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: We here present the Forward Planning Situated Protocol (FPSP), for scalable, energy efficient and fault tolerant data propagation in situated wireless sensor networks. To deal with the increased complexity of such deeply networked sensor systems, instead of emphasizing on a particular aspect of the services provided, i.e. either for low-energy periodic, or low-latency event-driven, or high-success query-based sensing, FPSP uses two novel mechanisms that allow the network operator to adjust the performance of the protocol in terms of energy, latency and success rate on a per-task basis. We emphasize on distributedness, direct or indirect interactions among relatively simple agents, flexibility and robustness.
The protocol operates by employing a series of plan & forward phases through which devices self-organize into forwarding groups that propagate data over discovered paths. FPSP performs a limited number of long range, high power data transmissions to collect information regarding the neighboring devices. The acquired information, allows to plan a (parameterizable long by {\"e}) sequence of short range, low power transmissions between nearby particles, based on certain optimization criteria. All particles that decide to respond (based on local criteria) to these long range transmissions enter the forwarding phase during which information is propagated via the acquired plan. Clearly, the duration of the forwarding phases is characterized by the parameter {\"e}, the transmission medium and the processing speed of the devices. In fact the parameter {\"e} provides a mechanism to adjust the protocol performance in terms of the latency--energy trade-off. By reducing {\"e} the latency is reduced at the cost of spending extra energy, while by increasing {\"e}, the energy dissipation is reduced but the latency is increased.
To control the success rate--energy trade-off, particles react locally on environment and context changes by using a set of rules that are based on response thresholds that relate individual-level plasticity with network-level resiliency, motivated by the nature-inspired method for dividing labor, a metaphor of social insect behavior for solving problems [1]. Each particle has an individual response threshold {\`E} that is related to the "local" density (as observed by the particle, [2]); particles engage in propagation of events when the level of the task-associated stimuli exceeds their thresholds. Let s be the intensity of a stimulus associated with a particular sensing task, set by the human authorities. We adopt the response function T_{{\`e}}(s) = s^{n}over s^{n} + {\`e}^{n}, the probability of performing the task as a function of s, where n > 1 determines the steepness of the threshold. Thus, when {\`e} is small (i.e. the network is sparse) then the response probability increases; when s increases (i.e. for critical sensing tasks) the response probability increases as well.
This role-based approach where a selective number of devices do the high cost planning and the rest of the network operates in a low cost state leads to systems that have increased energy efficiency and high fault-tolerance since these long range planning phases allow to bypass obstacles (where no sensors are available) or faulty sensors (that have been disabled due to power failure or other natural events).

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: We introduce a new modelling assumption for wireless sensor networks, that of node redeployment (addition of sensor devices during protocol evolution) and we extend the modelling assumption of heterogeneity (having sensor devices of various types). These two features further increase the highly dynamic nature of such networks and adaptation becomes a powerful technique for protocol design. Under these modelling assumptions, we design, implement and evaluate a new power conservation scheme for efficient data propagation. Our scheme is adaptive: it locally monitors the network conditions (density, energy) and accordingly adjusts the sleep-awake schedules of the nodes towards improved operation choices. The scheme is simple, distributed and does not require exchange of control messages between nodes.
Implementing our protocol in software we combine it with two well-known data propagation protocols and evaluate the achieved performance through a detailed simulation study using our extended version of the network simulator ns-2. We focus on highly dynamic scenarios with respect to network density, traffic conditions and sensor node resources. We propose a new general and parameterized metric capturing the trade-offs between delivery rate, energy efficiency and latency. The simulation findings demonstrate significant gains (such as more than doubling the success rate of the well-known Directed Diffusion propagation protocol) and good trade-offs achieved. Furthermore, the redeployment of additional sensors during network evolution and/or the heterogeneous deployment of sensors, drastically improve (when compared to ``equal total power" simultaneous deployment of identical sensors at the start) the protocol performance (i.e. the success rate increases up to four times} while reducing energy dissipation and, interestingly, keeping latency low).

