Renormalization group theory of percolation on pseudofractal simplicial and cell complexes

(11 pages,4 figures)(11 pages,4 figures)(11 pages,4 figures

Topological percolation on hyperbolic simplicial complexes

Simplicial complexes are increasingly used to understand the topology of complex systems as different as brain networks and social interactions. It is therefore of special interest to extend the study of percolation to simplicial complexes. Here we propose a topological theory of percolation for discrete hyperbolic simplicial complexes. Specifically we consider hyperbolic manifolds in dimension $d=2$ and $d=3$ formed by simplicial complexes, and we investigate their percolation properties in the presence of topological damage, i.e., when nodes, links, triangles or tetrahedra are randomly removed. We show that in $d=2$ simplicial complexes there are four topological percolation problems and in $d=3$, there are six. We demonstrate the presence of two percolation phase transitions characteristic of hyperbolic spaces for the different variants of topological percolation. While most of the known results on percolation in hyperbolic manifolds are in $d=2$, here we uncover the rich critical behavior of $d=3$ hyperbolic manifolds, and show that triangle percolation displays a Berezinskii-Kosterlitz-Thouless (BKT) transition. Finally we provide evidence that topological percolation can display a critical behavior that is unexpected if only node and link percolation are considered.Comment: (17 pages, 9 figures

Renormalization group for link percolation on planar hyperbolic manifolds

Network geometry is currently a topic of growing scientific interest as it opens the possibility to explore and interpret the interplay between structure and dynamics of complex networks using geometrical arguments. However the field is still in its infancy. In this work we investigate the role of network geometry in determining the nature of the percolation transition in planar hyperbolic manifolds. In Ref. [1], S. Boettcher, V. Singh, R. M. Ziff have shown that a special type of two-dimensional hyperbolic manifolds, the Farey graphs, display a discontinuous transition for ordinary link percolation. Here using the renormalization group we investigate the critical properties of link percolation on a wider class of two-dimensional hyperbolic deterministic and random manifolds constituting the skeletons of two-dimensional cell complexes. These hyperbolic manifolds are built iteratively by subsequently gluing $m$-polygons to single edges. We show that when the size $m$ of the polygons is drawn from a distribution $q_m$ with asymptotic power-law scaling $q_m\simeq Cm^{-\gamma}$ for $m\gg1$, different universality classes can be observed for different values of the power-law exponent $\gamma$. Interestingly the percolation transition is hybrid for $\gamma\in (3,4)$ and becomes continuous for $\gamma \in (2,3]$Comment: (19 pages, 9 figures

Impact of Single Links in Competitive Percolation -- How complex networks grow under competition

How a complex network is connected crucially impacts its dynamics and function. Percolation, the transition to extensive connectedness upon gradual addition of links, was long believed to be continuous but recent numerical evidence on "explosive percolation" suggests that it might as well be discontinuous if links compete for addition. Here we analyze the microscopic mechanisms underlying discontinuous percolation processes and reveal a strong impact of single link additions. We show that in generic competitive percolation processes, including those displaying explosive percolation, single links do not induce a discontinuous gap in the largest cluster size in the thermodynamic limit. Nevertheless, our results highlight that for large finite systems single links may still induce observable gaps because gap sizes scale weakly algebraically with system size. Several essentially macroscopic clusters coexist immediately before the transition, thus announcing discontinuous percolation. These results explain how single links may drastically change macroscopic connectivity in networks where links add competitively.Comment: non-final version, for final see Nature Physics homepag

How to share underground reservoirs

Many resources, such as oil, gas, or water, are extracted from porous soils and their exploration is often shared among different companies or nations. We show that the effective shares can be obtained by invading the porous medium simultaneously with various fluids. Partitioning a volume in two parts requires one division surface while the simultaneous boundary between three parts consists of lines. We identify and characterize these lines, showing that they form a fractal set consisting of a single thread spanning the medium and a surrounding cloud of loops. While the spanning thread has fractal dimension ${1.55\pm0.03}$, the set of all lines has dimension ${1.69\pm0.02}$. The size distribution of the loops follows a power law and the evolution of the set of lines exhibits a tricritical point described by a crossover with a negative dimension at criticality

