Abstract: We present a new dynamicgraph structure specifically suited
for large-scale transportation networks that provides simultaneously three
unique features: compactness, agility and dynamicity. We demonstrate
its practicality and superiority by conducting an experimental study for
shortest route planning in large-scale European and US road networks
with a few dozen millions of nodes and edges. Our approach is the first
one that concerns the dynamic maintenance of a large-scale graph with
ordered elements using a contiguous memory part, and which allows an
arbitrary online reordering of its elements.

Abstract: We investigate the problem of efficient data collection in wireless sensor networks where both the sensors and the sink move. We especially study the important, realistic case where the spatial distribution of sensors is non-uniform and their mobility is diverse and dynamic. The basic idea of our protocol is for the sink to benefit of the local information that sensors spread in the network as they move, in order
to extract current local conditions and accordingly adjust its trajectory. Thus, sensory motion anyway present in the network serves as a low cost replacement of network information propagation. In particular, we investigate two variations of our method: a)the greedy motion of the sink towards the region of highest density each time and b)taking into account the aggregate density in wider network regions. An extensive comparative evaluation to relevant data collection methods (both randomized and optimized deterministic), demonstrates that our approach achieves significant performance gains, especially in non-uniform placements (but also in uniform ones). In fact, the greedy version of our approach is more suitable in networks where the concentration regions appear in a spatially balanced manner, while the aggregate scheme is more appropriate in networks where the concentration areas are geographically correlated.

Abstract: We investigate the problem of efficient data collection in wireless sensor networks where both the sensors and the sink move. We especially study the important, realistic case where the spatial distribution of sensors is non-uniform and their mobility is diverse and dynamic. The basic idea of our protocol is for the sink to benefit of the local information that sensors spread in the network as they move, in order to extract current local conditions and accordingly adjust its trajectory. Thus, sensory motion anyway present in the network serves as a low cost replacement of network information propagation. In particular, we investigate two variations of our method: a) the greedy motion of the sink towards the region of highest density each time and b) taking into account the aggregate density in wider network regions. An extensive comparative evaluation to relevant data collection methods (both randomized and optimized deterministic), demonstrates that our approach achieves significant performance gains, especially in non-uniform placements (but also in uniform ones). In fact, the greedy version of our approach is more suitable in networks where the concentration regions appear in a spatially balanced manner, while the aggregate scheme is more appropriate in networks where the concentration areas are geographically correlated. We also investigate the case of multiple sinks by suggesting appropriate distributed coordination methods.