Abstract: We consider sensor networks where the sensor nodes are attached on entities that move in a highly dynamic, heterogeneous manner. To capture this mobility diversity we introduce a new network parameter, the direction-aware mobility
level, which measures how fast and close each mobile node is expected to get to the data destination (the sink). We then provide local, distributed data dissemination protocols
that adaptively exploit the node mobility to improve performance. In particular, "high" mobility is used as a low cost replacement for data dissemination (due to the ferrying of data), while in the case of "low" mobility either a) data propagation redundancy is increased (when highly mobile neighbors exist) or b) long-distance data transmissions are used (when the entire neighborhood is of low mobility) to accelerate data dissemination towards the sink. An extensive performance comparison to relevant methods from
the state of the art demonstrates signicant improvements i.e. latency is reduced by even 4 times while keeping energy dissipation and delivery success at very satisfactory levels.

Abstract: We investigate the problem of ecient wireless energy recharging in Wireless Rechargeable Sensor Networks (WRSNs). In
such networks a special mobile entity (called the Mobile Charger) traverses the network and wirelessly replenishes the energy
of sensor nodes. In contrast to most current approaches, we envision methods that are distributed, adaptive and use limited
network information. We propose three new, alternative protocols for ecient recharging, addressing key issues which we
identify, most notably (i) to what extent each sensor should be recharged (ii) what is the best split of the total energy between
the charger and the sensors and (iii) what are good trajectories the MC should follow. One of our protocols (
LRP
) performs
some distributed, limited sampling of the network status, while another one (
RTP
) reactively adapts to energy shortage alerts
judiciously spread in the network. As detailed simulations demonstrate, both protocols signicantly outperform known state
of the art methods, while their performance gets quite close to the performance of the global knowledge method (
GKP
) we
also provide, especially in heterogeneous network deployments.

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: 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: 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: Wireless Sensor Networks are complex systems consisting of a number of relatively simple autonomous sensing devices spread on a geographical area. The peculiarity of these devices lies on the constraints they face in relation to their energy reserves and their computational, storage and communication capabilities. The utility of these sensors is to measure certain environmental conditions and to detect critical events in relation to these measurements. Those events thereupon have to be reported to a specific central station namely the “sink”. This data propagation generally has the form of a hop-by-hop transmission. In this framework we work on distributed data propagation protocols which are taking into account the energy reserves of the sensors. In particular following the work of Chatzigiannakis et al. on the Probabilistic Forwarding Protocol (PFR) we present the distributed probabilistic protocol EFPFR, which favors transmission from the less depleted sensors in addition to favor transmissions close to the “optimal line”. This protocol is simple and relies only on local information for propagation decisions. Its main goal is to limit the total amount of energy dissipated per event and therefore to extend the network’s operation duration.

Abstract: In this work, we study the propagation of influence and computation in dynamic distributed computing 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: We study the problem of energy-balanced data propagation in wireless sensor networks. The energy balance property is crucial for maximizing the time the network is functional, by avoiding early energy depletion of a large portion of sensors. We propose a distributed, adaptive data propagation algorithm that exploits limited, local network density information for achieving energy-balance while at the same time
minimizing energy dissipation.
We investigate both uniform and heterogeneous sensor placement distributions. By a detailed experimental evaluation and comparison with well-known energy-balanced protocols, we show that our density-based protocol improves energy efficiency signicantly while also having better energy balance properties.
Furthermore, we compare the performance of our protocol with a centralized, o-line optimum solution derived by a linear program which maximizes the network lifetime and show that it achieves near-optimal performance for uniform sensor deployments.

Abstract: This chapter aims at presenting certain important aspects of the design of lightweight, event-driven algorithmic solutions for data dissemination in wireless sensor networks that provide support for reliable, efficient and concurrency-intensive operation. We wish to emphasize that efficient solutions at several levels are needed, e.g.~higher level energy efficient routing protools and lower level power management schemes. Furthermore, it is important to combine such different level methods into integrated protocols and approaches. Such solutions must be simple, distributed and local. Two useful algorithmic design principles are randomization (to trade-off efficiency and fault-tolerance) and adaptation (to adjust to high network dynamics towards improved operation). In particular, we provide a) a brief description of the technical specifications of state-of-the-art sensor devices b) a discussion of possible models used to abstract such networks, emphasizing heterogeneity, c) some representative power management schemes, and d) a presentation of some characteristic protocols for data propagation. Crucial efficiency properties of these schemes and protocols (and their combinations, in some cases) are investigated by both rigorous analysis and performance evaluations through large scale simulations.

