Search results for "random graph"
showing 10 items of 28 documents
Graph-theoretical derivation of brain structural connectivity
2020
Brain connectivity at the single neuron level can provide fundamental insights into how information is integrated and propagated within and between brain regions. However, it is almost impossible to adequately study this problem experimentally and, despite intense efforts in the field, no mathematical description has been obtained so far. Here, we present a mathematical framework based on a graph-theoretical approach that, starting from experimental data obtained from a few small subsets of neurons, can quantitatively explain and predict the corresponding full network properties. This model also changes the paradigm with which large-scale model networks can be built, from using probabilisti…
Molecular Diversity Required for the Formation of Autocatalytic Sets
2019
Systems chemistry deals with the design and study of complex chemical systems. However, such systems are often difficult to investigate experimentally. We provide an example of how theoretical and simulation-based studies can provide useful insights into the properties and dynamics of complex chemical systems, in particular of autocatalytic sets. We investigate the issue of the required molecular diversity for autocatalytic sets to exist in random polymer libraries. Given a fixed probability that an arbitrary polymer catalyzes the formation of other polymers, we calculate this required molecular diversity theoretically for two particular models of chemical reaction systems, and then verify …
Predicting mobile apps spread: An epidemiological random network modeling approach
2017
[EN] The mobile applications business is a really big market, growing constantly. In app marketing, a key issue is to predict future app installations. The influence of the peers seems to be very relevant when downloading apps. Therefore, the study of the evolution of mobile apps spread may be approached using a proper network model that considers the influence of peers. Influence of peers and other social contagions have been successfully described using models of epidemiological type. Hence, in this paper we propose an epidemiological random network model with realistic parameters to predict the evolution of downloads of apps. With this model, we are able to predict the behavior of an app…
Percolation on correlated random networks
2011
We consider a class of random, weighted networks, obtained through a redefinition of patterns in an Hopfield-like model and, by performing percolation processes, we get information about topology and resilience properties of the networks themselves. Given the weighted nature of the graphs, different kinds of bond percolation can be studied: stochastic (deleting links randomly) and deterministic (deleting links based on rank weights), each mimicking a different physical process. The evolution of the network is accordingly different, as evidenced by the behavior of the largest component size and of the distribution of cluster sizes. In particular, we can derive that weak ties are crucial in o…
Online shortest paths with confidence intervals for routing in a time varying random network
2018
International audience; The increase in the world's population and rising standards of living is leading to an ever-increasing number of vehicles on the roads, and with it ever-increasing difficulties in traffic management. This traffic management in transport networks can be clearly optimized by using information and communication technologies referred as Intelligent Transport Systems (ITS). This management problem is usually reformulated as finding the shortest path in a time varying random graph. In this article, an online shortest path computation using stochastic gradient descent is proposed. This routing algorithm for ITS traffic management is based on the online Frank-Wolfe approach.…
Community characterization of heterogeneous complex systems
2011
We introduce an analytical statistical method to characterize the communities detected in heterogeneous complex systems. By posing a suitable null hypothesis, our method makes use of the hypergeometric distribution to assess the probability that a given property is over-expressed in the elements of a community with respect to all the elements of the investigated set. We apply our method to two specific complex networks, namely a network of world movies and a network of physics preprints. The characterization of the elements and of the communities is done in terms of languages and countries for the movie network and of journals and subject categories for papers. We find that our method is ab…
Circular law for sparse random regular digraphs
2020
Fix a constant $C\geq 1$ and let $d=d(n)$ satisfy $d\leq \ln^{C} n$ for every large integer $n$. Denote by $A_n$ the adjacency matrix of a uniform random directed $d$-regular graph on $n$ vertices. We show that, as long as $d\to\infty$ with $n$, the empirical spectral distribution of appropriately rescaled matrix $A_n$ converges weakly in probability to the circular law. This result, together with an earlier work of Cook, completely settles the problem of weak convergence of the empirical distribution in directed $d$-regular setting with the degree tending to infinity. As a crucial element of our proof, we develop a technique of bounding intermediate singular values of $A_n$ based on studyi…
Spin Glasses on Thin Graphs
1995
In a recent paper we found strong evidence from simulations that the Isingantiferromagnet on ``thin'' random graphs - Feynman diagrams - displayed amean-field spin glass transition. The intrinsic interest of considering such random graphs is that they give mean field results without long range interactions or the drawbacks, arising from boundary problems, of the Bethe lattice. In this paper we reprise the saddle point calculations for the Ising and Potts ferromagnet, antiferromagnet and spin glass on Feynman diagrams. We use standard results from bifurcation theory that enable us to treat an arbitrary number of replicas and any quenched bond distribution. We note the agreement between the f…
Inferring networks from high-dimensional data with mixed variables
2014
We present two methodologies to deal with high-dimensional data with mixed variables, the strongly decomposable graphical model and the regression-type graphical model. The first model is used to infer conditional independence graphs. The latter model is applied to compute the relative importance or contribution of each predictor to the response variables. Recently, penalized likelihood approaches have also been proposed to estimate graph structures. In a simulation study, we compare the performance of the strongly decomposable graphical model and the graphical lasso in terms of graph recovering. Five different graph structures are used to simulate the data: the banded graph, the cluster gr…
Structure of eigenvectors of random regular digraphs
2018
Let $d$ and $n$ be integers satisfying $C\leq d\leq \exp(c\sqrt{\ln n})$ for some universal constants $c, C>0$, and let $z\in \mathbb{C}$. Denote by $M$ the adjacency matrix of a random $d$-regular directed graph on $n$ vertices. In this paper, we study the structure of the kernel of submatrices of $M-z\,{\rm Id}$, formed by removing a subset of rows. We show that with large probability the kernel consists of two non-intersecting types of vectors, which we call very steep and gradual with many levels. As a corollary, we show, in particular, that every eigenvector of $M$, except for constant multiples of $(1,1,\dots,1)$, possesses a weak delocalization property: its level sets have cardin…