Abstract: An ever growing emphasis is put nowadays in developing personalized journey planning and renewable mobility services in smart cities. These services combine means of scheduled-based public transport and electric vehicles or bikes, using crowdsourcing techniques for collecting real-time traffic information and for assessing the recommended routes. The goal is to develop an information system that will allow the fast, real-time computation of best routes.
The main challenges in developing such an information system are both technological and algorithmic. The technological challenge concerns the collection, storage, management, and updating of a huge volume of transport data that are usually time-dependent, and the provision (through these data) of personalized renewable mobility services in smartphones. This challenge is typically confronted by creating a cloud infrastructure that on the one hand will support the storage, management, and updating of data, while on the other hand it will handle the necessary data feed to the smartphone applications for providing the users with the requested best routes.
The algorithmic challenge concerns the development of innovative algorithms for the efficient provision of journey planning services in smartphones, based on data they will receive from the cloud infrastructure. These services guarantee the computation of realistic and useful best routes, as well as the updating of the precomputed (route and timetable) information, in case of delays of scheduled public transport vehicles, so that the users can online update their routes to destination. The goal is to develop an algorithmic basis for supporting modern renewable mobility services (information systems), such as "mobility on demand'' (where the next leg of a journey is decided in real-time) and "door-to-door'' personalized mobility, in urban scheduled-based public transport environments. Scheduled-based public transport information systems should not only compute in real-time end-user queries requesting best routes, but also to update the timetable information in case of delays.
The core algorithmic issues of mobility and journey planning (regarding the computation of optimal routes under certain criteria) in scheduled-based public transport systems concern the efficient solution of the fundamental earlier arrival (EA) problem (compute a journey from station S to station T minimizing the overall traveling time required to complete the journey), the minimum number of
transfers (MNT) problem (compute a journey from station S to station T minimizing the number of times a passenger is required to change vehicle), and the efficient updating of timetable information system in case of vehicle delays. The EA and MNT problems have been extensively studied in the literature under two main approaches: the array-based modeling (where the timetable is represented as an array) and the graph-based modeling (where the timetable is represented as a graph). Experimental results have shown so far that the array-based approaches are faster in terms of query time than graph-based ones, as they are able to better exploit data locality and do not rely on priority queues. On the other hand, the array-based approaches have not been theoretically or experimentally studied as far as the efficient updating of timetable information, in case of delays, is concerned.
In this thesis, new graph-based models are being developed that solve efficiently the aforementioned fundamental algorithmic mobility problems in urban scheduled-based public transport information systems, along with a mobile application (journey planner) running on Android-based smartphones that includes a service for the evaluation of the recommended routes by the users. In particular:
(a) An extensive comparative evaluation was conducted on graph-based dynamic models that represent big data volumes regarding their suitability for representing timetable information. The study confirmed that the realistic time-expanded model is the most suitable for representing timetable information.
(b) Two new graph-based models have been developed for representing timetable information (in a timetable information system), the reduced time-expanded model and the dynamic timetable model (DTM), both of which are more space-efficient with respect to the realistic time-expanded model. For both of the new models, new efficient algorithms were developed for fast answering of EA and MNT queries, as well as for updating the timetable information representation in case of delays.
(c)An experimental evaluation was conducted with the new graph-based models and their associated query and update algorithms on a set of 14 real-world scheduled-based transportation systems, including the metropolitan areas of Berlin, Athens, London, Rome, and Madrid. The experimental results showed that the query algorithms of the reduced time-expanded model are superior to those of the DTM model, while the reverse is true regarding the update algorithms. In addition, the experimental study showed that the query algorithms of the new graph-based models compete favorably with those of the best array-based models.
(d) A mobile, cloud-based, journey planner (information system) was developed whose core algorithmic engine builds upon the new graph-based models. The mobile application is accompanied by a service that allows the users to assess the recommended journeys. The journey planner demonstrates the practicality of the new graph-based models and their associated query and update algorithms.

Abstract: We have conducted an extensive experimental study on algorithms for fully dynamic transitive
closure. We have implemented the recent fully dynamic algorithms by King [1999], Roditty [2003],
Roditty and Zwick [2002, 2004], and Demetrescu and Italiano [2000, 2005] along with several variants
and compared them to pseudo fully dynamic and simple-minded algorithms developed in a
previous study [Frigioni et al. 2001].We tested and compared these implementations on random inputs,
synthetic (worst-case) inputs, and on inputs motivated by real-world graphs. Our experiments
reveal that some of the dynamic algorithms can really be of practical value in many situations.

Abstract: We have conducted an extensive experimental study on algorithms for fully dynamic transitive
closure. We have implemented the recent fully dynamic algorithms by King [1999], Roditty [2003],
Roditty and Zwick [2002, 2004], and Demetrescu and Italiano [2000, 2005] along with several variants
and compared them to pseudo fully dynamic and simple-minded algorithms developed in a
previous study [Frigioni et al. 2001].We tested and compared these implementations on random inputs,
synthetic (worst-case) inputs, and on inputs motivated by real-world graphs. Our experiments
reveal that some of the dynamic algorithms can really be of practical value in many situations.

Abstract: We perform an extensive experimental study of several dynamic algorithms for transitive closure. In particular, we implemented algorithms given by Italiano, Yellin, Cicerone et al., and two recent randomized algorithms by Henzinger and King. We propose a fine-tuned version of Italiano's algorithms as well as a new variant of them, both of which were always faster than any of the other implementations of the dynamic algorithms. We also considered simple-minded algorithms that were easy to implement and likely to be fast in practice. Wetested and compared the above implementations on random inputs, on non-random inputs that are worst-case inputs for the dynamic algorithms, and on an input motivated by a real-world graph.

