Skeleton and fractal scaling in complex networks

We find that the fractal scaling in a class of scale-free networks originates from the underlying tree structure called skeleton, a special type of spanning tree based on the edge betweenness centrality. The fractal skeleton has the property of the critical branching tree. The original fractal networks are viewed as a fractal skeleton dressed with local shortcuts. An in-silico model with both the fractal scaling and the scale-invariance properties is also constructed. The framework of fractal networks is useful in understanding the utility and the redundancy in networked systems.Comment: 4 pages, 2 figures, final version published in PR

Invasion Percolation Between two Sites

We investigate the process of invasion percolation between two sites (injection and extraction sites) separated by a distance r in two-dimensional lattices of size L. Our results for the non-trapping invasion percolation model indicate that the statistics of the mass of invaded clusters is significantly dependent on the local occupation probability (pressure) Pe at the extraction site. For Pe=0, we show that the mass distribution of invaded clusters P(M) follows a power-law P(M) ~ M^{-\alpha} for intermediate values of the mass M, with an exponent \alpha=1.39. When the local pressure is set to Pe=Pc, where Pc corresponds to the site percolation threshold of the lattice topology, the distribution P(M) still displays a scaling region, but with an exponent \alpha=1.02. This last behavior is consistent with previous results for the cluster statistics in standard percolation. In spite of these discrepancies, the results of our simulations indicate that the fractal dimension of the invaded cluster does not depends significantly on the local pressure Pe and it is consistent with the fractal dimension values reported for standard invasion percolation. Finally, we perform extensive numerical simulations to determine the effect of the lattice borders on the statistics of the invaded clusters and also to characterize the self-organized critical behavior of the invasion percolation process.Comment: 7 pages, 11 figures, submited for PR

Is subdiffusional transport slower than normal?

We consider anomalous non-Markovian transport of Brownian particles in viscoelastic fluid-like media with very large but finite macroscopic viscosity under the influence of a constant force field F. The viscoelastic properties of the medium are characterized by a power-law viscoelastic memory kernel which ultra slow decays in time on the time scale \tau of strong viscoelastic correlations. The subdiffusive transport regime emerges transiently for t<\tau. However, the transport becomes asymptotically normal for t>>\tau. It is shown that even though transiently the mean displacement and the variance both scale sublinearly, i.e. anomalously slow, in time, ~ F t^\alpha, ~ t^\alpha, 0<\alpha<1, the mean displacement at each instant of time is nevertheless always larger than one obtained for normal transport in a purely viscous medium with the same macroscopic viscosity obtained in the Markovian approximation. This can have profound implications for the subdiffusive transport in biological cells as the notion of "ultra-slowness" can be misleading in the context of anomalous diffusion-limited transport and reaction processes occurring on nano- and mesoscales

Optimal box-covering algorithm for fractal dimension of complex networks

The self-similarity of complex networks is typically investigated through computational algorithms the primary task of which is to cover the structure with a minimal number of boxes. Here we introduce a box-covering algorithm that not only outperforms previous ones, but also finds optimal solutions. For the two benchmark cases tested, namely, the E. Coli and the WWW networks, our results show that the improvement can be rather substantial, reaching up to 15% in the case of the WWW network.Comment: 5 pages, 6 figure

Extended microsatellite analysis in microsatellite stable, MSH2 and MLH1 mutation-negative HNPCC patients: Genetic reclassification and correlation with clinical features

Background: Hereditary nonpolyposis colorectal cancer (HNPCC) is an autosomal dominant disorder predisposing to predominantly colorectal cancer (CRC) and endometrial cancer frequently due to germline mutations in DNA mismatch repair (MMR) genes, mainly MLH1, MSH2 and also MSH6 in families seen to demonstrate an excess of endometrial cancer. As a consequence, tumors in HNPCC reveal alterations in the length of simple repetitive genomic sequences like poly-A, poly-T, CA or GT repeats (microsatellites) in at least 90% of the cases. Aim of the Study: The study cohort consisted of 25 HNPCC index patients ( 19 Amsterdam positive, 6 Bethesda positive) who revealed a microsatellite stable (MSS) - or low instable (MSI-L) - tumor phenotype with negative mutation analysis for the MMR genes MLH1 and MSH2. An extended marker panel (BAT40, D10S197, D13S153, D18S58, MYCL1) was analyzed for the tumors of these patients with regard to three aspects. First, to reconfirm the MSI-L phenotype found by the standard panel; second, to find minor MSIs which might point towards an MSH6 mutation, and third, to reconfirm the MSS status of hereditary tumors. The reconfirmation of the MSS status of tumors not caused by mutations in the MMR genes should allow one to define another entity of hereditary CRC. Their clinical features were compared with those of 150 patients with sporadic CRCs. Results: In this way, 17 MSS and 8 MSI-L tumors were reclassified as 5 MSS, 18 MSI-L and even 2 MSI-H ( high instability) tumors, the last being seen to demonstrate at least 4 instable markers out of 10. Among all family members, 87 malignancies were documented. The mean age of onset for CRCs was the lowest in the MSI-H-phenotyped patients with 40.5 +/- 4.9 years (vs. 47.0 +/- 14.6 and 49.8 +/- 11.9 years in MSI-L- and MSS-phenotyped patients, respectively). The percentage of CRC was the highest in families with MSS-phenotyped tumors (88%), followed by MSI-L-phenotyped ( 78%) and then by MSI-H-phenotyped (67%) tumors. MSS tumors were preferentially localized in the distal colon supposing a similar biologic behavior like sporadic CRC. MSH6 mutation analysis for the MSI-L and MSI-H patients revealed one truncating mutation for a patient initially with an MSS tumor, which was reclassified as MSI-L by analyzing the extended marker panel. Conclusion: Extended microsatellite analysis serves to evaluate the sensitivity of the reference panel for HNPCC detection and permits phenotype confirmation or upgrading. Additionally, it confirms the MSS status of hereditary CRCs not caused by the common mutations in the MMR genes and provides hints to another entity of hereditary CRC. Copyright (C) 2004 S. Karger AG, Basel

