Dynamic and Multi-functional Labeling Schemes

We investigate labeling schemes supporting adjacency, ancestry, sibling, and connectivity queries in forests. In the course of more than 20 years, the existence of $\log n + O(\log \log)$ labeling schemes supporting each of these functions was proven, with the most recent being ancestry [Fraigniaud and Korman, STOC '10]. Several multi-functional labeling schemes also enjoy lower or upper bounds of $\log n + \Omega(\log \log n)$ or $\log n + O(\log \log n)$ respectively. Notably an upper bound of $\log n + 5\log \log n$ for adjacency+siblings and a lower bound of $\log n + \log \log n$ for each of the functions siblings, ancestry, and connectivity [Alstrup et al., SODA '03]. We improve the constants hidden in the $O$-notation. In particular we show a $\log n + 2\log \log n$ lower bound for connectivity+ancestry and connectivity+siblings, as well as an upper bound of $\log n + 3\log \log n + O(\log \log \log n)$ for connectivity+adjacency+siblings by altering existing methods. In the context of dynamic labeling schemes it is known that ancestry requires $\Omega(n)$ bits [Cohen, et al. PODS '02]. In contrast, we show upper and lower bounds on the label size for adjacency, siblings, and connectivity of $2\log n$ bits, and $3 \log n$ to support all three functions. There exist efficient adjacency labeling schemes for planar, bounded treewidth, bounded arboricity and interval graphs. In a dynamic setting, we show a lower bound of $\Omega(n)$ for each of those families.Comment: 17 pages, 5 figure

Percolation and Connectivity on the Signal to Interference Ratio Graph

A wireless communication network is considered where any two nodes are connected if the signal-to-interference ratio (SIR) between them is greater than a threshold. Assuming that the nodes of the wireless network are distributed as a Poisson point process (PPP), percolation (unbounded connected cluster) on the resulting SIR graph is studied as a function of the density of the PPP. For both the path-loss as well as path-loss plus fading model of signal propagation, it is shown that for a small enough threshold, there exists a closed interval of densities for which percolation happens with non-zero probability. Conversely, for the path-loss model of signal propagation, it is shown that for a large enough threshold, there exists a closed interval of densities for which the probability of percolation is zero. Restricting all nodes to lie in an unit square, connectivity properties of the SIR graph are also studied. Assigning separate frequency bands or time-slots proportional to the logarithm of the number of nodes to different nodes for transmission/reception is sufficient to guarantee connectivity in the SIR graph.Comment: To appear in the Proceedings of the IEEE Conference on Computer Communications (INFOCOM 2012), to be held in Orlando Florida Mar. 201

The Cost of Global Broadcast in Dynamic Radio Networks

We study the single-message broadcast problem in dynamic radio networks. We show that the time complexity of the problem depends on the amount of stability and connectivity of the dynamic network topology and on the adaptiveness of the adversary providing the dynamic topology. More formally, we model communication using the standard graph-based radio network model. To model the dynamic network, we use a generalization of the synchronous dynamic graph model introduced in [Kuhn et al., STOC 2010]. For integer parameters $T\geq 1$ and $k\geq 1$, we call a dynamic graph $T$-interval $k$-connected if for every interval of $T$ consecutive rounds, there exists a $k$-vertex-connected stable subgraph. Further, for an integer parameter $\tau\geq 0$, we say that the adversary providing the dynamic network is $\tau$-oblivious if for constructing the graph of some round $t$, the adversary has access to all the randomness (and states) of the algorithm up to round $t-\tau$. As our main result, we show that for any $T\geq 1$, any $k\geq 1$, and any $\tau\geq 1$, for a $\tau$-oblivious adversary, there is a distributed algorithm to broadcast a single message in time $O\big(\big(1+\frac{n}{k\cdot\min\left\{\tau,T\right\}}\big)\cdot n\log^3 n\big)$. We further show that even for large interval $k$-connectivity, efficient broadcast is not possible for the usual adaptive adversaries. For a $1$-oblivious adversary, we show that even for any $T\leq (n/k)^{1-\varepsilon}$ (for any constant $\varepsilon>0$) and for any $k\geq 1$, global broadcast in $T$-interval $k$-connected networks requires at least $\Omega(n^2/(k^2\log n))$ time. Further, for a $0$ oblivious adversary, broadcast cannot be solved in $T$-interval $k$-connected networks as long as $T<n-k$.Comment: 17 pages, conference version appeared in OPODIS 201

