@article{Peixoto2014Graphtool,
    author = {Peixoto, Tiago P.},
    citeulike-article-id = {13701202},
    citeulike-linkout-0 = {http://dx.doi.org/10.6084/m9.figshare.1164194},
    citeulike-linkout-1 = {http://figshare.com/articles/graph\_tool/1164194},
    doi = {10.6084/m9.figshare.1164194},
    journal = {figshare},
    keywords = {all, complex, graph, network, networks, other, scientific-software},
    posted-at = {2015-08-13 07:34:10},
    priority = {0},
    title = {The graph-tool python library},
    url = {http://figshare.com/articles/graph\_tool/1164194},
    year = {2014}
}

@article{Fisher1967Theory,
    abstract = {The droplet or cluster theory of condensation is reviewed critically and extended. It is shown to imply that the condensation point is marked by a singularity of the thermodynamic potential as conjectured by Mayer. The singularity turns out to be an essential singularity at which all derivates of the thermodynamic variables remain finite. The theory also yields an understanding of the uniqueness of the critical point (in contrast to an extended critical region as Derby type of behaviour) and leads to relations between the various critical point singularities. A one-dimensional model is described with a Hamiltonian containing short-range many-body potentials. The exact solution of the model is sketched and shown to exhibit condensation and critical phenomena for suitable (fixed) potentials. The analysis confirms the conclusions of the cluster theory and thereby lends support to the validity of its underlying assumptions.},
    author = {Fisher, Michael E.},
    citeulike-article-id = {13317111},
    journal = {Physics},
    keywords = {percolation, phase\_transition},
    number = {5},
    pages = {255+},
    posted-at = {2015-04-24 16:42:07},
    priority = {0},
    title = {The Theory of Condensation and the Critical Point},
    volume = {3},
    year = {1967}
}

@article{Stauffer1979Scaling,
    abstract = {For beginners: This review tries to explain percolation through the cluster properties; it can also be used as an introduction to critical phenomena at other phase transitions for readers not familiar with scaling theory. In percolation each site of a periodic lattice is randomly occupied with probability p or empty with probability 1−p. An s-cluster is a group of s occupied sites connected by nearest-neighbor distances; the number of empty nearest neighbors of cluster sites is the perimeter t. For p above pc also one infinite cluster percolates through the lattice. How do the properties of s-clusters depend on s, and how do they feel the influence of the phase transition at p = pc? The answers to these questions are given by various methods (in particular computer simulations) and are interpreted by the so-called scaling theory of phase transitions. The results presented here suggest a qualitative difference of cluster structures above and below pc: Above pc some cluster properties suggest the existence of a cluster surface varying as in three dimensions, but below pc these ” surface” contributions are proportional to s. We suggest therefore that very large clusters above pc (but not at and below pc) behave like large clusters of Swiss cheese: Inspite of many internal holes the overall cluster shape is roughly spherical, similar to raindrops. For experts: Scaling theory suggests for large clusters near the percolation threshold pc that the average cluster numbers ns vary as s−\"{I}{\\AE}(z), with z ≡ (p − pc)s\"{I}. Analogously the average cluster perimeter is ts = s · (1 − p)/p + s\"{I} · \"{I}1(z), the average cluster radius Rs varies as s\"{I}v · R1(z), and the density profile Ds(r), which depends also on the distance r from the cluster center, varies as . These assumptions relate the seven critical exponents \^{I}±,\^{I}²,\^{I}³,\^{I}´,v,\"{I},\"{I} in d dimensions through the well-known five scaling laws , leaving only two exponents as independent variables to be fitted by ” experiment” and not predicted by scaling theory. For the lattice ” animals”, i.e. the number gst of geometrically different cluster configurations, a modified scaling assumption is derived: , with z ∝ (ac − t/s)s\"{I} and ac = (1 − pc)/pc. All these expressions are variants of the general scaling idea for second-order phase transitions that a function g(x,y) of two critical variables takes the homogeneous form {xcG}(x/yb) near the critical point, with two free exponents b and c and a scaling function G of a single variable. These assumptions, which may be regarded as generalizations of the Fisher droplet model, are tested ” experimentally” by Monte Carlo simulation, series expansion, renormalization group technique, and exact inequalities. In particular, detailed Monte Carlo evidence of Hoshen et al. and Leath and Reich is presented for the scaling of cluster numbers in two and three dimensions. If the cluster size s goes to infinity at fixed concentration p, not necessarily close to pc, three additional exponents \^{I}¾, \^{I}¸, \"{I}± are defined by: cluster numbers ∝ s−\^{I}¸ exp(−const · s\^{I}¾) and cluster radii ∝ s\"{I}±. These exponents are different on both sides of the phase transition; for example \^{I}¾(p < pc) = 1 and \^{I}¾(p > pc) = 1 − 1/d was found from inequalities, series and Monte Carlo data. The behavior of \^{I}¸ and of \"{I}±(p < pc) remains to be explained by scaling theory. This article does not cover experimental applications, correlation functions and ” classical” (mean field, Bethe lattice, effective medium) theories. For the reader to whom this abstract is too short and the whole article is too long we recommend sections 1 and 3.},
    author = {Stauffer, D.},
    citeulike-article-id = {643850},
    citeulike-linkout-0 = {http://dx.doi.org/10.1016/0370-1573(79)90060-7},
    citeulike-linkout-1 = {http://www.sciencedirect.com/science/article/B6TVP-46SXR69-DJ/2/7d5a261e703850c2bc6cc3f38152658a},
    doi = {10.1016/0370-1573(79)90060-7},
    journal = {Physics Reports},
    keywords = {percolation, phase\_transition},
    number = {1},
    pages = {1--74},
    posted-at = {2015-04-24 16:41:50},
    priority = {0},
    title = {Scaling theory of percolation clusters},
    url = {http://dx.doi.org/10.1016/0370-1573(79)90060-7},
    volume = {54},
    year = {1979}
}

