Paolo Frasca

Continuous-time quantized consensus: Convergence of Krasovskii solutions

This note studies a network of agents having continuous-time dynamics with quantized interactions and time-varying directed topology. Due to the discontinuity of the dynamics, solutions of the resulting ODE system are intended in the sense of Krasovskii. A limit connectivity graph is defined, which encodes persistent interactions between nodes: if such graph has a globally reachable node, Krasovskii solutions reach consensus (up to the quantizer precision) after a finite time. Under the additional assumption of a time-invariant topology, the convergence time is upper bounded by a quantity which depends on the network size and the quantizer precision. It is observed that the convergence time can be very large for solutions which stay on a discontinuity surface.

Average consensus by gossip algorithms with quantized communication

This work studies how the randomized gossip algorithm can solve the average consensus problem on networks with quantized communications. The algorithm is proved to converge to the average value, up to the size of the quantization bins, whenever the the graph is connected. Moreover, its speed of convergence is estimated.

Ergodic Randomized Algorithms and Dynamics Over Networks

Algorithms and dynamics over networks often involve randomization and randomization can induce oscillating dynamics that fail to converge in a deterministic sense. Under assumptions of independence across time and linearity of the updates, we show that the oscillations are ergodic if the expected dynamics is stable. We apply this result to three problems of network systems, namely, the estimation from relative measurements, the PageRank computation, and the dynamics of opinions in social networks. In these applications, the randomized dynamics is the asynchronous counterpart of a deterministic (stable) synchronous one. By ergodicity, the deterministic limit can be recovered via a time-averaging operation, which can be performed locally by each node of the network.

Gossips and prejudices: ergodic randomized dynamics in social networks

In this paper we study a new model of opinion dynamics in social networks, which has two main features. First, agents asynchronously interact in pairs, and these pairs are chosen according to a random process: following recent literature, we refer to this communication model as “gossiping”. Second, agents are not completely open-minded, but instead take into account their initial opinions, which may be thought of as their “prejudices”. In the literature, such agents are often called “stubborn”. We show that the opinions of the agents fail to converge, but persistently undergo ergodic oscillations, which asymptotically concentrate around a mean distribution of opinions. This mean value is exactly the limit of the synchronous dynamics of the expected opinions.

Existence and approximation of probability measure solutions to models of collective behaviors

In this paper we consider first order differential models of collective behaviors of groups of agents, based on the mass conservation equation. Models are formulated taking the spatial distribution of the agents as the main unknown, expressed in terms of a probability measure evolving in time. We develop an existence and approximation theory of the solutions to such models and we show that some recently proposed models of crowd and swarm dynamics fit our theoretic paradigm.

Effects of anisotropic interactions on the structure of animal groups

This paper proposes an agent-based model which reproduces different structures of animal groups. The shape and structure of the group is the effect of simple interaction rules among individuals: each animal deploys itself depending on the position of a limited number of close group mates. The proposed model is shown to produce clustered formations, as well as lines and V-like formations. The key factors which trigger the onset of different patterns are argued to be the relative strength of attraction and repulsion forces and, most important, the anisotropy in their application.

A probabilistic analysis of the average consensus algorithm with quantized communication

Abstract In the average consensus problem the states of a set of agents, linked according to a directed graph, have to be driven to their average. When the communication between neighbors is uniformly quantized, such a problem can not be exactly solved by a linear time-invariant algorithm. In this work, we propose a probabilistic estimate of the error from the agreement, in terms of the eigenvalues of the evolution matrix describing the algorithm.

Distributed randomized algorithms for opinion formation, centrality computation and power systems estimation: A tutorial overview ☆

In this tutorial paper, we study three specific applications: opinion formation in social networks, centrality measures in complex networks and estimation problems in large-scale power systems. These applications fall under a general framework which aims at the construction of algorithms for distributed computation over a network. The two key ingredients of randomization and time-averaging are used, together with a local gossip communication protocol, to obtain convergence of these distributed algorithms to the global synchronous dynamics.

Broadcast Gossip Averaging: Interference and Unbiasedness in Large Abelian Cayley Networks

In this paper, we study two related iterative randomized algorithms for distributed computation of averages. The first algorithm is the Broadcast Gossip Algorithm, in which at each iteration one randomly selected node broadcasts its own state to its neighbors. The second algorithm is a novel variation of the former, in which at each iteration every node is allowed to broadcast: hence, this algorithm, which we call Collision Broadcast Gossip Algorithm (CBGA), is affected by interference among messages. The performance of both algorithms is evaluated in terms of rate of convergence and asymptotical error: focusing on large Abelian Cayley networks, we highlight the role of topology and of design parameters. We show that on fully connected graphs the rate of convergence is bounded away from one, whereas the asymptotical error is bounded away from zero. On the contrary, on sparse graphs the rate of convergence goes to one and the asymptotical error goes to zero, as the size of the network grows larger. Our results also show that the performance of the CBGA is close to the performance of the BGA: this indicates the robustness of broadcast gossip algorithms to interferences.

Message Passing Optimization of Harmonic Influence Centrality

This paper proposes a new measure of node centrality in social networks, the Harmonic Influence Centrality (HIC), which emerges naturally in the study of social influence over networks. Using an intuitive analogy between social and electrical networks, we introduce a distributed message passing algorithm to compute the HIC of each node. Although its design is based on theoretical results which assume the network to have no cycle, the algorithm can also be successfully applied on general graphs.