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 distributed computing 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: DAP (Distributed Algorithms Platform) is a generic and homogeneous simulation environment aiming at the implementation, simulation, and testing of distributed algorithms for wired and wireless networks. In this work, we present its architecture, the most important design decisions, and discuss its distinct features and functionalities. DAP allows the algorithm designer to implement a distributedprotocol by creating his own customized environment, and programming in a standard programming language in a style very similar to that of a real-world application. DAP provides a graphical user interface that allows the designer to monitor and control the execution of simulations, visualize algorithms, as well as gather statistics and other information for their experimental analysis and testing.

Abstract: We work on an extension of the Population Protocol model of Angluin et al. that allows edges of the communication graph, G, to have states that belong to a constant size set. In this extension, the so called Mediated Population Protocol model (MPP), both uniformity and anonymity are preserved. We here study a simplified version of MPP, the Graph Decision Mediated Population Protocol model (GDM), in order to capture MPP's ability to decide (stably compute) graph languages (sets of communication graphs). To understand properties of the communication graph is an important step in almost any distributed system. We prove that any graph language is undecidable if we allow disconnected communication graphs. As a result, we focus on studying the computational limits of the GDM model in (at least) weakly connected communication graphs only and give several examples of decidable graph languages in this case. To do so, we also prove that the class of decidable graph languages is closed under complement, union and intersection operations. Node and edge parity, bounded out-degree by a constant, existence of a node with more incoming than outgoing neighbors and existence of some directed path of length at least k=O(1) are some examples of properties whose decidability is proven. To prove the decidability of graph languages we provide protocols (GDMs) for them and exploit the closure results. Finally, we prove the existence of symmetry in two specific communication (sub)graphs which we believe is the first step towards the proof of impossibility results in the GDM model. In particular, we prove that there exists no GDM, whose states eventually stabilize, to decide whether G contains some directed cycle of length 2 (2-cycle).

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: Wireless sensor networks are comprised of a vast number of devices, situated in an area of interest that self organize in a structureless network, in order to monitor/record/measure an environmental variable or phenomenon and subsequently to disseminate the data to the control center.
Here we present research focused on the development, simulation and evaluation of energy efficient algorithms, our basic goal is to minimize the energy consumption. Despite technology advances, the problem of energy use optimization remains valid since current and emerging hardware solutions fail to solve it.
We aim to reduce communication cost, by introducing novel techniques that facilitate the development of new algorithms. We investigated techniques of distributed adaptation of the operations of a protocol by using information available locally on every node, thus through local choices we improve overall performance. We propose techniques for collecting and exploiting limited local knowledge of the network conditions. In an energy efficient manner, we collect additional information which is used to achieve improvements such as forming energy efficient, low latency and fault tolerant paths to route data. We investigate techniques for managing mobility in networks where movement is a characteristic of the control center as well as the sensors. We examine methods for traversing and covering the network field based on probabilistic movement that uses local criteria to favor certain areas.
The algorithms we develop based on these techniques operate a) at low level managing devices, b) on the routing layer and c) network wide, achieving macroscopic behavior through local interactions. The algorithms are applied in network cases that differ in density, node distribution, available energy and also in fundamentally different models, such as under faults, with incremental node deployment and mobile nodes. In all these settings our techniques achieve significant gains, thus distinguishing their value as tools of algorithmic design.

