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Connections of a spectral nature are formed between Gibbs samplers and their blocked and collapsed variants. The solidarity principle of the spectral gap for full Gibbs samplers is generalized to different cycles and mixtures of Gibbs steps. This generalized solidarity principle is employed to establish that every cycle and mixture of Gibbs steps, which includes blocked Gibbs samplers and collapsed Gibbs samplers, inherits a spectral gap from a full Gibbs sampler. Exact relations between the spectra corresponding to blocked and collapsed variants of a Gibbs sampler are also established. An example is given to show that a blocked or collapsed Gibbs sampler does not in general inherit geometric ergodicity or a spectral gap from another blocked or collapsed Gibbs sampler.

Deterministic-scan and random-scan component-wise Markov chain Monte Carlo algorithms, such as Gibbs samplers and conditional Metropolis-Hastings, are popular approaches for sampling from multivariate distributions. A long-standing open question is to determine the conditions under which these algorithms have similar convergence rates. A block-wise contraction condition for the component-wise updates is used to establish a solidarity principle for the $L^2$ spectral gaps of the associated Markov chains. Specifically, under this condition, the spectral gaps of the random-scan and deterministic-scan versions of the Gibbs and component-wise chains are either simultaneously positive or simultaneously zero. Moreover, the spectral gaps differ by at most polynomial factors in the number of blocks. As an application of the general results, a deterministic-scan conditional Metropolis-adjusted Langevin algorithm (MALA) for multivariate Gaussian targets is studied. The block-wise contraction condition is combined with known spectral gap bounds for the random-scan Gibbs sampler to obtain a spectral gap bound that is polynomial in dimension. The result is used to clarify how the convergence rat

Solidarity is a crucial concept to understand social relations in societies. In this paper, we explore fine-grained solidarity frames to study solidarity towards women and migrants in German parliamentary debates between 1867 and 2022. Using 2,864 manually annotated text snippets (with a cost exceeding 18k Euro), we evaluate large language models (LLMs) like Llama 3, GPT-3.5, and GPT-4. We find that GPT-4 outperforms other LLMs, approaching human annotation quality. Using GPT-4, we automatically annotate more than 18k further instances (with a cost of around 500 Euro) across 155 years and find that solidarity with migrants outweighs anti-solidarity but that frequencies and solidarity types shift over time. Most importantly, group-based notions of (anti-)solidarity fade in favor of compassionate solidarity, focusing on the vulnerability of migrant groups, and exchange-based anti-solidarity, focusing on the lack of (economic) contribution. Our study highlights the interplay of historical events, socio-economic needs, and political ideologies in shaping migration discourse and social cohesion. We also show that powerful LLMs, if carefully prompted, can be cost-effective alternatives t

This study examines the evolution of a grassroots, volunteer-driven peer-to-peer (P2P) educational initiative from an emergency response to the 2023 Türkiye earthquake into a sustainable ecosystem that operated for over two years and supported 300+ middle-school learners with 40+ volunteer tutors. Employing an interpretive case study approach, we triangulated data from participant observation, focus groups, questionnaires, and collaborative visioning workshops to investigate the socio-technical dynamics enabling long-term resilience in a fully online, nonreciprocal far-peer tutoring setting. Our findings reveal that while age proximity fosters trust and open communication, it also poses challenges for tutors who must balance peer rapport with instructional authority. Volunteer engagement is driven primarily by intrinsic motives - educational impact and community belonging - while optional micro-earning is envisioned as a practical enabler for long-term sustainability. Tutees report significant gains in confidence, self-expression, and accelerated comprehension, attributing these outcomes to personalized, interactive sessions within a "family-like" safe space that combines academic

Migration has been a core topic in German political debate, from the postwar displacement of millions of expellees to labor migration and recent refugee movements. Studying political speech across such wide-ranging phenomena in depth has traditionally required extensive manual annotation, limiting analysis to small subsets of the data. Large language models (LLMs) offer a potential way to overcome this constraint. Using a theory-driven annotation scheme, we examine how well LLMs annotate subtypes of solidarity and anti-solidarity in German parliamentary debates and whether the resulting labels support valid downstream inference. We first provide a comprehensive evaluation of multiple LLMs, analyzing the effects of model size, prompting strategies, fine-tuning, historical versus contemporary data, and systematic error patterns. We find that the strongest models, especially GPT-5 and gpt-oss-120B, achieve human-level agreement on this task, although their errors remain systematic and bias downstream results. To address this issue, we combine soft-label model outputs with Design-based Supervised Learning (DSL) to reduce bias in long-term trend estimates. Beyond the methodological eval

South Korea's City P illustrates how lofty goals of digital solidarity can falter when challenged by local governance realities. Drawing on Hansmann's ownership theory, collaborative governance concepts, and platform cooperativism, we conducted a qualitative case study involving policy documents, independent assessments, and 11 in-depth interviews with residents, officials, and technology developers. Findings reveal a marked disconnect between the initiative's stated emphasis on community co-ownership and the actual power dynamics that largely favored government agencies and external firms. Although blockchain and integrated digital tools were meant to enhance transparency and inclusivity, stakeholders--especially elderly residents--experienced confusion and mistrust. We argue that genuine collaboration in digital solidarity economies requires not only robust technical designs but also culturally resonant ownership structures, substantive inclusion of local voices, and transparent governance mechanisms. The City P case underscores the necessity of addressing heterogeneous digital capacities, aligning funding and incentives with grassroots empowerment, and mitigating performative pa

As panoptical, AI-driven surveillance becomes a norm, everyone is impacted. In a reality where all people fall victim to these technologies, establishing links and solidarity is essential to fighting back. Two groups facing rising and targeted surveillance are workers and individuals impacted by the carceral system. Through preliminary data collection from a worker-surveillance lens, our findings reveal several cases of these surveillance infrastructures intersecting. Continuation of our work will involve collecting cases from a carceral-centered lens. Driven by a community-facing analysis of the overlap in the AI-driven surveillance experienced by workers and individuals impacted by the carceral system, we will facilitate discussions with restorative justice activists around cultivating solidarity and empowerment focused on the interconnected nature of workplace and carceral surveillance technologies.

