Markov models for fMRI correlation structure: is brain functional connectivity small world, or decomposable into networks?

Correlations in the signal observed via functional Magnetic Resonance Imaging (fMRI), are expected to reveal the interactions in the underlying neural populations through hemodynamic response. In particular, they highlight distributed set of mutually correlated regions that correspond to brain networks related to different cognitive functions. Yet graph-theoretical studies of neural connections give a different picture: that of a highly integrated system with small-world properties: local clustering but with short pathways across the complete structure. We examine the conditional independence properties of the fMRI signal, i.e. its Markov structure, to find realistic assumptions on the connectivity structure that are required to explain the observed functional connectivity. In particular we seek a decomposition of the Markov structure into segregated functional networks using decomposable graphs: a set of strongly-connected and partially overlapping cliques. We introduce a new method to efficiently extract such cliques on a large, strongly-connected graph. We compare methods learning different graph structures from functional connectivity by testing the goodness of fit of the model they learn on new data. We find that summarizing the structure as strongly-connected networks can give a good description only for very large and overlapping networks. These results highlight that Markov models are good tools to identify the structure of brain connectivity from fMRI signals, but for this purpose they must reflect the small-world properties of the underlying neural systems

Tight Bounds for Graph Problems in Insertion Streams

Despite the large amount of work on solving graph problems in the data stream model, there do not exist tight space bounds for almost any of them, even in a stream with only edge insertions. For example, for testing connectivity, the upper bound is O(n * log(n)) bits, while the lower bound is only Omega(n) bits. We remedy this situation by providing the first tight Omega(n * log(n)) space lower bounds for randomized algorithms which succeed with constant probability in a stream of edge insertions for a number of graph problems. Our lower bounds apply to testing bipartiteness, connectivity, cycle-freeness, whether a graph is Eulerian, planarity, H-minor freeness, finding a minimum spanning tree of a connected graph, and testing if the diameter of a sparse graph is constant. We also give the first Omega(n * k * log(n)) space lower bounds for deterministic algorithms for k-edge connectivity and k-vertex connectivity; these are optimal in light of known deterministic upper bounds (for k-vertex connectivity we also need to allow edge duplications, which known upper bounds allow). Finally, we give an Omega(n * log^2(n)) lower bound for randomized algorithms approximating the minimum cut up to a constant factor with constant probability in a graph with integer weights between 1 and n, presented as a stream of insertions and deletions to its edges. This lower bound also holds for cut sparsifiers, and gives the first separation of maintaining a sparsifier in the data stream model versus the offline model

Dynamic Graph Stream Algorithms in o(n)o(n) Space

In this paper we study graph problems in dynamic streaming model, where the input is defined by a sequence of edge insertions and deletions. As many natural problems require $\Omega(n)$ space, where $n$ is the number of vertices, existing works mainly focused on designing $\tilde{O}(n)$ space algorithms. Although sublinear in the number of edges for dense graphs, it could still be too large for many applications (e.g. $n$ is huge or the graph is sparse). In this work, we give single-pass algorithms beating this space barrier for two classes of problems. We present $o(n)$ space algorithms for estimating the number of connected components with additive error $\varepsilon n$ and $(1+\varepsilon)$-approximating the weight of minimum spanning tree, for any small constant $\varepsilon>0$. The latter improves previous $\tilde{O}(n)$ space algorithm given by Ahn et al. (SODA 2012) for connected graphs with bounded edge weights. We initiate the study of approximate graph property testing in the dynamic streaming model, where we want to distinguish graphs satisfying the property from graphs that are $\varepsilon$-far from having the property. We consider the problem of testing $k$-edge connectivity, $k$-vertex connectivity, cycle-freeness and bipartiteness (of planar graphs), for which, we provide algorithms using roughly $\tilde{O}(n^{1-\varepsilon})$ space, which is $o(n)$ for any constant $\varepsilon$. To complement our algorithms, we present $\Omega(n^{1-O(\varepsilon)})$ space lower bounds for these problems, which show that such a dependence on $\varepsilon$ is necessary.Comment: ICALP 201