Abstract: A temporal graph is, informally speaking, a graph that changes with time. When time is discrete and only the relationships between the participating entities may change and not the entities themselves, a temporal graph may be viewed as a sequence G1,G2 ,Gl of static graphs over the same (static) set of nodes V. Though static graphs have been extensively studied, for their temporal generalization we are still far from having a concrete set of structural and algorithmic principles. Recent research shows that many graph properties and problems become radically different and usually substantially more difficult when an extra time dimension in added to them. Moreover, there is already a rich and rapidly growing set of modern systems and applications that can be naturally modeled and studied via temporal graphs. This, further motivates the need for the development of a temporal extension of graphtheory. We survey here recent results on temporal graphs and temporal graph problems that have appeared in the Computer Science community.

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 investigate the practical merits of a parallel priority queue
through its use in the development of a fast and work-efficient parallel
shortest path algorithm, originally designed for an EREW PRAM. Our
study reveals that an efficient implementation on a real supercomputer
requires considerable effort to reduce the communication performance
(which in theory is assumed to take constant time). It turns out that the
most crucial part of the implementation is the mapping of the logical
processors to the physical processing nodes of the supercomputer. We
achieve the requested efficient mapping through a new graph-theoretic
result of independent interest: computing a Hamiltonian cycle on a directed
hyper-torus. No such algorithm was known before for the case of
directed hypertori. Our Hamiltonian cycle algorithm allows us to considerably
improve the communication cost and thus the overall performance
of our implementation.

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: We employ here the Probabilistic Method, a way of reasoning which shows existence of combinatorial structures and properties to prove refute conjectures. The radiocoloring problem (RCP) is the problem of assigning frequencies to the transmitters of a network so that transmitters of distance one get frequencies that di#er by at least two and any two transmitters of distance one get frequencies that di#er by at least one. The objective of an assignment may be to minimize the number of frequencies used (order) or the range of them (span). Here, we study the optimization version of RCP where the objective is to minimize the order. In graphtheory terms the problem is modelled by a variation of the vertex graph coloring problem. We investigate upper bounds for the minimum number of colors needed in a radiocoloring assignment of a graph G. We first provide an upper bound for the minimum number of colors needed to radiocolor a graph G of girth at most 7. Then, we study whether the minimum order of a radiocoloring assignment is determined by local conditions, i.e. by the minimum order radiocoloring assignment of some small subgraphs of it. We state a related conjecture which is analogous to a theorem of Molloy and Reed for the vertex coloring problem [12]. We then investigate whether the conjecture contradicts a Theorem of Molloy and Reed for the vertex coloring when applied on the graph G 2

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: A packet-switching network is stable if the number of packets in the network remains bounded at all times. A very natural question that arises in the context of stability properties of such networks is how network structure precisely affects these properties. In this work we embark on a systematic study of this question in the context of Adversarial Queueing Theory, which assumes that packets are adversarially injected into the network. We consider size, diameter, maximum vertex degree, minimum number of disjoint paths that cover all edges of the network and network subgraphs as crucial structural parameters of the network, and we present a comprehensive collection of structural results, in the form of stability and instability bounds on injection rate of the adversary for various greedy protocols: —Increasing the size of a network may result in dropping its instability bound. This is shown through a novel, yet simple and natural, combinatorial construction of a size-parameterized network on which certain compositions of greedy protocols are running. The convergence of the drop to 0.5 is found to be fast with and proportional to the increase in size. —Maintaining the size of a network small may already suffice to drop its instability bound to a substantially low value. This is shown through a construction of a FIFO network with size 22, which becomes unstable at rate 0.704. This represents the current state-of-the-art trade-off between network size and instability bound. —The diameter, maximum vertex degree and minimum number of edge-disjoint paths that cover a network may be used as control parameters for the stability bound of the network. This is shown through an improved analysis of the stability bound of any arbitrary FIFO network, which takes these parameters into account. —How much can network subgraphs that are forbidden for stability affect the instability bound? Through improved combinatorial constructions of networks and executions, we improve the state-of-the-art instability bound induced by certain known forbidden subgraphs on networks running a certain greedy protocol. —Our results shed more light and contribute significantly to a finer understanding of the impact of structural parameters on stability and instability properties of networks.

