arXiv subjects

Maya Paczuski

Publications and source records attributed to Maya Paczuski.

At least 19 recordsLinked to original sources

arXiv2012

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.

physics.soc-ph↗

arXiv2012

Agglomerative Percolation on Bipartite Networks: A Novel Type of Spontaneous Symmetry Breaking

Ordinary bond percolation (OP) can be viewed as a process where clusters grow by joining them pairwise, by adding links chosen randomly one by one from a set of predefined `virtual' links. In contrast, in agglomerative percolation (AP) clusters grow by choosing randomly a `target cluster' and joining it with all its neighbors, as defined by the same set of virtual links. Previous studies showed that AP is in different universality classes from OP for several types of (virtual) networks (linear chains, trees, Erdos-Renyi networks), but most surprising were the results for 2-d lattices: While AP on the triangular lattice was found to be in the OP universality class, it behaved completely differently on the square lattice. In the present paper we explain this striking violation of universality by invoking bipartivity. While the square lattice is a bipartite graph, the triangular lattice is not. In conformity with this we show that AP on the honeycomb and simple cubic (3-d) lattices -- both of which are bipartite -- are also not in the OP universality classes. More precisely, we claim that this violation of universality is basically due to a Z_2 symmetry that is spontaneously broken at the percolation threshold. We also discuss AP on bipartite random networks and suitable generalizations of AP on k-partite graphs.

cond-mat.stat-mech↗

arXiv2012

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.

cond-mat.dis-nn↗

arXiv2012

PageRank and rank-reversal dependence on the damping factor

PageRank (PR) is an algorithm originally developed by Google to evaluate the importance of web pages. Considering how deeply rooted Google's PR algorithm is to gathering relevant information or to the success of modern businesses, the question of rank-stability and choice of the damping factor (a parameter in the algorithm) is clearly important. We investigate PR as a function of the damping factor d on a network obtained from a domain of the World Wide Web, finding that rank-reversal happens frequently over a broad range of PR (and of d). We use three different correlation measures, Pearson, Spearman, and Kendall, to study rank-reversal as d changes, and show that the correlation of PR vectors drops rapidly as d changes from its frequently cited value, $d_0=0.85$. Rank-reversal is also observed by measuring the Spearman and Kendall rank correlation, which evaluate relative ranks rather than absolute PR. Rank-reversal happens not only in directed networks containing rank-sinks but also in a single strongly connected component, which by definition does not contain any sinks. We relate rank-reversals to rank-pockets and bottlenecks in the directed network structure. For the network studied, the relative rank is more stable by our measures around $d=0.65$ than at $d=d_0$.

physics.soc-ph↗

arXiv2011

Random Sequential Renormalization and Agglomerative Percolation in Networks: Application to Erd"os-R'enyi and Scale-free Graphs

We study the statistical behavior under random sequential renormalization(RSR) of several network models including Erd"os R'enyi (ER) graphs, scale-free networks and an annealed model (AM) related to ER graphs. In RSR the network is locally coarse grained by choosing at each renormalization step a node at random and joining it to all its neighbors. Compared to previous (quasi-)parallel renormalization methods [C.Song et.al], RSR allows a more fine-grained analysis of the renormalization group (RG) flow, and unravels new features, that were not discussed in the previous analyses. In particular we find that all networks exhibit a second order transition in their RG flow. This phase transition is associated with the emergence of a giant hub and can be viewed as a new variant of percolation, called agglomerative percolation. We claim that this transition exists also in previous graph renormalization schemes and explains some of the scaling laws seen there. For critical trees it happens as N/N0 -> 0 in the limit of large systems (where N0 is the initial size of the graph and N its size at a given RSR step). In contrast, it happens at finite N/N0 in sparse ER graphs and in the annealed model, while it happens for N/N0 -> 1 on scale-free networks. Critical exponents seem to depend on the type of the graph but not on the average degree and obey usual scaling relations for percolation phenomena. For the annealed model they agree with the exponents obtained from a mean-field theory. At late times, the networks exhibit a star-like structure in agreement with the results of Radicchi et. al. While degree distributions are of main interest when regarding the scheme as network renormalization, mass distributions (which are more relevant when considering 'supernodes' as clusters) are much easier to study using the fast Newman-Ziff algorithm for percolation, allowing us to obtain very high statistics.

cond-mat.stat-mech↗

arXiv2011

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.

cond-mat.stat-mech↗

arXiv2011

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.

physics.data-an↗

arXiv2011

Are Percolation Transitions always Sharpened by Making Networks Interdependent?

