arXiv subjects

Tiago P. Peixoto

Publications and source records attributed to Tiago P. Peixoto.

At least 55 records · Page 3Linked to original sources

arXiv2014

Hierarchical Block Structures and High-resolution Model Selection in Large Networks

Discovering and characterizing the large-scale topological features in empirical networks are crucial steps in understanding how complex systems function. However, most existing methods used to obtain the modular structure of networks suffer from serious problems, such as being oblivious to the statistical evidence supporting the discovered patterns, which results in the inability to separate actual structure from noise. In addition to this, one also observes a resolution limit on the size of communities, where smaller but well-defined clusters are not detectable when the network becomes large. This phenomenon occurs not only for the very popular approach of modularity optimization, which lacks built-in statistical validation, but also for more principled methods based on statistical inference and model selection, which do incorporate statistical validation in a formally correct way. Here we construct a nested generative model that, through a complete description of the entire network hierarchy at multiple scales, is capable of avoiding this limitation, and enables the detection of modular structure at levels far beyond those possible with current approaches. Even with this increased resolution, the method is based on the principle of parsimony, and is capable of separating signal from noise, and thus will not lead to the identification of spurious modules even on sparse networks. Furthermore, it fully generalizes other approaches in that it is not restricted to purely assortative mixing patterns, directed or undirected graphs, and ad hoc hierarchical structures such as binary trees. Despite its general character, the approach is tractable, and can be combined with advanced techniques of community detection to yield an efficient algorithm that scales well for very large networks.

physics.data-an↗

arXiv2014

Efficient Monte Carlo and greedy heuristic for the inference of stochastic block models

We present an efficient algorithm for the inference of stochastic block models in large networks. The algorithm can be used as an optimized Markov chain Monte Carlo (MCMC) method, with a fast mixing time and a much reduced susceptibility to getting trapped in metastable states, or as a greedy agglomerative heuristic, with an almost linear $O(N\ln^2N)$ complexity, where $N$ is the number of nodes in the network, independent on the number of blocks being inferred. We show that the heuristic is capable of delivering results which are indistinguishable from the more exact and numerically expensive MCMC method in many artificial and empirical networks, despite being much faster. The method is entirely unbiased towards any specific mixing pattern, and in particular it does not favor assortative community structures.

physics.data-an↗

arXiv2013

Entropy of stochastic blockmodel ensembles

Stochastic blockmodels are generative network models where the vertices are separated into discrete groups, and the probability of an edge existing between two vertices is determined solely by their group membership. In this paper, we derive expressions for the entropy of stochastic blockmodel ensembles. We consider several ensemble variants, including the traditional model as well as the newly introduced degree-corrected version [Karrer et al. Phys. Rev. E 83, 016107 (2011)], which imposes a degree sequence on the vertices, in addition to the block structure. The imposed degree sequence is implemented both as "soft" constraints, where only the expected degrees are imposed, and as "hard" constraints, where they are required to be the same on all samples of the ensemble. We also consider generalizations to multigraphs and directed graphs. We illustrate one of many applications of this measure by directly deriving a log-likelihood function from the entropy expression, and using it to infer latent block structure in observed data. Due to the general nature of the ensembles considered, the method works well for ensembles with intrinsic degree correlations (i.e. with entropic origin) as well as extrinsic degree correlations, which go beyond the block structure.

cond-mat.stat-mech↗

arXiv2013

Eigenvalue Spectra of Modular Networks

A large variety of dynamical processes that take place on networks can be expressed in terms of the spectral properties of some linear operator which reflects how the dynamical rules depend on the network topology. Often such spectral features are theoretically obtained by considering only local node properties, such as degree distributions. Many networks, however, possess large-scale modular structures that can drastically influence their spectral characteristics, and which are neglected in such simplified descriptions. Here we obtain in a unified fashion the spectrum of a large family of operators, including the adjacency, Laplacian and normalized Laplacian matrices, for networks with generic modular structure, in the limit of large degrees. We focus on the conditions necessary for the merging of the isolated eigenvalues with the continuous band of the spectrum, after which the planted modular structure can no longer be easily detected by spectral methods. This is a crucial transition point which determines when a modular structure is strong enough to affect a given dynamical process. We show that this transition happens in general at different points for the different matrices, and hence the detectability threshold can vary significantly depending on the operator chosen. Equivalently, the sensitivity to the modular structure of the different dynamical processes associated with each matrix will be different, given the same large-scale structure present in the network. Furthermore, we show that, with the exception of the Laplacian matrix, the different transitions coalesce into the same point for the special case where the modules are homogeneous, but separate otherwise.