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 contentionresolution 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 contentionresolution 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: Consider a network vulnerable to security attacks and equipped with defense mechanisms. How much is the loss in the provided security guarantees due to the selfish nature of attacks and defenses? The Price of Defense was recently introduced in [7] as a worst-case measure, over all associated Nash equilibria, of this loss. In the particular strategic game considered in [7], there are two classes of confronting randomized players on a graph G(V,E): v attackers, each choosing vertices and wishing to minimize the probability of being caught, and a single defender, who chooses edges and gains the expected number of attackers it catches. In this work, we continue the study of the Price of Defense. We obtain the following results: - The Price of Defense is at least |V| 2; this implies that the Perfect Matching Nash equilibria considered in [7] are optimal with respect to the Price of Defense, so that the lower bound is tight. - We define Defense-Optimal graphs as those admitting a Nash equilibrium that attains the (tight) lower bound of |V| 2. We obtain: A graph is Defense-Optimal if and only if it has a Fractional Perfect Matching. Since graphs with a Fractional Perfect Matching are recognizable in polynomial time, the same holds for Defense-Optimal graphs. We identify a very simple graph that is Defense-Optimal but has no Perfect Matching Nash equilibrium. - Inspired by the established connection between Nash equilibria and Fractional Perfect Matchings, we transfer a known bivaluedness result about Fractional Matchings to a certain class of Nash equilibria. So, the connection to Fractional GraphTheory may be the key to revealing the combinatorial structure of Nash equilibria for our network security game.

Abstract: Let M be a single s-t network of parallel links with load dependent latency functions shared by an infinite number of selfish users. This may yield a Nash equilibrium with unbounded Coordination Ratio [E. Koutsoupias, C. Papadimitriou, Worst-case equilibria, in: 16th Annual Symposium on Theoretical Aspects of Computer Science, STACS, vol. 1563, 1999, pp. 404-413; T. Roughgarden, E. Tardos, How bad is selfish routing? in: 41st IEEE Annual Symposium of Foundations of Computer Science, FOCS, 2000, pp. 93-102]. A Leader can decrease the coordination ratio by assigning flow {\'a}r on M, and then all Followers assign selfishly the (1-{\'a})r remaining flow. This is a Stackelberg Scheduling Instance(M,r,{\'a}),0≤{\'a}≤1. It was shown [T. Roughgarden, Stackelberg scheduling strategies, in: 33rd Annual Symposium on Theory of Computing, STOC, 2001, pp. 104-113] that it is weakly NP-hard to compute the optimal Leader's strategy. For any such network M we efficiently compute the minimum portion @b"M of flow r>0 needed by a Leader to induce M's optimum cost, as well as her optimal strategy. This shows that the optimal Leader's strategy on instances (M,r,@a>=@b"M) is in P. Unfortunately, Stackelberg routing in more general nets can be arbitrarily hard. Roughgarden presented a modification of Braess's Paradox graph, such that no strategy controlling {\'a}r flow can induce ≤1/{\'a} times the optimum cost. However, we show that our main result also applies to any s-t net G. We take care of the Braess's graph explicitly, as a convincing example. Finally, we extend this result to k commodities. A conference version of this paper has appeared in [A. Kaporis, P. Spirakis, The price of optimum in stackelberg games on arbitrary single commodity networks and latency functions, in: 18th annual ACM symposium on Parallelism in Algorithms and Architectures, SPAA, 2006, pp. 19-28]. Some preliminary results have also appeared as technical report in [A.C. Kaporis, E. Politopoulou, P.G. Spirakis, The price of optimum in stackelberg games, in: Electronic Colloquium on Computational Complexity, ECCC, (056), 2005].

Abstract: In this paper we study the threshold behavior of the fixed radius random graph model and its applications to the key management problem of sensor networks and, generally, for mobile ad-hoc networks. We show that this random graph model can realistically model the placement of nodes within a certain region and their interaction/sensing capabilities (i.e. transmission range, light sensing sensitivity etc.). We also show that this model can be used to define key sets for the network nodes that satisfy a number of good properties, allowing to set up secure communication with each other depending on randomly created sets of keys related to their current location. Our work hopes to inaugurate a study of key management schemes whose properties are related to properties of an appropriate random graph model and, thus, use the rich theory developed in the random graph literature in order to transfer ?good? properties of the graph model to the key sets of the nodes.
Partially supported by the IST Programme of the European Union under contact number IST-2005-15964 (AEOLUS) and the INTAS Programme under contract with Ref. No 04-77-7173 (Data Flow Systems: Algorithms and Complexity (DFS-AC)).