Abstract: A central problem in distributed computing 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: An ad hoc mobile network is a collection of mobile hosts, with wireless communication capabilities, forming a temporary network without the aid of any established fixed infrastructure. In such networks, topological connectivity is subject to frequent, unpredictable change. Our work focuses on networks with high rate of such changes to connectivity. For such dynamically changing networks we propose protocols which exploit the co-ordinated (by the protocol) motion of a small part of the network. We show that such protocols can be designed to work correctly and efficiently even in the case of arbitrary (but not malicious) movements of the hosts not affected by the protocol. We also propose a methodology for the analysis of the expected behavior of protocols for such networks, based on the assumption that mobile hosts (those whose motion is not guided by the protocol) conduct concurrent random walks in their motion space. In particular, our work examines the fundamental problem of communication and proposes distributed algorithms for it. We provide rigorous proofs of their correctness, and also give performance analyses by combinatorial tools. Finally, we have evaluated these protocols by experimental means.

Abstract: An ad hoc mobile network is a collection of mobile hosts, with wireless communication capabilities, forming a temporary network without the aid of any established fixed infrastructure. In such networks, topological connectivity is subject to frequent, unpredictable change. Our work focuses on networks with high rate of such changes to connectivity. For such dynamically changing networks we propose protocols which exploit the co-ordinated (by the protocol) motion of a small part of the network. We show that such protocols can be designed to work correctly and efficiently even in the case of arbitrary (but not malicious) movements of the hosts not affected by the protocol. We also propose a methodology for the analysis of the expected behavior of protocols for such networks, based on the assumption that mobile hosts (those whose motion is not guided by the protocol) conduct concurrent random walks in their motion space. In particular, our work examines the fundamental problem of communication and proposes distributed algorithms for it. We provide rigorous proofs of their correctness, and also give performance analyses by combinatorial tools. Finally, we have evaluated these protocols by experimental means.

Abstract: We propose a new data dissemination protocol for wireless sensor networks, that basically pulls some additional knowledge about the network in order to subsequently improve data forwarding towards the sink. This extra information is still local, limited and obtained in a distributed manner. This extra knowledge is acquired by only a small fraction of sensors thus the extra energy cost only marginally affects the overall protocol efficiency. The new protocol has low latency and manages to propagate data successfully even in the case of low densities. Furthermore, we study in detail the effect of failures and show that our protocol is very robust. In particular, we implement and evaluate the protocol using large scale simulation, showing that it significantly outperforms well known relevant solutions in the state of the art.

Abstract: We study the important problem of tracking moving
targets in wireless sensor networks. We try to overcome the
limitations of standard state of the art tracking methods based on
continuous location tracking, i.e. the high energy dissipation and
communication overhead imposed by the active participation of
sensors in the tracking process and the low scalability, especially
in sparse networks. Instead, our approach uses sensors in a
passive way: they only record and judiciously spread information
about observed target presence in their vicinity; this information
is then used by the (powerful) tracking agent to locate the target
by just following the traces left at sensors. Our protocol is greedy,
local, distributed, energy efﬁcient and very successful, in the
sense that (as shown by extensive simulations) the tracking agent
manages to quickly locate and follow the target; also, we achieve
good trade-offs between the energy dissipation and latency.

Abstract: We investigate the problem of ecient wireless energy recharging in Wireless Rechargeable Sensor Networks (WRSNs). In
such networks special mobile entities (called the Mobile Chargers) traverse the network and wirelessly replenish the energy
of sensor nodes. In contrast to most current approaches, we envision methods that are distributed and use limited network
information. We propose four new protocols for ecient recharging, addressing key issues which we identify, most notably (i)
what are good coordination procedures for the Mobile Chargers and (ii) what are good trajectories for the Mobile Chargers.
Two of our protocols (
DC,DCLK
) perform distributed, limited network knowledge coordination and charging, while two others
(
CC,CCGK
) perform centralized, global network knowledge coordination and charging. As detailed simulations demonstrate,
one of our distributedprotocols outperforms a known state of the art method, while its performance gets quite close to the
performance of the powerful centralized global knowledge method.

Abstract: We present key aspects (hardware, software, topology, networking) of SenseWall, an experimental sensor network test-bed we have created for the implementation and engineering of distributed sensor network algorithms. We then describe how SenseWall has been in particular used to implement two recent state of the art algorithms for energy balanced sensor data propagation. We elaborate on the issues and challenges created by the restrictions and particularities of the experimental test-bed and how we dealt with them. We also carry out a detailed performance evaluation comparing the energy balance protocols to two baseline protocols that include only either single hop or direct data transmissions.

