Graph-theoretic Perspective Research Articles

Formation Coordination with Connectivity Preservation in Presence of Obstacles

This paper presents a geometric approach to multi-robots group formation with connectivity preservation (from a graph-theoretic perspective) among group members. The controller demonstrates consistency among different formations, as well as stability while performing dynamic switching between formations. Inter-robots collision avoidance is delivered through formation preservation, while permitting high degree of formation re-adjustability. It has been proven that such formation approach would result into complete, isomorphic formations (with regards to its first and second isogonic) with edge connectivity Λ(G) = 1/4n(n – 1), and a unique, shortest connectivity link among group members. The complete connectivity along with the isomorphic property of the formations would, in essence, not only guarantee that the communication among the robotic agents will be preserved, but also relax the topological requirements for message passing among group members that might be needed while switching between different formations. In addition, the existence of the inter-robot shortest connectivity link at the group level, would ease the message routing once the information sharing among all the members of the group is necessary.

A graph-theoretic perspective on the links-to-concepts ratio expected in cognitive maps

Abstract Strategic options development and analysis (SODA) has maintained that it expects a links-to-concepts ratio of 1.15–1.20 in cognitive maps. This expectation is investigated from a graph-theoretic perspective in order to highlight two issues that the SODA literature has not mentioned. First, the ratio is impossible to achieve in tree-structured maps. Second, adherence to the expectation can result in minimally connected maps. Both issues are discussed with examples and calculations, and a conclusion is drawn that graph theory is a potentially rich, yet relatively untapped, source of insights for not only SODA, but for soft OR and systems thinking in general.

Controllability of Multi-Agent Systems from a Graph-Theoretic Perspective

In this work, we consider the controlled agreement problem for multi-agent networks, where a collection of agents take on leader roles while the remaining agents execute local, consensus-like protocols. Our aim is to identify reflections of graph-theoretic notions on system-theoretic properties of such systems. In particular, we show how the symmetry structure of the network, characterized in terms of its automorphism group, directly relates to the controllability of the corresponding multi-agent system. Moreover, we introduce network equitable partitions as a means by which such controllability characterizations can be extended to the multileader setting.

What Can Graph Theory Tell Us About Word Learning and Lexical Retrieval?

Graph theory and the new science of networks provide a mathematically rigorous approach to examine the development and organization of complex systems. These tools were applied to the mental lexicon to examine the organization of words in the lexicon and to explore how that structure might influence the acquisition and retrieval of phonological word-forms. Pajek, a program for large network analysis and visualization (V. Batagelj & A. Mvrar, 1998), was used to examine several characteristics of a network derived from a computerized database of the adult lexicon. Nodes in the network represented words, and a link connected two nodes if the words were phonological neighbors. The average path length and clustering coefficient suggest that the phonological network exhibits small-world characteristics. The degree distribution was fit better by an exponential rather than a power-law function. Finally, the network exhibited assortative mixing by degree. Some of these structural characteristics were also found in graphs that were formed by 2 simple stochastic processes suggesting that similar processes might influence the development of the lexicon. The graph theoretic perspective may provide novel insights about the mental lexicon and lead to future studies that help us better understand language development and processing.

A Graph-theoretic perspective on centrality

The concept of centrality is often invoked in social network analysis, and diverse indices have been proposed to measure it. This paper develops a unified framework for the measurement of centrality. All measures of centrality assess a node's involvement in the walk structure of a network. Measures vary along four key dimensions: type of nodal involvement assessed, type of walk considered, property of walk assessed, and choice of summary measure. If we cross-classify measures by type of nodal involvement (radial versus medial) and property of walk assessed (volume versus length), we obtain a four-fold polychotomization with one cell empty which mirrors Freeman's 1979 categorization. At a more substantive level, measures of centrality summarize a node's involvement in or contribution to the cohesiveness of the network. Radial measures in particular are reductions of pair-wise proximities/cohesion to attributes of nodes or actors. The usefulness and interpretability of radial measures depend on the fit of the cohesion matrix to the one-dimensional model. In network terms, a network that is fit by a one-dimensional model has a core-periphery structure in which all nodes revolve more or less closely around a single core. This in turn implies that the network does not contain distinct cohesive subgroups. Thus, centrality is shown to be intimately connected with the cohesive subgroup structure of a network.

