Complex networks

This chapter contains a brief introduction to complex networks, and in particular to small world and scale free networks. We show how to apply the replica method developed to analyse random matrices in statistical physics to calculate the spectral densities of the adjacency and Laplacian matrices of a scale free network. We use the effective medium approximation to treat networks with finite mean degree and discuss the local properties of random matrices associated with complex networks

The rank of diluted random graphs

We investigate the rank of the adjacency matrix of large diluted random graphs: for a sequence of graphs $(G_n)_{n\geq0}$ converging locally to a Galton--Watson tree $T$ (GWT), we provide an explicit formula for the asymptotic multiplicity of the eigenvalue 0 in terms of the degree generating function $\phi_*$ of $T$. In the first part, we show that the adjacency operator associated with $T$ is always self-adjoint; we analyze the associated spectral measure at the root and characterize the distribution of its atomic mass at 0. In the second part, we establish a sufficient condition on $\phi_*$ for the expectation of this atomic mass to be precisely the normalized limit of the dimension of the kernel of the adjacency matrices of $(G_n)_{n\geq 0}$. Our proofs borrow ideas from analysis of algorithms, functional analysis, random matrix theory and statistical physics.Comment: Published in at http://dx.doi.org/10.1214/10-AOP567 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Convexity-Increasing Morphs of Planar Graphs

We study the problem of convexifying drawings of planar graphs. Given any planar straight-line drawing of an internally 3-connected graph, we show how to morph the drawing to one with strictly convex faces while maintaining planarity at all times. Our morph is convexity-increasing, meaning that once an angle is convex, it remains convex. We give an efficient algorithm that constructs such a morph as a composition of a linear number of steps where each step either moves vertices along horizontal lines or moves vertices along vertical lines. Moreover, we show that a linear number of steps is worst-case optimal. To obtain our result, we use a well-known technique by Hong and Nagamochi for finding redrawings with convex faces while preserving y-coordinates. Using a variant of Tutte's graph drawing algorithm, we obtain a new proof of Hong and Nagamochi's result which comes with a better running time. This is of independent interest, as Hong and Nagamochi's technique serves as a building block in existing morphing algorithms.Comment: Preliminary version in Proc. WG 201

Formation Shape Control Based on Distance Measurements Using Lie Bracket Approximations

We study the problem of distance-based formation control in autonomous multi-agent systems in which only distance measurements are available. This means that the target formations as well as the sensed variables are both determined by distances. We propose a fully distributed distance-only control law, which requires neither a time synchronization of the agents nor storage of measured data. The approach is applicable to point agents in the Euclidean space of arbitrary dimension. Under the assumption of infinitesimal rigidity of the target formations, we show that the proposed control law induces local uniform asymptotic stability. Our approach involves sinusoidal perturbations in order to extract information about the negative gradient direction of each agent's local potential function. An averaging analysis reveals that the gradient information originates from an approximation of Lie brackets of certain vector fields. The method is based on a recently introduced approach to the problem of extremum seeking control. We discuss the relation in the paper

Bounds on the radius and status of graphs

Two classical concepts of centrality in a graph are the median and the center. The connected notions of the status and the radius of a graph seem to be in no relation. In this paper, however, we show a clear connection of both concepts, as they obtain their minimum and maximum values at the same type of tree graphs. Trees with fixed maximum degree and extremum radius and status, resp., are characterized. The bounds on radius and status can be transferred to general connected graphs via spanning trees. A new method of proof allows not only to regain results of Lin et al. on graphs with extremum status, but it allows also to prove analogous results on graphs with extremum radius

Avoiding the Global Sort: A Faster Contour Tree Algorithm

We revisit the classical problem of computing the \emph{contour tree} of a scalar field $f:\mathbb{M} \to \mathbb{R}$, where $\mathbb{M}$ is a triangulated simplicial mesh in $\mathbb{R}^d$. The contour tree is a fundamental topological structure that tracks the evolution of level sets of $f$ and has numerous applications in data analysis and visualization. All existing algorithms begin with a global sort of at least all critical values of $f$, which can require (roughly) $\Omega(n\log n)$ time. Existing lower bounds show that there are pathological instances where this sort is required. We present the first algorithm whose time complexity depends on the contour tree structure, and avoids the global sort for non-pathological inputs. If $C$ denotes the set of critical points in $\mathbb{M}$, the running time is roughly $O(\sum_{v \in C} \log \ell_v)$, where $\ell_v$ is the depth of $v$ in the contour tree. This matches all existing upper bounds, but is a significant improvement when the contour tree is short and fat. Specifically, our approach ensures that any comparison made is between nodes in the same descending path in the contour tree, allowing us to argue strong optimality properties of our algorithm. Our algorithm requires several novel ideas: partitioning $\mathbb{M}$ in well-behaved portions, a local growing procedure to iteratively build contour trees, and the use of heavy path decompositions for the time complexity analysis