This paper presents a comparative analysis of community unionism (CU) in two distinct historical and organizational contexts: the National Boot and Shoe Union (B\&S) in the 1920s and Unite Community in the 2010s--2020s. Using BERTopic for thematic modeling and cTF-IDF weighting, alongside word frequency analysis, the study examines the extent to which each union's discourse aligns with key features of CU -- such as coalition-building, grassroots engagement, and action beyond the workplace. The results reveal significant differences in thematic focus and discursive coherence. While Unite Community demonstrates stronger alignment with outward-facing, social justice-oriented themes, the B\&S corpus emphasizes internal administration, industrial relations, and member services -- reflecting a more traditional, servicing-oriented union model. The analysis also highlights methodological insights, demonstrating how modern NLP techniques can enhance the study of historical labor archives. Ultimately, the findings suggest that while both unions engage with community-related themes, their underlying models of engagement diverge significantly, challenging assumptions about the continui

Online communities are important spaces for members of marginalized groups to organize and support one another. To better understand the experiences of fat people -- a group whose marginalization often goes unrecognized -- in online communities, we conducted 12 semi-structured interviews with fat people. Our participants leveraged online communities to engage in consciousness raising around fat identity, learning to locate "the problem of being fat" not within themselves or their own bodies but rather in the oppressive design of the society around them. Participants were then able to use these communities to mitigate everyday experiences of anti-fatness, such as navigating hostile healthcare systems. However, to access these benefits, our participants had to navigate myriad sociotechnical harms, ranging from harassment to discriminatory algorithms. In light of these findings, we suggest that researchers and designers of online communities support selective fat visibility, consider fat people in the design of content moderation systems, and investigate algorithmic discrimination toward fat people. More broadly, we call on researchers and designers to contend with the social and mate

Gibbs samplers are preeminent Markov chain Monte Carlo algorithms used in computational physics and statistical computing. Yet, their most fundamental properties, such as relations between convergence characteristics of their various versions, are not well understood. In this paper we prove the solidarity of their spectral gaps: if any of the random scan or $d!$ deterministic scans has a~spectral gap then all of them have. Our methods rely on geometric interpretation of the Gibbs samplers as alternating projection algorithms and analysis of the rate of convergence in the von Neumann--Halperin method of cyclic alternating projections. In addition, we provide a quantitative result: if the spectral gap of the random scan Gibbs sampler scales polynomially with dimension, so does the spectral gap of any of the deterministic scans.

Despite their drawbacks and unintended consequences, social media networks have recently emerged as a crucial resource for individuals in distress, particularly during times of crisis. These platforms serve as a means to seek assistance and support, share reliable information, and appeal for action and solidarity. In this paper, we examine the online networks of support during the Russia-Ukraine conflict by analyzing four major social media networks: Twitter, Facebook, Instagram, and YouTube. Using a large dataset of 68 million posts, we explore the temporal patterns and interconnectedness between these platforms and online support websites. Our analysis highlights the prevalence of crowdsourcing and crowdfunding websites as the two main support platforms to mobilize resources and solicit donations, revealing their purpose and contents, and investigating different support-seeking and -receiving practices. Overall, our study underscores the potential of social media in facilitating online support in distressed environments through grassroots mobilization, contributing to the growing body of research on the positive impact of online platforms in promoting social good and protecting v

Agents may form coalitions. Each coalition shares its endowment among its agents by applying a sharing rule. The sharing rule induces a coalition formation problem by assuming that agents rank coalitions according to the allocation they obtain in the corresponding sharing problem. We characterize the sharing rules that induce a class of stable coalition formation problems as those that satisfy a natural axiom that formalizes the principle of solidarity. Thus, solidarity becomes a sufficient condition to achieve stability.

We introduce the well-established social scientific concept of social solidarity and its contestation, anti-solidarity, as a new problem setting to supervised machine learning in NLP to assess how European solidarity discourses changed before and after the COVID-19 outbreak was declared a global pandemic. To this end, we annotate 2.3k English and German tweets for (anti-)solidarity expressions, utilizing multiple human annotators and two annotation approaches (experts vs.\ crowds). We use these annotations to train a BERT model with multiple data augmentation strategies. Our augmented BERT model that combines both expert and crowd annotations outperforms the baseline BERT classifier trained with expert annotations only by over 25 points, from 58\% macro-F1 to almost 85\%. We use this high-quality model to automatically label over 270k tweets between September 2019 and December 2020. We then assess the automatically labeled data for how statements related to European (anti-)solidarity discourses developed over time and in relation to one another, before and during the COVID-19 crisis. Our results show that solidarity became increasingly salient and contested during the crisis. While