We investigate opinion dynamics in a fully-connected system, consisting of n agents, where one of the opinions, called correct, represents a piece of information to disseminate. One source agent initially holds the correct opinion and remains with this opinion throughout the execution. The goal of the remaining agents is to quickly agree on this correct opinion. At each round, one agent chosen uniformly at random is activated: unless it is the source, the agent pulls the opinions of l random agents and then updates its opinion according to some rule. We consider a restricted setting, in which agents have no memory and they only revise their opinions on the basis of those of the agents they currently sample. This setting encompasses very popular opinion dynamics, such as the voter model and best-of-k majority rules. Qualitatively speaking, we show that lack of memory prevents efficient convergence. Specifically, we prove that any dynamics requires Omega(n^2) expected time, even under a strong version of the model in which activated agents have complete access to the current configuration of the entire system, i.e., the case l=n. Conversely, we prove that the simple voter model (in which l=1) correctly solves the problem, while almost matching the aforementioned lower bound. These results suggest that, in contrast to symmetric consensus problems (that do not involve a notion of correct opinion), fast convergence on the correct opinion using stochastic opinion dynamics may require the use of memory.

We study parallel Load Balancing protocols for a client-server distributed model defined as follows. There is a set C of n clients and a set S of n servers where each client has (at most) a constant number d ≥ 1 of requests that must be assigned to some server. The client set and the server one are connected to each other via a fixed bipartite graph: the requests of client v can only be sent to the servers in its neighborhood N(v). The goal is to assign every client request so as to minimize the maximum load of the servers. In this setting, efficient parallel protocols are available only for dense topolgies. In particular, a simple symmetric, non-adaptive protocol achieving constant maximum load has been recently introduced by Becchetti et al [4] for regular dense bipartite graphs. The parallel completion time is O(log n) and the overall work is O(n), w.h.p. Motivated by proximity constraints arising in some client-server systems, we devise a simple variant of Becchetti et al's protocol [4] and we analyse it over almost-regular bipartite graphs where nodes may have neighborhoods of small size. In detail, we prove that, w.h.p., this new version has a cost equivalent to that of Becchetti et al's protocol (in terms of maximum load, completion time, and work complexity, respectively) on every almost-regular bipartite graph with degree Ω(log2n). Our analysis significantly departs from that in [4] for the original protocol and requires to cope with non-trivial stochastic-dependence issues on the random choices of the algorithmic process which are due to the worst-case, sparse topology of the underlying graph.

The Undecided-State Dynamics is a well-known protocol for distributed consensus. We analyze it in the parallel communication model on the complete graph for the binary case (every node can either support one of two possible colors, or be in the undecided state). An interesting open question is whether this dynamics always (i.e., starting from an arbitrary initial configuration) reaches consensus quickly (i.e., within a polylogarithmic number of rounds) in a complete graph with n nodes. Previous work in this setting only considers initial color configurations with no undecided nodes and a large bias (i.e., Θ(n)) towards the majority color. In this paper we present an unconditional analysis of the Undecided-State Dynamics that answers to the above question in the affirmative. We prove that, starting from any initial configuration, the process reaches a monochromatic configuration within O(log n) rounds, with high probability. This bound turns out to be tight. Our analysis also shows that, if the initial configuration has bias Ω(√(nlog n)), then the dynamics converges toward the initial majority color, with high probability.

Consensus and Broadcast are two fundamental problems in distributed computing, whose solutions have several applications. Intuitively, Consensus should be no harder than Broadcast, and this can be rigorously established in several models. Can Consensus be easier than Broadcast? In models that allow noiseless communication, we prove a reduction of (a suitable variant of) Broadcast to binary Consensus, that preserves the communication model and all complexity parameters such as randomness, number of rounds, communication per round, etc., while there is a loss in the success probability of the protocol. Using this reduction, we get, among other applications, the first logarithmic lower bound on the number of rounds needed to achieve Consensus in the uniform GOSSIP model on the complete graph. The lower bound is tight and, in this model, Consensus and Broadcast are equivalent. We then turn to distributed models with noisy communication channels that have been studied in the context of some bio-inspired systems. In such models, only one noisy bit is exchanged when a communication channel is established between two nodes, and so one cannot easily simulate a noiseless protocol by using error-correcting codes. An $\Omega(\epsilon^{-2} n)$ lower bound on the number of rounds needed for Broadcast is proved by Boczkowski et al. [PLOS Comp. Bio. 2018] in one such model (noisy uniform PULL, where $\epsilon$ is a parameter that measures the amount of noise). In such model, we prove a new $\Theta(\epsilon^{-2} n \log n)$ bound for Broadcast and a $\Theta(\epsilon^{-2} \log n)$ bound for binary Consensus, thus establishing an exponential gap between the number of rounds necessary for Consensus versus Broadcast.

