Information Spreading in Stationary Markovian Evolving Graphs

Markovian evolving graphs are dynamic-graph models where the links among a fixed set of nodes change during time according to an arbitrary Markovian rule. They are extremely general and they can well describe important dynamic-network scenarios. We study the speed of information spreading in the "stationary phase" by analyzing the completion time of the "flooding mechanism". We prove a general theorem that establishes an upper bound on flooding time in any stationary Markovian evolving graph in terms of its node-expansion properties. We apply our theorem in two natural and relevant cases of such dynamic graphs. "Geometric Markovian evolving graphs" where the Markovian behaviour is yielded by "n" mobile radio stations, with fixed transmission radius, that perform independent random walks over a square region of the plane. "Edge-Markovian evolving graphs" where the probability of existence of any edge at time "t" depends on the existence (or not) of the same edge at time "t-1". In both cases, the obtained upper bounds hold "with high probability" and they are nearly tight. In fact, they turn out to be tight for a large range of the values of the input parameters. As for geometric Markovian evolving graphs, our result represents the first analytical upper bound for flooding time on a class of concrete mobile networks.Comment: 16 page

Viral processes by random walks on random regular graphs

We study the SIR epidemic model with infections carried by $k$ particles making independent random walks on a random regular graph. Here we assume $k\leq n^{\epsilon}$, where $n$ is the number of vertices in the random graph, and $\epsilon$ is some sufficiently small constant. We give an edge-weighted graph reduction of the dynamics of the process that allows us to apply standard results of Erd\H{o}s-R\'{e}nyi random graphs on the particle set. In particular, we show how the parameters of the model give two thresholds: In the subcritical regime, $O(\ln k)$ particles are infected. In the supercritical regime, for a constant $\beta\in(0,1)$ determined by the parameters of the model, $\beta k$ get infected with probability $\beta$, and $O(\ln k)$ get infected with probability $(1-\beta)$. Finally, there is a regime in which all $k$ particles are infected. Furthermore, the edge weights give information about when a particle becomes infected. We exploit this to give a completion time of the process for the SI case.Comment: Published in at http://dx.doi.org/10.1214/13-AAP1000 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the Role of Mobility for Multi-message Gossip

We consider information dissemination in a large $n$-user wireless network in which $k$ users wish to share a unique message with all other users. Each of the $n$ users only has knowledge of its own contents and state information; this corresponds to a one-sided push-only scenario. The goal is to disseminate all messages efficiently, hopefully achieving an order-optimal spreading rate over unicast wireless random networks. First, we show that a random-push strategy -- where a user sends its own or a received packet at random -- is order-wise suboptimal in a random geometric graph: specifically, $\Omega(\sqrt{n})$ times slower than optimal spreading. It is known that this gap can be closed if each user has "full" mobility, since this effectively creates a complete graph. We instead consider velocity-constrained mobility where at each time slot the user moves locally using a discrete random walk with velocity $v(n)$ that is much lower than full mobility. We propose a simple two-stage dissemination strategy that alternates between individual message flooding ("self promotion") and random gossiping. We prove that this scheme achieves a close to optimal spreading rate (within only a logarithmic gap) as long as the velocity is at least $v(n)=\omega(\sqrt{\log n/k})$. The key insight is that the mixing property introduced by the partial mobility helps users to spread in space within a relatively short period compared to the optimal spreading time, which macroscopically mimics message dissemination over a complete graph.Comment: accepted to IEEE Transactions on Information Theory, 201

A graph-based mathematical morphology reader

This survey paper aims at providing a "literary" anthology of mathematical morphology on graphs. It describes in the English language many ideas stemming from a large number of different papers, hence providing a unified view of an active and diverse field of research

Randomized rumor spreading in dynamic graphs

International audienceWe consider the well-studied rumor spreading model in which nodes contact a random neighbor in each round in order to push or pull the rumor. Unlike most previous works which focus on static topologies, we look at a dynamic graph model where an adversary is allowed to rewire the connections between vertices before each round, giving rise to a sequence of graphs, G1, G2, . . . Our first result is a bound on the rumor spreading time in terms of the conductance of those graphs. We show that if the degree of each node does not change much during the protocol (that is, by at most a constant factor), then the spread completes within t rounds for some t such that the sum of conductances of the graphs G1 up to Gt is O(log n). This result holds even against an adaptive adversary whose decisions in a round may depend on the set of informed vertices before the round, and implies the known tight bound with conductance for static graphs. Next we show that for the alternative expansion measure of vertex expansion, the situation is different. An adaptive adversary can delay the spread of rumor significantly even if graphs are regular and have high expansion, unlike in the static graph case where high expansion is known to guarantee fast rumor spreading. However, if the adversary is oblivious, i.e., the graph sequence is decided before the protocol begins, then we show that a bound close to the one for the static case holds for any sequence of regular graphs

