The two-star model: exact solution in the sparse regime and condensation transition

The $2$-star model is the simplest exponential random graph model that displays complex behavior, such as degeneracy and phase transition. Despite its importance, this model has been solved only in the regime of dense connectivity. In this work we solve the model in the finite connectivity regime, far more prevalent in real world networks. We show that the model undergoes a condensation transition from a liquid to a condensate phase along the critical line corresponding, in the ensemble parameters space, to the Erd\"os-R\'enyi graphs. In the fluid phase the model can produce graphs with a narrow degree statistics, ranging from regular to Erd\"os-R\'enyi graphs, while in the condensed phase, the "excess" degree heterogeneity condenses on a single site with degree $\sim\sqrt{N}$. This shows the unsuitability of the two-star model, in its standard definition, to produce arbitrary finitely connected graphs with degree heterogeneity higher than Erd\"os-R\'enyi graphs and suggests that non-pathological variants of this model may be attained by softening the global constraint on the two-stars, while keeping the number of links hardly constrained.Comment: 20 pages, 3 figure

The role of idiotypic interactions in the adaptive immune system: a belief-propagation approach

In this work we use belief-propagation techniques to study the equilibrium behaviour of a minimal model for the immune system comprising interacting T and B clones. We investigate the effect of the so-called idiotypic interactions among complementary B clones on the system's activation. Our result shows that B-B interactions increase the system's resilience to noise, making clonal activation more stable, while increasing the cross-talk between different clones. We derive analytically the noise level at which a B clone gets activated, in the absence of cross-talk, and find that this increases with the strength of idiotypic interactions and with the number of T cells signalling the B clone. We also derive, analytically and numerically, via population dynamics, the critical line where clonal cross-talk arises. Our approach allows us to derive the B clone size distribution, which can be experimentally measured and gives important information about the adaptive immune system response to antigens and vaccination.Comment: 37 pages, 18 figure

Supersymmetric quenched complexity in the Sherrington-Kirkpatrick model

By using the BRST supersymmetry we compute the quenched complexity of the TAP states in the SK model. We prove that the BRST complexity is equal to the Legendre transform of the static free energy with respect to the largest replica symmetry breaking point of its overlap matrix

Extensive load in multitasking associative networks

We use belief-propagation techniques to study the equilibrium behavior of a bipartite spin-glass, with interactions between two sets of $N$ and $P = \alpha N$ spins. Each spin has a finite degree, i.e.\ number of interaction partners in the opposite set; an equivalent view is then of a system of $N$ neurons storing $P$ diluted patterns. We show that in a large part of the parameter space of noise, dilution and storage load, delimited by a critical surface, the network behaves as an extensive parallel processor, retrieving all $P$ patterns {\it in parallel} without falling into spurious states due to pattern cross-talk and typical of the structural glassiness built into the network. Our approach allows us to consider effects beyond those studied in replica theory so far, including pattern asymmetry and heterogeneous dilution. Parallel extensive retrieval is more robust for homogeneous degree distributions, and is not disrupted by biases in the distributions of the spin-glass links

Recent advances in understanding and managing diverticulitis.

In the past few decades, the increasing socioeconomic burden of acute diverticulitis (AD) has become evident, and with the growth of the population age, this significant economic impact will likely continue to rise. Furthermore, recent evidence showed an increased rate of hospital admissions especially evident among women and younger individuals. The natural history and pathophysiology of this clinical condition is still to be fully defined, and efforts continue to be made in the identification of risk factors and the establishment of relative preventive strategies. The actual therapeutic strategies aimed to modulate gut microbiota, such as rifaximin or probiotics, or to reduce mucosal inflammation, such as mesalazine, present a relatively poor efficacy for both the prevention of the first AD episode (primary prevention) and its recurrence (secondary prevention). In the last few years, the main goal achieved has been in the management of AD in that uncomplicated AD can, to a larger extent, be managed in an outpatient setting with no or little supportive therapy, a strategy that will certainly impact on the health costs of this disease. The problem of AD recurrence remains a topic of debate. The aim of this review is to present updated evidence on AD epidemiology and relative open clinical questions and to analyze in detail predisposing and protective factors with an attempt to integrate their possible modes of action into the several pathogenic mechanisms that have been suggested to contribute to this multifactorial disease. A unifying hypothesis dealing with the colonic luminal and extra-luminal microenvironments separately is provided. Finally, evidence-based changes in therapeutic management will be summarized. Because of an ascertained multifactorial pathogenesis of uncomplicated and complicated AD, it is probable that a single ‘causa prima’ will not be identifiable, and a better stratification of patients could allow one to pursue tailored therapeutic algorithm strategies

Price of information in games of chance

We consider a game where $N$ players bet on the outcome of a biased coin and share the entry fees pot if successful. We assume that one player holds information about past outcomes of the game, which they may either use exclusively to improve their betting strategy or offer to sell to another player. We determine analytically the optimal price curves for the data seller and the prospective buyer. We find a sharp transition in the number $N$ of players that separates a phase where the transaction is always profitable for the seller from one where it may not be. In both phases, different regimes are possible, depending on the "quality" of information being put up for sale: we observe symbiotic regimes, where both parties collude effectively to rig the game in their favor, competitive regimes, where the transaction is unappealing to the data holder as it overly favors a competitor for scarce resources, and even prey-predator regimes, where the data holder is eager to give away bad-quality data to undercut a competitor. Our framework can be generalized to more complex settings and constitutes a flexible tool to address the rich and timely problem of pricing information in games of chance.Comment: 18 pages, 6 figure

Retrieving Infinite Numbers of Patterns in a Spin-Glass Model of Immune Networks

The similarity between neural and immune networks has been known for decades, but so far we did not understand the mechanism that allows the immune system, unlike associative neural networks, to recall and execute a large number of memorized defense strategies {\em in parallel}. The explanation turns out to lie in the network topology. Neurons interact typically with a large number of other neurons, whereas interactions among lymphocytes in immune networks are very specific, and described by graphs with finite connectivity. In this paper we use replica techniques to solve a statistical mechanical immune network model with `coordinator branches' (T-cells) and `effector branches' (B-cells), and show how the finite connectivity enables the system to manage an extensive number of immune clones simultaneously, even above the percolation threshold. The system exhibits only weak ergodicity breaking, so that both multiple antigen defense and homeostasis can be accomplished.Comment: Editor's Choice 201