Consider the following asynchronous, opportunistic communication model over a graph $G$: in each round, one edge is activated uniformly and independently at random and (only) its two endpoints can exchange messages and perform local computations. Under this model, we study the following random process: The first time a vertex is an endpoint of an active edge, it chooses a random number, say $\pm 1$ with probability $1/2$; then, in each round, the two endpoints of the currently active edge update their values to their average. We show that, if $G$ exhibits a two-community structure (for example, two expanders connected by a sparse cut), the values held by the nodes will collectively reflect the underlying community structure over a suitable phase of the above process, allowing efficient and effective recovery in important cases. In more detail, we first provide a first-moment analysis showing that, for a large class of almost-regular clustered graphs that includes the stochastic block model, the expected values held by all but a negligible fraction of the nodes eventually reflect the underlying cut signal. We prove this property emerges after a mixing period of length $\mathcal O(n\log n)$. We further provide a second-moment analysis for a more restricted class of regular clustered graphs that includes the regular stochastic block model. For this case, we are able to show that most nodes can efficiently and locally identify their community of reference over a suitable time window. This results in the first opportunistic protocols that approximately recover community structure using only polylogarithmic work per node. Even for the above class of regular graphs, our second moment analysis requires new concentration bounds on the product of certain random matrices that are technically challenging and possibly of independent interest.

•Three error sources were investigated in estimating particle size distribution by CHDF.•Small error in diameter calibration can produce large deviation in the estimated average diameter.•Error in particle refractive index can affect the estimated mode number concentration.•Instrumental broadening correction can produce PSDs with underestimated width and distorted shape.•Simulated and experimental examples were used to quantify the errors.

Inspired by the increasing interest in self-organizing social opportunistic networks, we investigate the problem of distributed detection of unknown communities in dynamic random graphs. As a formal framework, we consider the dynamic version of the well-studied Planted Bisection Model $\mbox{dyn-}\mathcal{G}(n,p,q)$ where the node set [n] of the network is partitioned into two unknown communities and, at every time step, each possible edge (u,v) is active with probability p if both nodes belong to the same community, while it is active with probability q (with q < < p) otherwise. We also consider a time-Markovian generalization of this model. We propose a distributed protocol based on the popular Label Propagation Algorithm and prove that, when the ratio p/q is larger than n b (for an arbitrarily small constant b > 0), the protocol finds the right “planted” partition in O(logn) time even when the snapshots of the dynamic graph are sparse and disconnected (i.e. in the case p = Θ(1/n)).

We study the information spreading yielded by the (Parsimonious) k-Flooding Protocol in geometric Mobile Ad-Hoc Networks. We consider n agents on a square of side length L performing independent random walks with move radius ρ. At any time step, every active agent v informs every non-informed agent which is within distance R from v (R 0 is the transmission radius). An agent is only active for the next k time steps following the one in which has been informed and, after that, she is removed. At the initial time step, a source agent is informed and we look at the completion time of the protocol, i.e., the first time step (if any) in which all agents are informed. The presence of removed agents makes this process much more complex than the (standard) flooding and no analytical results are available over any explicit mobility model. We prove optimal bounds on the completion time depending on the parameters n, L, R, and ρ. The obtained bounds hold with high probability. Our method of analysis provides a clear picture of the dynamic shape of the information spreading (or infection wave) over the time.

Performance bounds for opportunistic networks have been derived in a number of recent papers for several key quantities, such as the expected delivery time of a unicast message, or the ooding time, i.e., the time needed to deliver a message to all nodes in the network. However, to the best of our knowledge, none of the existing results is based on \realistic mobility models, where \realistic refers to a mobility model which is able to accurately reproduce the power law+exponential tail dichotomy of the pairwise node inter-meeting time distribution which has been observed in several real world traces. The contributions of this paper are three-fold: rst, we present a simple link model { called the Home-MEG model { for opportunistic networks based on the notion of a home location derived in previous works, and we show through extensive comparison with realworld traces that the Home-MEG model is \realistic. Second, we use the Home-MEG model to analyze ooding time in opportunistic networks, presenting two upper bounds on ooding time that assume dierent initial conditions for the existence of opportunistic links. Finally, we show that our bounds improve existing ones exponentially, revealing that ooding time in opportunistic networks can be much faster than that predicted by existing bounds based on less realistic mobility models.