Abstract: We investigate the impact of multiple, mobile sinks on
efficient data collection in wireless sensor networks. To
improve performance, our protocol design focuses on minimizing
overlaps of sink trajectories and balancing the service load
among the sinks. To cope with high network dynamics, placement
irregularities and limited network knowledge we propose three different
protocols: a) a centralized one, that explicitly equalizes spatial coverage;
this protocol assumes strong modeling assumptions, and also serves as a kind
of performance lower bound in uniform networks of low dynamics b)
a distributedprotocol based on mutual avoidance of sinks c) a clustering
protocol that distributively groups service areas towards balancing the load per sink.
Our simulation findings demonstrate significant gains in latency, while keeping the success
rate and the energy dissipation at very satisfactory levels even under
high network dynamics and deployment heterogeneity.

Abstract: We propose a new data dissemination protocol for wireless sensor networks, that basically pulls some additional knowledge about the network in order to subsequently improve data forwarding towards the sink. This extra information is still local, limited and obtained in a distributed manner. This extra knowledge is acquired by only a small fraction of sensors thus the extra energy cost only marginally affects the overall protocol efficiency. The new protocol has low latency and manages to propagate data successfully even in the case of low densities. Furthermore, we study in detail the effect of failures and show that our protocol is very robust. In particular, we implement and evaluate the protocol using large scale simulation, showing that it significantly outperforms well known relevant solutions in the state of the art.

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 distributedprotocol 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 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: Recent rapid developments in micro-electro-mechanical systems
(MEMS), wireless communications and digital electronics have already
led to the development of tiny, low-power, low-cost sensor devices.
Such devices integrate sensing, limited data processing and restricted
communication capabilities.
Each sensor device individually might have small utility, however the
effective distributed co-ordination of large numbers of such devices can
lead to the efficient accomplishment of large sensing tasks. Large numbers
of sensors can be deployed in areas of interest (such as inaccessible
terrains or disaster places) and use self-organization and collaborative
methods to form an ad-hoc network.
We note however that the efficient and robust realization of such large,
highly-dynamic, complex, non-conventional networking environments is
a challenging technological and algorithmic task, because of the unique
characteristics and severe limitations of these devices.
This talk will present and discuss several important aspects of the
design, deployment and operation of sensor networks. In particular, we
provide a brief description of the technical specifications of state-of-theart
sensor, a discussion of possible models used to abstract such networks,
a discussion of some key algorithmic design techniques (like randomization,
adaptation and hybrid schemes), a presentation of representative
protocols for sensor networks, for important problems including data
propagation, collision avoidance and energy balance and an evaluation
of crucial performance properties (correctness, efficiency, fault-tolerance)
of these protocols, both with analytic and simulation means.

Abstract: Let n atomic players be routing their unsplitable flow on mresources.
When each player has the option to drop her current resource and select a better
one, and this option is exercised sequentially and unilaterally, then a Nash Equilibrium
(NE) will be eventually reached. Acting sequentially, however, is unrealistic
in large systems. But, allowing concurrency, with an arbitrary number of
players updating their resources at each time point, leads to an oscillation away
from NE, due to big groups of players moving simultaneously and due to nonsmooth
resource cost functions. In this work, we validate experimentally simple
concurrent protocols that are realistic, distributed and myopic yet are scalable, require
only information local at each resource and, still, are experimentally shown
to quickly reach a NE for a range of arbitrary cost functions.

