Abstract: Many of the network security protocols employed today utilize symmetric block ciphers (DES, AES and CAST etc). The majority of the symmetric block ciphers implement the crucial substitution operation using look up tables, called substitution boxes. These structures should be highly nonlinear and have bit dispersal, i.e. avalanche, properties in order to render the cipher with resistant to cryptanalysis attempts, such as linear and differential cryptanalysis. Highly secure substitution boxes can be constructed using particular Boolean functions as components that have certain mathematical properties which enhance the robustness of the whole cryptoalgorithm. However, enforcing these properties on SBoxes is a highly computationally intensive task. In this paper, we present a distributed algorithm and its implementation on a computing cluster that accelerates the construction of secure substitution boxes with good security properties. It is fully parametric since it can employ any class of Boolean functions with algorithmically definable properties and can construct SBoxes of arbitrary sizes. We demonstrate the efficiency of the distributed algorithm implementation compared to its sequential counterpart, in a number of experiments.

Abstract: In this work we present the architecture and implementation of WebDust, a software platform for managing multiple, heterogeneous (both in terms of software and hardware), geographically disparate sensor networks. We describe in detail the main concepts behind its design, and basic aspects of its implementation, including the services provided to end-users and developers. WebDust uses a peer-to-peer substrate, based on JXTA, in order to unify multiple sensor networks installed in various geographic areas. We aim at providing a software framework that will permit developers to deal with the new and critical aspects that networks of sensors and tiny devices bring into global computing, and to provide a coherent set of high level services, design rules and technical recommendations, in order to be able to develop the envisioned applications of global sensor networks. Furthermore, we give an overview of a deployed distributed testbed, consisting of a total 56 nodes and describing in more detail two specific testbed sites and the integration of the related software and hardware technologies used for its operation with our platform. Finally, we describe the design and implementation of an interface option provided to end-users, based on the popular Google Earth application.

Abstract: We study the problem of data propagation in sensor networks,
comprised of a large number of very small and low-cost nodes,
capable of sensing, communicating and computing. The distributed
co-operation of such nodes may lead to the accomplishment of large
sensing tasks, having useful applications in practice. We present
a new protocol for data propagation towards a control center
(``sink") that avoids flooding by probabilistically favoring
certain (``close to optimal") data transmissions.
This protocol is very simple to implement in sensor devices
and operates under total absence
of co-ordination between sensors. We consider a network model of randomly deployed sensors of sufficient density.
As shown by a geometry analysis,
the protocol is correct, since it always propagates data
to the sink, under ideal network conditions (no failures). Using
stochastic processes, we show that the protocol is very energy efficient. Also, when part of the network is inoperative, the
protocol manages to propagate data very close to the sink, thus in
this sense it is robust. We finally present and discuss
large-scale experimental findings validating the analytical
results.

Abstract: We study the problem of data propagation in sensor networks,
comprised of a large number of very small and low-cost nodes,
capable of sensing, communicating and computing. The distributed
co-operation of such nodes may lead to the accomplishment of large
sensing tasks, having useful applications in practice. We present a new protocol for data propagation towards a control center ("sink") that avoids flooding by probabilistically favoring certain ("close to optimal") data transmissions. Motivated by certain applications and also as a starting point for a rigorous analysis, we study here lattice-shaped sensor networks. We however show that this lattice shape emerges even in randomly deployed sensor networks of sufficient sensor density. Our work is inspired and builds upon the directed diffusion paradigm.
This protocol is very simple to implement in sensor devices, uses only local information and operates under total absence of co-ordination between sensors. We consider a network model of randomly deployed sensors of sufficient density. As shown by a geometry analysis, the protocol is correct, since it always propagates data to the sink, under ideal network conditions (no failures). Using stochastic processes, we show that the protocol is very energy efficient. Also, when part of the network is inoperative, the protocol manages to propagate data very close to the sink, thus in this sense it is robust. We finally present and discuss large-scale experimental findings validating the analytical results.