We study a model for coupled networks introduced recently by Buldyrev et al., Nature 464, 1025 (2010), where each node has to be connected to others via two types of links to be viable. Removing a critical fraction of nodes leads to a percolation transition that has been claimed to be more abrupt than that for uncoupled networks. Indeed, it was found to be discontinuous in all cases studied. Using an efficient new algorithm we verify that the transition is discontinuous for coupled Erdos-Renyi networks, but find it to be continuous for fully interdependent diluted lattices. In 2 and 3 dimension, the order parameter exponent $β$ is larger than in ordinary percolation, showing that the transition is less sharp, i.e. further from discontinuity, than for isolated networks. Possible consequences for spatially embedded networks are discussed.

physics.data-an↗

arXiv2011

Exact solutions for mass-dependent irreversible aggregations

We consider the mass-dependent aggregation process (k+1)X -> X, given a fixed number of unit mass particles in the initial state. One cluster is chosen proportional to its mass and is merged into one either with k-neighbors in one dimension, or -- in the well-mixed case -- with k other clusters picked randomly. We find the same combinatorial exact solutions for the probability to find any given configuration of particles on a ring or line, and in the well-mixed case. The mass distribution of a single cluster exhibits scaling laws and the finite size scaling form is given. The relation to the classical sum kernel of irreversible aggregation is discussed.

physics.data-an↗

arXiv2011

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 $N_0^ν\lesssim N<N_0$, 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 $ν=1/2$ is derived using random walk arguments. The degree distribution becomes broader under successive renormalization -- reaching a power law, $p_k\sim 1/k^γ$ with $γ=2$ 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 $N_0^{ν_{star}}$ with $ν_{star}\approx 1/4$.

cond-mat.stat-mech↗

arXiv2011

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 $N^{-η}$ with $η> 0$. We find different positive values of $β$ (defined via $< s_{\rm max}/N > \sim (p-p_c)^β$ for infinite systems) for each model, showing that they are all in different universality classes. In contrast, the exponent $Θ$ (defined such that observables are homogeneous functions of $(p-p_c)N^Θ$) is close to -- or even equal to -- 1/2 for all models.

cond-mat.dis-nn↗

arXiv2011

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.

physics.data-an↗

arXiv2011

Clustering Drives Assortativity and Community Structure in Ensembles of Networks

Clustering, assortativity, and communities are key features of complex networks. We probe dependencies between these attributes and find that ensembles with strong clustering display both high assortativity by degree and prominent community structure, while ensembles with high assortativity are much less biased towards clustering or community structure. Further, clustered networks can amplify small homophilic bias for trait assortativity. This marked asymmetry suggests that transitivity, rather than homophily, drives the standard nonsocial/social network dichotomy.

physics.soc-ph↗

arXiv2010

Edge direction and the structure of networks

Directed networks are ubiquitous and are necessary to represent complex systems with asymmetric interactions---from food webs to the World Wide Web. Despite the importance of edge direction for detecting local and community structure, it has been disregarded in studying a basic type of global diversity in networks: the tendency of nodes with similar numbers of edges to connect. This tendency, called assortativity, affects crucial structural and dynamic properties of real-world networks, such as error tolerance or epidemic spreading. Here we demonstrate that edge direction has profound effects on assortativity. We define a set of four directed assortativity measures and assign statistical significance by comparison to randomized networks. We apply these measures to three network classes---online/social networks, food webs, and word-adjacency networks. Our measures (i) reveal patterns common to each class, (ii) separate networks that have been previously classified together, and (iii) expose limitations of several existing theoretical models. We reject the standard classification of directed networks as purely assortative or disassortative. Many display a class-specific mixture, likely reflecting functional or historical constraints, contingencies, and forces guiding the system's evolution.