Sensor networks is one of the most relevant concrete examples of dynamic networks. Their dynamic behaviour is mainly due to the presence of node/link faults and node mobility. The aim of this chapter is to survey a new approach to study such dynamic networks, recently introduced in [16, 15, 18, 17, 19]. The major novelty of this approach relies on two basic issues. 1. The dynamic network is modeled as an evolving graph whose topology changes at every time according to some law/adversary. Both worst-case adversarial scenarios and graphs that evolve randomly are deeply studied. 2. This new approach provides a general framework where it is possible to determine the speed of information spreading from an analytical point view. Does the dynamic unknown behaviour of sensor networks always slow-down the speed of information spreading? Which is the real impact of this dynamic behaviour on the completion time of some basic communication protocols? Can unknown random node-mobility be exploited to asymptotically speed-up information spreading? This new general approach provides some clean mathematical answers to the above fundamental questions.

The Min Energy Broadcast problem consists in assigning transmission ranges to the nodes of an ad-hoc network in order to guarantee a directed spanning tree from a given source node and, at the same time, to minimize the energy consumption (i.e. the energy cost) yielded by the range assignment. Min Energy Broadcast is known to be NP-hard. We consider random-grid networks where nodes are chosen independently at random from the n points of a √n x √n square grid in the plane. The probability of the existence of a node at a given point of the grid does depend on that point, that is, the probability distribution can be non-uniform. By using information-theoretic arguments, we prove a lower bound (1 -- µ) n/À on the energy cost of any feasible solution for this problem. Then, we provide an efficient solution of energy cost not larger than 1.1204n/À. Finally, we present a fully-distributed protocol that constructs a broadcast range assignment of energy cost not larger than 8n, thus still yielding constant approximation. The energy load is well balanced and, at the same time, the work complexity (i.e. the energy due to all message transmissions of the protocol) is asymptotically optimal. The completion time of the protocol is only a O(log n) factor slower than the optimum. The approximation quality of our distributed solution is also experimentally evaluated. All bounds hold with probability at least 1 -- 1/n¸(1).

We consider the problem of computing an optimal range assignment in a wireless network which allows a specified source station to perform a broadcast operation. In particular, we analyze a polynomial-time algorithm proposed by Wieselthier, Nguyen, and Ephremides, which is based on the computation of a standard minimum spanning tree, and we prove that this algorithm computes a solution whose performance ratio is bounded by 20 in the case in which both the dimension d and the gradient are equal to 2. Moreover, we show how this approximation algorithm can be generalized to the case in which d, for any dimension d, and we prove a lower bound on its performance ratio, which is exponential with respect to d. A preliminary version of this paper appeared in Proc. of STACS’01 [CCP+01]. yDipartimento di Matematica, Universita degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica, I-00133 Roma, Italy. Email: clementi@mat.uniroma2.it zDipartimento di Sistemi e Informatica, Universita degli Studi di Firenze, Via C. Lombroso 6/17, I-50134 Firenze, Italy. Email: piluc@dsi.unifi.it xInstitut fur Teoretische Informatik, Swiss Federal Institute of Technology Zurich (ETH), Clausiustrasse 49, CH-8092 Zurich, Switzerland. Email: penna@inf.ethz.ch {Dipartimento di Sistemi e Informatica, Universita degli Studi di Firenze, Via C. Lombroso 6/17, I-50134 Firenze, Italy. Email: rossig@dsi.unifi.it kDipartimento di Matematica, Universita degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica, I-00133 Roma, Italy. Email: vocca@mat.uniroma2.it

The d-Dimh-hops MST problem is defined as follows: given a set S of points in the d-dimensional Euclidean space and s∈S, find a minimum-cost spanning tree for S rooted at s with height at most h. We investigate the problem for any constant h and d>0. We prove the first nontrivial lower bound on the solution cost for almost all Euclidean instances (i.e. the lower bound holds with high probability). Then we introduce an easy-to-implement, fast divide et impera heuristic and we prove that its solution cost matches the lower bound.

We consider static ad hoc wireless networks whose nodes, equipped with the same initial battery charge, may dynamically change their transmission range. When a node v transmits with range r(v), its battery charge is decreased by β r(v) 2 , where β > 0 is a fixed constant. The goal is to provide a range assignment schedule that maximizes the number of broadcast operations from a given source (this number is denoted by the length of the schedule). This maximization problem, denoted by Max LifeTime, is known to be NP-hard and the best algorithm yields worst-case approximation ratio Θ (log n), where n is the number of nodes of the network. We consider random geometric instances formed by selecting n points independently and uniformly at random from a square of side length √n in the euclidean plane. We present an efficient algorithm that constructs a range assignment schedule having length not smaller than 1/12 of the optimum with high probability. Then we design an efficient distributed version of the above algorithm, where nodes initially know n and their own position only. The resulting schedule guarantees the same approximation ratio achieved by the centralized version, thus, obtaining the first distributed algorithm having provably good performance for this problem.