@article{Broadbent1957Percolation,
    abstract = {The paper studies, in a general way, how the random properties of a 'medium' influence the percolation of a 'fluid' through it. The treatment diifers from conventional diffusion theory, in which it is the random properties of the fluid that matter. Fluid and medium bear general interpretations: for example, solute diffusing through solvent, electrons migrating over an atomic lattice, molecules penetrating a porous solid, disease infecting a community, etc.},
    author = {Broadbent, S. R. and Hammersley, J. M.},
    citeulike-article-id = {6527688},
    citeulike-linkout-0 = {http://journals.cambridge.org/action/displayAbstract?fromPage=online\&aid=2048852},
    citeulike-linkout-1 = {http://dx.doi.org/10.1017/s0305004100032680},
    doi = {10.1017/s0305004100032680},
    journal = {Mathematical Proceedings of the Cambridge Philosophical Society},
    keywords = {classic\_paper, percolation},
    month = jul,
    number = {03},
    pages = {629--641},
    posted-at = {2015-04-24 16:41:35},
    priority = {0},
    title = {{Percolation processes. I. Crystals and mazes}},
    url = {http://dx.doi.org/10.1017/s0305004100032680},
    volume = {53},
    year = {1957}
}

@article{Stockmayer1943Theory,
    abstract = {The most probable distributions of molecular sizes are calculated for certain types of branched‐chain polymers. The results represent an extension of the previous work of Flory, who showed that very large polymeric molecules appear suddenly at a critical extent of reaction, which is predicted to occur very nearly at the experimentally observed gel point. This transition from liquid to gel is shown to be analagous to the condensation of a saturated vapor. It is believed that the size distributions obtained herein will aid in a study of viscosity‐molecular weight relationships in branched‐chain polymers.},
    author = {Stockmayer, Walter H.},
    citeulike-article-id = {5989115},
    citeulike-linkout-0 = {http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal\&id=JCPSA6000011000002000045000001\&idtype=cvips\&gifs=yes},
    citeulike-linkout-1 = {http://link.aip.org/link/?JCP/11/45},
    citeulike-linkout-2 = {http://dx.doi.org/10.1063/1.1723803},
    doi = {10.1063/1.1723803},
    journal = {The Journal of Chemical Physics},
    keywords = {classic\_paper, percolation},
    number = {2},
    pages = {45--55},
    posted-at = {2015-04-24 16:40:56},
    priority = {0},
    title = {{Theory of Molecular Size Distribution and Gel Formation in Branched-Chain Polymers}},
    url = {http://dx.doi.org/10.1063/1.1723803},
    volume = {11},
    year = {1943}
}