Abstract: Wireless sensor networks are comprised of a vast number of
ultra-small autonomous computing, communication and sensing devices,
with restricted energy and computing capabilities, that co-operate
to accomplish a large sensing task. Such networks can be very useful
in practice, e.g.~in the local monitoring of ambient conditions and
reporting them to a control center. In this paper we propose a
distributed group key establishment protocol that uses mobile agents
(software) and is particularly suitable for energy constrained,
dynamically evolving ad-hoc networks. Our approach totally avoids
the construction and the maintenance of a distributed structure that
reflects the topology of the network. Moreover, it trades-off
complex message exchanges by performing some amount of additional
local computations in order to be applicable at dense and dynamic
sensor networks. The extra computations are simple for the devices
to implement and are evenly distributed across the participants of
the network leading to good energy balance. We evaluate the
performance of our protocol in a simulated environment and compare
our results with existing group key establishment protocols. The
security of the protocol is based on the Diffie-Hellman problem and
we used in our experiments its elliptic curve analog. Our findings
basically indicate the feasibility of implementing our protocol in
real sensor network devices and highlight the advantages and
disadvantages of each approach given the available technology and
the corresponding efficiency (energy, time) criteria.

Abstract: As a result of recent significant technological advances, a new computing and communication environment, Mobile Ad Hoc Networks (MANET), is about to enter the mainstream. A multitude of critical aspects, including mobility, severe limitations and limited reliability, create a new set of crucial issues and trade-offs that must be carefully taken into account in the design of robust and efficient algorithms for these environments. The communication among mobile hosts is one among the many issues that need to be resolved efficiently before MANET becomes a commodity.
In this paper, we propose to discuss the communication problem in MANET as well as present some characteristic techniques for the design, the analysis and the performance evaluation of distributed communication protocols for mobile ad hoc networks. More specifically, we propose to review two different design techniques. While the first type of protocols tries to create and maintain routing paths among the hosts, the second set of protocols uses a randomly moving subset of the hosts that acts as an intermediate pool for receiving and delivering messages. We discuss the main design choices for each approach, along with performance analysis of selected protocols.

Abstract: We focus on the problem of computing approximate Nash equilibria and well-supported approximate Nash equilibria in random bimatrix games, where each player's payoffs are bounded and independent random variables, not necessarily identically distributed, but with common expectations. We show that the completely mixed uniform strategy profile, i.e. the combination of mixed strategies (one per player) where each player plays with equal probability each one of her available pure strategies, is an almost Nash equilibrium for random bimatrix games, in the sense that it is, with high probability, an {\aa}-well-supported Nash equilibrium where {\aa} tends to zero as n tends to infinity.

Abstract: In this work, we study the propagation of influence and computation in dynamic distributedcomputing systems that are possibly disconnected at every instant. We focus on a synchronous message-passing communication model with broadcast and bidirectional links. Our network dynamicity assumption is a worst-case dynamicity controlled by an adversary scheduler, which has received much attention recently. We replace the usual (in worst-case dynamic networks) assumption that the network is connected at every instant by minimal temporal connectivity conditions. Our conditions only require that another causal influence occurs within every time window of some given length. Based on this basic idea, we define several novel metrics for capturing the speed of information spreading in a dynamic network. We present several results that correlate these metrics. Moreover, we investigate termination criteria in networks in which an upper bound on any of these metrics is known. We exploit our termination criteria to provide efficient (and optimal in some cases) protocols that solve the fundamental counting and all-to-all token dissemination (or gossip) problems.

Abstract: In this chapter, our focus is on computational network analysis from a theoretical point of view. In particular, we study the \emph{propagation of influence and computation in dynamic distributedcomputing systems}. We focus on a \emph{synchronous message passing} communication model with bidirectional links. Our network dynamicity assumption is a \emph{worst-case dynamicity} controlled by an adversary scheduler, which has received much attention recently. We first study the fundamental \emph{naming} and \emph{counting} problems (and some variations) in
networks that are \emph{anonymous}, \emph{unknown}, and possibly dynamic. Network dynamicity is modeled here by the \emph{1-interval connectivity model}, in which communication is synchronous and a (worst-case) adversary
chooses the edges of every round subject to the condition that each instance is connected. We then replace this quite strong assumption by minimal \emph{temporal connectivity} conditions. These conditions only require that \emph{another causal influence occurs within every time-window of some given length}. Based on this basic idea we define several novel metrics for capturing the speed of information spreading in a dynamic network. We present several results that correlate these metrics. Moreover, we investigate \emph{termination criteria} in networks in which an upper bound on any of these metrics is known. We exploit these termination criteria to provide efficient (and optimal in some cases) protocols that solve the fundamental \emph{counting} and \emph{all-to-all token dissemination} (or \emph{gossip}) problems. Finally, we propose another model of worst-case temporal connectivity, called \emph{local
communication windows}, that assumes a fixed underlying communication network and restricts the adversary to allow communication between local neighborhoods in every time-window of some fixed length. We prove some basic properties and provide a protocol for counting in this model.