Landscape Connectivity: A Graph-Theoretic Perspective

Ecologists are familiar with two data structures commonly used to represent landscapes. Vector-based maps delineate land cover types as polygons, while raster lattices represent the landscape as a grid. Here we adopt a third lattice data structure, the graph. A graph represents a landscape as a set of nodes (e.g., habitat patches) connected to some degree by edges that join pairs of nodes functionally (e.g., via dispersal). Graph theory is well developed in other fields, including geography (transportation networks, routing applications, siting problems) and computer science (circuitry and network optimization). We present an overview of basic elements of graph theory as it might be applied to issues of connectivity in heterogeneous landscapes, focusing especially on applications of metapopulation theory in conservation biology. We develop a general set of analyses using a hypothetical landscape mosaic of habitat patches in a nonhabitat matrix. Our results suggest that a simple graph construct, the minimum spanning tree, can serve as a powerful guide to decisions about the relative importance of individual patches to overall landscape connectivity. We then apply this approach to an actual conservation scenario involving the threatened Mexican Spotted Owl (Strix occidentalis lucida). Simulations with an incidence-function metapopulation model suggest that population persistence can be maintained despite substantial losses of habitat area, so long as the minimum spanning tree is protected. We believe that graph theory has considerable promise for applications concerned with connectivity and ecological flows in general. Because the theory is already well developed in other disciplines, it might be brought to bear immediately on pressing ecological applications in conservation biology and landscape ecology.

Vertex models on Feynman diagrams

We discuss the statistical mechanics of vertex models on both generic (“thin”) and planar (“fat”) random graphs. Such models can be formulated as the N → 1 and N → ∞ limits of N × N complex matrix models, respectively. From the graph theoretic perspective one is using matrix model and field theory inspired methods to count various classes of directed graphs. For the thin random graphs we use saddle point methods to solve the models in the thermodynamic, large number of vertices limit and note that, as in the case of the eight-vertex model on the square lattice, various other models such as the Ising model appear as particular limits. The generic solution of the fat graph model is rather more elusive, but we show that for several choices of the couplings the models can be reduced to eigenvalue integrals and their critical behaviour deduced.

Wittenburg's formulation of multibody dynamics equations from a graph-theoretic perspective

In the past 20 years, many multibody or “self-formulating” computer programs have been developed for the simulation of dynamic mechanical systems. These programs are based on algorithms that enable the computer to formulate, automatically, the equations of motion for a specified system, given only a description of the system as input; numerical methods can be applied to the resulting differential-algebraic equations in order to determine the time response of the system. Whether explicitly or implicitly, all of these multibody algorithms use concepts from linear graph theory in some form or another. The “vector-network technique” represents a direct application of graph-theoretic methods to dynamic mechanical systems. In contrast, Wittenburg's well-known formalism for creating the equations of motion appears, at first glance, to use linear graph theory for the sole purpose of representing the system topology. The goal of this paper is to examine in detail the relationship between these two methodologies. In particular, the authors will establish the equivalence of the equations derived using Wittenburg's formulation, and those arising out of a formal graph-theoretic approach.

On Multi-Label Linear Interval Routing Schemes

We consider linear interval routing schemes studied by [3,5] from a graph-theoretic perspective. We examine how the number of linear intervals needed to obtain shortest path routings in networks is affected by the product, join and composition operations on graphs. This approach allows us to generalize some of the results of [3,5] concerning the minimum number of intervals needed to achieve shortest path routings in certain special classes of networks. We also establish an Ω(n 1/3 ) lower bound on the minimum number of intervals needed to achieve shortest path routings in the network considered.

EFFICIENT ALGORITHMS FOR CONSTRUCTING PROPER HIGHER ORDER SPATIAL LAG OPERATORS*

ABSTRACT. This paper extends the work of Blommestein and Koper (1992)–BK–on the construction of higher‐order spatial lag operators without redundant and circular paths. For the case most relevant in spatial econometrics and spatial statistics, i.e., when contiguity between two observations (locations) is defined in a simple binary fashion, some deficiencies of the BK algorithms are outlined, corrected and an improvement suggested. In addition, three new algorithms are introduced and compared in terms of performance for a number of empirical contiguity structures. Particular attention is paid to a graph theoretic perspective on spatial lag operators and to the most efficient data structures for the storage and manipulation of spatial lags. The new forward iterative algorithm which uses a list form rather than a matrix to store the spatial lag information is shown to be several orders of magnitude faster than the BK solution. This allows the computation of proper higher‐order spatial lags “on the fly” for even moderately large data sets such as 3,111 contiguous U. S. counties, which is not practical with the other algorithms.