physics.data-an↗

arXiv2013

Spontaneous centralization of control in a network of company ownerships

We introduce a model for the adaptive evolution of a network of company ownerships. In a recent work it has been shown that the empirical global network of corporate control is marked by a central, tightly connected "core" made of a small number of large companies which control a significant part of the global economy. Here we show how a simple, adaptive "rich get richer" dynamics can account for this characteristic, which incorporates the increased buying power of more influential companies, and in turn results in even higher control. We conclude that this kind of centralized structure can emerge without it being an explicit goal of these companies, or as a result of a well-organized strategy.

physics.soc-ph↗

arXiv2013

Parsimonious module inference in large networks

We investigate the detectability of modules in large networks when the number of modules is not known in advance. We employ the minimum description length (MDL) principle which seeks to minimize the total amount of information required to describe the network, and avoid overfitting. According to this criterion, we obtain general bounds on the detectability of any prescribed block structure, given the number of nodes and edges in the sampled network. We also obtain that the maximum number of detectable blocks scales as $\sqrt{N}$, where $N$ is the number of nodes in the network, for a fixed average degree $ $. We also show that the simplicity of the MDL approach yields an efficient multilevel Monte Carlo inference algorithm with a complexity of $O(τN\log N)$, if the number of blocks is unknown, and $O(τN)$ if it is known, where $τ$ is the mixing time of the Markov chain. We illustrate the application of the method on a large network of actors and films with over $10^6$ edges, and a dissortative, bipartite block structure.

physics.data-an↗

arXiv2013

No need for conspiracy: Self-organized cartel formation in a modified trust game

We investigate the dynamics of a trust game on a mixed population where individuals with the role of buyers are forced to play against a predetermined number of sellers, whom they choose dynamically. Agents with the role of sellers are also allowed to adapt the level of value for money of their products, based on payoff. The dynamics undergoes a transition at a specific value of the strategy update rate, above which an emergent cartel organization is observed, where sellers have similar values of below optimal value for money. This cartel organization is not due to an explicit collusion among agents; instead it arises spontaneously from the maximization of the individual payoffs. This dynamics is marked by large fluctuations and a high degree of unpredictability for most of the parameter space, and serves as a plausible qualitative explanation for observed elevated levels and fluctuations of certain commodity prices.

q-fin.GN↗

arXiv2012

Evolution of robust network topologies: Emergence of central backbones

We model the robustness against random failure or intentional attack of networks with arbitrary large-scale structure. We construct a block-based model which incorporates --- in a general fashion --- both connectivity and interdependence links, as well as arbitrary degree distributions and block correlations. By optimizing the percolation properties of this general class of networks, we identify a simple core-periphery structure as the topology most robust against random failure. In such networks, a distinct and small "core" of nodes with higher degree is responsible for most of the connectivity, functioning as a central "backbone" of the system. This centralized topology remains the optimal structure when other constraints are imposed, such as a given fraction of interdependence links and fixed degree distributions. This distinguishes simple centralized topologies as the most likely to emerge, when robustness against failure is the dominant evolutionary force.

physics.soc-ph↗

arXiv2012

Emergence of robustness against noise: A structural phase transition in evolved models of gene regulatory networks