Abstract: An ad-hoc mobile network is a collection of mobile hosts, with wireless communication capability, forming a temporary network without the aid of any established fixed infrastructure. In such a (dynamically changing) network it is not at all easy to avoid broadcasting (and flooding).
In this paper we propose, theoretically analyse and experimentally validate a new and efficient protocol for pairwise communication. The protocol exploits the co-ordinated motion of a small part of the network (i.e. it is a semi-compulsory protocol) in order to provide to various senders and receivers an efficient support for message passing. Our implementation platform is the LEDA system and we have tested the protocol for three classes of graphs (grids, random graphs and bipartite multi-stage graphs) each ing a different ?motion topology?.
Our theoretical analysis (based on properties of random walks) and our experimental measurements indicate that only a small fraction of the mobile stations are enough to be exploited by the support in order to achieve very fast communication between any pair of mobile stations.

Abstract: This paper addresses the problem of counting the size of a network where (i) processes have the same identifiers (anonymous nodes) and (ii) the et-
work topology constantly changes (dynamic network). Changes are riven by a powerful adversary that can look at internal process states and add and remove edges in order to contrast the convergence of the algorithm to the correct count. The paper proposes two leader-based counting algorithms. Such algorithms are based on a technique that mimics an energy-transfer between network nodes. The first algorithm assumes that the adversary cannot generate either disconnected network graphs or network graphs where nodes have degree greater than D. In such algorithm, the leader can count the size of the network and detect the counting termination in a finite time (i.e., conscious counting algorithm). The second algorithm assumes that the adversary only keeps the network graph connected at any time and we prove that the leader can still converge to a correct count in a finite number of rounds, but it is not conscious when this convergence happens.

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: In this work we investigate the problem of communication among mobile hosts, one of the most fundamental problems in ad-hoc mobile networks that is at the core of many algorithms. Our work investigates the extreme case of total absence of any fixed network backbone or centralized administration, instantly forming networks based only on mobile hosts with wireless communication capabilities, where topological connectivity is subject to frequent, unpredictable change.
For such dynamically changing networks we propose a set of protocols which exploit the coordinated (by the protocol) motion of a small part of the network in order to manage network operations. We show that such protocols can be designed to work correctly and efficiently for communication by avoiding message flooding. Our protocols manage to establish communication between any pair of mobile hosts in small, a-priori guaranteed expected time bounds. Our results exploit and further develop some fundamental properties of random walks in finite graph.
Apart from studying the general case, we identify two practical and interesting cases of ad-hoc mobile networks: a) hierarchical ad-hoc networks, b) highly changing ad-hoc networks, for which we propose protocols that efficiently deal with the problem of basic communication.
We have conducted a set of extensive experiments, comprised of thousands of mobile hosts in order to validate the theoretical results and show that our protocols achieve very efficient communication under different scenaria.

Abstract: Wireless sensor networks are a recently introduced category of ad hoc computer networks, which are comprised by nodes of small size and limited computing and energy resources. Such nodes are able of measuring physical properties such as temperature, humidity, etc., wireless communication between each other and in some cases interaction with their surrounding environments (through the use of electromechanical parts).
As these networks have begun to be widely available (in terms of cost and commercial hardware availability), their field of application and philosophy of use is constantly evolving. We have numerous examples of their applications, ranging from monitoring the biodiversity of a specific outdoor area to structural health monitoring of bridges, and also networks ranging from few tens of nodes to even thousands of nodes.
In this PhD thesis we investigated the following basic research lines related to wireless sensor networks:
a) their simulation,
b) the development of data propagation protocols suited to such networks and their evaluation through simulation,
c) the modelling of ``hostile'' circumstances (obstacles) during their operation and evaluation of their impact through simulation,
d) the development of a sensor network management application.
Regarding simulation, we initially placed an emphasis to issues such as the effective simulation of networks of several thousands of nodes, and in that respect we developed a network simulator (simDust), which is extendable through the addition of new data propagation protocols and visualization capabilities. This simulator was used to evaluate the performance of a number of characteristic data propagation protocols for wireless sensor networks. Furthermore, we developed a new protocol (VRTP) and evaluated its performance against other similar protocols. Our studies show that the new protocol, that uses dynamic changes of the transmission range of the network nodes, performs better in certain cases than other related protocols, especially in networks containing obstacles and in the case of non-homogeneous placement of nodes.
Moreover, we emphasized on the addition of ``realistic'' conditions to the simulation of such protocols, that have an adversarial effect on their operation. Our goal was to introduce a model for obstacles that adds little computational overhead to a simulator, and also study the effect of the inclusion of such a model on data propagation protocols that use geographic information (absolute or relative). Such protocols are relatively sensitive to dynamic topology changes and network conditions. Through our experiments, we show that the inclusion of obstacles during simulation can have a significant effect on these protocols.
Finally, regarding applications, we initially proposed an architecture (WebDust/ShareSense), for the management of such networks, that would provide basic capabilities of managing such networks and developing applications above it. Features that set it apart are the capability of managing multiple heterogeneous sensor networks, openess, the use of a peer-to-peer architecture for the interconnection of multiple sensor network. A large part of the proposed architecture was implemented, while the overall architecture was extended to also include additional visualization capabilities.