Abstract: We study the partially eponymous model of distributed computation, which simultaneously
generalizes the anonymous and the eponymous models. In this model, processors have
identities, which are neither necessarily all identical (as in the anonymous model) nor
necessarily unique (as in the eponymous model). In a decision problem formalized as a
relation, processors receive inputs and seek to reach outputs respecting the relation. We
focus on the partially eponymous ring, and we shall consider the computation of circularly
symmetric relations on it. We consider sets of rings where all rings in the set have the same
multiset of identity multiplicities.
We distinguish between solvability and computability: in solvability, processors are
required to always reach outputs respecting the relation; in computability, they must
do so whenever this is possible, and must otherwise report impossibility.
We present a topological characterization of solvability for a relation on a set of rings,
which can be expressed as an efficiently checkable, number-theoretic predicate.
We present a universal distributed algorithm for computing a relation on a set of
rings; it runs any distributed algorithm for constructing views, followed by local steps.
We derive, as our main result, a universal upper bound on the message complexity to
compute a relation on a set of rings; this bound demonstrates a graceful degradation
with the Least Minimum Base, a parameter indicating the degree of least possible
eponymity for a set of rings. Thereafter, we identify two cases where a relation can be
computed on a set of rings, with rings of size n, with an efficient number of O .n lg n/
messages.

Abstract: We consider a synchronous distributed system with n processes that communicate through a dynamic network guaranteeing 1-interval connectivity i.e., the network topology graph might change at each interval while keeping the graph connected at any time. The processes belonging to the distributed system are identified through a set of labels L = {l1 , l2 . . . , lk } (with 1 ≤ k < n). In this challenging system model, the paper addresses the following problem: ”counting the number of processes with the same label”. We provide a distributed algorithm that is able solve the problem based on the notion of energy transfer. Each process owns a fixed energy charge, and tries to discharge itself exchanging, at each round, at most half of its charge with neighbors. The paper also discusses when such counting is possible in the presence of failures. Counting processes with the same label in dynamic networks with homonyms is of great importance because it is as difficult as computing generic aggregating functions.

Abstract: Wireless sensor networks are comprised of a vast number of ultra-small autonomous computing, communication and sensing devices, with restricted energy and computing capabilities, that co-operate to accomplish a large sensing task. Such networks can be very useful in practice, e.g.~in the local monitoring of ambient conditions and reporting them to a control center. In this paper we propose a new lightweight, distributed group key establishment protocol suitable for such energy constrained networks. Our approach basically trade-offs complex message exchanges by performing some amount of additional local computations. The extra computations are simple for the devices to implement and are evenly distributed across the participants of the network leading to good energy balance. We evaluate the performance our protocol in comparison to existing group key establishment protocols both in simulated and real environments. The intractability of all protocols is based on the Diffie-Hellman problem and we used its elliptic curve analog in our experiments. Our findings basically indicate the feasibility of implementing our protocol in real sensor network devices and highlight the advantages and disadvantages of each approach given the available technology and the corresponding efficiency (energy, time) criteria.

Abstract: Wireless sensor networks are comprised of a vast number of ultra-small autonomous computing, communication and sensing devices, with restricted energy and computing capabilities, that co-operate to accomplish a large sensing task. Such networks can be very useful in practice, e.g.~in the local monitoring of ambient conditions and reporting them to a control center. In this paper we propose a new lightweight, distributed group key establishment protocol suitable for such energy constrained networks. Our approach basically trade-offs complex message exchanges by performing some amount of additional local computations. The extra computations are simple for the devices to implement and are evenly distributed across the participants of the network leading to good energy balance. We evaluate the performance our protocol in comparison to existing group key establishment protocols both in simulated and real environments. The intractability of all protocols is based on the Diffie-Hellman problem and we used its elliptic curve analog in our experiments. Our findings basically indicate the feasibility of implementing our protocol in real sensor network devices and highlight the advantages and disadvantages of each approach given the available technology and the corresponding efficiency (energy, time) criteria.