Abstract: Wireless Sensor Networks (WSNs) constitute a recent and promising new
technology that is widely applicable. Due to the applicability of this
technology and its obvious importance for the modern distributed
computational world, the formal scientific foundation of its inherent laws
becomes essential. As a result, many new computational models for WSNs
have been proposed. Population Protocols (PPs) are a special category of
such systems. These are mainly identified by three distinctive
characteristics: the sensor nodes (agents) move passively, that is, they
cannot control the underlying mobility pattern, the available memory to
each agent is restricted, and the agents interact in pairs. It has been
proven that a predicate is computable by the PP model iff it is
semilinear. The class of semilinear predicates is a fairly small class. In
this work, our basic goal is to enhance the PP model in order to improve
the computational power. We first make the assumption that not only the
nodes but also the edges of the communication graph can store restricted
states. In a complete graph of n nodes it is like having added O(n2)
additional memory cells which are only read and written by the endpoints
of the corresponding edge. We prove that the new model, called Mediated
Population Protocol model, can operate as a distributed nondeterministic
Turing machine (TM) that uses all the available memory. The only
difference from a usual TM is that this one computes only symmetric
languages. More formally, we establish that a predicate is computable by
the new model iff it is symmetric and belongs to NSPACE(n2). Moreover, we
study the ability of the new model to decide graph languages (for general
graphs). The next step is to ignore the states of the edges and provide
another enhancement straight away from the PP model. The assumption now is
that the agents are multitape TMs equipped with infinite memory, that can
perform internal computation and interact with other agents, and we define
space-bounded computations. We call this the Passively mobile Machines
model. We prove that if each agent uses at most f(n) memory for f(n)={\`U}(log
n) then a predicate is computable iff it is symmetric and belongs to
NSPACE(nf(n)). We also show that this is not the case for f(n)=o(log n).
Based on these, we show that for f(n)={\`U}(log n) there exists a space
hierarchy like the one for classical symmetric TMs. We also show that the
latter is not the case for f(n)=o(loglog n), since here the corresponding
class collapses in the class of semilinear predicates and finally that for
f(n)={\`U}(loglog n) the class becomes a proper superset of semilinear
predicates. We leave open the problem of characterizing the classes for
f(n)={\`U}(loglog n) and f(n)=o(log n).

Abstract: An ad-hoc mobile network is a collection of mobile hosts, with
wireless communication capabilities, forming a temporary network
without the aid of any established fixed infrastructure.
In such networks, topological connectivity is subject to frequent,
unpredictable change. Our work focuses on networks with high
rate of such changes to connectivity. For such dynamic changing
networks we propose protocols which exploit the co-ordinated
(by the protocol) motion of a small part of the network.
We show that such protocols can be designed to work
correctly and efficiently even in the case of arbitrary (but not
malicious) movements of the hosts not affected by the protocol.
We also propose a methodology for the analysis of the expected
behaviour of protocols for such networks, based on the assumption that mobile hosts (whose motion is not guided by
the protocol) conduct concurrent random walks in their
motion space.
Our work examines some fundamental problems such as pair-wise
communication, election of a leader and counting, and proposes
distributed algorithms for each of them. We provide their
proofs of correctness, and also give rigorous analysis by
combinatorial tools and also via experiments.

Abstract: We investigate the problem of how to achieve energy balanced data propagation in distributed wireless sensor networks. The energy balance property guarantees that the average per sensor energy dissipation is the same for all sensors in the network, throughout the execution of the data propagation protocol. This property is crucial for prolonging the network lifetime, by avoiding early energy depletion of sensors.
We survey representative solutions from the state of the art. We first present a basic algorithm that in each step probabilistically decides whether to propagate data one-hop towards the final destination (the sink), or to send it directly to the sink. This randomized choice trades-off the (cheap, but slow) one-hop transmissions with the direct transmissions to the sink, which are more expensive but bypass the bottleneck region around the sink and propagate data fast. By a detailed analysis using properties of stochastic processes and recurrence relations we precisely estimate (even in closed form) the probability for each propagation option necessary for energy balance.
The fact (shown by our analysis) that direct (expensive) transmissions to the sink are needed only rarely, shows that our protocol, besides energy balanced, is also energy efficient. We then enhance this basic result by surveying some recent findings including a generalized algorithm and demonstrating the optimality of this two-way probabilistic data propagation, as well as providing formal proofs of the energy optimality of the energy balance property.

