Spectra of random stochastic matrices and relaxation in complex systems

Abstract

We compute spectra of large stochastic matrices W , defined on sparse random graphs, where edges (i, j) of the graph are given positive random weights Wij > 0 in such a fashion that column sums are normalized to one. We compute spectra of such matrices both in the thermodynamic limit, and for single large instances. The structure of the graphs and the distribution of the non-zero edge weights Wij are largely arbitrary, as long as the mean vertex degree remains finite in the thermodynamic limit and the Wij satisfy a detailed balance condition. Knowing the spectra of stochastic matrices is tantamount to knowing the complete spectrum of relaxation times of stochastic processes described by them, so our results should have many interesting applications for the description of relaxation in complex systems. Our approach allows to disentangle contributions to the spectral density related to extended and localized states, respectively, allowing to differentiate between time-scales associated with transport processes and those associated with the dynamics of local rearrangements. There are numerous processes, both natural and artificial, which can be understood in terms of random walks on complex networks [1][2][3], including the spread of diseases in social networks [4,5], the transmission of information in communication networks (e.g.[6]), search algorithms [7,8], the out-of-equilibrium dynamics of glassy systems at low temperatures as described in terms of hopping between long-lived states in state space [9][10][11], the dynamics of major conformational changes in macro-molecules [12], or cell-signalling through protein-protein interaction networks [13], to name but a few. For reviews that cover several of these topics, see e.g. [14][15][16].The purpose of the present letter is to use random matrix theory to contribute to the understanding of systems of this type. We compute spectra of transition matrices for discrete Markov chains describing stochastic dynamics in complex systems. We construct these in terms of sparse random graphs in such a way that an edge (i, j) in a graph corresponds to a possible transition j → i, with the edge weight W ij > 0 quantifying the associated transitions probability, requiring i W ij = 1 for all j. We are interested in the limit, where the number N of possible states becomes large, with the average number of possible transitions at each state remaining finite in the thermodynamic limit (N → ∞).Given a time-dependent probability vector p(t) = (p i (t)), we have an evolution equation of the formThe condition W ij ≥ 0 for all (i, j) and the column sum constraint together entail that the spectrum of W is contained in the unit disc of the complex plane, σ(W ) ⊆ {z; |z| ≤ 1}. If W satisfies a detailed balance condition with an equilibrium distribution, p i = p eq i , such that W ij p j = W ji p i for all pairs (i, j), then W can be symmetrized by a similarity transformation -Our main interest here is the relation between eigenvalues of W and relaxation times of the Markov chain it describes...

A random walk perspective on hide-and-seek games

“…To proceed it is advantageous [24] to rescale variables x i / √ Γ i → x i . Keeping the same symbols for the rescaled variables, we have…”

Glassy dynamics on networks: local spectra and return probabilities

“…Summarizing, P τ (t) exhibits an initial exponential decay typical of the mean field limit for t τ , followed by a power law behaviour with a T -dependent exponent for t τ ; the latter arises from deep minima surrounding the departing node. Note that the crossover point from the exponential decay to the power law regime occurs at a value of P τ (t) that decreases as τ increases, which is directly related to the fact that the power law tail does not just depend on the scaled time t/τ (see (30)). Both features can be seen in figure 5, where we compare evaluations of P τ (t) for different values of τ : in the left plot the x-axis is scaled by τ , which delivers a collapse of the exponential decay at short times, while in the right plot we have similarly scaled the y-axis to show that the prefactor of the tail is indeed proportional to τ as the approximation (30) predicts.…”

“…Next we analyse what (37) says about the long time behaviour of F (t), by considerinĝ F (s) for small s. From equation (30) one obtains the following approximation for the leading singular small s-behaviour of the return probability in Laplace space 4…”

“…This approach exploits the local tree-like structure of networks with sparse connectivity and follows in this analogous applications to the spectral analysis of symmetric random matrices; see e.g. [27][28][29][30][31] or [32,33] for a rigorous discussion. We used two relevant limits as benchmarks: the mean field (MF) limit where the average connectivity diverges in the thermodynamic limit of infinite system size, thus giving the original Bouchaud trap model, and the infinite temperature or random walk (RW) limit, where the energy landscape no longer plays a role.…”

“…One has generallyPτ (0) −Pτ (s) = ∞ 0 dt Pτ (t)(1 − e −st ). The integrand becomes dominated by large t for small s; substituting the tail estimate(30) and integrating by parts then gives(38).…”