Abstract: A central problem in distributedcomputing and telecommunications
is the establishment of common knowledge between two computing
entities. An immediate use of such common knowledge is in the
initiation of a secure communication session between two entities
since the two entities may use this common knowledge in order to
produce a secret key for use with some symmetric cipher.
%
The dynamic establishment of shared information (e.g. secret key)
between two entities is particularly important in networks with no
predetermined structure such as wireless mobile ad-hoc networks.
In such networks, nodes establish and terminate communication
sessions dynamically with other nodes which may have never been
encountered before in order to somehow exchange information which
will enable them to subsequently communicate in a secure manner.
%
In this paper we give and theoretically analyze a protocol that
enables two entities initially possessing a string each to
securely eliminate inconsistent bit positions, obtaining strings
with a larger percentage of similarities. This can help the nodes
establish a shared set of bits and use it as a key with some
shared key encryption scheme.

Abstract: When one engineers distributed algorithms, some special characteristics
arise that are different from conventional (sequential or parallel)
computing paradigms. These characteristics include: the need for either a
scalable real network environment or a platform supporting a simulated
distributed environment; the need to incorporate asynchrony, where arbitrarya
synchrony is hard, if not impossible, to implement; and the generation
of “difficult” input instances which is a particular challenge. In this
work, we identifys ome of the methodological issues required to address
the above characteristics in distributed algorithm engineering and illustrate
certain approaches to tackle them via case studies. Our discussion
begins byad dressing the need of a simulation environment and how asynchronyis
incorporated when experimenting with distributed algorithms.
We then proceed bys uggesting two methods for generating difficult input
instances for distributed experiments, namelya game-theoretic one and another
based on simulations of adversarial arguments or lower bound proofs.
We give examples of the experimental analysis of a pursuit-evasion protocol
and of a shared memorypro blem in order to demonstrate these ideas.
We then address a particularlyi nteresting case of conducting experiments
with algorithms for mobile computing and tackle the important issue of
motion of processes in this context. We discuss the two-tier principle as
well as a concurrent random walks approach on an explicit representation
of motions in ad-hoc mobile networks, which allow at least for averagecase
analysis and measurements and may give worst-case inputs in some
cases. Finally, we discuss a useful interplay between theory and practice
that arise in modeling attack propagation in networks.

Abstract: Wireless sensor networks are comprised of a vast number of ultra-small fully autonomous computing, communication and sensing devices, with very restricted energy and computing capabilities, which co-operate to accomplish a large sensing task. Such networks can be very useful in practice in applications that require fine-grain monitoring of physical environment subjected to critical conditions (such as inaccessible terrains or disaster places). Very large numbers of sensor devices can be deployed in areas of interest and use self-organization and collaborative methods to form deeply networked environments. Features including the huge number of sensor devices involved, the severe power, computational and memory limitations, their dense deployment and frequent failures, pose new design and implementation aspects. The efficient and robust realization of such large, highly-dynamic, complex, non-conventional environments is a challenging algorithmic and technological task. In this work we consider certain important aspects of the design, deployment and operation of distributed algorithms for data propagation in wireless sensor networks and discuss some characteristic protocols, along with an evaluation of their performance.

Abstract: Counting items in a distributed system, and estimating the cardinality of multisets in particular,
is important for a large variety of applications and a fundamental building block for emerging Internet-scale information systems. Examples of such applications range from optimizing query access plans in peer-to-peer data sharing, to computing the significance (rank/score) of data items in distributed information retrieval. The general formal problem addressed in this article is computing the network-wide distinct number of items with some property (e.g., distinct files with file name
containing “spiderman”) where each node in the network holds an arbitrary subset, possibly overlapping the subsets of other nodes. The key requirements that a viable approach must satisfy are:
(1) scalability towards very large network size, (2) efficiency regarding messaging overhead, (3) load
balance of storage and access, (4) accuracy of the cardinality estimation, and (5) simplicity and easy
integration in applications. This article contributes the DHS (Distributed Hash Sketches) method
for this problem setting: a distributed, scalable, efficient, and accurate multiset cardinality estimator.
DHSis based on hash sketches for probabilistic counting, but distributes the bits of each counter
across network nodes in a judicious manner based on principles of Distributed Hash Tables, paying
careful attention to fast access and aggregation as well as update costs. The article discusses various
design choices, exhibiting tunable trade-offs between estimation accuracy, hop-count efficiency, and
load distribution fairness. We further contribute a full-fledged, publicly available, open-source implementation of all our methods, and a comprehensive experimental evaluation for various settings.

