We study the minority-opinion dynamics over a fully-connected network of n nodes with binary opinions. Upon activation, a node receives a sample of opinions from a limited number of neighbors chosen uniformly at random. Each activated node then adopts the opinion that is least common within the received sample.

Opinion diffusion is a crucial phenomenon in social networks, often underlying the way in which a collection of agents develops a consensus on relevant decisions. Voter models are well-known theoretical models to study opinion spreading in social networks and structured populations. Their simplest version assumes that an updating agent will adopt the opinion of a neighboring agent chosen at random. These models allow us to study, for example, the probability that a certain opinion will fixate into a consensus opinion, as well as the expected time it takes for a consensus opinion to emerge. Standard voter models are oblivious to the opinions held by the agents involved in the opinion adoption process. We propose and study a context-dependent opinion spreading process on an arbitrary social graph, in which the probability that an agent abandons opinion a in favor of opinion b depends on both a and b. We discuss the relations of the model with existing voter models and then derive theoretical results for both the fixation probability and the expected consensus time for two opinions, for both the synchronous and the asynchronous update models.

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 opinion dynamics in multi-agent networks when a bias toward one of two possible opinions exists, for example reflecting a status quo versus a superior alternative. Our aim is to investigate the combined effect of bias, network structure, and opinion dynamics on the convergence of the system of agents as a whole. Models of such evolving processes can easily become analytically intractable. In this paper, we consider a simple yet mathematically rich setting, in which all agents initially share an initial opinion representing the status quo. The system evolves in steps. In each step, one agent selected uniformly at random follows an underlying update rule to revise its opinion on the basis of those held by its neighbors, but with a probabilistic bias towards the superior alternative. We analyze convergence of the resulting process under well-known update rules. The framework we propose is simple and modular, but at the same time complex enough to highlight a nonobvious interplay between topology and underlying update rule.

We study expansion and information diffusion properties of dynamic networks, i.e., networks whose topologies evolve over time as nodes enter or leave the system and edges are continuously created or destroyed. In this scenario, we investigate flooding as a basic information diffusion mechanism. We are interested in models that are likely to result in sparse networks, i.e., in networks containing$O(n)$edges, with$n$the number of nodes that are present at any given time of interest, with a focus on models in which edges are created randomly according to simple probabilistic mechanisms, rather than according to carefully designed distributed algorithms. In this perspective, in all models we consider, upon joining the network, a node connects to$d=O(1)$random nodes currently in the system. On the other hand, an edge remains alive as long as both its endpoints are. For the case in which edges that fail (because one endpoint left the network) are not replaced, we show that, although the network is likely to contain$\Omega_{d}(n)$isolated nodes, flooding still informs a fraction$1-\exp(-\Omega(d))$of the nodes in time$\mathrm{O}(\log n)$with large, constant probability. Moreover, we are able to show, that at any given time, the graph exhibits a “large-set expansion” property. We further investigate models that exhibit edge regeneration, meaning that, whenever an edge$(v, w)$established by$v$fails because$w$leaves the network, it is replaced by a new random edge$(v, z)$. We show that models with edge regeneration result in evolving networks that, at any given time, are vertex expanders with high probability, so that flooding takes$\mathrm{O}(\log n)$time. The above results hold both for a simplfied streaming model of node churn and in a more realistic, continuous-time setting, in which the interval between two consecutive node arrivals follows a Poisson distribution, while nodes' lifetimes follow an exponential distribution. Previous work considered models in which either the vertex set is fixed or edges are established according to more or less sophisticated algorithms. Our motivation for studying models with simple and random edge creation mechanisms is to move one step further towards models that may eventually capture key aspects of the formation of social or peer-to-peer networks.

We investigate opinion dynamics in multi-agent networks when there exists a bias toward one of two possible opinions; for example, reflecting a status quo vs a superior alternative. Starting with all agents sharing an initial opinion representing the status quo, the system evolves in steps. In each step, one agent selected uniformly at random adopts with some probability a the superior opinion, and with probability 1 - a it follows an underlying update rule to revise its opinion on the basis of those held by its neighbors. We analyze the convergence of the resulting process under two well-known update rules, namely majority and voter. The framework we propose exhibits a rich structure, with a nonobvious interplay between topology and underlying update rule. For example, for the voter rule we show that the speed of convergence bears no significant dependence on the underlying topology, whereas the picture changes completely under the majority rule, where network density negatively affects convergence. We believe that the model we propose is at the same time simple, rich, and modular, affording mathematical characterization of the interplay between bias, underlying opinion dynamics, and social structure in a unified setting.

