5,315 research outputs found
Finding community structure in networks using the eigenvectors of matrices
We consider the problem of detecting communities or modules in networks,
groups of vertices with a higher-than-average density of edges connecting them.
Previous work indicates that a robust approach to this problem is the
maximization of the benefit function known as "modularity" over possible
divisions of a network. Here we show that this maximization process can be
written in terms of the eigenspectrum of a matrix we call the modularity
matrix, which plays a role in community detection similar to that played by the
graph Laplacian in graph partitioning calculations. This result leads us to a
number of possible algorithms for detecting community structure, as well as
several other results, including a spectral measure of bipartite structure in
networks and a new centrality measure that identifies those vertices that
occupy central positions within the communities to which they belong. The
algorithms and measures proposed are illustrated with applications to a variety
of real-world complex networks.Comment: 22 pages, 8 figures, minor corrections in this versio
Partitioning Regular Polygons into Circular Pieces I: Convex Partitions
We explore an instance of the question of partitioning a polygon into pieces,
each of which is as ``circular'' as possible, in the sense of having an aspect
ratio close to 1. The aspect ratio of a polygon is the ratio of the diameters
of the smallest circumscribing circle to the largest inscribed disk. The
problem is rich even for partitioning regular polygons into convex pieces, the
focus of this paper. We show that the optimal (most circular) partition for an
equilateral triangle has an infinite number of pieces, with the lower bound
approachable to any accuracy desired by a particular finite partition. For
pentagons and all regular k-gons, k > 5, the unpartitioned polygon is already
optimal. The square presents an interesting intermediate case. Here the
one-piece partition is not optimal, but nor is the trivial lower bound
approachable. We narrow the optimal ratio to an aspect-ratio gap of 0.01082
with several somewhat intricate partitions.Comment: 21 pages, 25 figure
Reformulating Space Syntax: The Automatic Definition and Generation of Axial Lines and Axial Maps
Space syntax is a technique for measuring the relative accessibility of different locations in a spatial system which has been loosely partitioned into convex spaces.These spaces are approximated by straight lines, called axial lines, and the topological graph associated with their intersection is used to generate indices of distance, called integration, which are then used as proxies for accessibility. The most controversial problem in applying the technique involves the definition of these lines. There is no unique method for their generation, hence different users generate different sets of lines for the same application. In this paper, we explore this problem, arguing that to make progress, there need to be unambiguous, agreed procedures for generating such maps. The methods we suggest for generating such lines depend on defining viewsheds, called isovists, which can be approximated by their maximum diameters,these lengths being used to form axial maps similar to those used in space syntax. We propose a generic algorithm for sorting isovists according to various measures,approximating them by their diameters and using the axial map as a summary of the extent to which isovists overlap (intersect) and are accessible to one another. We examine the fields created by these viewsheds and the statistical properties of the maps created. We demonstrate our techniques for the small French town of Gassin used originally by Hillier and Hanson (1984) to illustrate the theory, exploring different criteria for sorting isovists, and different axial maps generated by changing the scale of resolution. This paper throws up as many problems as it solves but we believe it points the way to firmer foundations for space syntax
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