Power laws in complex graphs: parsimonious generative models, similarity testing algorithms, and the origins

This dissertation mainly discussed topics related to power law graphs, including graph similarity testing algorithms and power law generative models. For graph similarity testing, we proposed a method based on the mathematical theory of diffusion over manifolds using random walks over graphs. We show that our method not only distinguishes between graphs with different degree distributions, but also graphs with the same degree distributions. We compare the undirected power law graphs generated by Barabasi-Albert model and directed power law graphs generated by Krapivsky\u27s model to the random graphs generated by Erdos-Renyi model. We also compare power law graphs generated by four different generative models with the same degree distribution. The comparison results show that, our method performs better compared to the traditional features like eigenvalue spectrum and degree distributions. To study the generative mechanism of bivariate power law data in social networks, we use Poisson Counter Driven Stochastic Differential Equation (PCSDE) models as mathematical tool. We propose three types of bivariate PCSDE models. We study the tail dependence of the models and compare the models to real data in social networks. Type 1 model with Markov on-off modulation generates tail dependence coefficient (TDC) with values either zero or one; while the Type 2 model with coupled growth has the values between zero and one. The first two types of models do not fit the real data in distribution. Type 3 model keeps the shared Poisson counter in Type 1, but uses independent Brownian motion components instead of independent Poisson counters. We show that second Type 3 model with $0\u3c\gamma\u3c1$ has fractional TDC and synthetic data fits the real data in distribution. We study the applications of our proposed bivariate models. At first, the connection between Type 3 model to the existing network growing models is discussed. By connecting the two, our model explains why correlated bivariate power law in directed growing networks. The idea of exponential growth and random stopping can also be used to explain the existence of power law in many other natural or man-made phenomenons. We show that bivariate power law data also exists in natural images. We propose a new generative model for self-similar images based on our second Type 3 model

Swamp ecology in a dynamic coastal landscape: an investigation through field study and simulation modeling

Increased flooding, nutrient and sediment deprivation, and saltwater intrusion have been implicated as probable causes of coastal swamp deterioration in the Mississippi Delta. An understanding of the interactive effects of these factors is required to enable successful planning of wetland restoration activities. I used field data collected from 2000 till 2005 at forty study sites to characterize the baseline conditions of the Maurepas swamp. I used a cluster analysis to identify four swamp habitat clusters, and characterized the clusters on the basis of soil properties, salinity, basal area, stem density, and other tree-related variables. ANOVA and related statistical techniques showed that three of the four habitat clusters exhibited tree biomass and densities indicative of flooding stress, and one cluster showed high tree mortality in response to salt-water intrusion. I then developed a two-species individual-based forest succession model (IBM) of a coastal swamp. The IBM followed the weekly growth, mortality, and reproduction of individuals of Taxodium distichum and Nyssa aquatica trees in a 1-km2 spatial grid, using historical time-series of stage and salinity data as inputs. IBM simulations predicted that increased flooding leads to swamps with reduced basal areas and stem densities, while increased salinity (~1-3 psu) resulted in lower basal areas. The IBM showed a tendency to overestimate wood production and the dominance of T. distichum in comparison to field data. Lastly, I compared the predictions of the IBM and a widely-used landscape model. I used salinity and flooding conditions simulated by the landscape model in eight of its 1-km2 cells as input to the IBM, and compared both models’ predictions of habitat change over 100 years. The models showed good agreement in their predictions of marsh persistence and swamp to marsh conversion. The IBM, however, showed higher sensitivity to changes in both salinity and flooding than the landscape model, and never predicted swamp persistence. The next generation of models for forecasting coastal habitat change in the Mississippi Delta will likely be a combination of the individual-based and landscape models used in this dissertation