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 Ω(ε^{-2} n) lower bound is proved by Boczkowski et al. [PLOS Comp. Bio. 2018] on the convergence time of binary Broadcast in one such model (noisy uniform PULL), where ε is a parameter that measures the amount of noise). We prove an O(ε^{-2} log n) upper bound on the convergence time of binary Consensus in such model, thus establishing an exponential complexity gap between Consensus versus Broadcast. We also prove our upper bound above is tight and this implies, for binary Consensus, a further strong complexity gap between noisy uniform PULL and noisy uniform PUSH. Finally, we show a Θ(ε^{-2} n log n) bound for Broadcast in the noisy uniform PULL.

It follows from the Marcus-Spielman-Srivastava proof of the Kadison-Singer conjecture that if G = (V, E) is a Δ-regular dense expander then there is an edge-induced subgraph H = (V, Eh) of G of constant maximum degree which is also an expander. As with other consequences of the MSS theorem, it is not clear how one would explicitly construct such a subgraph. We show that such a subgraph (although with quantitatively weaker expansion and near-regularity properties than those predicted by MSS) can be constructed with high probability in linear time, via a simple algorithm. Our algorithm allows a distributed implementation that runs in O(log n) rounds and does O(n) total work with high probability. The analysis of the algorithm is complicated by the complex dependencies that arise between edges and between choices made in different rounds. We sidestep these difficulties by following the combinatorial approach of counting the number of possible random choices of the algorithm which lead to failure. We do so by a compression argument showing that such random choices can be encoded with a non-trivial compression. Our algorithm bears some similarity to the way agents construct a communication graph in a peer-to-peer network, and, in the bipartite case, to the way agents select servers in blockchain protocols.

Given an underlying graph, we consider the following dynamics: Initially, each node locally chooses a value in $\{-1,1\}$, uniformly at random and independently of other nodes. Then, in each consecutive round, every node updates its local value to the average of the values held by its neighbors, at the same time applying an elementary, local clustering rule that only depends on the current and the previous values held by the node. We prove that the process resulting from this dynamics produces a clustering that exactly or approximately (depending on the graph) reflects the underlying cut in logarithmic time, under various graph models that exhibit a sparse balanced cut, including the stochastic block model. We also prove that a natural extension of this dynamics performs community detection on a regularized version of the stochastic block model with multiple communities. Rather surprisingly, our results provide rigorous evidence for the ability of an extremely simple and natural dynamics to perform community detection, a computational problem which is nontrivial even in a centralized setting.

Spectral techniques have proved amongst the most effective approaches to graph clustering. However, in general they require explicit computation of the main eigenvectors of a suitable matrix (usually the Laplacian matrix of the graph). Recent work (e.g., Becchetti et al., SODA 2017) suggests that observing the temporal evolution of the power method applied to an initial random vector may, at least in some cases, provide enough information on the space spanned by the first two eigenvectors, so as to allow recovery of a hidden partition without explicit eigenvector computations. While the results of Becchetti et al. apply to perfectly balanced partitions and/or graphs that exhibit very strong forms of regularity, we extend their approach to graphs containing a hidden k partition and characterized by a milder form of volume-regularity. We show that the class of k-volume regular graphs is the largest class of undirected (possibly weighted) graphs whose transition matrix admits k “stepwise” eigenvectors (i.e., vectors that have constant entries over the components corresponding to the same set of the hidden partition). To obtain this result, we highlight a connection between volume regularity and lumpability of Markov chains. Moreover, we prove that if the stepwise eigenvectors are those associated to the first k largest eigenvalues of the transition matrix of a random walk on the graph and the gap between the k-th and the (k+1)-th eigenvalues is sufficiently large, the Averaging dynamics of Becchetti et al. recovers the underlying community structure of the graph in logarithmic time, with high probability.

We study the following synchronous process that we call repeated balls-into-bins. The process is started by assigning n balls to n bins in an arbitrary fashion. In every subsequent round, one ball is extracted from each non-empty bin according to some fixed strategy (random, FIFO, etc), and re-assigned to one of the n bins uniformly at random. We define a configuration legitimate if its maximum load is $$\mathcal {O}(\log n)$$ . We prove that, starting from any configuration, the process converges to a legitimate configuration in linear time and then only takes on legitimate configurations over a period of length bounded by any polynomial in n, with high probability (w.h.p.). This implies that the process is self-stabilizing and that every ball traverses all bins within $$\mathcal {O}(n\log ^2 n)$$ rounds, w.h.p.

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.

