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arXiv2026

Convergence and acceleration of a nonlinear fixed-point iteration for computing the Fitness Centrality of general graphs

We establish the global convergence of the (non-homogeneous) Fitness Centrality algorithm for general graphs, deriving an explicit convergence bound for the corresponding fixed-point iteration. Furthermore, we show how the convergence can be dramatically improved by Anderson acceleration and by switching to Newton's method once a sufficiently good approximation to the fixed point has been found. The efficacy of this strategy is illustrated by numerical experiments on different types of graphs.

math.NA↗

arXiv2026

Whose Judgments Count? Representation Gaps in Crowdsourced Content Moderation Produce Unequal Protection from Perceived Toxicity

Content moderation is a central form of digital governance, yet people disagree over what content should be removed from shared online spaces. While platforms aggregate human judgments to build moderation systems, it remains unclear how this process shapes which users are protected from content they perceive as toxic. We address this gap by combining large-scale judgment data with counterfactual simulations that trace how the demographic composition of moderator pools shapes the distribution of protection across users. Applying this framework to removal judgments from 16,221 U.S. respondents evaluating 102,463 comments from Twitter, Reddit, and 4chan, we find demographic heterogeneities in moderation demand. We further reveal a consistent pattern of in-group protection: reductions in perceived toxicity accrue disproportionately to users who share the demographic identities of the moderator pool. Crucially, moderator pools that mirror the demographic composition of self-identified moderators on Prolific widen these disparities relative to a nationally representative baseline, while even fully representative pools fail to ensure equal protection: Black and LGB users remain underprotected unless they are represented well beyond their population share. These findings show that unequal protection from perceived toxicity can arise structurally from the aggregation of stratified removal standards, making the demographic composition of moderation inputs a key determinant of who is protected online.

cs.SI↗

arXiv2026

Centrality Measures in Temporal Networks: A Critical and Comparative Survey

Temporal networks offer a suitable representation for complex systems in which interactions vary over time, such as communication, transportation, and social networks. Identifying influential nodes in such networks is more challenging than in static graphs because node importance depends not only on network structure but also on the timing and ordering of interactions. Although many temporal centrality measures have been proposed, the literature remains fragmented, with limited consensus on their comparative performance and applicability. This paper presents a critical and comparative survey of centrality measures in temporal networks. We review both temporal extensions of classical centrality metrics and measures specifically designed for temporal graphs, and propose a functional taxonomy that categorizes existing approaches according to the primary mechanism through which influence is quantified in temporal networks. The proposed taxonomy organizes temporal centrality measures into interaction-based, path-based, walk-based, spectral-based and robustness-based categories, providing a unified perspective on their underlying principles. In addition, we provide comparative insights to help select appropriate temporal centrality measures under different network characteristics and application settings. To complement the survey, we conduct experiments on multiple real-world temporal network datasets. The measures are evaluated through influence spreading experiments using epidemic diffusion models, ranking consistency analysis based on Kendall's rank correlation, and runtime complexity analysis to assess computational efficiency and scalability. Finally, we highlight key open challenges and future research directions, including scalability for million-sized networks and the need for standardized evaluation frameworks.

cs.SI↗

arXiv2026

Core-periphery identification in massive networks

Modern networks can be huge with millions or even billions of nodes and edges. Thus, algorithms must be capable of scaling to such large networks in order to be practically useful. In this work, we are interested in developing an algorithm to identify core-periphery structure in massive networks. Core-periphery structure is a meso-scale feature where nodes are grouped into a densely connected core or sparsely connected periphery. To identify such structures in large networks, we propose a divide-and-conquer algorithm. The key feature of our algorithm is leveraging the edge list representation of the network, instead of the adjacency matrix, as it tends to be faster and makes a more efficient use of memory. We apply the proposed algorithm to synthetic and real-world data, notably demonstrating its performance on a real-world network with almost 14 million edges without loading the entire network into memory.

cs.SI↗

arXiv2026

On graphs with modularity zero or near-zero

It is known that complete graphs and complete multipartite graphs have modularity zero. We show that the least number of edges we may delete from the complete graph $K_n$ to obtain a graph with non-zero modularity is $\lfloor n/2\rfloor +1$. Similarly we determine the least number of edges we may delete from or add to a complete bipartite graph to reach non-zero modularity. We give some corresponding results for complete multipartite graphs, and a short proof that complete multipartite graphs have modularity zero. We also analyse the modularity of very dense random graphs, and in particular we find that there is a transition to modularity zero when the average degree of the complementary graph drops below 1. Finally we consider some natural variants of the definition of modularity; and investigate which graphs have corresponding modularity value 0, and the least number of edges we may delete from the complete graph $K_n$ to obtain a graph with non-zero modularity.

