number of site animals (polyominoes) on the square lattice up to size 46. Analysis of the resulting series yields an improved estimate, τ = 4.062570(8), for the growth constant of lattice animals and confirms to a very high degree of certainty that the generating function has a logarithmic divergence. We prove the bound τ> 3.90318. We also calculate the radius of gyration of both lattice animals and polygons enumerated by area. The analysis of the radius of gyration series yields the estimate ν = 0.64115(5), for both animals and polygons enumerated by area. The mean perimeter of polygons of area n is also calculated. A number of new amplitude estimates are given. The enumeration of lattice animals is a classical combinatorial problem of great interest both intrinsically and as a paradigm of recreational mathematics [1]. A lattice animal is a finite set of nearest neighbour sites on a lattice. The fundamental problem is the calculation of the number of animals, bn, containing n sites. In the physics literature lattice animals are very often called clusters due to their very close relationship to percolation problems [2]. Series
Polygonal polyominoes on the square lattice
 J. PHYS. A: MATH. GEN. 34 (2001) 3721–3733
, 2001
"... We study a proper subset of polyominoes, called polygonal polyominoes, which are defined to be selfavoiding polygons containing any number of holes, each of which is a selfavoiding polygon. The staircase polygon subset, with staircase holes, is also discussed. The internal holes have no common ver ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We study a proper subset of polyominoes, called polygonal polyominoes, which are defined to be selfavoiding polygons containing any number of holes, each of which is a selfavoiding polygon. The staircase polygon subset, with staircase holes, is also discussed. The internal holes have no common vertices with each other, nor any common vertices with the surrounding polygon. There are no ‘holeswithinholes’. We use the finitelattice method to count the number of polygonal polyominoes on the square lattice. Series have been derived for both the perimeter and area generating functions. It is known that while the critical point is unchanged by a finite number of holes, when the number of holes is unrestricted the critical point changes. The area generating function coefficients grow exponentially, with a growth constant greater than that for polygons with a finite number of holes, but less than that of polyominoes. We provide an estimate for this growth constant and prove that it is strictly less than that for polyominoes. Also, we prove that, enumerating by perimeter, the generating function of polygonal polyominoes has zero radius of convergence and furthermore we calculate the dominant asymptotics of its coefficients using rigorous bounds.
unknown title
, 2008
"... least resistance path to the analysis of unstructured overlay networks ..."
(Show Context)
v Quantifying Stickiness in 2D AreaPreserving Maps by Means of Recurrence Plots
"... To Phil, for letting me find my way, helping me find confidence through group meeting talks, and being patient with my naive and stubborn questions. To Norbert Marwan, for providing indispensable programming skills and alwaysprompt responses. To Marco Thiel and Carmen Romano, for being patient and ..."
Abstract
 Add to MetaCart
(Show Context)
To Phil, for letting me find my way, helping me find confidence through group meeting talks, and being patient with my naive and stubborn questions. To Norbert Marwan, for providing indispensable programming skills and alwaysprompt responses. To Marco Thiel and Carmen Romano, for being patient and gracious those two days in Aberdeen, Scotland. To Ibere Caldas, Zwinglio Guimaraes, and Ricardo Luiz Viana, for being constantly helpful and encouraging during our fruitful convsersations. To Richard Hazeltine, for always being approachable and helpful. Special thanks also go to: Pete and Laura, dedicated parents that are always there; Ester, for continual support; Leila, for answering any question in the universe I had and always helping out with concise and fruitful answers; sister Katie, for piquing my interest in science in the first place; Karl, for being a dedicated and encouraging friend all these years; Kate and Matt, for the many conversations that keep the mind curious; Walter Stroup, for constant reminders through gushing enthusiasm about the wonders of science and teaching. And, finally, to my grandparents, for always being curious about and supportive of my varied interests.
Statistical properties of the set of sites visited by the twodimensional random walk
, 1996
"... Abstract. We study the support (i.e. the set of visited sites) of a tstep random walk on a twodimensional square lattice in the large t limit. A broad class of global properties, M.t/, of the support is considered, including for example the number, S.t/, of its sites; the length of its boundary; t ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. We study the support (i.e. the set of visited sites) of a tstep random walk on a twodimensional square lattice in the large t limit. A broad class of global properties, M.t/, of the support is considered, including for example the number, S.t/, of its sites; the length of its boundary; the number of islands of unvisited sites that it encloses; the number of such islands of given shape, size, and orientation; and the number of occurrences in space of specific local patterns of visited and unvisited sites. On a finite lattice we determine the scaling functions that describe the averages, M.t/, on appropriate lattice sizedependent time scales. On an infinite lattice we first observe that the M.t / all increase with t as t = logk t, where k is an Mdependent positive integer. We then consider the class of random processes constituted by the fluctuations around average 1M.t/. We show that, to leading order as t gets large, these fluctuations are all proportional to a single universal random process, .t/, normalized to 2.t / D 1. For t! 1 the probability law of .t / tends to that of Varadhan’s renormalized local time of selfintersections. An implication is that in the long time limit all 1M.t / are proportional to 1S.t/. 1. Introduction and
Territorial Developments Based on Graffiti: a Statistical Mechanics Approach
, 2014
"... We study the wellknown sociological phenomenon of gang aggregation and territory formation through an interacting agent system defined on a lattice. We introduce a twogang Hamiltonian model where agents have red or blue affiliation but are otherwise indistinguishable. In this model, all interactio ..."
Abstract
 Add to MetaCart
(Show Context)
We study the wellknown sociological phenomenon of gang aggregation and territory formation through an interacting agent system defined on a lattice. We introduce a twogang Hamiltonian model where agents have red or blue affiliation but are otherwise indistinguishable. In this model, all interactions are indirect and occur only via graffiti markings, onsite as well as on nearest neighbor locations. We also allow for gang proliferation and graffiti suppression. Within the context of this model, we show that gang clustering and territory formation may arise under specific parameter choices and that a phase transition may occur between well–mixed, possibly dilute configurations and well separated, clustered ones. Using methods from statistical mechanics, we study the phase transition between these two qualitatively different scenarios. In the mean–fields rendition of this model, we identify parameter regimes where the transition is first or second order. In all cases, we have found that the transitions are a consequence solely of the gang to graffiti couplings, implying that direct gang to gang interactions are not strictly necessary for gang territory formation; in particular, graffiti may be the sole driving force behind gang clustering. We further discuss possible sociological – as well as ecological – ramifications of our results.