Local Treewidth of Random and Noisy Graphs with Applications to Stopping Contagion in Networks

We study the notion of local treewidth in sparse random graphs: the maximum treewidth over all k-vertex subgraphs of an n-vertex graph. When k is not too large, we give nearly tight bounds for this local treewidth parameter; we also derive nearly tight bounds for the local treewidth of noisy trees, trees where every non-edge is added independently with small probability. We apply our upper bounds on the local treewidth to obtain fixed parameter tractable algorithms (on random graphs and noisy trees) for edge-removal problems centered around containing a contagious process evolving over a network. In these problems, our main parameter of study is k, the number of initially "infected" vertices in the network. For the random graph models we consider and a certain range of parameters the running time of our algorithms on n-vertex graphs is 2^o(k) poly(n), improving upon the 2^?(k) poly(n) performance of the best-known algorithms designed for worst-case instances of these edge deletion problems

Local treewidth of random and noisy graphs with applications to stopping contagion in networks

We study the notion of local treewidth in sparse random graphs: the maximum treewidth over all $k$-vertex subgraphs of an $n$-vertex graph. When $k$ is not too large, we give nearly tight bounds for this local treewidth parameter; we also derive tight bounds for the local treewidth of noisy trees, trees where every non-edge is added independently with small probability. We apply our upper bounds on the local treewidth to obtain fixed parameter tractable algorithms (on random graphs and noisy trees) for edge-removal problems centered around containing a contagious process evolving over a network. In these problems, our main parameter of study is $k$, the number of "infected" vertices in the network. For a certain range of parameters the running time of our algorithms on $n$-vertex graphs is $2^{o(k)}\textrm{poly}(n)$, improving upon the $2^{\Omega(k)}\textrm{poly}(n)$ performance of the best-known algorithms designed for worst-case instances of these edge deletion problems

On Approximating Target Set Selection

We study the Target Set Selection (TSS) problem introduced by Kempe, Kleinberg, and Tardos (2003). This problem models the propagation of influence in a network, in a sequence of rounds. A set of nodes is made "active" initially. In each subsequent round, a vertex is activated if at least a certain number of its neighbors are (already) active. In the minimization version, the goal is to activate a small set of vertices initially - a seed, or target, set - so that activation spreads to the entire graph. In the absence of a sublinear-factor algorithm for the general version, we provide a (sublinear) approximation algorithm for the bounded-round version, where the goal is to activate all the vertices in r rounds. Assuming a known conjecture on the hardness of Planted Dense Subgraph, we establish hardness-of-approximation results for the bounded-round version. We show that they translate to general Target Set Selection, leading to a hardness factor of n^(1/2-epsilon) for all epsilon > 0. This is the first polynomial hardness result for Target Set Selection, and the strongest conditional result known for a large class of monotone satisfiability problems. In the maximization version of TSS, the goal is to pick a target set of size k so as to maximize the number of nodes eventually active. We show an n^(1-epsilon) hardness result for the undirected maximization version of the problem, thus establishing that the undirected case is as hard as the directed case. Finally, we demonstrate an SETH lower bound for the exact computation of the optimal seed set

Predictors of contraction and expansion of area of occupancy for British birds

Copyright © 2014 The Author(s) Published by the Royal SocietyGeographical range dynamics are driven by the joint effects of abiotic factors, human ecosystem modifications, biotic interactions and the intrinsic organismal responses to these. However, the relative contribution of each component remains largely unknown. Here, we compare the contribution of life-history attributes, broad-scale gradients in climate and geographical context of species’ historical ranges, as predictors of recent changes in area of occupancy for 116 terrestrial British breeding birds (74 contractors, 42 expanders) between the early 1970s and late 1990s. Regional threat classifications demonstrated that the species of highest conservation concern showed both the largest contractions and the smallest expansions. Species responded differently to climate depending on geographical distribution—northern species changed their area of occupancy (expansion or contraction) more in warmer and drier regions, whereas southern species changed more in colder and wetter environments. Species with slow life history (larger body size) tended to have a lower probability of changing their area of occupancy than species with faster life history, whereas species with greater natal dispersal capacity resisted contraction and, counterintuitively, expansion. Higher geographical fragmentation of species' range also increased expansion probability, possibly indicating a release from a previously limiting condition, for example through agricultural abandonment since the 1970s. After accounting statistically for the complexity and nonlinearity of the data, our results demonstrate two key aspects of changing area of occupancy for British birds: (i) climate is the dominant driver of change, but direction of effect depends on geographical context, and (ii) all of our predictors generally had a similar effect regardless of the direction of the change (contraction versus expansion). Although we caution applying results from Britain's highly modified and well-studied bird community to other biogeographic regions, our results do indicate that a species' propensity to change area of occupancy over decadal scales can be explained partially by a combination of simple allometric predictors of life-history pace, average climate conditions and geographical context.Australian Research CouncilIntegrated Program of IC&DTFCT (Fundação para a Ciência e a Tecnologia