math.CO↗

arXiv2026

Walk based Laplacians for Modeling Diffusion on Complex Networks

We develop a novel framework for modeling diffusion on complex networks by constructing Laplacian-like operators based on walks around a graph. Our approach introduces a parametric family of walk-based Laplacians that naturally incorporate memory effects by excluding or downweighting backtracking trajectories, where walkers immediately revisit nodes. The framework includes: (i) walk-based Laplacians that count all traversals in the network; (ii) nonbacktracking variants that eliminate immediate reversals; and (iii) backtrack-downweighted variants that provide a continuous interpolation between these two regimes. We establish that these operators extend the definition of the standard Laplacian and also preserve some of its properties. We present efficient algorithms using Krylov subspace methods for computing them, ensuring applicability of our proposed framework to large networks. Extensive numerical experiments on real-world networks validate the modeling flexibility of our approach and demonstrate the computational efficiency of the proposed algorithms, including GPU acceleration.

cs.SI↗

arXiv2026

Adaptive Epidemic Dynamics on Hypergraphs with Group-Level Immunization and Rewiring

Understanding how higher-order social structures shape epidemic spreading requires models that couple group interactions with adaptive behavior. We introduce an adaptive simplicial susceptible-infected-susceptible (s-SIS) model on d-uniform hypergraphs, where both node states and hyperedge activity co-evolve in response to local infection pressure. Hyperedges represent group interactions of fixed size and dynamically reduce their activity through a feedback mechanism in highly infected environments. Within this framework, we design two classes of hyperedge-level interventions: (i) risk-driven immunization, combining spontaneous, activity-based isolation with targeted deactivation guided by hyperedge infection pressure, and (ii) structural rewiring, which reconstructs group structures either randomly or via degree-preferential attachment. By extending the microscopic Markov chain approximation to higher-order interactions, we derive analytical conditions for the existence and stability of both endemic and disease-free stationary states. Our analysis shows that adaptive hyperedge feedback can induce discontinuous phase transitions, nonlinear epidemic thresholds, and bistable regimes in which sufficiently high initial prevalence drives the system to a disease-free equilibrium. Extensive Monte Carlo simulations support the theory and confirm that targeted immunization and degree-preferential rewiring substantially suppress epidemic prevalence, outperforming random strategies. These results demonstrate that higher-order interactions and adaptive group-level responses fundamentally reshape epidemic bifurcations and suggest principles for designing effective intervention policies in complex social systems.

physics.soc-ph↗

arXiv2026

Structural Bias Beyond Homophily: A Study of Fairness in Link Prediction

Graph link prediction (LP) plays a critical role in socially impactful applications such as job recommendation and friendship formation, making fairness a critical concern in this task. While many fairness-aware methods manipulate graph structures to mitigate prediction disparities, the topological biases inherent to social graphs remain poorly understood and are consistently conflated with homophily alone. In this work, we study the relationship between structural biases and fairness outcomes in LP. To this end, we formalize a taxonomy of topological bias measures and introduce a graph generation method producing a diverse corpus of synthetic graphs with controlled structural properties. Using this corpus, we show empirically that fairness outcomes are strongly correlated with graph topology, and that current fairness-aware methods remain sensitive to structural biases beyond homophily. These findings highlight the need for structurally grounded evaluations in fair graph learning.

cs.LG↗

arXiv2025

Opinion Dynamics Models for Sentiment Evolution in Weibo Blogs

Online social media platforms enable influencers to distribute content and quickly capture audience reactions, significantly shaping their promotional strategies and advertising agreements. Understanding how sentiment dynamics and emotional contagion unfold among followers is vital for influencers and marketers, as these processes shape engagement, brand perception, and purchasing behavior. While sentiment analysis tools effectively track sentiment fluctuations, dynamical models explaining their evolution remain limited, often neglecting network structures and interactions both among blogs and between their topic-focused follower groups. In this study, we tracked influential tech-focused Weibo bloggers over six months, quantifying follower sentiment from text-mined feedback. By treating each blogger's audience as a single "macro-agent", we find that sentiment trajectories follow the principle of iterative averaging -- a foundational mechanism in many dynamical models of opinion formation, a theoretical framework at the intersection of social network analysis and dynamical systems theory. The sentiment evolution aligns closely with opinion-dynamics models, particularly modified versions of the classical French-DeGroot model that incorporate delayed perception and distinguish between expressed and private opinions. The inferred influence structures reveal interdependencies among blogs that may arise from homophily, whereby emotionally similar users subscribe to the same blogs and collectively shape the shared sentiment expressed within these communities.

