An Algebra for Patterns on a Complex, I

Abstract

We propose a notion of continuous path for locally finite metric spaces, taking inspiration from the recent development of A-theory for locally finite connected graphs. We use this notion of continuity to derive an analogue in Z 2 of the Jordan curve theorem and to extend to a quite large class of locally finite metric spaces (containing all finite metric spaces) an inequality for the ℓ p -distortion of a metric space that has been recently proved by Pierre-Nicolas Jolissaint and Alain Valette for finite connected graphs.2010 MSC: Primary 52A01; Secondary 46L36.

Foundations of a Connectivity Theory for Simplicial Complexes

“…such that h n+1 i 1 i n r = f n i 1 i n and h n+1 i 1 i n 2t − r = g n i 1 i n . Pictorially, the homotopy is easy to see.…”

“…Then f is a homotopy equivalence. In particular, if we view 1 as a subgraph of 2 , then it is a homotopy retract of 2 , with retraction the unique map g from 2 to 1 …”

“…Simplicial complexes have proven very useful in studying dynamic processes in the network [4]. A first analysis tool, called Q-analysis, was developed by the British physicist R. Atkin [1,2], who modeled social networks by simplicial complexes, with the simplices representing highly connected subnetworks. He then studied clusters of simplices any two of which were connected by a chain of simplices in which successive ones share faces of certain dimensions.…”

Homotopy theory of graphs

“…(See the example on p. 101 of [4].) The theory is based on an approach proposed by R. Atkin [1,2]; hence the letter "A." This is not to be mistaken with the A 1 -homotopy theory of schemes by Voevodsky [9].…”

Simplicial Complexes of Networks and Their Statistical Properties

“…The first is the dimension of the complex. The next one is the so called an f -vector (also known as the second structure vector ) [2], [3], [4], [5], which is an integer vector with dim(K) + 1 components, the i-th one being equal to the number of i-dimensional simplices in K. An invariant is also a Q-vector (f irst structure vector), an integer vector of the same length as the f -vector, whose i-th component is equal to the number of i-connectivity classes. The structure vector, illustrated in Fig.…”