Given a set S of points (stations) located in the d-dim. Euclidean space and a rootb ∈S, the h-hops Convergecast problem asks to find for a minimal energy-cost range assignment which allows to perform the converge-cast primitive (i.e. node accumulation) towards b in at most h hops. For this problem no polynomial time algorithm is known even for h = 2. The main goal of this work is the design of an efficient distributed heuristic (i.e. protocol) and the analysis (both theoretical and experimental) of its expected solution cost. In particular, we introduce an efficient parameterized randomized protocol for h-hops Convergecast and we analyze it on random instances created by placing n points uniformly at random in a d-cube of side length L. We prove that for h = 2, its expected approximation ratio is bounded by some constant factor. Finally, for h = 3,..., 8, we provide a wide experimental study showing that our protocol has very good performances when compared with previously introduced (centralized) heuristics.

We prove an optimal bound for the function L(n, m, ε) that gives the worst-case circuit-size complexity to approximate partial boolean functions having n inputs and domain size m within degree at least ε. Our bound applies to any partial boolean function and any approximation degree, completing the study of boolean function approximation introduced in [15]. We also provide the approximation degree (i.e. the value ε) achieved by polynomial size circuits on a ‘random’ boolean function. Our results give a new upper bound for the hardness function h(f), the function denoting the minimum value l for which there exists a circuit of size at most l that approximates a boolean function f with degree at least 1/l [14]. The contribution in the proof of the upper bound for L(n, m, ε) can be viewed as a set of technical results that globally show how boolean linear operators are “well” distributed over the class of 4-regular domains. We show how to apply this property to approximate partial boolean functions on general domains.

Given a positive number δ ∈ (0,1), a subset H ⊑ {0,1}n is a δ-Hitting Set for a class R of boolean functions with n inputs if, for any function f ∈ R such that Pr (f=1)≥δ, there exists an element h ∈ H such that f(h)=1. Our paper presents a new deterministic method to efficiently construct δ-Hitting Set for the class of systems (i.e. logical conjunctions) of boolean linear functions. Systems of boolean linear functions can be considered as the algebraic generalization of boolean combinatorial rectangular functions, the only significative example for which an efficient deterministic construction of Hitting Sets were previously known. In the restricted case of boolean rectangular functions, our method (even though completely different) achieves equivalent results to those obtained in [11]. Our results also gives an upper bound on the minimum cardinality of solution covers for the class of systems of linear equations defined over a finite field. Furthermore, as preliminary result, we show a new upper bound on the circuit complexity of integer monotone functions generalizing previous results obtained in [12].

We show that hitting sets can derandomize any probabilistic, two-sided error algorithm. This gives a positive answer to a fundamental open question in probabilistic algorithms. More precisely, we present a polynomial time deterministic algorithm which uses any given hitting set to approximate the fractions of 1's in the output of any boolean circuit of polynomial size. This new algorithm implies that if a quick hitting set generator with logarithmic price exists then BPP = P. Furthermore, we generalize this result by showing that the existence of a quick hitting set generator with price k implies that BPTIME(t) \(\subseteq DTIME(2^{O(k(t^{O(1)} ))} )\). The existence of quick hitting set generators is thus a new weaker sufficient condition to obtain BPP = P; this can be considered as another strong indication that the gap between probabilistic and deterministic computational power is not large.

Studying the computational complexity of the Configuration Reachabilty Problem (CREP) is a good way to investigate properties of a given discrete deterministic dynamical system like a Toroidal Cellular Automaton (TCA). We study CREP for two natural weakly predictable classes of TCA: the booleandisjunctive class and the additive one. For the first class, we reduce CREP to a path problem on a strongly connected digraph and we show a polynomialtime algorithm for this problem. Some consequences of this result on arbitrary TCA are also analysed. For the second class, we show that CREP is not easier than computing the vectorial version of the Discrete Log Problem (DLP). However, we also show CREP is unlikely to be N P-complete. To do this, we prove that CREP is in Co-AM[2], where AM[2] is the class of problems with a constant round interactive protocol [1, 13]. CREP is unlikely to be N P-complete (unless the polynomial time hierarchy collapses), then follows by the results of [3]. All such results hold even when multidimensional and/or non homogeneous TCA arise. As a global consequence, we argue that the structure of the weakly predictable class resembles the N P one.