Given an underlying graph, we consider the following dynamics: Initially, each node locally chooses a value in {-1,1}, uniformly at random and independently of other nodes. Then, in each consecutive round, every node updates its local value to the average of the values held by its neighbors, at the same time applying an elementary, local clustering rule that only depends on the current and the previous values held by the node. We prove that the process resulting from this dynamics produces a clustering that exactly or approximately (depending on the graph) reflects the underlying cut in logarithmic time, under various graph models that exhibit a sparse balanced cut, including the stochastic block model. We also prove that a natural extension of this dynamics performs community detection on a regularized version of the stochastic block model with multiple communities. Rather surprisingly, our results provide rigorous evidence for the ability of an extremely simple and natural dynamics to address a computational problem that is non-trivial even in a centralized setting. Distributed Algorithms, Averaging Dynamics, Community Detection, Spectral Analysis, Stochastic Block Models.

The Undecided-State Dynamics is a well-known protocol that achieves Consensus in distributed systems formed by a set of n anonymous nodes interacting via a communication network. We consider this dynamics in the parallel PULL communication model on the complete graph for the binary case, i.e., when every node can either support one of two possible colors or stay in the undecided state. Previous work in this setting only considers initial color configurations with no undecided nodes and a large bias (i.e., Theta(n)) towards the majority color. A interesting open question here is whether this dynamics reaches consensus quickly, i.e. within a polylogarithmic number of rounds. In this paper we present an unconditional analysis of the Undecided-State Dynamics which answers to the above question in the affirmative. Our analysis shows that, starting from any initial configuration, the Undecided-State Dynamics reaches a monochromatic configuration within O(log^2 n) rounds, with high probability (w.h.p.). Moreover, we prove that if the initial configuration has bias Omega(sqrt(n log n)), then the dynamics converges toward the initial majority color within O(log n) round, w.h.p. At the heart of our approach there is a new analysis of the symmetry-breaking phase that the process must perform in order to escape from (almost-)unbiased configurations. Previous symmetry-breaking analysis of consensus dynamics essentially concern sequential communication models (such as Population Protocols) and/or symmetric updated rules (such as majority rules).

We consider the following distributed consensus problem: Each node in a complete communication network of size n initially holds an opinion, which is chosen arbitrarily from a finite set Σ. The system must converge toward a consensus state in which all, or almost all nodes, hold the same opinion. Moreover, this opinion should be valid, i.e., it should be one among those initially present in the system. This condition should be met even in the presence of a malicious adversary who can modify the opinions of a bounded subset of nodes, adaptively chosen in every round. We consider the 3-majority dynamics: At every round, every node pulls the opinion from three random neighbors and sets his new opinion to the majority one (ties are broken arbitrarily). Let k be the number of valid opinions. We show that, if k ≤ nα, where α is a suitable positive constant, the 3-majority dynamics converges in time polynomial in k and log n with high probability even in the presence of an adversary who can affect up to nodes at each round. Previously, the convergence of the 3-majority protocol was known for |Σ| = 2 only, with an argument that is robust to adversarial errors. On the other hand, no anonymous, uniform-gossip protocol that is robust to adversarial errors was known for |Σ| > 2.

We study Plurality Consensus in the Model over a network of n anonymous agents. Each agent supports an initial opinion or color. We assume that at the onset, the number of agents supporting the plurality color exceeds that of the agents supporting any other color by a sufficiently-large bias, though the initial plurality itself might be very far from absolute majority. The goal is to provide a protocol that, with high probability, brings the system into the configuration in which all agents support the (initial) plurality color. We consider the Undecided-State Dynamics, a well-known protocol which uses just one more state (the undecided one) than those necessary to store colors. We show that the speed of convergence of this protocol depends on the initial color configuration as a whole, not just on the gap between the plurality and the second largest color community. This dependence is best captured by a novel notion we introduce, namely, the monochromatic distance md which measures the distance of the initial color configuration from the closest monochromatic one. In the complete graph, we prove that, for a wide range of the input parameters, this dynamics converges within O(md log n) rounds. We prove that this upper bound is almost tight in the strong sense: Starting from any color configuration , the convergence time is Ω(md). Finally, we adapt the Undecided-State Dynamics to obtain a fast, random walk-based protocol for plurality consensus on regular expanders. This protocol converges in O(md polylog(n)) rounds using only polylog(n) local memory. A key-ingredient to achieve the above bounds is a new analysis of the maximum node congestion that results from performing n parallel random walks on regular expanders. All our bounds hold with high probability.