Abstract: Top-k query processing is a fundamental building block for efficient ranking in a large number of applications. Efficiency is a central issue, especially for distributed settings, when the data is spread across different nodes in a network. This paper introduces novel optimization methods for top-k aggregation queries in such distributed environments. The optimizations can be applied to all algorithms that fall into the frameworks of the prior TPUT and KLEE methods. The optimizations address three degrees of freedom: 1) hierarchically grouping input lists into top-k operator trees and optimizing the tree structure, 2) computing data-adaptive scan depths for different input sources, and 3) data-adaptive sampling of a small subset of input sources in scenarios with hundreds or thousands of query-relevant network nodes. All optimizations are based on a statistical cost model that utilizes local synopses, e.g., in the form of histograms, efficiently computed convolutions, and estimators based on order statistics. The paper presents comprehensive experiments, with three different real-life datasets and using the ns-2 network simulator for a packet-level simulation of a large Internet-style network.

Abstract: Andrews et al. [Automatic method for hiding latency in high bandwidth networks, in: Proceedings of the ACM Symposium on Theory of Computing, 1996, pp. 257–265; Improved methods for hiding latency in high bandwidth networks, in: Proceedings of the Eighth Annual ACM Symposium on Parallel Algorithms and Architectures, 1996, pp. 52–61] introduced a number of techniques for automatically hiding latency when performing simulations of networks with unit delay links on networks with arbitrary unequal delay links. In their work, they assume that processors of the host network are identical in computational power to those of the guest network being simulated. They further assume that the links of the host are able to pipeline messages, i.e., they are able to deliver P packets in time O(P+d) where d is the delay on the link.
In this paper we examine the effect of eliminating one or both of these assumptions. In particular, we provide an efficient simulation of a linear array of homogeneous processors connected by unit-delay links on a linear array of heterogeneous processors connected by links with arbitrary delay. We show that the slowdown achieved by our simulation is optimal. We then consider the case of simulating cliques by cliques; i.e., a clique of heterogeneous processors with arbitrary delay links is used to simulate a clique of homogeneous processors with unit delay links. We reduce the slowdown from the obvious bound of the maximum delay link to the average of the link delays. In the case of the linear array we consider both links with and without pipelining. For the clique simulation the links are not assumed to support pipelining.
The main motivation of our results (as was the case with Andrews et al.) is to mitigate the degradation of performance when executing parallel programs designed for different architectures on a network of workstations (NOW). In such a setting it is unlikely that the links provided by the NOW will support pipelining and it is quite probable the processors will be heterogeneous. Combining our result on clique simulation with well-known techniques for simulating shared memory PRAMs on distributed memory machines provides an effective automatic compilation of a PRAM algorithm on a NOW.

Abstract: We call radiation at a point of a wireless network the total amount of electromagnetic quantity (energy or power density) the point is exposed to. The impact of radiation can be high and we believe it is worth studying and control; towards radiation aware wireless networking we take (for the first time in the study of this aspect) a distributedcomputing, algorithmic approach. We exemplify this line of research by focusing on sensor networks, studying the minimum radiation path problem of finding the lowest radiation trajectory of a person moving from a source to a destination point in the network region. For this problem, we sketch the main ideas behind a linear program that can provide a tight approximation of the optimal solution, and then we discuss three heuristics that can lead to low radiation paths. We also plan to investigate the impact of diverse node mobility to the heuristics' performance.

Abstract: We consider the following distributed optimization problem: three agents i =
1; 2; 3 are each presented with a load drawn independently from the same known prior distribution.
Then each agent decides on which of two available bins to put her load. Each bin has
capacity �, and the objective is to find a distributed protocol that minimizes the probability
that an overflow occurs (or, equivalently, maximizes the winning probability).
In this work, we focus on the cases of full information and local information, depending on
whether each agent knows the loads of both other agents or not. Furthermore, we distinguish
between the cases where the agents are allowed to follow different decision rules (eponymous
model) or not (anonymous model ). We assume no communication among agents.
First, we present optimal protocols for the full information case, for both the anonymous and
the eponymous model.
For the local information, anonymous case, we show that the winning probability is upper
bounded by 0.622 in the case where the input loads are drawn from the uniform distribution.
Motivated by [3], we present a general method for computing the optimal single-threshold protocol
for any continuous distribution, and we apply this method to the case of the exponential
distribution.
Finally, we show how to compute, in exponential time, an optimal protocol for the local
information, eponymous model for the case where the input loads are drawn from a discretevalued,
bounded distribution.

