Critical dynamics of the k-core pruning process

We present the theory of the k-core pruning process (progressive removal of nodes with degree less than k) in uncorrelated random networks. We derive exact equations describing this process and the evolution of the network structure, and solve them numerically and, in the critical regime of the process, analytically. We show that the pruning process exhibits three different behaviors depending on whether the mean degree of the initial network is above, equal to, or below the threshold _c corresponding to the emergence of the giant k-core. We find that above the threshold the network relaxes exponentially to the k-core. The system manifests the phenomenon known as "critical slowing down", as the relaxation time diverges when tends to _c. At the threshold, the dynamics become critical characterized by a power-law relaxation (1/t^2). Below the threshold, a long-lasting transient process (a "plateau" stage) occurs. This transient process ends with a collapse in which the entire network disappears completely. The duration of the process diverges when tends to _c. We show that the critical dynamics of the pruning are determined by branching processes of spreading damage. Clusters of nodes of degree exactly k are the evolving substrate for these branching processes. Our theory completely describes this branching cascade of damage in uncorrelated networks by providing the time dependent distribution function of branching. These theoretical results are supported by our simulations of the $k$-core pruning in Erdos-Renyi graphs.Comment: 12 pages, 10 figure

Finite size scaling of the de Almeida-Thouless instability in random sparse networks

We study, in random sparse networks, finite size scaling of the spin glass susceptibility $\chi_{\rm SG}$, which is a proper measure of the de Almeida-Thouless (AT) instability of spin glass systems. Using a phenomenological argument regarding the band edge behavior of the Hessian eigenvalue distribution, we discuss how $\chi_{\rm SG}$ is evaluated in infinitely large random sparse networks, which are usually identified with Bethe trees, and how it should be corrected in finite systems. In the high temperature region, data of extensive numerical experiments are generally in good agreement with the theoretical values of $\chi_{\rm SG}$ determined from the Bethe tree. In the absence of external fields, the data also show a scaling relation $\chi_{\rm SG}=N^{1/3}F(N^{1/3}|T-T_c|/T_c)$, which has been conjectured in the literature, where $T_c$ is the critical temperature. In the presence of external fields, on the other hand, the numerical data are not consistent with this scaling relation. A numerical analysis of Hessian eigenvalues implies that strong finite size corrections of the lower band edge of the eigenvalue distribution, which seem relevant only in the presence of the fields, are a major source of inconsistency. This may be related to the known difficulty in using only numerical methods to detect the AT instability.Comment: 8 figures, 2 table

Firing statistics and correlations in spiking neurons: a level-crossing approach

We present a time-dependent level-crossing theory for linear dynamical systems perturbed by colored Gaussian noise. We apply these results to approximate the firing statistics of conductance-based integrate-and-fire neurons receiving excitatory and inhibitory Poissonian inputs. Analytical expressions are obtained for three key quantities characterizing the neuronal response to time-varying inputs: the mean firing rate, the linear response to sinusoidally-modulated inputs, and the pairwise spike-correlation for neurons receiving correlated inputs. The theory yields tractable results that are shown to accurately match numerical simulations, and provides useful tools for the analysis of interconnected neuronal populations

Evolution of opinions on social networks in the presence of competing committed groups

Public opinion is often affected by the presence of committed groups of individuals dedicated to competing points of view. Using a model of pairwise social influence, we study how the presence of such groups within social networks affects the outcome and the speed of evolution of the overall opinion on the network. Earlier work indicated that a single committed group within a dense social network can cause the entire network to quickly adopt the group's opinion (in times scaling logarithmically with the network size), so long as the committed group constitutes more than about 10% of the population (with the findings being qualitatively similar for sparse networks as well). Here we study the more general case of opinion evolution when two groups committed to distinct, competing opinions $A$ and $B$, and constituting fractions $p_A$ and $p_B$ of the total population respectively, are present in the network. We show for stylized social networks (including Erd\H{o}s-R\'enyi random graphs and Barab\'asi-Albert scale-free networks) that the phase diagram of this system in parameter space $(p_A,p_B)$ consists of two regions, one where two stable steady-states coexist, and the remaining where only a single stable steady-state exists. These two regions are separated by two fold-bifurcation (spinodal) lines which meet tangentially and terminate at a cusp (critical point). We provide further insights to the phase diagram and to the nature of the underlying phase transitions by investigating the model on infinite (mean-field limit), finite complete graphs and finite sparse networks. For the latter case, we also derive the scaling exponent associated with the exponential growth of switching times as a function of the distance from the critical point.Comment: 23 pages: 15 pages + 7 figures (main text), 8 pages + 1 figure + 1 table (supplementary info