Recovering the Long Range Links in Augmented Graphs

The augmented graph model, as introduced by Kleinberg (STOC 2000), is an appealing model for analyzing navigability in social networks. Informally, this model is defined by a pair (H,phi), where H is a graph in which inter-node distances are supposed to be easy to compute or at least easy to estimate. This graph is "augmented" by links, called long range links, which are selected according to the probability distribution phi. The augmented graph model enables the analysis of greedy routing in augmented graphs G in (H,phi). In greedy routing, each intermediate node handling a message for a target t selects among all its neighbors in G the one that is the closest to t in H and forwards the message to it. This paper addresses the problem of checking whether a given graph G is an augmented graph. It answers part of the questions raised by Kleinberg in his Problem 9 (Int. Congress of Math. 2006). More precisely, given G in (H,phi), we aim at extracting the base graph H and the long range links R out of G. We prove that if H has a high clustering coefficient and bounded doubling dimension, then a simple algorithm enables to partition the edges of G into two sets H' and R' such that E(H) is included in H' and the edges in H'\E(H) are of small stretch, i.e., the map H is not perturbed too greatly by undetected long range links remaining in H'. The perturbation is actually so small that we can prove that the expected performances of greedy routing in G using the distances in H' are close to the expected performances of greedy routing in (H,phi). Although this latter result may appear intuitively straightforward, since H' is included in E(H), it is not, as we also show that routing with a map more precise than H may actually damage greedy routing significantly. Finally, we show that in absence of a hypothesis regarding the high clustering coefficient, any structural attempt to extract the long range links will miss the detection of at least $\Omega(n^{5\epsilon}/\log n)$ long range links of stretch at least $\Omega(n^{1/5-\epsilon})$ for any $0<\epsilon<1/5$, and thus the map H cannot be recovered with good accuracy. To sum up, we solve Kleinberg's Problem 9 in the sense that we show that reconstructing augmented graphs is achievable if and only if the base graph has a high clustering coefficient

Robust Temporally Coherent Laplacian Protrusion Segmentation of 3D Articulated Bodies

In motion analysis and understanding it is important to be able to fit a suitable model or structure to the temporal series of observed data, in order to describe motion patterns in a compact way, and to discriminate between them. In an unsupervised context, i.e., no prior model of the moving object(s) is available, such a structure has to be learned from the data in a bottom-up fashion. In recent times, volumetric approaches in which the motion is captured from a number of cameras and a voxel-set representation of the body is built from the camera views, have gained ground due to attractive features such as inherent view-invariance and robustness to occlusions. Automatic, unsupervised segmentation of moving bodies along entire sequences, in a temporally-coherent and robust way, has the potential to provide a means of constructing a bottom-up model of the moving body, and track motion cues that may be later exploited for motion classification. Spectral methods such as locally linear embedding (LLE) can be useful in this context, as they preserve "protrusions", i.e., high-curvature regions of the 3D volume, of articulated shapes, while improving their separation in a lower dimensional space, making them in this way easier to cluster. In this paper we therefore propose a spectral approach to unsupervised and temporally-coherent body-protrusion segmentation along time sequences. Volumetric shapes are clustered in an embedding space, clusters are propagated in time to ensure coherence, and merged or split to accommodate changes in the body's topology. Experiments on both synthetic and real sequences of dense voxel-set data are shown. This supports the ability of the proposed method to cluster body-parts consistently over time in a totally unsupervised fashion, its robustness to sampling density and shape quality, and its potential for bottom-up model constructionComment: 31 pages, 26 figure

Adaptive Network Dynamics and Evolution of Leadership in Collective Migration

The evolution of leadership in migratory populations depends not only on costs and benefits of leadership investments but also on the opportunities for individuals to rely on cues from others through social interactions. We derive an analytically tractable adaptive dynamic network model of collective migration with fast timescale migration dynamics and slow timescale adaptive dynamics of individual leadership investment and social interaction. For large populations, our analysis of bifurcations with respect to investment cost explains the observed hysteretic effect associated with recovery of migration in fragmented environments. Further, we show a minimum connectivity threshold above which there is evolutionary branching into leader and follower populations. For small populations, we show how the topology of the underlying social interaction network influences the emergence and location of leaders in the adaptive system. Our model and analysis can describe other adaptive network dynamics involving collective tracking or collective learning of a noisy, unknown signal, and likewise can inform the design of robotic networks where agents use decentralized strategies that balance direct environmental measurements with agent interactions.Comment: Submitted to Physica D: Nonlinear Phenomen