cs.SI↗

arXiv2025

YTLive: A Dataset of Real-World YouTube Live Streaming Sessions

Live streaming plays a major role in today's digital platforms, supporting entertainment, education, social media, etc. However, research in this field is limited by the lack of large, publicly available datasets that capture real-time viewer behavior at scale. To address this gap, we introduce YTLive, a public dataset focused on YouTube Live. Collected through the YouTube Researcher Program over May and June 2024, YTLive includes more than 507000 records from 12156 live streams, tracking concurrent viewer counts at five-minute intervals along with precise broadcast durations. We describe the dataset design and collection process and present an initial analysis of temporal viewing patterns. Results show that viewer counts are higher and more stable on weekends, especially during afternoon hours. Shorter streams attract larger and more consistent audiences, while longer streams tend to grow slowly and exhibit greater variability. These insights have direct implications for adaptive streaming, resource allocation, and Quality of Experience (QoE) modeling. YTLive offers a timely, open resource to support reproducible research and system-level innovation in live streaming. The dataset is publicly available at github.

cs.MM↗

arXiv2025

Symmetry-driven embedding of networks in hyperbolic space

Hyperbolic models are known to produce networks with properties observed empirically in most network datasets, including heavy-tailed degree distribution, high clustering, and hierarchical structures. As a result, several embeddings algorithms have been proposed to invert these models and assign hyperbolic coordinates to network data. Current algorithms for finding these coordinates, however, do not quantify uncertainty in the inferred coordinates. We present BIGUE, a Markov chain Monte Carlo (MCMC) algorithm that samples the posterior distribution of a Bayesian hyperbolic random graph model. We show that the samples are consistent with current algorithms while providing added credible intervals for the coordinates and all network properties. We also show that some networks admit two or more plausible embeddings, a feature that an optimization algorithm can easily overlook.

stat.CO↗

arXiv2025

Security Science (SecSci), Basic Concepts and Mathematical Foundations

This textbook compiles the lecture notes from security courses taught at Oxford in the 2000s, at Royal Holloway in the 2010s, and currently in Hawaii. The early chapters are suitable for a first course in security. The middle chapters have been used in advanced courses. Towards the end there are also some research problems.

cs.CR↗

arXiv2025

Join the Chat: How Curiosity Sparks Participation in Telegram Groups

This study delves into the mechanisms that spark user curiosity driving active engagement within public Telegram groups. By analyzing approximately 6 million messages from 29,196 users across 409 groups, we identify and quantify the key factors that stimulate users to actively participate (i.e., send messages) in group discussions. These factors include social influence, novelty, complexity, uncertainty, and conflict, all measured through metrics derived from message sequences and user participation over time. After clustering the messages, we apply explainability techniques to assign meaningful labels to the clusters. This approach uncovers macro categories representing distinct curiosity stimulation profiles, each characterized by a unique combination of various stimuli. Social influence from peers and influencers drives engagement for some users, while for others, rare media types or a diverse range of senders and media sparks curiosity. Analyzing patterns, we found that user curiosity stimuli are mostly stable, but, as the time between the initial message increases, curiosity occasionally shifts. A graph-based analysis of influence networks reveals that users motivated by direct social influence tend to occupy more peripheral positions, while those who are not stimulated by any specific factors are often more central, potentially acting as initiators and conversation catalysts. These findings contribute to understanding information dissemination and spread processes on social media networks, potentially contributing to more effective communication strategies.

cs.SI↗

arXiv2022

Hypergraph reconstruction from noisy pairwise observations

The network reconstruction task aims to estimate a complex system's structure from various data sources such as time series, snapshots, or interaction counts. Recent work has examined this problem in networks whose relationships involve precisely two entities-the pairwise case. Here we investigate the general problem of reconstructing a network in which higher-order interactions are also present. We study a minimal example of this problem, focusing on the case of hypergraphs with interactions between pairs and triplets of vertices, measured imperfectly and indirectly. We derive a Metropolis-Hastings-within-Gibbs algorithm for this model and use the algorithms to highlight the unique challenges that come with estimating higher-order models. We show that this approach tends to reconstruct empirical and synthetic networks more accurately than an equivalent graph model without higher-order interactions.

