No abstract

Consider a graph where each of the n nodes is in one of two possible states. Herein, we analyze the synchronous k-majority dynamics, where nodes sample k neighbors uniformly at random with replacement and adopt the majority state among the nodes in the sample (potential ties are broken uniformly at random). This class of dynamics generalizes other well-known dynamics, e.g., voter and 3-majority, which have been studied in the literature as distributed algorithms for consensus.We consider a biased communication model : whenever nodes sample a neighbor they see state σ with some probability p, regardless of the state of the sampled node, and its true state with probability 1 − p. Differently from previous works where specific graph topologies-typically characterized by good expansion properties-are considered, our analysis only requires the graphs to be sufficiently dense, i.e., to have minimum degree ω(log n), without any further topological assumption.In this setting we prove two phase transition phenomena, both occurring with high probability, depending on the bias p and on the initial unbalance toward state σ. More in detail, we prove that for every k ≥ 3 there exists a p k such that if p > p k the process reaches in O(1) rounds a σ-almost-consensus, i.e., a configuration where a fraction 1 − γ of the volume is in state σ, for any arbitrarily-small positive constant γ. On the other hand, if p < p k , we look at random initial configurations in which every node is in state σ with probability 1−q independently of the others. We prove that there exists a constant q p,k such that if q < q p,k then a σ-almost-consensus is still reached in O(1) rounds, while, if q > q p,k , the process spends n ω(1) rounds in a metastable phase where the fraction of volume in state σ is around a constant value depending only on p and k.Finally we also investigate, in such a biased setting, the differences and similarities between k-majority and other closely-related dynamics, namely voter and deterministic majority.

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.

This contribution draws on the voices and reflections from young people as co-researchers in the Growing-Up Under Covid-19 project – a longitudinal ethnographic action research project to document, share, and respond to impacts of the pandemic on different spheres of young people’s lives. The research was conducted entirely online over 18 months in seven countries and has involved youth-led approaches to research, including video diaries and the use of artefacts and visual material to convey their experiences and support reflection and dialogue across research groups and with external stakeholders. In this contribution, the young co-researchers reflect on their rationale for using different visual media and why this was important for them. They also reflect on the significance of the representations in the visual images and how these images communicate how young people’s understanding of COVID and its impact on young people has changed (or given new meaning to) and how this in turn has given rise to particular responses and opportunities for young people. The article draws on examples of different visual forms selected by young people in Singapore, Italy, Lebanon, and the UK nations, including video, drawing, photography, and crafts. These different media and links to videos were included in the accompanying document. The contribution explores the different narratives and meanings behind the visuals, using the words of young people themselves, interspersed with narration from the adult researchers.

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 are constant over each 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 eigenvalues 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.