Network Topology Mapping from Partial Virtual Coordinates and Graph Geodesics

For many important network types (e.g., sensor networks in complex harsh environments and social networks) physical coordinate systems (e.g., Cartesian), and physical distances (e.g., Euclidean), are either difficult to discern or inapplicable. Accordingly, coordinate systems and characterizations based on hop-distance measurements, such as Topology Preserving Maps (TPMs) and Virtual-Coordinate (VC) systems are attractive alternatives to Cartesian coordinates for many network algorithms. Herein, we present an approach to recover geometric and topological properties of a network with a small set of distance measurements. In particular, our approach is a combination of shortest path (often called geodesic) recovery concepts and low-rank matrix completion, generalized to the case of hop-distances in graphs. Results for sensor networks embedded in 2-D and 3-D spaces, as well as a social networks, indicates that the method can accurately capture the network connectivity with a small set of measurements. TPM generation can now also be based on various context appropriate measurements or VC systems, as long as they characterize different nodes by distances to small sets of random nodes (instead of a set of global anchors). The proposed method is a significant generalization that allows the topology to be extracted from a random set of graph shortest paths, making it applicable in contexts such as social networks where VC generation may not be possible.Comment: 17 pages, 9 figures. arXiv admin note: substantial text overlap with arXiv:1712.1006

Multiscale Bayesian State Space Model for Granger Causality Analysis of Brain Signal

Modelling time-varying and frequency-specific relationships between two brain signals is becoming an essential methodological tool to answer heoretical questions in experimental neuroscience. In this article, we propose to estimate a frequency Granger causality statistic that may vary in time in order to evaluate the functional connections between two brain regions during a task. We use for that purpose an adaptive Kalman filter type of estimator of a linear Gaussian vector autoregressive model with coefficients evolving over time. The estimation procedure is achieved through variational Bayesian approximation and is extended for multiple trials. This Bayesian State Space (BSS) model provides a dynamical Granger-causality statistic that is quite natural. We propose to extend the BSS model to include the \`{a} trous Haar decomposition. This wavelet-based forecasting method is based on a multiscale resolution decomposition of the signal using the redundant \`{a} trous wavelet transform and allows us to capture short- and long-range dependencies between signals. Equally importantly it allows us to derive the desired dynamical and frequency-specific Granger-causality statistic. The application of these models to intracranial local field potential data recorded during a psychological experimental task shows the complex frequency based cross-talk between amygdala and medial orbito-frontal cortex. Keywords: \`{a} trous Haar wavelets; Multiple trials; Neuroscience data; Nonstationarity; Time-frequency; Variational methods The published version of this article is Cekic, S., Grandjean, D., Renaud, O. (2018). Multiscale Bayesian state-space model for Granger causality analysis of brain signal. Journal of Applied Statistics. https://doi.org/10.1080/02664763.2018.145581

ARE EXPORTS CAUSING GROWTH? EVIDENCE ON INTERNATIONAL TRADE EXPANSION IN CUBA, 1960-2004

Economic development in Cuban economy in the last 50 years has been involved in the so called socialist revolution time. In the external sector, the COMECON arrangements have determined its international specialization trade pattern and balance of payments position until 1989. When the Berlin Wall fell down, Cuban economy collapsed showing the malfunctions of the previous external regulated period. In this paper, we analyzed the role of exports as an engine of economic growth in Cuba considering essential events in its commercial policy-making in the long period from 1960 to 2004. Our results show that the export led growth (ELG) hypothesis is not an appealing phenomenon. Causality proofs on the basis of error correction and augmented level VAR modellings show the imperious necesssity to import for the Cuban development. The inclusion of imports not only evidences the weakness in the feedback and interrelation between economic growth and exports but also their expansion has been precisely causing growth in most of the considered periods.Cuba, Export-led Growth, commercial agreements effects, cointegration, causality, error correction and augmented VAR modelling

Edge Augmentation on Disconnected Graphs via Eigenvalue Elevation

The graph-theoretical task of determining most likely inter-community edges based on disconnected subgraphs' intra-community connectivity is proposed. An algorithm is developed for this edge augmentation task, based on elevating the zero eigenvalues of graph's spectrum. Upper bounds for eigenvalue elevation amplitude and for the corresponding augmented edge density are derived and are authenticated with simulation on random graphs. The algorithm works consistently across synthetic and real networks, yielding desirable performance at connecting graph components. Edge augmentation reverse-engineers graph partition under different community detection methods (Girvan-Newman method, greedy modularity maximization, label propagation, Louvain method, and fluid community), in most cases producing inter-community edges at >50% frequency.Comment: 6 pages, 3 figure