Abstract: Recent rapid technological developments have led to the
development of tiny, low-power, low-cost sensors. Such devices
integrate sensing, limited data processing and communication
capabilities.The effective distributed collaboration
of large numbers of such devices can lead to the efficient
accomplishment of large sensing tasks.
This talk focuses on several aspects of energy efficiency.
Two protocols for data propagation are studied: the first
creates probabilistically optimized redundant data transmissions
to combine energy efficiency with fault tolerance,
while the second guarantees (in a probabilistic way) the
same per sensor energy dissipation, towards balancing the
energy load and prolong the lifetime of the network.
A third protocol (in fact a power saving scheme) is also
presented, that directly and adaptively affects power dissipation
at each sensor. This “lower level” scheme can be
combined with data propagation protocols to further improve
energy efficiency.

Abstract: In this work we present three new distributed, probabilistic data propagation protocols for Wireless Sensor Networks which aim at maximizing the network's operational life and improve its performance. The keystone of these protocols' design is fairness which declares that fair portions of network's work load should be assigned to each node, depending on their role in the system. All the three protocols, EFPFR, MPFR and TWIST, emerged from the study of the rigorously analyzed protocol PFR. Its design elements were identified and improvements were suggested and incorporated into the introduced protocols. The experiments conducted show that our proposals manage to improve PFR's performance in terms of success rate, total amount of energy saved, number of alive sensors and standard deviation of the energy left. Indicatively we note that while PFR's success rate is 69.5%, TWIST is achieving 97.5% and its standard deviation of energy is almost half of that of PFR.

Abstract: Data propagation in wireless sensor
networks is usually performed as a multihop process.
Thus,
To deliver a single
message, the resources of many sensor nodes are used and
a lot of energy is spent.
Recently, a novel approach is catching momentum because of important applications;
that of having a mobile sink move inside the network area and collect
the data with low energy cost.
Here we extend this line of research by proposing and evaluating three new protocols.
Our protocols are novel in
a) investigating the impact of having {many} mobile sinks
b) in weak models with restricted mobility, proposing and evaluating
a mix of static and mobile sinks and c) proposing a distributedprotocol that tends to {equally spread the sinks} in the network to
further improve performance.
Our protocols are simple, based on randomization and assume locally
obtainable information. We perform an extensive evaluation via simulation; our
findings demonstrate that our solutions scale very well with respect to the number of sinks
and significantly reduce energy consumption and delivery delay.

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: In this work, we study protocols so that populations of distributed processes can construct networks. In order to highlight the basic principles of distributed network construction, we keep the model minimal in all respects. In particular, we assume finite-state processes that all begin from the same initial state and all execute the same protocol. Moreover, we assume pairwise interactions between the processes that are scheduled by a fair adversary. In order to allow processes to construct networks, we let them activate and deactivate their pairwise connections. When two processes interact, the protocol takes as input the states of the processes and the state of their connection and updates all of them. Initially all connections are inactive and the goal is for the processes, after interacting and activating/deactivating connections for a while, to end up with a desired stable network. We give protocols (optimal in some cases) and lower bounds for several basic network construction problems such as spanning line, spanning ring, spanning star, and regular network. The expected time to convergence of our protocols is analyzed under a uniform random scheduler. Finally, we prove several universality results by presenting generic protocols that are capable of simulating a Turing Machine (TM) and exploiting it in order to construct a large class of networks. We additionally show how to partition the population into k supernodes, each being a line of log k nodes, for the largest such k. This amount of local memory is sufficient for the supernodes to obtain unique names and exploit their names and their memory to realize nontrivial constructions.