Pooled analysis of WHO Surgical Safety Checklist use and mortality after emergency laparotomy

Background The World Health Organization (WHO) Surgical Safety Checklist has fostered safe practice for 10 years, yet its place in emergency surgery has not been assessed on a global scale. The aim of this study was to evaluate reported checklist use in emergency settings and examine the relationship with perioperative mortality in patients who had emergency laparotomy. Methods In two multinational cohort studies, adults undergoing emergency laparotomy were compared with those having elective gastrointestinal surgery. Relationships between reported checklist use and mortality were determined using multivariable logistic regression and bootstrapped simulation. Results Of 12 296 patients included from 76 countries, 4843 underwent emergency laparotomy. After adjusting for patient and disease factors, checklist use before emergency laparotomy was more common in countries with a high Human Development Index (HDI) (2455 of 2741, 89.6 per cent) compared with that in countries with a middle (753 of 1242, 60.6 per cent; odds ratio (OR) 0.17, 95 per cent c.i. 0.14 to 0.21, P <0001) or low (363 of 860, 422 per cent; OR 008, 007 to 010, P <0.001) HDI. Checklist use was less common in elective surgery than for emergency laparotomy in high-HDI countries (risk difference -94 (95 per cent c.i. -11.9 to -6.9) per cent; P <0001), but the relationship was reversed in low-HDI countries (+121 (+7.0 to +173) per cent; P <0001). In multivariable models, checklist use was associated with a lower 30-day perioperative mortality (OR 0.60, 0.50 to 073; P <0.001). The greatest absolute benefit was seen for emergency surgery in low- and middle-HDI countries. Conclusion Checklist use in emergency laparotomy was associated with a significantly lower perioperative mortality rate. Checklist use in low-HDI countries was half that in high-HDI countries.Peer reviewe

Global variation in anastomosis and end colostomy formation following left-sided colorectal resection

Background End colostomy rates following colorectal resection vary across institutions in high-income settings, being influenced by patient, disease, surgeon and system factors. This study aimed to assess global variation in end colostomy rates after left-sided colorectal resection. Methods This study comprised an analysis of GlobalSurg-1 and -2 international, prospective, observational cohort studies (2014, 2016), including consecutive adult patients undergoing elective or emergency left-sided colorectal resection within discrete 2-week windows. Countries were grouped into high-, middle- and low-income tertiles according to the United Nations Human Development Index (HDI). Factors associated with colostomy formation versus primary anastomosis were explored using a multilevel, multivariable logistic regression model. Results In total, 1635 patients from 242 hospitals in 57 countries undergoing left-sided colorectal resection were included: 113 (6·9 per cent) from low-HDI, 254 (15·5 per cent) from middle-HDI and 1268 (77·6 per cent) from high-HDI countries. There was a higher proportion of patients with perforated disease (57·5, 40·9 and 35·4 per cent; P < 0·001) and subsequent use of end colostomy (52·2, 24·8 and 18·9 per cent; P < 0·001) in low- compared with middle- and high-HDI settings. The association with colostomy use in low-HDI settings persisted (odds ratio (OR) 3·20, 95 per cent c.i. 1·35 to 7·57; P = 0·008) after risk adjustment for malignant disease (OR 2·34, 1·65 to 3·32; P < 0·001), emergency surgery (OR 4·08, 2·73 to 6·10; P < 0·001), time to operation at least 48 h (OR 1·99, 1·28 to 3·09; P = 0·002) and disease perforation (OR 4·00, 2·81 to 5·69; P < 0·001). Conclusion Global differences existed in the proportion of patients receiving end stomas after left-sided colorectal resection based on income, which went beyond case mix alone