physics.soc-ph↗

arXiv2010

The Interacting Branching Process as a Simple Model of Innovation

We describe innovation in terms of a generalized branching process. Each new invention pairs with any existing one to produce a number of offspring, which is Poisson distributed with mean p. Existing inventions die with probability p/τat each generation. In contrast to mean field results, no phase transition occurs; the chance for survival is finite for all p > 0. For τ= \infty, surviving processes exhibit a bottleneck before exploding super-exponentially - a growth consistent with a law of accelerating returns. This behavior persists for finite τ. We analyze, in detail, the asymptotic behavior as p \to 0.

physics.soc-ph↗

arXiv2010

Sequence alignment, mutual information, and dissimilarity measures for constructing phylogenies

Existing sequence alignment algorithms use heuristic scoring schemes which cannot be used as objective distance metrics. Therefore one relies on measures like the p- or log-det distances, or makes explicit, and often simplistic, assumptions about sequence evolution. Information theory provides an alternative, in the form of mutual information (MI) which is, in principle, an objective and model independent similarity measure. MI can be estimated by concatenating and zipping sequences, yielding thereby the "normalized compression distance". So far this has produced promising results, but with uncontrolled errors. We describe a simple approach to get robust estimates of MI from global pairwise alignments. Using standard alignment algorithms, this gives for animal mitochondrial DNA estimates that are strikingly close to estimates obtained from the alignment free methods mentioned above. Our main result uses algorithmic (Kolmogorov) information theory, but we show that similar results can also be obtained from Shannon theory. Due to the fact that it is not additive, normalized compression distance is not an optimal metric for phylogenetics, but we propose a simple modification that overcomes the issue of additivity. We test several versions of our MI based distance measures on a large number of randomly chosen quartets and demonstrate that they all perform better than traditional measures like the Kimura or log-det (resp. paralinear) distances. Even a simplified version based on single letter Shannon entropies, which can be easily incorporated in existing software packages, gave superior results throughout the entire animal kingdom. But we see the main virtue of our approach in a more general way. For example, it can also help to judge the relative merits of different alignment algorithms, by estimating the significance of specific alignments.

q-bio.GN↗

arXiv2010

Attractor and Basin Entropies of Random Boolean Networks Under Asynchronous Stochastic Update

We introduce a numerical method to study random Boolean networks with asynchronous stochas- tic update. Each node in the network of states starts with equal occupation probability and this probability distribution then evolves to a steady state. Nodes left with finite occupation probability determine the attractors and the sizes of their basins. As for synchronous update, the basin entropy grows with system size only for critical networks, where the distribution of attractor lengths is a power law. We determine analytically the distribution for the number of attractors and basin sizes for frozen networks with connectivity K = 1.

cond-mat.stat-mech↗

arXiv2009

Clustering Phase Transitions and Hysteresis: Pitfalls in Constructing Network Ensembles

Ensembles of networks are used as null models in many applications. However, simple null models often show much less clustering than their real-world counterparts. In this paper, we study a model where clustering is enhanced by means of a fugacity term as in the Strauss (or "triangle") model, but where the degree sequence is strictly preserved -- thus maintaining the quenched heterogeneity of nodes found in the original degree sequence. Similar models had been proposed previously in [R. Milo et al., Science 298, 824 (2002)]. We find that our model exhibits phase transitions as the fugacity is changed. For regular graphs (identical degrees for all nodes) with degree k > 2 we find a single first order transition. For all non-regular networks that we studied (including Erdos - Renyi and scale-free networks) we find multiple jumps resembling first order transitions, together with strong hysteresis. The latter transitions are driven by the sudden emergence of "cluster cores": groups of highly interconnected nodes with higher than average degrees. To study these cluster cores visually, we introduce q-clique adjacency plots. We find that these cluster cores constitute distinct communities which emerge spontaneously from the triangle generating process. Finally, we point out that cluster cores produce pitfalls when using the present (and similar) models as null models for strongly clustered networks, due to the very strong hysteresis which effectively leads to broken ergodicity on realistic time scales.

cond-mat.stat-mech↗