We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Underwater sensor nodes will find applications in oceanographic data collection, pollution monitoring, offshore exploration, disaster prevention, assisted navigation and tactical surveillance applications. Moreover, unmanned or autonomous underwater vehicles (UUVs, AUVs), equipped with sensors, will enable the exploration of natural undersea resources and gathering of scientific data in collaborative monitoring missions. Underwater acoustic networking is the enabling technology for these applications. Underwater networks consist of a variable number of sensors and vehicles that are deployed to perform collaborative monitoring tasks over a given area. In this paper, several fundamental key aspects of underwater acoustic communications are investigated. Different architectures for two-dimensional and three-dimensional underwater sensor networks are discussed, and the characteristics of the underwater channel are detailed. The main challenges for the development of efficient networking solutions posed by the underwater environment are detailed and a cross-layer approach to the integration of all communication functionalities is suggested. Furthermore, open research issues are discussed and possible solution approaches are outlined. � 2005 Published by Elsevier B.V.

/pdf/underwater-acoustic-sensor-networks-research-challenges-15fjl1gvhc.pdf

Underwater acoustic sensor networks: research challenges

The major challenge in designing wireless sensor networks (WSNs) is the support of the functional, such as data latency, and the non-functional, such as data integrity, requirements while coping with the computation, energy and communication constraints. Careful node placement can be a very effective optimization means for achieving the desired design goals. In this paper, we report on the current state of the research on optimized node placement in WSNs. We highlight the issues, identify the various objectives and enumerate the different models and formulations. We categorize the placement strategies into static and dynamic depending on whether the optimization is performed at the time of deployment or while the network is operational, respectively. We further classify the published techniques based on the role that the node plays in the network and the primary performance objective considered. The paper also highlights open problems in this area of research.

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Strategies and techniques for node placement in wireless sensor networks: A survey

In this paper, architectures for two-dimensional and three-dimensional underwater sensor networks are discussed. A detailed overview on the current solutions for medium access control, network, and transport layer protocols are given and open research issues are discussed.

/pdf/state-of-the-art-in-protocol-research-for-underwater-1at0sx7th8.pdf

State-of-the-art in protocol research for underwater acoustic sensor networks

Underwater wireless communications can be carried out through acoustic, radio frequency (RF), and optical waves. Compared to its bandwidth limited acoustic and RF counterparts, underwater optical wireless communications (UOWCs) can support higher data rates at low latency levels. However, the severe aquatic channel conditions (e.g., absorption, scattering, turbulence, etc.) pose great challenges for UOWCs and significantly reduce the attainable communication ranges, which necessitates efficient networking and localization solutions. Therefore, we provide a comprehensive survey on the challenges, advances, and prospects of underwater optical wireless networks (UOWNs) from a layer by layer perspective which includes: (1) Physical layer issues including propagation characteristics, channel modeling, and modulation techniques (2) Data link layer problems covering link configurations, link budgets, performance metrics, and multiple access schemes; (3) Network layer topics containing relaying techniques and potential routing algorithms; (4) Transport layer subjects such as connectivity, reliability, flow and congestion control; (5) Application layer goals, and (6) Localization and its impacts on UOWN layers. Finally, we outline the open research challenges and point out the prospective directions for underwater optical wireless communications, networking, and localization studies.

/pdf/underwater-optical-wireless-communications-networking-and-4b2y1mfv1m.pdf

Underwater Optical Wireless Communications, Networking, and Localization: A Survey

We analyze various critical transmitting/sensing ranges for connectivity and coverage in three-dimensional sensor networks. As in other large-scale complex systems, many global parameters of sensor networks undergo phase transitions. For a given property of the network, there is a critical threshold, corresponding to the minimum amount of the communication effort or power expenditure by individual nodes, above (respectively, below) which the property exists with high (respectively, a low) probability. For sensor networks, properties of interest include simple and multiple degrees of connectivity/coverage. First, we investigate the network topology according to the region of deployment, the number of deployed sensors, and their transmitting/sensing ranges. More specifically, we consider the following problems: assume that n nodes, each capable of sensing events within a radius of r, are randomly and uniformly distributed in a 3-dimensional region R of volume V, how large must the sensing range R/sub SENSE/ be to ensure a given degree of coverage of the region to monitor? For a given transmission range R/sub TRANS/, what is the minimum (respectively, maximum) degree of the network? What is then the typical hop diameter of the underlying network? Next, we show how these results affect algorithmic aspects of the network by designing specific distributed protocols for sensor networks.

https://hal.archives-ouvertes.fr/docs/00/02/74/71/PDF/three-dim.pdf

Extremal properties of three-dimensional sensor networks with applications

Given a set $\xi=\{H_1,H_2,...\}$ of connected non acyclic graphs, a $\xi$-free graph is one which does not contain any member of $% \xi$ as copy. Define the excess of a graph as the difference between its number of edges and its number of vertices. Let ${\gr{W}}_{k,\xi}$ be theexponential generating function (EGF for brief) of connected $\xi$-free graphs of excess equal to $k$ ($k \geq 1$). For each fixed $\xi$, a fundamental differential recurrence satisfied by the EGFs ${\gr{W}}_{k,\xi}$ is derived. We give methods on how to solve this nonlinear recurrence for the first few values of $k$ by means of graph surgery. We also show that for any finite collection $\xi$ of non-acyclic graphs, the EGFs ${\gr{W}}_{k,\xi}$ are always rational functions of the generating function, $T$, of Cayley's rooted (non-planar) labelled trees. From this, we prove that almost all connected graphs with $n$ nodes and $n+k$ edges are $\xi$-free, whenever $k=o(n^{1/3})$ and $|\xi| < \infty$ by means of Wright's inequalities and saddle point method. Limiting distributions are derived for sparse connected $\xi$-free components that are present when a random graph on $n$ nodes has approximately $\frac{n}{2}$ edges. In particular, the probability distribution that it consists of trees, unicyclic components, $...$, $(q+1)$-cyclic components all $\xi$-free is derived. Similar results are also obtained for multigraphs, which are graphs where self-loops and multiple-edges are allowed.

Forbidden Subgraphs in Connected Graphs

https://hal.archives-ouvertes.fr/hal-00003325/document

Forbidden subgraphs in connected graphs

https://hal.archives-ouvertes.fr/hal-00153895/document

On the growth of components with non-fixed excesses

Denote by an $l$-component a connected graph with $l$ edges more than vertices. We prove that the expected number of creations of $(l+1)$-component, by means of adding a new edge to an $l$-component in a randomly growing graph with $n$ vertices, tends to 1 as $l,n$ tends to $\infty$ but with $l = o(n^{1/4})$. We also show, under the same conditions on $l$ and $n$, that the expected number of vertices that ever belong to an $l$-component is $\sim (12l)^{1/3} n^{2/3}$.

Vlady Ravelomanana

Papers

Extremal properties of three-dimensional sensor networks with applications

Forbidden Subgraphs in Connected Graphs

Forbidden subgraphs in connected graphs

On the growth of components with non-fixed excesses

On the growth of components with non fixed excesses