A random journey through the math of gambling

The laws of chance are often subtle and deceptive. This is why games of chance work. People are convinced that they obey seemingly intuitive laws, while the underlying mathematical structure reveals a different and more complex reality. This article is a brief and rigorous journey through the implications that the mathematical laws governing stochastic processes have on gambling. It addresses a specific process, the random walk, and analyze some instances of fair and unfair games by highlighting the fallacy of many of our intuitions and beliefs. The paper gradually moves from the analysis of the random walk properties to a comprehensive description of the ruin problem. The introduction of the idea of transient and persistent states concludes the discussion. Much emphasis is placed on concrete examples and on the numerical values, in particular of the involved probabilities, and the interpretation of the results is always more central than the demonstrative technical details, which are nevertheless available to the reader

Taxonomy of Cohesion Coefficients for Weighted and Directed Multilayer Networks

Clustering and closure coefficients are among the most widely applied indicators in the description of the topological structure of a network. Many distinct definitions have been proposed over time, particularly in the case of weighted networks, where the choice of the weight attributed to the triangles is a crucial aspect. In the present work, in the framework of weighted directed multilayer networks, we extend the classical clustering and closure coefficients through the introduction of the clumping coefficient, which generalizes them to incomplete triangles of any type. We then organize the class of these coefficients in a systematic taxonomy in the more general context of weighted directed multilayer networks. Such cohesion coefficients have also been adapted to the different scales that characterize a multilayer network, in order to grasp their structure from different perspectives. We also show how the tensor formalism allows incorporating the new definitions, as well as all those existing in the literature, in a single unified writing, in such a way that a suitable choice of the involved adjacency tensors allows obtaining each of them. Finally, through some applications to simulated networks, we show the effectiveness of the proposed coefficients in capturing different peculiarities of the network structure on different scales

A Novel Self-Adaptive SIS Model Based on the Mutual Interaction between a Graph and its Line Graph

We propose a new paradigm to design a network-based self-adaptive epidemic model that relies on the interplay between the network and its line graph. We implement this proposal on a Susceptible-Infected-Susceptible model in which both nodes and edges are considered susceptible and their respective probabilities of being infected result in a real-time re-modulation of the weights of both the graph and its line graph. The new model can be considered as an appropriate perturbation of the standard Susceptible-Infected-Susceptible model, and the coupling between the graph and its line graph is interpreted as a reinforcement factor that fosters diffusion through a continuous adjustment of the parameters involved. We study the existence and stability conditions of the endemic and disease-free states for general network topologies. Moreover, we introduce, through the asymptotic values in the endemic steady states, a new type of eigenvector centrality where the score of a node depends on both the neighboring nodes and the edges connected to it. We also investigate the properties of this new model on some specific synthetic graphs, such as cycle, regular, and star graphs. Finally, we perform a series of numerical simulations and prove their effectiveness in capturing some empirical evidence on behavioral adoption mechanisms

Multi-criteria community detection in International Trade Network

Understanding the community structure has great importance for economic analysis. Communities are characterized by properties different from those of both the individual node and the whole network and they affect various processes on the network. We combine community detection with specific topological indicators. As a result, a new weighted network is constructed by the original one, in which weights are determined taking into account all the topological indicators in a multi-criteria approach. We introduce a new algorithm to detect communities by solving the NP-hard CP-problem

Local balance of signed networks: Definition and application to reveal historical events in international relations

Alliances and conflicts represent important features of complex systems like international relations. Such relations create a time-evolving signed network, where each node contributes in a unique manner to the global balance of the system. Therefore, a local index mathematically quantifying such a property becomes valuable to understand complex signed networks . In this work, we introduce a local balance index for signed networks. We analyze its mathematical foundations and unique structural properties, differentiating it from existing local vertex invariants. We also establish a novel methodology linking changes in a nation's local balance to historical events. By scrutinizing the time series of local balance for countries between 1816 and 2014, we detect and categorize major historic events based on balance fluctuations. This approach harmonizes quantitative and qualitative analyses, and combined with the theory of "balance of power" is able to build up a new mixed approach to history based on network theory.Comment: 33 pages, 15 figure

Inspiratory muscle training and its effect on indices of physiological and perceived stress during incremental walking exercise in normobaric hypoxia

This study evaluated the effects of inspiratory muscle training (IMT) on inspiratory muscle fatigue (IMF) and physiological and perceptual responses during trekking-specific exercise. An 8-week IMT program was completed by 21 males (age 32.4 ± 9.61 years, VO2peak 58.8 ± 6.75 mL/kg/min) randomised within matched pairs to either the IMT group (n = 11) or the placebo group [(P), n = 9]. Twice daily, participants completed 30 (IMT) or 60 (P) inspiratory efforts using a Powerbreathe initially set at a resistance of 50% (IMT) or used at 15% (P) of maximal inspiratory pressure (MIP) throughout. A loaded (12.5 kg) 39-minute incremental walking protocol (3–5 km/hour and 1–15% gradient) was completed in normobaric hypoxia (PIO2 = 110 mmHg, 3000 m) before and after training. MIP increased from 164 to 188 cmH2O (18%) and from 161 to 171 cmH2O (6%) in the IMT and P groups (P = 0.02). The 95% CI for IMT showed a significant improvement in MIP (5.21±43.33 cmH2O), but not for P. IMF during exercise (MIP) was*5%, showing no training effect for either IMT or P (P = 0.23). Rating of perceived exertion (RPE) was consistently reduced (*1) throughout exercise following training for IMT, but not for P (P = 0.03). The mean blood lactate concentration during exercise was significantly reduced by 0.26 and 0.15 mmol/L in IMT and P (P = 0.00), with no differences between groups (P = 0.34). Rating of dyspnoea during exercise decreased (*0.4) following IMT but increased (*0.3) following P (P = 0.01). IMT may attenuate the increased physiological and perceived exercise stress experienced during normobaric hypoxia, which may benefit moderate altitude expedition