Abstract: We study here the problem of determining the majority type in an arbitrary connected network, each vertex of which has initially two possible types. The vertices may have a few additional possible states and can interact in pairs only if they share an edge. Any (population) protocol is required to stabilize in the initial majority. We first present and analyze a protocol with 4 states per vertex that always computes the initial majority value, under any fair scheduler. As we prove, this protocol is optimal, in the sense that there is no population protocol that always computes majority with fewer than 4 states per vertex. However this does not rule out the existence of a protocol with 3 states per vertex that is correct with high probability. To this end, we examine a very natural majority protocol with 3 states per vertex, introduced in [Angluin et al. 2008] where its performance has been analyzed for the clique graph. We study the performance of this protocol in arbitrary networks. We prove that, when the two initial states are put uniformly at random on the vertices, this protocol converges to the initial majority with probability higher than the probability of converging to the initial minority. In contrast, we present an infinite family of graphs, on which the protocol can fail whp, even when the difference between the initial majority and the initial minority is n−Θ(lnn). We also present another infinite family of graphs in which the protocol of Angluin et al. takes an expected exponential time to converge. These two negative results build upon a very positive result concerning the robustness of the protocol on the clique. Surprisingly, the resistance of the clique to failure causes the failure in general graphs. Our techniques use new domination and coupling arguments for suitably defined processes whose dynamics capture the antagonism between the states involved.

Abstract: Using a set of geometric containers to speed up shortest path queries in a weighted graph has been proven a useful tool for dealing with large sparse graphs. Given a layout of a graph G=(V,E), we store, for each edge (u,v)set membership, variantE, the bounding box of all nodes tset membership, variantV for which a shortest u-t-path starts with (u,v). Shortest path queries can then be answered by DijkstraImage restricted to edges where the corresponding bounding box contains the target.
In this paper, we present new algorithms as well as an empirical study for the dynamic case of this problem, where edge weights are subject to change and the bounding boxes have to be updated. We evaluate the quality and the time for different update strategies that guarantee correct shortest paths in an interesting application to railway information systems, using real-world data from six European countries.

Abstract: Many efforts have been done in the last years to model public transport timetables in order to
find optimal routes. The proposed models can be classified into two types: those representing the
timetable as an array, and those representing it as a graph. The array-based models have been
shown to be very effective in terms of query time, while the graph-based models usually answer
queries by computing shortest paths, and hence they are suitable to be used in combination with
speed-up techniques developed for road networks.
In this paper, we focus on the dynamic behavior of graph-based models considering the case
where transportation systems are subject to delays with respect to the given timetable. We
make three contributions: (i) we give a simplified and optimized update routine for the wellknown
time-expanded model along with an engineered query algorithm; (ii) we propose a new
graph-based model tailored for handling dynamic updates; (iii) we assess the effectiveness of
the proposed models and algorithms by an experimental study, which shows that both models
require negligible update time and a query time which is comparable to that required by some
array-based models.