We study the following synchronous process that we call repeated balls-into-bins. The process is started by assigning n balls to n bins in an arbitrary way. Then, in every subsequent round, one ball is chosen according to some fixed strategy (random, FIFO, etc) from each non-empty bin, and re-assigned to one of the n bins uniformly at random. This process corresponds to a non-reversible Markov chain and our aim is to study its self-stabilization properties with respect to the maximum(bin) load and some related performance measures. We define a configuration (i.e., a state) legitimate if its maximum load is O(log n). We first prove that, starting from any legitimate configuration, the process will only take on legitimate configurations over a period of length bounded by any polynomial in n, with high probability (w.h.p.). Further we prove that, starting from any configuration, the process converges to a legitimate configuration in linear time, w.h.p. This implies that the process is self-stabilizing w.h.p. and, moreover, that every ball traverses all bins in O(n log2 n) rounds, w.h.p. The latter result can also be interpreted as an almost tight bound on the cover time for the problem of parallel resource assignment in the complete graph.

We study a Plurality Consensus process in which each of n anonymous agents of a communication network supports an initial opinion (a colorchosen from a finite set [k]) and, at every time step, he can revise his color according to a random sample of neighbors.

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 flooding time (a measure of how fast information spreads). However, to the best of our knowledge, none of the existing results is derived under a mobility model which is able to reproduce the power law+exponential tail dichotomy of the pairwise node intercontact time distribution which has been observed in traces of several real opportunistic networks. The contributions of this paper are two-fold: first, we present a simple pairwise contact model—called the Home-MEG model—for opportunistic networks based on the observation made in previous work that pairs of nodes in the network tend to meet in very few, selected locations (home locations); this contact model is shown to be able to faithfully reproduce the power law+exponential tail dichotomy of intercontact time. Second, we use the Home-MEG model to analyze flooding time in opportunistic networks, presenting asymptotic bounds on flooding time that assume different initial conditions for the existence of opportunistic links. By comparing asymptotic bounds with the results of simulations performed using a realistic human mobility model, we demonstrate the capability of the proposed Home-MEG model to faithfully predict the speed of information spreading in large-scale opportunistic networks. Finally, our bounds provide some analytical evidences that the speed of information spreading in opportunistic networks can be much faster than that predicted by simple geometric mobility models.

Opportunistic mobile ad hoc networks (MANETs) are a special class of sparse and disconnected MANETs where data communication exploits sporadic contact opportunities among nodes. We consider opportunistic MANETs where nodes move independently at random over a square of the plane. Nodes exchange data if they are at a distance at mostrwithin each other, wherer> 0 is the node transmission radius. The flooding time is the number of time-steps required to broadcast a message from a source node to every node of the network. Flooding time is an important measure of how fast information can spread in dynamic networks. We derive the first upper bound on the flooding time, which is a decreasing function of the maximal speed of the nodes. The bound holds with high probability, and it is nearly tight. Our bound shows that, thanks to node mobility, even when the network is sparse and disconnected, information spreading can be fast.

Logit Dynamics [Blume, Games and Economic Behavior, 1993] is a randomized best response dynamics for strategic games: at every time step a player is selected uniformly at random and she chooses a new strategy according to a probability distribution biased toward strategies promising higher payoffs. This process defines an ergodic Markov chain, over the set of strategy profiles of the game, whose unique stationary distribution is the long-term equilibrium concept for the game. However, when the mixing time of the chain is large (e.g., exponential in the number of players), the stationary distribution loses its appeal as equilibrium concept, and the transient phase of the Markov chain becomes important. In several cases it happens that on a time-scale shorter than mixing time the chain is “quasi-stationary”, meaning that it stays close to some small set of the state space, while in a time-scale multiple of the mixing time it jumps from one quasi-stationary configuration to another; this phenomenon is usually called “metastability”. In this paper we give a quantitative definition of “metastable probability distributions” for a Markov chain and we study the metastability of the Logit dynamics for some classes of coordination games. In particular, we study no-risk-dominant coordination games on the clique (which is equivalent to the well-known Glauber dynamics for the Ising model) and coordination games on a ring (both the risk-dominant and no-risk-dominant case). We also describe a simple “artificial” game that highlights the distinctive features of our metastability notion based on distributions.

