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This paper treats the problem of the merging of formations, where the underlying model of a formation is graphical. We first analyze the rigidity and persistence of meta-formations, which are formations obtained by connecting several rigid…
By an undirected rigid formation of mobile autonomous agents is meant a formation based on graph rigidity in which each pair of "neighboring" agents is responsible for maintaining a prescribed target distance between them. In a recent paper…
The graphical notion of effective resistance has found wide-ranging applications in many areas of pure mathematics, applied mathematics and control theory. By the nature of its construction, effective resistance can only be computed in…
Graphs are commonly used to encode relationships among entities, yet their abstractness makes them difficult to analyze. Node-link diagrams are popular for drawing graphs, and force-directed layouts provide a flexible method for node…
We revisit the concept of minimal rigidity as applied to soft repulsive, frictionless sphere packings in two-dimensions with the introduction of the jamming graph. Minimal rigidity is a purely combinatorial property encoded via Laman's…
We define persistent homology groups over any set of spaces which have inclusions defined so that the corresponding directed graph between the spaces is acyclic, as well as along any subgraph of this directed graph. This method…
Graphs are a basic tool for the representation of modern data. The richness of the topological information contained in a graph goes far beyond its mere interpretation as a one-dimensional simplicial complex. We show how topological…
In weighted graphs the shortest path between two nodes is often reached through an indirect path, out of all possible connections, leading to structural redundancies which play key roles in the dynamics and evolution of complex networks. We…
We propose a geometric approach to distance-based formation control modeled on a minimum-norm lifting of Riemannian gradient descent in edge-space to node-space. This yields a unified family of controllers, including the classical gradient…
The functional demands of robotic systems often require completing various tasks or behaviors under the effect of disturbances or uncertain environments. Of increasing interest is the autonomy for dynamic robots, such as multirotors, motor…
In this paper, we develop new methods for the analysis of decentralized control systems and we apply them to formation control problems. The basic set-up consists of a system with multiple agents corresponding to the nodes of a graph whose…
This paper introduces the notion of weak rigidity to characterize a framework by pairwise inner products of inter-agent displacements. Compared to distance-based rigidity, weak rigidity requires fewer constrained edges in the graph to…
Let $\{D_M\}_{M\geq 0}$ be the $n$-vertex random directed graph process, where $D_0$ is the empty directed graph on $n$ vertices, and subsequent directed graphs in the sequence are obtained by the addition of a new directed edge uniformly…
Graph rigidity theory studies the capability of a graph embedded in the Euclidean space to constrain its global geometric shape via local constraints among nodes and edges, and has been widely exploited in network localization and formation…
The notions of $r$-robustness and $(r,s)$-robustness of a network have been earlier introduced in the literature to achieve resilient consensus in the presence of misbehaving agents. However, while higher robustness levels enable networks…
This paper studies the problem of stabilizing target formations specified by inter-neighbor bearings with relative position measurements. While the undirected case has been studied in the existing works, this paper focuses on the case where…
The theory of multidimensional persistent homology was initially developed in the discrete setting, and involved the study of simplicial complexes filtered through an ordering of the simplices. Later, stability properties of…
We develop a method to calculate the persistent currents and their spatial distribution (and transport properties) on graphs made of quasi-1D diffusive wires. They are directly related to the field derivatives of the determinant of a matrix…
In Part I of this work we defined a generalization of the concept of effective resistance to directed graphs, and we explored some of the properties of this new definition. Here, we use the theory developed in Part I to compute effective…
Persistent homology is a topological data analysis tool that has been widely generalized, extending its scope beyond the field of topology. Among its extensions, steady and ranging persistence were developed to study a wide variety of graph…