Abstract: In this work, we study protocols (i.e. distributed algorithms) so that populations of distributed processes can construct networks. In order to highlight the basic principles of distributed network construction we keep the model minimal in all respects. In particular, we assume finite-state processes that all begin from the same initial state and all execute the same protocol (i.e. the system is homogeneous). Moreover, we assume pairwise interactions between the processes that are scheduled by an adversary. The only constraint on the adversary scheduler is that it must be fair, intuitively meaning that it must assign to every reachable configuration of the system a non-zero probability to occur. In order to allow processes to construct networks, we let them activate and deactivate their pairwise connections. When two processes interact, the protocol takes as input the states of the processes and the state of their connection and updates all of them. In particular, in every interaction, the protocol may activate an inactive connection, deactivate an active one, or leave the state of a connection unchanged. Initially all connections are inactive and the goal is for the processes, after interacting and activating/deactivating connections for a while, to end up with a desired stable network (i.e. one that does not change any more). We give protocols (optimal in some cases) and lower bounds for several basic network construction problems such as spanning line, spanning ring, spanning star, and regular network. We provide proofs of correctness for all of our protocols and analyze the expected time to convergence of most of them under a uniform random scheduler that selects the next pair of interacting processes uniformly at random from all such pairs. Finally, we prove several universality results by presenting generic protocols that are capable of simulating a Turing Machine (TM) and exploiting it in order to construct a large class of networks. Our universality protocols use a subset of the population (waste) in order to distributedly construct there a TM able to decide a graph class in some given space. Then, the protocols repeatedly construct in the rest of the population (useful space) a graph equiprobably drawn from all possible graphs. The TM works on this and accepts if the presented graph is in the class. We additionally show how to partition the population into k supernodes, each being a line of log k nodes, for the largest such k. This amount of local memory is sufficient for the supernodes to obtain unique names and exploit their names and their memory to realize nontrivial constructions. Delicate composition and reinitialization issues have to be solved for these general constructions to work.

Abstract: We work on an extension of the Population Protocol model of Angluin et al. that allows edges of the communication graph, G, to have states that belong to a constant size set. In this extension, the so called Mediated Population Protocol model (MPP), both uniformity and anonymity are preserved. We study here a simplified version of MPP in order to capture MPP's ability to stably compute graph properties. To understand properties of the communication graph is an important step in almost any distributed system. We prove that any graph property is not computable if we allow disconnected communication graphs. As a result, we focus on studying (at least) weakly connected communication graphs only and give several examples of computable properties in this case. To do so, we also prove that the class of computable properties is closed under complement, union and intersection operations. Node and edge parity, bounded out-degree by a constant, existence of a node with more incoming than outgoing neighbors, and existence of some directed path of length at least k=O(1) are some examples of properties whose computability is proven. Finally, we prove the existence of symmetry in two specific communication graphs and, by exploiting this, we prove that there exists no protocol, whose states eventually stabilize, to determine whether G contains some directed cycle of length 2.

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 extend the population protocol model with a cover-time service that informs a walking state every time it covers the whole network. This represents a known upper bound on the cover time of a random walk. The cover-time service allows us to introduce termination into population protocols, a capability that is crucial for any distributed system. By reduction to an oracle-model we arrive at a very satisfactory lower bound on the computational power of the model: we prove that it is at least as strong as a Turing Machine of space log n with input commutativity, where n is the number of nodes in the network. We also give a log n-space, but nondeterministic this time, upper bound. Finally, we prove interesting similarities of this model to linear bounded automata.

Abstract: We extend the population protocol model with a cover-time service that informs a walking state every time it covers the whole network. This is simply a known upper bound on the cover time of a random walk. This allows us to introduce termination into population protocols, a capability that is crucial for any distributed system. By reduction to an oracle-model we arrive at a very satisfactory lower bound on the computational power of the model: we prove that it is at least as strong as a Turing Machine of space logn with input commutativity, where n is the number of nodes in the network. We also give a logn-space, but nondeterministic this time, upper bound. Finally, we prove interesting similarities of this model to linear bounded automata.

Abstract: In this paper we describe a new simulation platform for complex wireless sensor networks that operate a collection of distributed algorithms and network protocols. Simulating such systems is complicated because of the need to coordinate different network layers and debug protocol stacks, often with very different interfaces, options, and fidelities. Our platform (which we call WSNGE) is a flexible and extensible environment that provides a highly scalable simulator with unique characteristics. It focuses on user friendliness, providing every function in both scriptable and visual way, allowing the researcher to define simulations and view results in an easy to use graphical environment. Unlike other solutions, WSNGE does not distinguish between different scenario types, allowing multiple different protocols to run at the same time. It enables rich online interaction with running simulations, allowing parameters, topologies or the whole scenario to be altered at any point in time.