Abstract We study deterministic fault‐tolerant gossiping protocols in geometric radio networks. Node and link faults may happen during every time‐slot of the protocol's execution. We first consider the model where every node can send at most one message per time‐slot. We provide a protocol that completes gossiping inO(nΔ) time (wherenis the number of nodes and Δ is the maximal in‐degree) and has message complexityO(n2). Both bounds are then shown to be optimal. Second, we consider the model where messages can be arbitrarily combined and sent in one time‐slot. We give a protocol working in optimal completion timeO(DΔ) (whereDis the maximal source eccentricity) and message complexityO(Dn). © 2012 Wiley Periodicals, Inc. NETWORKS, Vol. 2012

We present the first general bounds on the mixing time of logit dynamics for wide classes of strategic games. The logit dynamics describes the behaviour of a complex system whose individual components act "selfishly" and keep responding according to some partial ("noisy") knowledge of the system. In particular, we prove nearly tight bounds for potential games and games with dominant strategies. Our results show that, for potential games, the mixing time is upper and lower bounded by an "exponential" in the inverse of the noise and in the maximum potential difference. Instead, for games with dominant strategies, the mixing time cannot grow arbitrarily with the inverse of the noise. Finally, we refine our analysis for a subclass of potential games called "graphical" coordination games and we give evidence that the mixing time strongly depends on the structure of the underlying graph. Games in this class have been previously studied in Physics and, more recently, in Computer Science in the context of diffusion of new technologies.

Markovian evolving graphs are dynamic-graph models where the links among a fixed set of nodes change during time according to an arbitrary Markovian rule. They are extremely general and they can well describe important dynamic-network scenarios. We study the speed of information spreading in the stationary phase by analyzing the completion time of the flooding mechanism. We prove a general theorem that establishes an upper bound on flooding time in any stationary Markovian evolving graph in terms of its node-expansion properties. We apply our theorem in two natural and relevant cases of such dynamic graphs. Geometric Markovian evolving graphs where the Markovian behaviour is yielded by n mobile radio stations, with fixed transmission radius, that perform independent random walks over a square region of the plane. Edge-Markovian evolving graphs where the probability of existence of any edge at time t depends on the existence (or not) of the same edge at time t-1. In both cases, the obtained upper bounds hold with high probability and they are nearly tight. In fact, they turn out to be tight for a large range of the values of the input parameters. As for geometric Markovian evolving graphs, our result represents the first analytical upper bound for flooding time on a class of concrete mobile networks.

=1We introduce stochastic time-dependency in evolving graphs: starting from an initial graph, at every time step, every edge changes its state (existing or not) according to a two-state Markovian process with probabilities p (edge birth-rate) and q (edge death-rate). If an edge exists at time t, then, at time $t+1$, it dies with probability q. If instead the edge does not exist at time t, then it will come into existence at time $t+1$ with probability p. Such an evolving graph model is a wide generalization of time-independent dynamic random graphs [A. E. F. Clementi, A. Monti, F. Pasquale, and R. Silvestri, J. Comput. System Sci., 75 (2009), pp. 213–220] and will be called edge-Markovian evolving graphs. We investigate the speed of information spreading in such evolving graphs. We provide nearly tight bounds (which in fact turn out to be tight for a wide range of probabilities p and q) on the completion time of the flooding mechanism aiming to broadcast a piece of information from a source node to all nodes. In particular, we provide i) a tight characterization of the class of edge-Markovian evolving graphs where flooding time is constant and, thus, it does not asymptotically depend on the initial graph; ii) a tight characterization of the class of edge-Markovian evolving graphs where flooding time does not asymptotically depend on the edge death-rate q. An interesting consequence of our results is that information spreading can be fast even if the graph, at every time step, is very sparse and disconnected. Furthermore, our bounds imply that the flooding time can be exponentially shorter than the mixing time of the edge-Markovian graph.

Markovian evolving graphs are dynamic-graph models where the links among a fixed set of nodes change during time according to an arbitrary Markovian rule. They are extremely general and they can well describe important dynamic-network scenarios. We study the speed of information spreading in the stationary phase by analyzing the completion time of the flooding mechanism. We prove a general theorem that establishes an upper bound on flooding time in any stationary Markovian evolving graph in terms of its node-expansion properties. We apply our theorem in two natural and relevant cases of such dynamic graphs: edge-Markovian evolving graphs where the probability of existence of any edge at time t depends on the existence (or not) of the same edge at time t-1; geometric Markovian evolving graphs where the Markovian behaviour is yielded by n mobile radio stations, with fixed transmission radius, that perform n independent random walks over a square region of the plane. In both cases, the obtained upper bounds are shown to be nearly tight and, in fact, they turn out to be tight for a large range of the values of the input parameters.

We introduce stochastic time-dependency in evolving graphs: starting from an arbitrary initial edge probability distribution, at every time step, every edge changes its state (existing or not) according to a two-state Markovian process with probabilities p (edge birth-rate) and q (edge death-rate). If an edge exists at time t then, at time t+1, it dies with probability q. If instead the edge does not exist at time t, then it will come into existence at time t+1 with probability p.

We study the completion time of distributed broadcast protocols in dynamic radio networks. The dynamic network is modelled by means of adversaries: we consider two of them that somewhat are the extremal cases.