@article{Flory1941Molecular,
    author = {Flory, Paul J.},
    citeulike-article-id = {13317167},
    citeulike-linkout-0 = {http://dx.doi.org/10.1021/ja01856a061},
    citeulike-linkout-1 = {http://pubs.acs.org/doi/abs/10.1021/ja01856a061},
    doi = {10.1021/ja01856a061},
    journal = {Journal of the American Chemical Society},
    keywords = {classic\_paper, percolation},
    number = {11},
    pages = {3083--3090},
    posted-at = {2015-04-24 16:40:37},
    priority = {0},
    title = {{Molecular Size Distribution in Three Dimensional Polymers. I. Gelation}},
    url = {http://dx.doi.org/10.1021/ja01856a061},
    volume = {63},
    year = {1941}
}

@book{Hunt2014Percolation,
    address = {Cham, Switzerland},
    author = {Hunt, Allen and Ewing, Robert and Ghanbarian, Behzad},
    citeulike-article-id = {13315992},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-319-03771-4},
    doi = {10.1007/978-3-319-03771-4},
    edition = {Third},
    isbn = {978-3-319-03770-7},
    keywords = {percolation, textbook},
    posted-at = {2015-04-24 16:40:17},
    priority = {0},
    publisher = {Springer},
    series = {Lecture Notes in Physics},
    title = {Percolation Theory for Flow in Porous Media},
    url = {http://dx.doi.org/10.1007/978-3-319-03771-4},
    volume = {880},
    year = {2014}
}

@article{Agresti1998Approximate,
    abstract = {For interval estimation of a proportion, coverage probabilities tend to be too large for "exact" confidence intervals based on inverting the binomial test and too small for the interval based on inverting the Wald large-sample normal test (i.e., sample proportion ± z-score × estimated standard error). Wilson's suggestion of inverting the related score test with null rather than estimated standard error yields coverage probabilities close to nominal confidence levels, even for very small sample sizes. The 95\% score interval has similar behavior as the adjusted Wald interval obtained after adding two "successes" and two "failures" to the sample. In elementary courses, with the score and adjusted Wald methods it is unnecessary to provide students with awkward sample size guidelines.},
    author = {Agresti, Alan and Coull, Brent A.},
    citeulike-article-id = {1060968},
    citeulike-linkout-0 = {http://dx.doi.org/10.2307/2685469},
    citeulike-linkout-1 = {http://www.jstor.org/stable/2685469},
    doi = {10.2307/2685469},
    journal = {The American Statistician},
    keywords = {statistics},
    number = {2},
    pages = {119--126},
    posted-at = {2014-10-17 17:50:45},
    priority = {0},
    publisher = {American Statistical Association},
    title = {{Approximate Is Better than "Exact" for Interval Estimation of Binomial Proportions}},
    url = {http://dx.doi.org/10.2307/2685469},
    volume = {52},
    year = {1998}
}