Abstract: A fundamental approach in finding efficiently best routes or optimal itineraries in traffic information
systems is to reduce the search space (part of graph visited) of the most commonly used
shortest path routine (Dijkstra¢s algorithm) on a suitably defined graph. We investigate reduction
of the search space while simultaneously retaining data structures, created during a preprocessing
phase, of size linear (i.e., optimal) to the size of the graph. We show that the search space of
Dijkstra¢s algorithm can be significantly reduced by extracting geometric information from a given
layout of the graph and by encapsulating precomputed shortest-path information in resulted geometric
objects (containers). We present an extensive experimental study comparing the impact of
different types of geometric containers using test data from real-world traffic networks. We also
present new algorithms as well as an empirical study for the dynamic case of this problem, where
edge weights are subject to change and the geometric containers have to be updated and show that
our new methods are two to three times faster than recomputing everything from scratch. Finally,
in an appendix, we discuss the software framework that we developed to realize the implementations
of all of our variants of Dijkstra¢s algorithm. Such a framework is not trivial to achieve as our
goal was to maintain a common code base that is, at the same time, small, efficient, and flexible,
as we wanted to enhance and combine several variants in any possible way.

Abstract: Dynamicgraph algorithms have been extensively studied in the last two
decades due to their wide applicabilityin manycon texts. Recently, several
implementations and experimental studies have been conducted investigating
the practical merits of fundamental techniques and algorithms. In most
cases, these algorithms required sophisticated engineering and fine-tuning
to be turned into efficient implementations. In this paper, we surveysev -
eral implementations along with their experimental studies for dynamic
problems on undirected and directed graphs. The former case includes
dynamic connectivity, dynamic minimum spanning trees, and the sparsification
technique. The latter case includes dynamic transitive closure and
dynamic shortest paths. We also discuss the design and implementation of
a software libraryfor dynamicgraph algorithms.

Abstract: We describe algorithms for finding shortest paths and distances in outerplanar and planar digraphs
that exploit the particular topology of the input graph. An important feature of our algorithms is that they can
work in a dynamic environment, where the cost of any edge can be changed or the edge can be deleted. In the
case of outerplanar digraphs, our data structures can be updated after any such change in only logarithmic time.
A distance query is also answered in logarithmic time. In the case of planar digraphs, we give an interesting
tradeoff between preprocessing, query, and update times depending on the value of a certain topological
parameter of the graph. Our results can be extended to n-vertex digraphs of genus O.n1¡"/ for any " > 0.

Abstract: In this work, we study the impact of the dynamic changing of the network link capacities on the stability properties of packet-switched networks. Especially, we consider the Adversarial, Quasi-Static Queuing Theory model, where each link capacity may take on only two possible (integer) values, namely 1 and C>1 under a (w,\~{n})-adversary. We obtain the following results:
• Allowing such dynamic changes to the link capacities of a network with just ten nodes that uses the LIS (Longest-in-System) protocol for contention–resolution results in instability at rates View the MathML source and for large enough values of C.
• The combination of dynamically changing link capacities with compositions of contention–resolution protocols on network queues suffices for similar instability bounds: The composition of LIS with any of SIS (Shortest-in-System), NTS (Nearest-to-Source), and FTG (Furthest-to-Go) protocols is unstable at rates View the MathML source for large enough values of C.
• The instability bound of the network subgraphs that are forbidden for stability is affected by the dynamic changes to the link capacities: we present improved instability bounds for all the directed subgraphs that were known to be forbidden for stability on networks running a certain greedy protocol.

Abstract: In this book chapter we will consider key establishment protocols for wireless sensor networks.
Several protocols have been proposed in the literature for the establishment of a shared group key for wired networks.
The choice of a protocol depends whether the key is established by one of the participants (and then transported to the other(s)) or agreed among the participants, and on the underlying cryptographic mechanisms (symmetric or asymmetric). Clearly, the design of key establishment protocols for sensor networks must deal with different problems and challenges that do not exist in wired networks. To name a few, wireless links are particularly vulnerable to eavesdropping, and that sensor devices can be captured (and the secrets they contain can be compromised); in many upcoming wireless sensor networks, nodes cannot rely on the presence of an online trusted server (whereas most standardized authentication and key establishment protocols do rely on such a server).
In particular, we will consider five distributed group key establishment protocols. Each of these protocols applies a different algorithmic technique that makes it more suitable for (i) static sensor networks, (ii) sensor networks where nodes enter sleep mode (i.e. dynamic, with low rate of updates on the connectivity graph) and (iii) fully dynamic networks where nodes may even be mobile. On the other hand, the common factor for all five protocols is that they can be applied in dynamic groups (where members can be excluded or added) and provide forward and backward secrecy. All these protocols are based on the Diffie-Hellman key exchange algorithm and constitute natural extensions of it in the multiparty case.