Abstract: In this paper we present the design of a simulator platform called FUSE (Fast Universal Simulator Engine). The term Universal means that the Engine can be adapted easily to different domains and be used for varying simulation needs, although our main target is simulation of distributed algorithms in distributedcomputing environments. The Engine is Fast in the sense that the simulation overhead is minimal and very large systems can be simulated. We discuss the architecture and the design decisions that form the basis of these features. We also describe the functionality that is provided to its users (e.g., monitoring, statistics, etc.).

Abstract: The possibilities offered by utilizing sensors and pervasive computing technologies for creating large-scale multiplayer games are discussed in this chapter. Such game installations constitute a new social form of play taking place in public spaces. A main characteristic is the need to scale to a large number of users and engage players located simultaneously in dispersed areas, thus connected both on a local and Internet level. Fun in Numbers is a platform for developing and playing mobile, locative and collaborative distributed games and interactive installations, based on the participation of large numbers of people and their movement in the physical space. Players interact with each other using a wide range of hardware devices that are either generic (smartphones) or specific (sensor devices A set of related fundamental issues drawn upon the experience from several public events, where the FinN platform supported as many as 50 local users at the same time, is hereby presented.

Abstract: The promises inherent in users coming together to form data
sharing network communities, bring to the foreground new problems formulated
over such dynamic, ever growing, computing, storage, and networking
infrastructures. A key open challenge is to harness these highly
distributed resources toward the development of an ultra scalable, efficient
search engine. From a technical viewpoint, any acceptable solution
must fully exploit all available resources dictating the removal of any
centralized points of control, which can also readily lead to performance
bottlenecks and reliability/availability problems. Equally importantly,
however, a highly distributed solution can also facilitate pluralism in informing
users about internet content, which is crucial in order to preclude
the formation of information-resource monopolies and the biased visibility
of content from economically-powerful sources. To meet these challenges,
the work described here puts forward MINERVA{\^a}{\"i}¿½{\"i}¿½, a novel search
engine architecture, designed for scalability and efficiency. MINERVA{\^a}{\"i}¿½{\"i}¿½
encompasses a suite of novel algorithms, including algorithms for creating
data networks of interest, placing data on network nodes, load balancing,
top-k algorithms for retrieving data at query time, and replication algorithms
for expediting top-k query processing. We have implemented the
proposed architecture and we report on our extensive experiments with
real-world, web-crawled, and synthetic data and queries, showcasing the
scalability and efficiency traits of MINERVA{\^a}{\"i}¿½{\"i}¿½.

Abstract: This research further investigates the recently introduced
(in [4]) paradigm of radiation awareness in ambient environments with abundant heterogeneous wireless networking
from a distributedcomputing perspective. We call radiation
at a point of a wireless network the total amount of electromagnetic quantity the point is exposed to; our denition incorporates the eect of topology as well as the time domain
and environment aspects. Even if the impact of radiation to
human health remains largely unexplored and controversial,
we believe it is worth trying to understand and control, in
a way that does not decrease much the quality of service
oered to users of the wireless network.
In particular, we here focus on the fundamental problem
of ecient data propagation in wireless sensor networks, try-
ing to keep latency low while maintaining at low levels the
radiation cumulated by wireless transmissions. We rst propose greedy and oblivious routing heuristics that are radiation aware. We then combine them with temporal back-o
schemes that use local properties of the network (e.g. number of neighbours, distance from sink) in order to spread" radiation in a spatio-temporal way. Our proposed radiation
aware routing heuristics succeed to keep radiation levels low,
while not increasing latency.

Abstract: We focus on the problem of computing approximate Nash equilibria and well-supported approximate Nash equilibria in random bimatrix games, where each player's payoffs are bounded and independent random variables, not necessarily identically distributed, but with almost common expectations. We show that the completely mixed uniform strategy profile, i.e., the combination of mixed strategies (one per player) where each player plays with equal probability each one of her available pure strategies, is with high probability a TeX -Nash equilibrium and a TeX -well supported Nash equilibrium, where n is the number of pure strategies available to each player. This asserts that the completely mixed, uniform strategy profile is an almost Nash equilibrium for random bimatrix games, since it is, with high probability, an ϵ-well-supported Nash equilibrium where ϵ tends to zero as n tends to infinity.