@article{Cameron2011On,
    abstract = {I present a critical review of techniques for estimating confidence intervals on binomial population proportions inferred from success counts in small to intermediate samples. Population proportions arise frequently as quantities of interest in astronomical research; for instance, in studies aiming to constrain the bar fraction, active galactic nucleus fraction, supermassive black hole fraction, merger fraction, or red sequence fraction from counts of galaxies exhibiting distinct morphological features or stellar populations. However, two of the most widely-used techniques for estimating binomial confidence intervals — the 'normal approximation' and the Clopper \& Pearson approach — are liable to misrepresent the degree of statistical uncertainty present under sampling conditions routinely encountered in astronomical surveys, leading to an ineffective use of the experimental data (and, worse, an inefficient use of the resources expended in obtaining that data). Hence, I provide here an overview of the fundamentals of binomial statistics with two principal aims: (I) to reveal the ease with which (Bayesian) binomial confidence intervals with more satisfactory behaviour may be estimated from the quantiles of the beta distribution using modern mathematical software packages (e.g. r, matlab, mathematica, idl, python); and (ii) to demonstrate convincingly the major flaws of both the 'normal approximation' and the Clopper \& Pearson approach for error estimation.},
    author = {Cameron, Ewan},
    citeulike-article-id = {13336801},
    citeulike-linkout-0 = {http://journals.cambridge.org/action/displayAbstract?fromPage=online\&aid=8794731},
    citeulike-linkout-1 = {http://dx.doi.org/10.1071/as10046},
    doi = {10.1071/as10046},
    journal = {Publications of the Astronomical Society of Australia},
    keywords = {statistics},
    pages = {128--139},
    posted-at = {2014-10-17 17:50:25},
    priority = {0},
    title = {{On the Estimation of Confidence Intervals for Binomial Population Proportions in Astronomy: The Simplicity and Superiority of the Bayesian Approach}},
    url = {http://dx.doi.org/10.1071/as10046},
    volume = {28},
    year = {2011}
}

@article{DasGupta2001Interval,
    abstract = {We revisit the problem of interval estimation of a binomial proportion. The erratic behavior of the coverage probability of the standard Wald confidence interval has previously been remarked on in the literature (Blyth and Still, Agresti and Coull, Santner and others). We begin by showing that the chaotic coverage properties of the Wald interval are far more persistent than is appreciated. Furthermore, common textbook prescriptions regarding its safety are misleading and defective in several respects and cannot be trusted.

This leads us to consideration of alternative intervals. A number of natural alternatives are presented, each with its motivation and context. Each interval is examined for its coverage probability and its length. Based on this analysis, we recommend the Wilson interval or the equal-tailed Jeffreys prior interval for small n and the interval suggested in Agresti and Coull for larger n. We also provide an additional frequentist justification for use of the Jeffreys interval.},
    author = {DasGupta, Anirban and Cai, Tony T. and Brown, Lawrence D.},
    citeulike-article-id = {7557924},
    citeulike-linkout-0 = {http://dx.doi.org/10.1214/ss/1009213286},
    doi = {10.1214/ss/1009213286},
    journal = {Statistical Science},
    keywords = {statistics},
    number = {2},
    pages = {101--133},
    posted-at = {2014-10-17 17:47:21},
    priority = {0},
    title = {{Interval Estimation for a Binomial Proportion}},
    url = {http://dx.doi.org/10.1214/ss/1009213286},
    volume = {16},
    year = {2001}
}

@book{Wasserman2004All,
    author = {Wasserman, Larry},
    citeulike-article-id = {13336822},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-0-387-21736-9},
    doi = {10.1007/978-0-387-21736-9},
    isbn = {978-1-4419-2322-6},
    keywords = {statistics, textbook},
    posted-at = {2014-10-17 17:45:56},
    priority = {2},
    publisher = {Springer New York},
    title = {All of Statistics},
    url = {http://dx.doi.org/10.1007/978-0-387-21736-9},
    year = {2004}
}

