Agglomerative Percolation in Two Dimensions

We study a process termed "agglomerative percolation" (AP) in two dimensions. Instead of adding sites or bonds at random, in AP randomly chosen clusters are linked to all their neighbors. As a result the growth process involves a diverging length scale near a critical point. Picking target clusters with probability proportional to their mass leads to a runaway compact cluster. Choosing all clusters equally leads to a continuous transition in a new universality class for the square lattice, while the transition on the triangular lattice has the same critical exponents as ordinary percolation.Comment: Paper and supplementary figures and discussio

Irreversible Aggregation and Network Renormalization

Irreversible aggregation is revisited in view of recent work on renormalization of complex networks. Its scaling laws and phase transitions are related to percolation transitions seen in the latter. We illustrate our points by giving the complete solution for the probability to find any given state in an aggregation process $(k+1)X\to X$, given a fixed number of unit mass particles in the initial state. Exactly the same probability distributions and scaling are found in one dimensional systems (a trivial network) and well-mixed solutions. This reveals that scaling laws found in renormalization of complex networks do not prove that they are self-similar.Comment: 4 pages, 2 figure

Percolation Theory on Interdependent Networks Based on Epidemic Spreading

We consider percolation on interdependent locally treelike networks, recently introduced by Buldyrev et al., Nature 464, 1025 (2010), and demonstrate that the problem can be simplified conceptually by deleting all references to cascades of failures. Such cascades do exist, but their explicit treatment just complicates the theory -- which is a straightforward extension of the usual epidemic spreading theory on a single network. Our method has the added benefits that it is directly formulated in terms of an order parameter and its modular structure can be easily extended to other problems, e.g. to any number of interdependent networks, or to networks with dependency links.Comment: 6 pages, 5 figure

Discontinuous Percolation Transitions in Epidemic Processes, Surface Depinning in Random Media and Hamiltonian Random Graphs

Discontinuous percolation transitions and the associated tricritical points are manifest in a wide range of both equilibrium and non-equilibrium cooperative phenomena. To demonstrate this, we present and relate the continuous and first order behaviors in two different classes of models: The first are generalized epidemic processes (GEP) that describe in their spatially embedded version - either on or off a regular lattice - compact or fractal cluster growth in random media at zero temperature. A random graph version of GEP is mapped onto a model previously proposed for complex social contagion. We compute detailed phase diagrams and compare our numerical results at the tricritical point in d = 3 with field theory predictions of Janssen et al. [Phys. Rev. E 70, 026114 (2004)]. The second class consists of exponential ("Hamiltonian", or formally equilibrium) random graph models and includes the Strauss and the 2-star model, where 'chemical potentials' control the densities of links, triangles or 2-stars. When the chemical potentials in either graph model are O(logN), the percolation transition can coincide with a first order phase transition in the density of links, making the former also discontinuous. Hysteresis loops can then be of mixed order, with second order behavior for decreasing link fugacity, and a jump (first order) when it increases

Explosive Percolation is Continuous, but with Unusual Finite Size Behavior

We study four Achlioptas type processes with "explosive" percolation transitions. All transitions are clearly continuous, but their finite size scaling functions are not entire holomorphic. The distributions of the order parameter, the relative size $s_{\rm max}/N$ of the largest cluster, are double-humped. But -- in contrast to first order phase transitions -- the distance between the two peaks decreases with system size $N$ as $N^{-\eta}$ with $\eta > 0$. We find different positive values of $\beta$ (defined via $< s_{\rm max}/N > \sim (p-p_c)^\beta$ for infinite systems) for each model, showing that they are all in different universality classes. In contrast, the exponent $\Theta$ (defined such that observables are homogeneous functions of $(p-p_c)N^\Theta$) is close to -- or even equal to -- 1/2 for all models.Comment: 4 pages (including 4 figures), plus 7 pages of supplementary materia

Evaluation of Mast Cell and Blood Vessel Density in Inflammatory Periapical Lesions

