Separators for sphere-packings and nearest neighbor graphs

Miller, Gary L.; Teng, Shang‐Hua; Thurston, William P.; Vavasis, Stephen A.

doi:10.1145/256292.256294

Cited by 196 publications

(208 citation statements)

References 45 publications

Supporting

Mentioning

207

Contrasting

Order By: Relevance

“…Solving these problems simultaneously yields complete sets of isocontacting packings of hard spheres and of isoenergetic states of sticky hard spheres. These in turn have applications to physical problems ranging from crystal-lization and jamming [3-7, 12-15, 18] to cluster physics [11,22,26] to liquid structure [16,[27][28][29] to protein folding [30,31], as well as engineering applications such as circuit design [32] and error-correcting codes [33].…”

Section: Introductionmentioning

confidence: 99%

Structure of finite sphere packings via exact enumeration: Implications for colloidal crystal nucleation

2012

View full text Add to dashboard Cite

We analyze the geometric structure and mechanical stability of a complete set of isostatic and hyperstatic sphere packings obtained via exact enumeration. The number of nonisomorphic isostatic packings grows exponentially with the number of spheres N , and their diversity of structure and symmetry increases with increasing N and decreases with increasing hyperstaticity H ≡ Nc − NISO, where Nc is the number of pair contacts and NISO = 3N − 6. Maximally contacting packings are in general neither the densest nor the most symmetric. Analyses of local structure show that the fraction f of nuclei with order compatible with the bulk (RHCP) crystal decreases sharply with increasing N due to a high propensity for stacking faults, 5-and near-5-fold symmetric structures, and other motifs that preclude RHCP order. While f increases with increasing H, a significant fraction of hyperstatic nuclei for N as small as 11 retain non-RHCP structure. Classical theories of nucleation that consider only spherical nuclei, or only nuclei with the same ordering as the bulk crystal, cannot capture such effects. Our results provide an explanation for the failure of classical nucleation theory for hard-sphere systems of N < ∼ 10 particles; we argue that in this size regime, it is essential to consider nuclei of unconstrained geometry. Our results are also applicable to understanding kinetic arrest and jamming in systems that interact via hard-core-like repulsive and short-ranged attractive interactions.

show abstract

Section: Introductionmentioning

confidence: 99%

Structure of finite sphere packings via exact enumeration: Implications for colloidal crystal nucleation

2012

View full text Add to dashboard Cite

show abstract

“…This has been generalized in various directions: to graphs embedded in a surface of bounded genus [15], graphs with a forbidden minor [1], intersection graphs of balls in R d [26], intersection graphs of Jordan regions [10], and intersection graphs of convex sets in the plane [10]. Our main result is a separator theorem for string graphs.…”

Section: The Weight Of a Subset S ⊆ V Denoted By W(s) Is Definedmentioning

confidence: 98%

A Separator Theorem for String Graphs and Its Applications

Fox

Pach

2009

WALCOM: Algorithms and Computation

View full text Add to dashboard Cite

A string graph is the intersection graph of a collection of continuous arcs in the plane. It is shown that any string graph with m edges can be separated into two parts of roughly equal size by the removal of O(m 3/4 √ log m) vertices. This result is then used to deduce that every string graph with n vertices and no complete bipartite subgraph K t,t has at most c t n edges, where c t is a constant depending only on t. Another application shows that locally tree-like string graphs are globally tree-like: for any > 0, there is an integer g( ) such that every string graph with n vertices and girth at least g( ) has at most (1 + )n edges.

show abstract

“…For instance, the k-nearest neighbor graph has Δ = (c d + 1)k, where c d is a constant called the kissing number [11,Cor. 3.2.3].…”

Section: A Stabilization Under Finite Maximum Degreementioning

confidence: 99%

Limit laws for random spatial graphical models

Anandkumar¹,

Yukich

Willsky³

2010

2010 IEEE International Symposium on Information Theory

View full text Add to dashboard Cite

Abstract-We consider spatial graphical models on random Euclidean points, applicable for data in sensor and social networks. We establish limit laws for general functions of the graphical model such as the mean value, the entropy rate etc. as the number of nodes goes to infinity under certain conditions. These conditions require the corresponding Gibbs measure to be spatially mixing and for the random graph of the model to satisfy a certain localization property known as stabilization. Graphs such the k nearest neighbor graph and the geometric disc graph belong to the class of stabilizing graphs. Intuitively, these conditions require the data at each node not to have strong dependence on data and positions of nodes far away. Finally, it is shown that spatial mixing of the Gibbs measure on a random graph holds when a suitably defined degree-dependent (but otherwise independent) node percolation does not have a giant component.

show abstract

Separators for sphere-packings and nearest neighbor graphs

Cited by 196 publications

References 45 publications

Structure of finite sphere packings via exact enumeration: Implications for colloidal crystal nucleation

Structure of finite sphere packings via exact enumeration: Implications for colloidal crystal nucleation

A Separator Theorem for String Graphs and Its Applications

Limit laws for random spatial graphical models

Contact Info

Product

Resources

About