@book{Stauffer1994Introduction,
    abstract = {Percolation theory deals with clustering, criticality, diffusion, fractals, phase transitions and disordered systems. It provides a quantitative model for understanding these phenomena, and therefore provides a theoretical, statistical background to many physical and natural science disciplines. This book explains the basic theory for the graduate while also reaching into the specialized fields of disordered systems and renormalization groups. Readers are expected to be able to handle some fundamental mathematical procedures, such as integration and differentiation of single variable functions, probability and statistics. Computer programming experience in Fortran is also useful. While percolation is treated as a fundamentally physical concept, its relevance to various natural and living systems is addressed. Much of the book deals with systems lying close to the critical point phase transition point, where the subject is at its most interesting and sensitive. This book should be of value to all those who deal with systems which exhibit critical points and phase transition behavior.},
    address = {London},
    author = {Stauffer, Dietrich and Aharony, Amnon},
    citeulike-article-id = {2698119},
    citeulike-linkout-0 = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20\&\#38;path=ASIN/0748402535},
    citeulike-linkout-1 = {http://www.amazon.ca/exec/obidos/redirect?tag=citeulike09-20\&amp;path=ASIN/0748402535},
    citeulike-linkout-10 = {http://www.librarything.com/isbn/0748402535},
    citeulike-linkout-11 = {http://www.worldcat.org/oclc/31776610},
    citeulike-linkout-2 = {http://www.amazon.de/exec/obidos/redirect?tag=citeulike01-21\&amp;path=ASIN/0748402535},
    citeulike-linkout-3 = {http://www.amazon.fr/exec/obidos/redirect?tag=citeulike06-21\&amp;path=ASIN/0748402535},
    citeulike-linkout-4 = {http://www.amazon.jp/exec/obidos/ASIN/0748402535},
    citeulike-linkout-5 = {http://www.amazon.co.uk/exec/obidos/ASIN/0748402535/citeulike00-21},
    citeulike-linkout-6 = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20\&path=ASIN/0748402535},
    citeulike-linkout-7 = {http://www.worldcat.org/isbn/0748402535},
    citeulike-linkout-8 = {http://books.google.com/books?vid=ISBN0748402535},
    citeulike-linkout-9 = {http://www.amazon.com/gp/search?keywords=0748402535\&index=books\&linkCode=qs},
    edition = {Second},
    isbn = {0748402535},
    keywords = {percolation, phase\_transition, textbook},
    posted-at = {2014-10-15 10:52:01},
    priority = {0},
    publisher = {Taylor \& Francis},
    title = {Introduction to Percolation Theory},
    url = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20\&path=ASIN/0748402535},
    year = {1994}
}

@article{Newman2001Fast,
    abstract = {We describe in detail an efficient algorithm for studying site or bond percolation on any lattice. The algorithm can measure an observable quantity in a percolation system for all values of the site or bond occupation probability from zero to one in an amount of time that scales linearly with the size of the system. We demonstrate our algorithm by using it to investigate a number of issues in percolation theory, including the position of the percolation transition for site percolation on the square lattice, the stretched exponential behavior of spanning probabilities away from the critical point, and the size of the giant component for site percolation on random graphs.},
    archivePrefix = {arXiv},
    author = {Newman, M. E. J. and Ziff, R. M.},
    citeulike-article-id = {3373958},
    citeulike-linkout-0 = {http://arxiv.org/abs/cond-mat/0101295},
    citeulike-linkout-1 = {http://arxiv.org/pdf/cond-mat/0101295},
    citeulike-linkout-2 = {http://dx.doi.org/10.1103/physreve.64.016706},
    citeulike-linkout-3 = {http://link.aps.org/abstract/PRE/v64/i1/e016706},
    citeulike-linkout-4 = {http://link.aps.org/pdf/PRE/v64/i1/e016706},
    doi = {10.1103/physreve.64.016706},
    eprint = {cond-mat/0101295},
    journal = {Physical Review E},
    keywords = {monte\_carlo, percolation, statistical\_physics},
    number = {1},
    pages = {016706+},
    posted-at = {2014-10-15 10:16:04},
    priority = {0},
    title = {Fast Monte Carlo algorithm for site or bond percolation},
    url = {http://dx.doi.org/10.1103/physreve.64.016706},
    volume = {64},
    year = {2001}
}

@book{Binder2010Monte,
    address = {Berlin, Heidelberg},
    author = {Binder, Kurt and Heermann, Dieter W.},
    citeulike-article-id = {13318034},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-642-03163-2},
    doi = {10.1007/978-3-642-03163-2},
    isbn = {978-3-642-03162-5},
    keywords = {monte\_carlo, phase\_transition, simulation, statistical\_physics, textbook},
    posted-at = {2014-10-15 10:12:02},
    priority = {0},
    publisher = {Springer},
    title = {Monte Carlo Simulation in Statistical Physics},
    url = {http://dx.doi.org/10.1007/978-3-642-03163-2},
    year = {2010}
}