Introduction: Radicular cystsand periapical granulomas are the most common periapical inflammatory lesions. However, the role of cellular immunity and microvessels in their pathogenesis remains unknown. The aim of this study was to evaluate the mast cell density (MCD), mircovessel density (MVD) and investigating the correlation between their densities with each other in the above mentioned lesions.Materials & Methods: In this descriptive cross-sectional study, 40 paraffin blocks of mentioned lesions were selected from achieves of School of Dentistry, Babol University of Medical Sciences. Three sections were prepared from each block and stained by hematoxylin-eosin, toluidine blue, and immunohistochemically for CD34 to determine the score of inflammation, presence of mast cells and degranulatedmast cells (DMCs), and MVD, respectively. The correlation between MCD and either inflammatory infiltrate or MVD was evaluated. Data analyzed by t student, Mann-Whitney and Spearman test.Results: Mast cells were present in all periapical inflammatory lesions; 15.4±14.8 for MCD, 7.2±6.1 for DMCs, and the ratio of DMCs to total number of MCs was 0.354±0.166 and 14.8+4.44 for blood vessel density in radicular cyst and 8.52±6.75, 2.91±2.1, 0.196±0.194 and 13±8.02 in periapical granulomas, respectively. There was a positive correlation between MCD and MVD in radicular cyst (P=0.03, r=0.341), but not in periapical granulomas (P=0.6, r=0.124). MCD and MVD increased with the score of inflammation in radicular cyst (P=0.001, r=0.7) and periapical granuloma (P=0.012, r=0.54).Conclusion: Mast cells and microvessels play a role in pathogenesis of periapical inflammatory lesions. In this study, the density of mast cells and DMCs in radicular cyst was higher than periapical granulomas, but no difference was observed regarding MVD in periapical inflammatory lesions. It seems that the relationship between MCD and MVD is different based on the clinical stage of periapical inflammatory lesions

Association between single nucleotide polymorphisms in the PI3K/AKT/mTOR pathway and bladder cancer risk in a sample of Iranian population

In the past few years several investigations have focused on the role of PI3K/AKT/mTOR pathway and its deregulations in different cancers. This study aimed to examine genetic polymorphisms of this pathway in bladder cancer (BC). In this case-control study, 235 patients with pathologically confirmed bladder cancer and 254 control subjects were examined. PIK3CA, AKT1 and mTOR variants were analyzed using polymerase chain reaction-restriction fragment length polymorphism (PCR-RFLP). The findings proposed that the PIK3CA rs6443624 SNP significantly decreased the risk of BC (OR=0.44, 95 % CI=0.30-0.65, p<0.0001 CA vs CC; OR=0.35, 95 % CI=0.16-0.78, p=0.0107, AA vs CC; OR=0.60, 95 % CI=0.46-0.79, p=0.0002, A vs T). The AKT1 rs2498801 variant is associated with a decreased risk of BC (OR=0.57, 95 % CI=0.39-0.82, p=0.003, AG vs AA; OR=0.74, 95 % CI=0.56-0.97, p=0.032, G vs A) while, AKT1 rs1130233 polymorphism considerably increased the risk of BC (OR=3.70, 95 % CI=2.52-5.43, p<0.0001, GA vs GG; OR=5.81, 95 % CI=1.53-21.97, p=0.010, AA vs GG; OR=2.71, 95 % CI=1.98-3.70, p<0.0001, A vs G). Additionally, mTOR rs2295080 variant notably increased the risk of BC (OR=2.25, 95 % CI=1.50-3.38, p<0.0001, GT vs GG; OR=4.75, 95 % CI=2.80-8.06, p<0.0001, TT vs GG; OR=3.10, 95 % CI=2.34-4.10, p<0.0001, T vs G). None of the other examined polymorphisms (AKT1 rs1130214, AKT1 rs3730358, mTOR rs1883965) revealed significant association with BC. In conclusion, our findings suggest that PIK3CA rs6443624, AKT1 rs2498801, AKT1 rs1130233, as well mTOR rs2295080 polymorphism may be related to bladder cancer development in a sample of Iranian population. Validation of our findings in larger sample sizes of different ethnicities would provide evidence on the role of variants of PI3K/AKT/mTOR pathway in developing BC