We investigate the evolution of Boolean networks subject to a selective pressure which favors robustness against noise, as a model of evolved genetic regulatory systems. By mapping the evolutionary process into a statistical ensemble and minimizing its associated free energy, we find the structural properties which emerge as the selective pressure is increased and identify a phase transition from a random topology to a "segregated core" structure, where a smaller and more densely connected subset of the nodes is responsible for most of the regulation in the network. This segregated structure is very similar qualitatively to what is found in gene regulatory networks, where only a much smaller subset of genes --- those responsible for transcription factors --- is responsible for global regulation. We obtain the full phase diagram of the evolutionary process as a function of selective pressure and the average number of inputs per node. We compare the theoretical predictions with Monte Carlo simulations of evolved networks and with empirical data for Saccharomyces cerevisiae and Escherichia coli.

cond-mat.dis-nn↗

arXiv2012

The behavior of noise-resilient Boolean networks with diverse topologies

The dynamics of noise-resilient Boolean networks with majority functions and diverse topologies is investigated. A wide class of possible topological configurations is parametrized as a stochastic blockmodel. For this class of networks, the dynamics always undergoes a phase transition from a non-ergodic regime, where a memory of its past states is preserved, to an ergodic regime, where no such memory exists and every microstate is equally probable. Both the average error on the network, as well as the critical value of noise where the transition occurs are investigated analytically, and compared to numerical simulations. The results for "partially dense" networks, comprised of relatively few, but dynamically important nodes, which have a number of inputs which greatly exceeds the average for the entire network, give very general upper bounds on the maximum resilience against noise attainable on globally sparse systems.

cond-mat.dis-nn↗

arXiv2010

Trust transitivity in social networks

Non-centralized recommendation-based decision making is a central feature of several social and technological processes, such as market dynamics, peer-to-peer file-sharing and the web of trust of digital certification. We investigate the properties of trust propagation on networks, based on a simple metric of trust transitivity. We investigate analytically the percolation properties of trust transitivity in random networks with arbitrary degree distribution, and compare with numerical realizations. We find that the existence of a non-zero fraction of absolute trust (i.e. entirely confident trust) is a requirement for the viability of global trust propagation in large systems: The average pair-wise trust is marked by a discontinuous transition at a specific fraction of absolute trust, below which it vanishes. Furthermore, we perform an extensive analysis of the Pretty Good Privacy (PGP) web of trust, in view of the concepts introduced. We compare different scenarios of trust distribution: community- and authority-centered. We find that these scenarios lead to sharply different patterns of trust propagation, due to the segregation of authority hubs and densely-connected communities. While the authority-centered scenario is more efficient, and leads to higher average trust values, it favours weakly-connected "fringe" nodes, which are directly trusted by authorities. The community-centered scheme, on the other hand, favours nodes with intermediate degrees, in detriment of the authorities and its "fringe" peers.

physics.soc-ph↗

arXiv2010

Boolean networks with robust and reliable trajectories

We construct and investigate Boolean networks that follow a given reliable trajectory in state space, which is insensitive to fluctuations in the updating schedule, and which is also robust against noise. Robustness is quantified as the probability that the dynamics return to the reliable trajectory after a perturbation of the state of a single node. In order to achieve high robustness, we navigate through the space of possible update functions by using an evolutionary algorithm. We constrain the networks to having the minimum number of connections required to obtain the reliable trajectory. Surprisingly, we find that robustness always reaches values close to 100 percent during the evolutionary optimization process. The set of update functions can be evolved such that it differs only slightly from that of networks that were not optimized with respect to robustness. The state space of the optimized networks is dominated by the basin of attraction of the reliable trajectory.

physics.bio-ph↗

arXiv2010

The phase diagram of random Boolean networks with nested canalizing functions

We obtain the phase diagram of random Boolean networks with nested canalizing functions. Using the annealed approximation, we obtain the evolution of the number $b_t$ of nodes with value one, and the network sensitivity $λ$, and we compare with numerical simulations of quenched networks. We find that, contrary to what was reported by Kauffman et al. [Proc. Natl. Acad. Sci. 2004 101 49 17102-7], these networks have a rich phase diagram, were both the "chaotic" and frozen phases are present, as well as an oscillatory regime of the value of $b_t$. We argue that the presence of only the frozen phase in the work of Kauffman et al. was due simply to the specific parametrization used, and is not an inherent feature of this class of functions. However, these networks are significantly more stable than the variants where all possible Boolean functions are allowed.