Abstract: In this Phd thesis,, we try to use formal logic and threshold phenomena that asymptotically emerge with certainty in order to build new trust models and to evaluate the existing one. The departure point of our work is that dynamic, global computing systems are not amenable to a static viewpoint of the trust concept, no matter how this concept is formalized. We believe that trust should be a statistical, asymptotic concept to be studied in the limit as the system's components grow according to some growth rate. Thus, our main goal is to define trust as an emerging system property that ``appears'' or "disappears" when a set of properties hold, asymptotically with probability$ 0$ or $1$ correspondingly . Here we try to combine first and second order logic in order to analyze the trust measures of specific network models. Moreover we can use formal logic in order to determine whether generic reliability trust models provide a method for deriving trust between peers/entities as the network's components grow. Our approach can be used in a wide range of applications, such as monitoring the behavior of peers, providing a measure of trust between them, assessing the level of reliability of peers in a network. Wireless sensor networks are comprised of a vast number of ultra-small autonomous computing, communication and sensing devices, with restricted energy and computing capabilities, that co-operate to accomplish a large sensing task. Sensor networks can be very useful in practice. Such systems should at least guarantee the confidentiality and integrity of the information reported to the controlling authorities regarding the realization of environmental events. Therefore, key establishment is critical for the protection in wireless sensor networks and the prevention of adversaries from attacking the network. Finally in this dissertation we also propose three distributed group key establishment protocols suitable for such energy constrained networks. This dissertation is composed of two parts. Part I develops the theory of the first and second order logic of graphs - their definition, and the analysis of their properties that are expressible in the {\em first order language} of graphs. In part II we introduce some new distributed group key establishment protocols suitable for sensor networks. Several key establishment schemes are derived and their performance is demonstrated.

Abstract: We consider the important problem of energy balanced data propagation in wireless sensor networks and we extend and generalize
previous works by allowing adaptive energy assignment. We consider the data gathering problem where data are generated by the sensors and
must be routed toward a unique sink. Sensors route data by either sending the data directly to the sink or in a multi-hop fashion by delivering
the data to a neighbouring sensor. Direct and neighbouring transmissions require different levels of energy consumption. Basically, the protocols balance the energy consumption among the sensors by computing the adequate ratios of direct and neighbouring transmissions. An abstract model of energy dissipation as a random walk is proposed, along with rigorous performance analysis techniques. Two efficient distributed algorithms are presented and analysed, by both rigorous means and simulation.
The first one is easy to implement and fast to execute. The protocol assumes that sensors know a-priori the rate of data they generate.
The sink collects and processes all these information in order to compute the relevant value of the protocol parameter. This value is transmitted
to the sensors which individually compute their optimal ratios of direct and neighbouring transmissions. The second protocol avoids the necessary a-priori knowledge of the data rate generated by sensors by inferring the relevant information from the observation of the data paths.
Furthermore, this algorithm is based on stochastic estimation methods and is adaptive to environmental changes.

Abstract: We consider a security problem on a distributed network.
We assume a network whose nodes are vulnerable to infection
by threats (e.g. viruses), the attackers. A system security
software, the defender, is available in the system. However,
due to the network¢s size, economic and performance reasons,
it is capable to provide safety, i.e. clean nodes from
the possible presence of attackers, only to a limited part of
it. The objective of the defender is to place itself in such a
way as to maximize the number of attackers caught, while
each attacker aims not to be caught.
In [7], a basic case of this problem was modeled as a
non-cooperative game, called the Edge model. There, the
defender could protect a single link of the network. Here,
we consider a more general case of the problem where the
defender is able to scan and protect a set of k links of the
network, which we call the Tuple model. It is natural to expect
that this increased power of the defender should result
in a better quality of protection for the network. Ideally,
this would be achieved at little expense on the existence and
complexity of Nash equilibria (profiles where no entity can
improve its local objective unilaterally by switching placements
on the network).
In this paper we study pure and mixed Nash equilibria
in the model. In particular, we propose algorithms for computing
such equilibria in polynomial time and we provide a
polynomial-time transformation of a special class of Nash
equilibria, called matching equilibria, between the Edge
model and the Tuple model, and vice versa. Finally, we
establish that the increased power of the defender results in
higher-quality protection of the network.