Random Sequential Renormalization of Networks I: Application to Critical Trees

We introduce the concept of Random Sequential Renormalization (RSR) for arbitrary networks. RSR is a graph renormalization procedure that locally aggregates nodes to produce a coarse grained network. It is analogous to the (quasi-)parallel renormalization schemes introduced by C. Song {\it et al.} (Nature {\bf 433}, 392 (2005)) and studied more recently by F. Radicchi {\it et al.} (Phys. Rev. Lett. {\bf 101}, 148701 (2008)), but much simpler and easier to implement. In this first paper we apply RSR to critical trees and derive analytical results consistent with numerical simulations. Critical trees exhibit three regimes in their evolution under RSR: (i) An initial regime $N_0^{\nu}\lesssim N<N_0$, where $N$ is the number of nodes at some step in the renormalization and $N_0$ is the initial size. RSR in this regime is described by a mean field theory and fluctuations from one realization to another are small. The exponent $\nu=1/2$ is derived using random walk arguments. The degree distribution becomes broader under successive renormalization -- reaching a power law, $p_k\sim 1/k^{\gamma}$ with $\gamma=2$ and a variance that diverges as $N_0^{1/2}$ at the end of this regime. Both of these results are derived based on a scaling theory. (ii) An intermediate regime for $N_0^{1/4}\lesssim N \lesssim N_0^{1/2}$, in which hubs develop, and fluctuations between different realizations of the RSR are large. Crossover functions exhibiting finite size scaling, in the critical region $N\sim N_0^{1/2} \to \infty$, connect the behaviors in the first two regimes. (iii) The last regime, for $1 \ll N\lesssim N_0^{1/4}$, is characterized by the appearance of star configurations with a central hub surrounded by many leaves. The distribution of sizes where stars first form is found numerically to be a power law up to a cutoff that scales as $N_0^{\nu_{star}}$ with $\nu_{star}\approx 1/4$

Sampling properties of directed networks

For many real-world networks only a small "sampled" version of the original network may be investigated; those results are then used to draw conclusions about the actual system. Variants of breadth-first search (BFS) sampling, which are based on epidemic processes, are widely used. Although it is well established that BFS sampling fails, in most cases, to capture the IN-component(s) of directed networks, a description of the effects of BFS sampling on other topological properties are all but absent from the literature. To systematically study the effects of sampling biases on directed networks, we compare BFS sampling to random sampling on complete large-scale directed networks. We present new results and a thorough analysis of the topological properties of seven different complete directed networks (prior to sampling), including three versions of Wikipedia, three different sources of sampled World Wide Web data, and an Internet-based social network. We detail the differences that sampling method and coverage can make to the structural properties of sampled versions of these seven networks. Most notably, we find that sampling method and coverage affect both the bow-tie structure, as well as the number and structure of strongly connected components in sampled networks. In addition, at low sampling coverage (i.e. less than 40%), the values of average degree, variance of out-degree, degree auto-correlation, and link reciprocity are overestimated by 30% or more in BFS-sampled networks, and only attain values within 10% of the corresponding values in the complete networks when sampling coverage is in excess of 65%. These results may cause us to rethink what we know about the structure, function, and evolution of real-world directed networks.Comment: 21 pages, 11 figure

A Novel Approach to Discontinuous Bond Percolation Transition

We introduce a bond percolation procedure on a $D$-dimensional lattice where two neighbouring sites are connected by $N$ channels, each operated by valves at both ends. Out of a total of $N$, randomly chosen $n$ valves are open at every site. A bond is said to connect two sites if there is at least one channel between them, which has open valves at both ends. We show analytically that in all spatial dimensions, this system undergoes a discontinuous percolation transition in the $N\to \infty$ limit when $\gamma =\frac{\ln n}{\ln N}$ crosses a threshold. It must be emphasized that, in contrast to the ordinary percolation models, here the transition occurs even in one dimensional systems, albeit discontinuously. We also show that a special kind of discontinuous percolation occurs only in one dimension when $N$ depends on the system size.Comment: 6 pages, 6 eps figure