Globally and locally minimal weight spanning tree networks

Abstract

The competition between local and global driving forces is significant in a wide variety of naturally occurring branched networks. We have investigated the impact of a global minimization criterion versus a local one on the structure of spanning trees. To do so, we consider two spanning tree structures—the generalized minimal spanning tree (GMST) defined by Dror et al. (Eur. J. Oper. Res. 120 (2000) 583) and an analogous structure based on the invasion percolation network, which we term the generalized invasive spanning tree (GIST). In general, these two structures represent extremes of global and local optimality, respectively. Structural characteristics are compared between the GMST and GIST for a fixed lattice. In addition, we demonstrate a method for creating a series of structures which enable one to span the range between these two extremes. Two structural characterizations, the occupied edge density (i.e., the fraction of edges in the graph that are included in the tree) and the tortuosity of the arcs in the trees, are shown to correlate well with the degree to which an intermediate structure resembles the GMST or GIST. Both characterizations are straightforward to determine from an image and are potentially useful tools in the analysis of the formation of network structures.

Introduction

The purpose of the present research is to detail a new method by which information extracted from a single, fixed network structure can be utilized to understand the physical processes which guided the formation of that structure. There are a variety of structures in nature and biology whose temporal development is difficult to observe. Accordingly, the principal data from which to understand the physics that drove the formation of these structures is the final structure itself.

An excellent example of the use of a final network structure to study the underlying physics is the work of Rodriguez–Iturbe and Rinaldo on river basins [2]. Detailed investigations of the structure of river basins combined with a variety of simulation and theoretical analysis support the conclusion that minimal energy dissipation is the driving force (both global and local) in the structure of river basins [3], [4], [5]. Similarly, natural complex branching patterns are observed in systems as diverse as retinal neurons [6], dielectric breakdown [7] and human vasculature [8]. Another recent example is the network of invading cells in malignant brain tumors observed in vitro [9].

All of these problems can be mapped to the language of spanning trees. For example, in the case of invading tumor cells, the tumor cells form branched chains, i.e., tree structures. The brain offers these invading cells a variety of pathways they can invade along (such as blood vessel and white fiber tracts) which may be interpreted as the edges of an underlying graph, with the various resistances along these pathways playing the role of edge weights. In many of these cases, the underlying physics behind the formation of the observed patterns are only beginning to be understood. The work presented here offers a useful tool in studying the driving forces in the formation of these structures.

Here we consider the class of structures called spanning trees. Formally, spanning trees are defined on graphs and, in the most basic definition, are a loopless, connected set of edges that connect all of the nodes in the underlying graph (see Fig. 1). Many different spanning trees can be generated for any given graph. Therefore, it is possible to introduce minimization criteria on the spanning-tree problem and select only those trees which satisfy the criteria. Thus, spanning trees represent an excellent test case for investigating the relation between individual structures and the minimization criteria that govern their formation.

A broadly useful class of spanning trees (for examples see [10], [11], [12]), is the minimal weight spanning tree (MST) [13], [14]. The MST is defined on an underlying graph whose edges each have some weight assigned to them. The MST is then the spanning tree (a subset of the edges in the underlying graph) that minimizes the total weight of the edges it includes. The minimal weight spanning tree represents a structure whose formation is guided by a global optimization principle. It is also possible to define other types of criteria for spanning trees. For example, it is possible to define a spanning tree such that only the lowest weight edges at each node are used (a detailed discussion of such a class of structures follows), giving a system with purely local criteria. Other types of criteria can also be imposed, such as the degree-constrained minimum spanning tree [15], [16], but they are not considered here.

One of the structures we study in this paper is the generalized minimal spanning tree (GMST), proposed by Dror et al. [1]. The GMST is useful in considering problems in which there are relevant length scales longer than a single edge. For example, a biological system is characterized by the diameter of a cell (mapped to a graph edge) as well as the length scale of diffusion in the system, which might be several cell diameters. As the name suggests, the GMST is a generalization of the MST. The GMST is defined on a graph in which the nodes have been partitioned into groups. The spanning condition for the GMST is redefined (relative to the MST) such that instead of requiring that every node in the graph be included in the tree, the inclusion of at least one node from each group is required. The GMST structure is the tree that meets this definition of spanning and minimizes the total weight of the edges it includes. When each group contains only one node, the GMST reduces to the MST.

The second class of trees considered in the present work is our generalization of the invasion percolation network [17] that we call the generalized invasive spanning tree (GIST). The invasion percolation network begins with a connected cluster of edges (in the simplest case, this could be just one edge). This cluster then “invades” the remaining edges by taking one edge from the boundary of the cluster and including it in the cluster. The edge that is included is the single edge, of those on the cluster boundary, with the lowest weight. Additional edges are then included, one at a time, in the same fashion until the cluster percolates (spans) across the system. The generalization of the invasion percolation network to the GIST, in analogy to that of the GMST, partitions the nodes into groups and modifies the percolation condition, such that one node from each group must be spanned.

For graphs in which each group is a single node (i.e., those graphs for which the GMST reduces to the MST), the GIST reduces to an acyclic invasion percolation network (i.e., an invasion percolation network without loops). It has been shown that the acyclic invasion percolation network is identical to the MST [19], [18]. Thus for graphs in which each group is a single node, the GMST and GIST are equivalent structures. Because of this equivalence, it is necessary to consider the GMST and GIST with groups of more than one node, rather than only the MST and invasion percolation, to understand the relation between local and global minimization criteria on tree structures.

The GMST and GIST structures were chosen because they generally offer extremes of global and local criteria. Both classes of trees have criteria which dictate the weight of edges chosen. The GMST structures choose the edges that minimize the total weight of the structure, even if that forces a higher weight edge to be chosen locally. In contrast, the GIST structures include the lowest weight edge locally, even if this results in a higher total weight for the entire tree. Except in the case of single-node groups noted above, each criterion results in a different final structure (though by definition both yield spanning trees). By comparing these structures, the effect of each type of criteria can be identified. Moreover, we provide a method to change a GIST structure incrementally into a more globally optimal GMST-like structure. This allows various structural features to be observed as a function of the degree to which either criterion is imposed. These intermediate structures can then serve as benchmarks for comparison when a real image is analyzed.

This paper is organized as follows. Section 2 is comprised of a description of the GMST and GIST structures. It also contains brief summaries of the protocols used to form these trees, as well as methods for transitioning from the GIST towards the GMST. Section 3 contains basic statistical descriptions of the structures generated for a given set of graph realizations. Section 4 introduces the statistical measurements of edge density and tortuosity and outlines how they may be used to study an experimentally observed image. Finally, Section 5 has some concluding remarks.