physics.bio-ph↗

arXiv2010

Spatiotemporal correlations of aftershock sequences

Aftershock sequences are of particular interest in seismic research since they may condition seismic activity in a given region over long time spans. While they are typically identified with periods of enhanced seismic activity after a large earthquake as characterized by the Omori law, our knowledge of the spatiotemporal correlations between events in an aftershock sequence is limited. Here, we study the spatiotemporal correlations of two aftershock sequences form California (Parkfield and Hector Mine) using the recently introduced concept of "recurrent" events. We find that both sequences have very similar properties and that most of them are captured by the space-time epidemic-type aftershock sequence (ETAS) model if one takes into account catalog incompleteness. However, the stochastic model does not capture the spatiotemporal correlations leading to the observed structure of seismicity on small spatial scales.

physics.geo-ph↗

arXiv2010

Redundancy and error resilience in Boolean Networks

We consider the effect of noise in sparse Boolean Networks with redundant functions. We show that they always exhibit a non-zero error level, and the dynamics undergoes a phase transition from non-ergodicity to ergodicity, as a function of noise, after which the system is no longer capable of preserving a memory if its initial state. We obtain upper-bounds on the critical value of noise for networks of different sparsity.

physics.bio-ph↗

arXiv2009

Boolean networks with reliable dynamics

We investigated the properties of Boolean networks that follow a given reliable trajectory in state space. A reliable trajectory is defined as a sequence of states which is independent of the order in which the nodes are updated. We explored numerically the topology, the update functions, and the state space structure of these networks, which we constructed using a minimum number of links and the simplest update functions. We found that the clustering coefficient is larger than in random networks, and that the probability distribution of three-node motifs is similar to that found in gene regulation networks. Among the update functions, only a subset of all possible functions occur, and they can be classified according to their probability. More homogeneous functions occur more often, leading to a dominance of canalyzing functions. Finally, we studied the entire state space of the networks. We observed that with increasing systems size, fixed points become more dominant, moving the networks close to the frozen phase.

physics.bio-ph↗

arXiv2008

Noise in random Boolean networks

We investigate the effect of noise on Random Boolean Networks. Noise is implemented as a probability $p$ that a node does not obey its deterministic update rule. We define two order parameters, the long-time average of the Hamming distance between a network with and without noise, and the average frozenness, which is a measure of the extent to which a node prefers one of the two Boolean states. We evaluate both order parameters as function of the noise strength, finding a smooth transition from deterministic ($p=0$) to fully stochastic ($p=1/2$) dynamics for networks with $K\le2$, and a first order transition at $p=0$ for $K>2$. Most of the results obtained by computer simulation are also derived analytically. The average Hamming distance can be evaluated using the annealed approximation. In order to obtain the distribution of frozenness as function of the noise strength, more sophisticated self-consistent calculations had to be performed. This distribution is a collection of delta peaks for K=1, and it has a fractal sructure for $K>1$, approaching a continuous distribution in the limit $K\gg1$.

physics.bio-ph↗

arXiv2008

Network of recurrent events for the Olami-Feder-Christensen model

We numerically study the dynamics of a discrete spring-block model introduced by Olami, Feder and Christensen (OFC) to mimic earthquakes and investigate to which extent this simple model is able to reproduce the observed spatiotemporal clustering of seismicty. Following a recently proposed method to characterize such clustering by networks of recurrent events [Geophys. Res. Lett. {\bf 33}, L1304, 2006], we find that for synthetic catalogs generated by the OFC model these networks have many non-trivial statistical properties. This includes characteristic degree distributions -- very similar to what has been observed for real seismicity. There are, however, also significant differences between the OFC model and earthquake catalogs indicating that this simple model is insufficient to account for certain aspects of the spatiotemporal clustering of seismicity.

cond-mat.dis-nn↗