The effects of hemodynamic lag on functional connectivity and behavior after stroke

Stroke disrupts the brain's vascular supply, not only within but also outside areas of infarction. We investigated temporal delays (lag) in resting state functional magnetic resonance imaging signals in 130 stroke patients scanned two weeks, three months and 12 months post stroke onset. Thirty controls were scanned twice at an interval of three months. Hemodynamic lag was determined using cross-correlation with the global gray matter signal. Behavioral performance in multiple domains was assessed in all patients. Regional cerebral blood flow and carotid patency were assessed in subsets of the cohort using arterial spin labeling and carotid Doppler ultrasonography. Significant hemodynamic lag was observed in 30% of stroke patients sub-acutely. Approximately 10% of patients showed lag at one-year post-stroke. Hemodynamic lag corresponded to gross aberrancy in functional connectivity measures, performance deficits in multiple domains and local and global perfusion deficits. Correcting for lag partially normalized abnormalities in measured functional connectivity. Yet post-stroke FC-behavior relationships in the motor and attention systems persisted even after hemodynamic delays were corrected. Resting state fMRI can reliably identify areas of hemodynamic delay following stroke. Our data reveal that hemodynamic delay is common sub-acutely, alters functional connectivity, and may be of clinical importance

Absolutely Continuous Spectrum for Random Schroedinger Operators on the Bethe Strip

The Bethe Strip of width $m$ is the cartesian product \B\times\{1,...,m\}, where \B is the Bethe lattice (Cayley tree). We prove that Anderson models on the Bethe strip have "extended states" for small disorder. More precisely, we consider Anderson-like Hamiltonians \;H_\lambda=\frac12 \Delta \otimes 1 + 1 \otimes A + \lambda \Vv on a Bethe strip with connectivity $K \geq 2$, where $A$ is an $m\times m$ symmetric matrix, \Vv is a random matrix potential, and $\lambda$ is the disorder parameter. Given any closed interval $I\subset (-\sqrt{K}+a_{\mathrm{max}},\sqrt{K}+a_{\mathrm{min}})$, where $a_{\mathrm{min}}$ and $a_{\mathrm{max}}$ are the smallest and largest eigenvalues of the matrix $A$, we prove that for $\lambda$ small the random Schr\"odinger operator $\;H_\lambda$ has purely absolutely continuous spectrum in $I$ with probability one and its integrated density of states is continuously differentiable on the interval $I$

Redividing the Cake

A heterogeneous resource, such as a land-estate, is already divided among several agents in an unfair way. It should be re-divided among the agents in a way that balances fairness with ownership rights. We present re-division protocols that attain various trade-off points between fairness and ownership rights, in various settings differing in the geometric constraints on the allotments: (a) no geometric constraints; (b) connectivity --- the cake is a one-dimensional interval and each piece must be a contiguous interval; (c) rectangularity --- the cake is a two-dimensional rectangle or rectilinear polygon and the pieces should be rectangles; (d) convexity --- the cake is a two-dimensional convex polygon and the pieces should be convex. Our re-division protocols have implications on another problem: the price-of-fairness --- the loss of social welfare caused by fairness requirements. Each protocol implies an upper bound on the price-of-fairness with the respective geometric constraints.Comment: Extended IJCAI 2018 version. Previous name: "How to Re-Divide a Cake Fairly

Spatiotemporal Regularity in Networks with Stochastically Varying Links

In this work we investigate time varying networks with complex dynamics at the nodes. We consider two scenarios of network change in an interval of time: first, we have the case where each link can change with probability pt, i.e. the network changes occur locally and independently at each node. Secondly we consider the case where the entire connectivity matrix changes with probability pt, i.e. the change is global. We show that network changes, occurring both locally and globally, yield an enhanced range of synchronization. When the connections are changed slowly (i.e. pt is low) the nodes display nearly synchronized intervals interrupted by intermittent unsynchronized chaotic bursts. However when the connections are switched quickly (i.e. pt is large), the intermittent behavior quickly settles down to a steady synchronized state. Furthermore we find that the mean time taken to reach synchronization from generic random initial states is significantly reduced when the underlying links change more rapidly. We also analyze the probabilistic dynamics of the system with changing connectivity and the stable synchronized range thus obtained is in broad agreement with those observed numerically.Comment: 15 pages, 8 figures, Keywords: Complex Networks, Temporal Networks, Synchronization, Coupled Map Lattic