Abstract: Evolutionary dynamics have been traditionally studied in the context of homogeneous populations, mainly described by the Moran process [15]. Recently, this approach has been generalized in [13] by arranging individuals on the nodes of a network (in general, directed). In this setting, the existence of directed arcs enables the simulation of extreme phenomena, where the fixation probability of a randomly placed mutant (i.e. the probability that the offsprings of the mutant eventually spread over the whole population) is arbitrarily small or large. On the other hand, undirected networks (i.e. undirected graphs) seem to have a smoother behavior, and thus it is more challenging to find suppressors/amplifiers of selection, that is, graphs with smaller/greater fixation probability than the complete graph (i.e. the homogeneous population). In this paper we focus on undirected graphs. We present the first class of undirected graphs which act as suppressors of selection, by achieving a fixation probability that is at most one half of that of the complete graph, as the number of vertices increases. Moreover, we provide some generic upper and lower bounds for the fixation
probability of general undirected graphs. As our main contribution, we introduce the natural alternative of the model proposed in [13]. In our new evolutionary model, all individuals interact simultaneously and the result is a compromise between aggressive and non-aggressive individuals. That is, the behavior of the individuals in our new model and in the model of [13] can be interpreted as an “aggregation” vs. an “all-or-nothing” strategy, respectively. We prove that our new model of mutual influences admits a potential function, which guarantees the convergence of the system for any graph topology and any initial fitness vector of the individuals. Furthermore, we prove fast convergence to the stable state for the case of the complete graph, as well as we provide almost tight bounds on the limit fitness of the individuals. Apart from being important on its own, this new evolutionary model appears to be useful also in the abstract modeling of control mechanisms over invading populations in networks. We demonstrate this by introducing and analyzing two alternative control approaches, for which we bound the time needed to stabilize to the “healthy” state of the system.

Abstract: In this work, we study the impact of dynamically changing
link capacities on the delay bounds of LIS (Longest-In-
System) and SIS (Shortest-In-System) protocols on specific
networks (that can be modelled as Directed Acyclic Graphs-
DAGs) and stability bounds of greedy contention-resolution
protocols running on arbitrary networks under the Adversarial
Queueing Theory. Especially, we consider the model
of dynamic capacities, where each link capacity may take
on integer values from [1, C] withC > 1, under a (w, \~{n})-
adversary.

Abstract: In this work, we study the impact of dynamically changing link capacities on the delay bounds of LIS (Longest-In-System) and SIS (Shortest-In-System) protocols on specific networks (that can be modelled as Directed Acyclic Graphs (DAGs)) and stability bounds of greedy contention–resolution protocols running on arbitrary networks under the Adversarial Queueing Theory. Especially, we consider the model of dynamic capacities, where each link capacity may take on integer values from [1,C] with C>1, under a (w,\~{n})-adversary. We show that the packet delay on DAGs for LIS is upper bounded by O(iw\~{n}C) and lower bounded by {\`U}(iw\~{n}C) where i is the level of a node in a DAG (the length of the longest path leading to node v when nodes are ordered by the topological order induced by the graph). In a similar way, we show that the performance of SIS on DAGs is lower bounded by {\`U}(iw\~{n}C), while the existence of a polynomial upper bound for packet delay on DAGs when SIS is used for contention–resolution remains an open problem. We prove that every queueing network running a greedy contention–resolution protocol is stable for a rate not exceeding a particular stability threshold, depending on C and the length of the longest path in the network.

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 problem of preprocessing an n-vertex digraph with real edge weights so that
subsequent queries for the shortest path or distance between any two vertices can be efficiently answered. We
give algorithms that depend on the treewidth of the input graph. When the treewidth is a constant, our algorithms
can answer distance queries in O ({\'a}(n) ) time after O.n/ preprocessing. This improves upon previously known
results for the same problem.We also give a dynamic algorithm which, after a change in an edge weight, updates
the data structure in time O.n¯ /, for any constant 0 < ¯ < 1. Furthermore, an algorithm of independent interest
is given: computing a shortest path tree, or finding a negative cycle in linear time.