Emergence of fractal behavior in condensation-driven aggregation

We investigate a model in which an ensemble of chemically identical Brownian particles are continuously growing by condensation and at the same time undergo irreversible aggregation whenever two particles come into contact upon collision. We solved the model exactly by using scaling theory for the case whereby a particle, say of size $x$, grows by an amount $\alpha x$ over the time it takes to collide with another particle of any size. It is shown that the particle size spectra of such system exhibit transition to dynamic scaling $c(x,t)\sim t^{-\beta}\phi(x/t^z)$ accompanied by the emergence of fractal of dimension $d_f={{1}\over{1+2\alpha}}$. One of the remarkable feature of this model is that it is governed by a non-trivial conservation law, namely, the $d_f^{th}$ moment of $c(x,t)$ is time invariant regardless of the choice of the initial conditions. The reason why it remains conserved is explained by using a simple dimensional analysis. We show that the scaling exponents $\beta$ and $z$ are locked with the fractal dimension $d_f$ via a generalized scaling relation $\beta=(1+d_f)z$.Comment: 8 pages, 6 figures, to appear in Phys. Rev.

Multifractal Analysis of the Coupling Space of Feed-Forward Neural Networks

Random input patterns induce a partition of the coupling space of feed-forward neural networks into different cells according to the generated output sequence. For the perceptron this partition forms a random multifractal for which the spectrum $f(\alpha)$ can be calculated analytically using the replica trick. Phase transition in the multifractal spectrum correspond to the crossover from percolating to non-percolating cell sizes. Instabilities of negative moments are related to the VC-dimension.Comment: 10 pages, Latex, submitted to PR

Critical scaling in standard biased random walks

The spatial coverage produced by a single discrete-time random walk, with asymmetric jump probability $p\neq 1/2$ and non-uniform steps, moving on an infinite one-dimensional lattice is investigated. Analytical calculations are complemented with Monte Carlo simulations. We show that, for appropriate step sizes, the model displays a critical phenomenon, at $p=p_c$. Its scaling properties as well as the main features of the fragmented coverage occurring in the vicinity of the critical point are shown. In particular, in the limit $p\to p_c$, the distribution of fragment lengths is scale-free, with nontrivial exponents. Moreover, the spatial distribution of cracks (unvisited sites) defines a fractal set over the spanned interval. Thus, from the perspective of the covered territory, a very rich critical phenomenology is revealed in a simple one-dimensional standard model.Comment: 4 pages, 4 figure

Distinguishing cancerous from non-cancerous cells through analysis of electrical noise

Since 1984, electric cell-substrate impedance sensing (ECIS) has been used to monitor cell behavior in tissue culture and has proven sensitive to cell morphological changes and cell motility. We have taken ECIS measurements on several cultures of non-cancerous (HOSE) and cancerous (SKOV) human ovarian surface epithelial cells. By analyzing the noise in real and imaginary electrical impedance, we demonstrate that it is possible to distinguish the two cell types purely from signatures of their electrical noise. Our measures include power-spectral exponents, Hurst and detrended fluctuation analysis, and estimates of correlation time; principal-component analysis combines all the measures. The noise from both cancerous and non-cancerous cultures shows correlations on many time scales, but these correlations are stronger for the non-cancerous cells.Comment: 8 pages, 4 figures; submitted to PR

On the Threshold of Intractability

We study the computational complexity of the graph modification problems Threshold Editing and Chain Editing, adding and deleting as few edges as possible to transform the input into a threshold (or chain) graph. In this article, we show that both problems are NP-complete, resolving a conjecture by Natanzon, Shamir, and Sharan (Discrete Applied Mathematics, 113(1):109--128, 2001). On the positive side, we show the problem admits a quadratic vertex kernel. Furthermore, we give a subexponential time parameterized algorithm solving Threshold Editing in $2^{O(\surd k \log k)} + \text{poly}(n)$ time, making it one of relatively few natural problems in this complexity class on general graphs. These results are of broader interest to the field of social network analysis, where recent work of Brandes (ISAAC, 2014) posits that the minimum edit distance to a threshold graph gives a good measure of consistency for node centralities. Finally, we show that all our positive results extend to the related problem of Chain Editing, as well as the completion and deletion variants of both problems