cs.SI↗

arXiv2021

Detecting the ultra low dimensionality of real networks

Reducing dimension redundancy to find simplifying patterns in high-dimensional datasets and complex networks has become a major endeavor in many scientific fields. However, detecting the dimensionality of their latent space is challenging but necessary to generate efficient embeddings to be used in a multitude of downstream tasks. Here, we propose a method to infer the dimensionality of networks without the need for any a priori spatial embedding. Due to the ability of hyperbolic geometry to capture the complex connectivity of real networks, we detect ultra low dimensionality far below values reported using other approaches. We applied our method to real networks from different domains and found unexpected regularities, including: tissue-specific biomolecular networks being extremely low dimensional; brain connectomes being close to the three dimensions of their anatomical embedding; and social networks and the Internet requiring slightly higher dimensionality. Beyond paving the way towards an ultra efficient dimensional reduction, our findings help address fundamental issues that hinge on dimensionality, such as universality in critical behavior.

physics.soc-ph↗

arXiv2018

Foursquare to The Rescue: Predicting Ambulance Calls Across Geographies

Understanding how ambulance incidents are spatially distributed can shed light to the epidemiological dynamics of geographic areas and inform healthcare policy design. Here we analyze a longitudinal dataset of more than four million ambulance calls across a region of twelve million residents in the North West of England. With the aim to explain geographic variations in ambulance call frequencies, we employ a wide range of data layers including open government datasets describing population demographics and socio-economic characteristics, as well as geographic activity in online services such as Foursquare. Working at a fine level of spatial granularity we demonstrate that daytime population levels and the deprivation status of an area are the most important variables when it comes to predicting the volume of ambulance calls at an area. Foursquare check-ins on the other hand complement these government sourced indicators, offering a novel view to population nightlife and commercial activity locally. We demonstrate how check-in activity can provide an edge when predicting certain types of emergency incidents in a multi-variate regression model.

physics.soc-ph↗

arXiv2017

Semantic homophily in online communication: evidence from Twitter

People are observed to assortatively connect on a set of traits. This phenomenon, termed assortative mixing or sometimes homophily, can be quantified through assortativity coefficient in social networks. Uncovering the exact causes of strong assortative mixing found in social networks has been a research challenge. Among the main suggested causes from sociology are the tendency of similar individuals to connect (often itself referred as homophily) and the social influence among already connected individuals. An important question to researchers and in practice can be tackled, as we present here: understanding the exact mechanisms of interplay between these tendencies and the underlying social network structure. Namely, in addition to the mentioned assortativity coefficient, there are several other static and temporal network properties and substructures that can be linked to the tendencies of homophily and social influence in the social network and we herein investigate those. Concretely, we tackle a computer-mediated \textit{communication network} (based on Twitter mentions) and a particular type of assortative mixing that can be inferred from the semantic features of communication content that we term \textit{semantic homophily}. Our work, to the best of our knowledge, is the first to offer an in-depth analysis on semantic homophily in a communication network and the interplay between them. We quantify diverse levels of semantic homophily, identify the semantic aspects that are the drivers of observed homophily, show insights in its temporal evolution and finally, we present its intricate interplay with the communication network on Twitter. By analyzing these mechanisms we increase understanding on what are the semantic aspects that shape and how they shape the human computer-mediated communication.

physics.soc-ph↗

arXiv2015

Everyday the Same Picture: Popularity and Content Diversity

Facebook is flooded by diverse and heterogeneous content, from kittens up to music and news, passing through satirical and funny stories. Each piece of that corpus reflects the heterogeneity of the underlying social background. In the Italian Facebook we have found an interesting case: a page having more than $40K$ followers that every day posts the same picture of a popular Italian singer. In this work, we use such a page as a control to study and model the relationship between content heterogeneity on popularity. In particular, we use that page for a comparative analysis of information consumption patterns with respect to pages posting science and conspiracy news. In total, we analyze about $2M$ likes and $190K$ comments, made by approximately $340K$ and $65K$ users, respectively. We conclude the paper by introducing a model mimicking users selection preferences accounting for the heterogeneity of contents.

cs.SI↗