Abstract: We consider the problem of preprocessing an n-vertex digraph with real edge weights so that subsequent queries for the shortest path or distance between any two vertices can be efficiently answered. We give parallel algorithms for the EREW PRAM model of computation that depend on the treewidth of the input graph. When the treewidth is a constant, our algorithms can answer distance queries in O({\'a}(n)) time using a single processor, after a preprocessing of O(log2n) time and O(n) work, where {\'a}(n) is the inverse of Ackermann's function. The class of constant treewidth graphs contains outerplanar graphs and series-parallel graphs, among others. To the best of our knowledge, these are the first parallel algorithms which achieve these bounds for any class of graphs except trees. We also give a dynamic algorithm which, after a change in an edge weight, updates our data structures in O(log n) time using O(n{\^a}) work, for any constant 0 < {\^a} < 1. Moreover, we give an algorithm of independent interest: computing a shortest path tree, or finding a negative cycle in O(log2n) time using O(n) work.

Abstract: Geographic routing is becoming the protocol of choice for
many sensor network applications. The current state of the art is unsatisfactory:
some algorithms are very efficient, however they require a
preliminary planarization of the communication graph. Planarization induces
overhead and is thus not realistic for some scenarios such as the
case of highly dynamic network topologies. On the other hand, georouting
algorithms which do not rely on planarization have fairly low success
rates and fail to route messages around all but the simplest obstacles.
To overcome these limitations, we propose the GRIC geographic routing
algorithm. It has absolutely no topology maintenance overhead, almost
100% delivery rates (when no obstacles are added), bypasses large convex
obstacles, finds short paths to the destination, resists link failure
and is fairly simple to implement. The case of hard concave obstacles
is also studied; such obstacles are hard instances for which performance
diminishes.

Abstract: In this work, we study the impact of the dynamic changing of the network link capacities on the stability properties of packet-switched networks. Especially, we consider the Adversarial, Quasi-Static Queuing Theory model, where each link capacity may take on only two possible (integer) values, namely 1 and C>1 under a (w,\~{n})-adversary. We obtain the following results:
• Allowing such dynamic changes to the link capacities of a network with just ten nodes that uses the LIS (Longest-in-System) protocol for contention–resolution results in instability at rates View the MathML source and for large enough values of C.
• The combination of dynamically changing link capacities with compositions of contention–resolution protocols on network queues suffices for similar instability bounds: The composition of LIS with any of SIS (Shortest-in-System), NTS (Nearest-to-Source), and FTG (Furthest-to-Go) protocols is unstable at rates View the MathML source for large enough values of C.
• The instability bound of the network subgraphs that are forbidden for stability is affected by the dynamic changes to the link capacities: we present improved instability bounds for all the directed subgraphs that were known to be forbidden for stability on networks running a certain greedy protocol.

Abstract: We study here dynamic antagonism in a fixed network, represented as a graph $G$ of $n$ vertices. In particular, we consider the case of $k \leq n$ particles walking randomly independently around the network. Each particle belongs to exactly one of two antagonistic species, none of which can give birth to children. When two particles meet, they are engaged in a (sometimes mortal) local fight. The outcome of the fight depends on the species to which the particles belong. Our problem is \emph{to predict} (i.e. to compute) the eventual chances of species survival. We prove here that this can indeed be done in \emph{expected polynomial time on the size of the network}, provided that the network is \emph{undirected}.

Abstract: We study the existence and tractability of a notion of approximate
equilibria in bimatrix games, called well supported approximate
Nash Equilibria (SuppNE in short).We prove existence of "−SuppNE for
any constant " 2 (0, 1), with only logarithmic support sizes for both players.
Also we propose a polynomial–time construction of SuppNE, both
for win lose and for arbitrary (normalized) bimatrix games. The quality
of these SuppNE depends on the girth of the Nash Dynamics graph in
the win lose game, or a (rounded–off) win lose image of the original normalized
game. Our constructions are very successful in sparse win lose
games (ie, having a constant number of (0, 1)−elements in the bimatrix)
with large girth in the Nash Dynamics graph. The same holds also for
normalized games whose win lose image is sparse with large girth.
Finally we prove the simplicity of constructing SuppNE both in random
normalized games and in random win lose games. In the former case we
prove that the uniform full mix is an o(1)−SuppNE, while in the case
of win lose games, we show that (with high probability) there is either a
PNE or a 0.5-SuppNE with support sizes only 2.