186 research outputs found
Bootstrap percolation in strong products of graphs
Given a graph and assuming that some vertices of are infected, the
-neighbor bootstrap percolation rule makes an uninfected vertex infected
if has at least infected neighbors. The -percolation number,
, of is the minimum cardinality of a set of initially infected
vertices in such that after continuously performing the -neighbor
bootstrap percolation rule each vertex of eventually becomes infected. In
this paper, we consider percolation numbers of strong products of graphs. If
is the strong product of connected
graphs, we prove that as soon as and .
As a dichotomy, we present a family of strong products of connected graphs
with the -percolation number arbitrarily large. We refine these
results for strong products of graphs in which at least two factors have at
least three vertices. In addition, when all factors have at least three
vertices we prove that
for all , and we again get a dichotomy, since there exist families
of strong products of graphs such that their -percolation numbers
are arbitrarily large. While if both and have at
least three vertices, we also characterize the strong prisms
for which this equality holds. Some of the results naturally extend to infinite
graphs, and we briefly consider percolation numbers of strong products of
two-way infinite paths.Comment: 21 page
On interval number in cycle convexity
International audienceRecently, Araujo et al. [Manuscript in preparation, 2017] introduced the notion of Cycle Convexity of graphs. In their seminal work, they studied the graph convexity parameter called hull number for this new graph convexity they proposed, and they presented some of its applications in Knot theory. Roughly, the tunnel number of a knot embedded in a plane is upper bounded by the hull number of a corresponding planar 4-regular graph in cycle convexity. In this paper, we go further in the study of this new graph convexity and we study the interval number of a graph in cycle convexity. This parameter is, alongside the hull number, one of the most studied parameters in the literature about graph convexities. Precisely, given a graph G, its interval number in cycle convexity, denoted by , is the minimum cardinality of a set S ⊆ V (G) such that every vertex w ∈ V (G) \ S has two distinct neighbors u, v ∈ S such that u and v lie in same connected component of G[S], i.e. the subgraph of G induced by the vertices in S.In this work, first we provide bounds on and its relations to other graph convexity parameters, and explore its behavior on grids. Then, we present some hardness results by showing that deciding whether ≤ k is NP-complete, even if G is a split graph or a bounded-degree planar graph, and that the problem is W[2]-hard in bipartite graphs when k is the parameter. As a consequence, we obtainthat cannot be approximated up to a constant factor in the classes of split graphs and bipartite graphs (unless P = N P ).On the positive side, we present polynomial-time algorithms to compute for outerplanar graphs, cobipartite graphs and interval graphs. We also present fixed-parameter tractable (FPT) algorithms to compute it for (q, q − 4)-graphs when q is the parameter and for general graphs G when parameterized either by the treewidth or the neighborhood diversity of G.Some of our hardness results and positive results are not known to hold for related graph convexities and domination problems. We hope that the design of our new reductions and polynomial-time algorithms can be helpful in order to advance in the study of related graph problems
Two Biological Applications of Optimal Control to Hybrid Differential Equations and Elliptic Partial Differential Equations
In this dissertation, we investigate optimal control of hybrid differential equations and elliptic partial differential equations with two biological applications. We prove the existence of an optimal control for which the objective functional is maximized. The goal is to characterize the optimal control in terms of the solution of the opti- mality system. The optimality system consists of the state equations coupled with the adjoint equations. To obtain the optimality system we differentiate the objective functional with respect to the control. This process is applied to studying two prob- lems: one is a type of hybrid system involving ordinary differential equations and a discrete time feature. We apply our approach to a tick-transmitted disease model in which the tick dynamics changes seasonally while hosts have continuous dynam- ics. The goal is to maximize disease-free ticks and minimize infected ticks through an optimal control strategy of treatment with acaricide. The other is a semilinear elliptic partial differential equation model for fishery harvesting. We consider two objective functionals: maximizing the yield and minimizing the cost or variation in the fishing effort (control). Existence, necessary conditions and uniqueness for the optimal control for both problems are established. Numerical examples are given to illustrate the results
Estimating Diffusion Network Structures: Recovery Conditions, Sample Complexity & Soft-thresholding Algorithm
Information spreads across social and technological networks, but often the
network structures are hidden from us and we only observe the traces left by
the diffusion processes, called cascades. Can we recover the hidden network
structures from these observed cascades? What kind of cascades and how many
cascades do we need? Are there some network structures which are more difficult
than others to recover? Can we design efficient inference algorithms with
provable guarantees?
Despite the increasing availability of cascade data and methods for inferring
networks from these data, a thorough theoretical understanding of the above
questions remains largely unexplored in the literature. In this paper, we
investigate the network structure inference problem for a general family of
continuous-time diffusion models using an -regularized likelihood
maximization framework. We show that, as long as the cascade sampling process
satisfies a natural incoherence condition, our framework can recover the
correct network structure with high probability if we observe
cascades, where is the maximum number of parents of a node and is the
total number of nodes. Moreover, we develop a simple and efficient
soft-thresholding inference algorithm, which we use to illustrate the
consequences of our theoretical results, and show that our framework
outperforms other alternatives in practice.Comment: To appear in the 31st International Conference on Machine Learning
(ICML), 201
Irreversible 2-conversion set in graphs of bounded degree
An irreversible -threshold process (also a -neighbor bootstrap
percolation) is a dynamic process on a graph where vertices change color from
white to black if they have at least black neighbors. An irreversible
-conversion set of a graph is a subset of vertices of such that
the irreversible -threshold process starting with black eventually
changes all vertices of to black. We show that deciding the existence of an
irreversible 2-conversion set of a given size is NP-complete, even for graphs
of maximum degree 4, which answers a question of Dreyer and Roberts.
Conversely, we show that for graphs of maximum degree 3, the minimum size of an
irreversible 2-conversion set can be computed in polynomial time. Moreover, we
find an optimal irreversible 3-conversion set for the toroidal grid,
simplifying constructions of Pike and Zou.Comment: 18 pages, 12 figures; journal versio
50 years of first passage percolation
We celebrate the 50th anniversary of one the most classical models in
probability theory. In this survey, we describe the main results of first
passage percolation, paying special attention to the recent burst of advances
of the past 5 years. The purpose of these notes is twofold. In the first
chapters, we give self-contained proofs of seminal results obtained in the '80s
and '90s on limit shapes and geodesics, while covering the state of the art of
these questions. Second, aside from these classical results, we discuss recent
perspectives and directions including (1) the connection between Busemann
functions and geodesics, (2) the proof of sublinear variance under 2+log
moments of passage times and (3) the role of growth and competition models. We
also provide a collection of (old and new) open questions, hoping to solve them
before the 100th birthday.Comment: 160 pages, 17 figures. This version has updated chapters 3-5, with
expanded and additional material. Small typos corrected throughou
Learning graphical models from the Glauber dynamics
In this paper we consider the problem of learning undirected graphical models from data generated according to the Glauber dynamics. The Glauber dynamics is a Markov chain that sequentially updates individual nodes (variables) in a graphical model and it is frequently used to sample from the stationary distribution (to which it converges given sufficient time). Additionally, the Glauber dynamics is a natural dynamical model in a variety of settings. This work deviates from the standard formulation of graphical model learning in the literature, where one assumes access to i.i.d. samples from the distribution. Much of the research on graphical model learning has been directed towards finding algorithms with low computational cost. As the main result of this work, we establish that the problem of reconstructing binary pairwise graphical models is computationally tractable when we observe the Glauber dynamics. Specifically, we show that a binary pairwise graphical model on p nodes with maximum degree d can be learned in time f(d)p[superscript 3] log p, for a function f(d), using nearly the information-theoretic minimum possible number of samples. There is no known algorithm of comparable efficiency for learning arbitrary binary pairwise models from i.i.d. samples.National Science Foundation (U.S.) (Grant CMMI-1335155)National Science Foundation (U.S.) (Grant CNS-1161964)United States. Army Research Office. Multidisciplinary University Research Initiative (Award W911NF-11-1-0036
On the Voting Time of the Deterministic Majority Process
In the deterministic binary majority process we are given a simple graph
where each node has one out of two initial opinions. In every round, every node
adopts the majority opinion among its neighbors. By using a potential argument
first discovered by Goles and Olivos (1980), it is known that this process
always converges in rounds to a two-periodic state in which every node
either keeps its opinion or changes it in every round.
It has been shown by Frischknecht, Keller, and Wattenhofer (2013) that the
bound on the convergence time of the deterministic binary majority
process is indeed tight even for dense graphs. However, in many graphs such as
the complete graph, from any initial opinion assignment, the process converges
in just a constant number of rounds.
By carefully exploiting the structure of the potential function by Goles and
Olivos (1980), we derive a new upper bound on the convergence time of the
deterministic binary majority process that accounts for such exceptional cases.
We show that it is possible to identify certain modules of a graph in order
to obtain a new graph with the property that the worst-case
convergence time of is an upper bound on that of . Moreover, even
though our upper bound can be computed in linear time, we show that, given an
integer , it is NP-hard to decide whether there exists an initial opinion
assignment for which it takes more than rounds to converge to the
two-periodic state.Comment: full version of brief announcement accepted at DISC'1
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