arXiv subjects

Reimer Kühn

Publications and source records attributed to Reimer Kühn.

At least 19 recordsLinked to original sources

arXiv2025

Localization and top eigenvalue detection

The detection of the top eigenvalue and its corresponding eigenvector in ensembles of random matrices has significant applications across various fields. An existing method, based on the linear stability of a complementary set of cavity equations, has been successful in identifying the top eigenvalue when the associated eigenvector is extended. However, this approach fails when the eigenvector is localized. In this work, we adapt the real-valued cavity method to address this limitation by introducing a novel criterion that exploits the constraints of the cavity equations to detect the top eigenvalue in systems with a localized top eigenvector. Our results are validated using the Anderson model as a paradigmatic example.

cond-mat.dis-nn↗

arXiv2024

The joint distribution of first return times and of the number of distinct sites visited by a 1D random walk before returning to the origin

We present analytical results for the joint probability distribution $P(T_{FR}=t,S=s)$ of first return (FR) times t and of the number of distinct sites s visited by a random walk (RW) on a one dimensional lattice before returning to the origin. The RW on a one dimensional lattice is recurrent, namely the probability to return to the origin is $P_{R}=1$. However the mean $\langle T_{FR}\rangle$ of the distribution $P(T_{FR}=t)$ of first return times diverges. Similarly, the mean $\langle S\rangle$ of the distribution $P(S=s)$ of the number of distinct sites visited before returning to the origin also diverges. The joint distribution $P(T_{FR}=t,S=s)$ provides a formulation that controls these divergences and accounts for the interplay between the kinetic and geometric properties of first return trajectories. We calculate the conditional distributions $P(T_{FR}=t|S=s)$ and $P(S=s|T_{FR}=t)$. We find that the conditional expectation value of first return times of trajectories that visit s distinct sites is ${\mathbb E}[T_{FR}|S=s]=\frac{2}{3}(s^2+s+1)$, and the variance is $Var(T_{FR}|S=s)=\frac{4}{45}(s-1)(s+2)(s^2+s-1)$. We also find that in the asymptotic limit, the conditional expectation value of the number of distinct sites visited by an RW that first returns to the origin at time $t=2n$ is ${\mathbb E}[S|T_{FR}=2n] \simeq \sqrt{\pi n}$, and the variance is $Var(S|T_{FR}=2n) \simeq \pi\left(\frac{\pi}{3}-1\right)n$. These results go beyond the important recent results of Klinger et al. [{\it Phys. Rev. E} {\bf 105}, 034116 (2022)], who derived a closed form expression for the generating function of the joint distribution, but did not go further to extract an explicit expression for the joint distribution itself. The joint distribution provides useful insight on the efficiency of random search processes, in which the aim is to cover as many sites as possible in a given number of steps.

cond-mat.stat-mech↗

arXiv2023

The distribution of the number of cycles in directed and undirected random 2-regular graphs

We present analytical results for the distribution of the number of cycles in directed and undirected random 2-regular graphs (2-RRGs) consisting of $N$ nodes. In directed 2-RRGs each node has one inbound link and one outbound link, while in undirected 2-RRGs each node has two undirected links. Since all the nodes are of degree $k=2$, the resulting networks consist of cycles. These cycles exhibit a broad spectrum of lengths, where the average length of the shortest cycle in a random network instance scales with $\ln N$, while the length of the longest cycle scales with $N$. The number of cycles varies between different network instances in the ensemble, where the mean number of cycles $\langle S \rangle$ scales with $\ln N$. Here we present exact analytical results for the distribution $P_N(S=s)$ of the number of cycles $s$ in ensembles of directed and undirected 2-RRGs, expressed in terms of the Stirling numbers of the first kind. In both cases the distributions converge to a Poisson distribution in the large $N$ limit. The moments and cumulants of $P_N(S=s)$ are also calculated. The statistical properties of directed 2-RRGs are equivalent to the combinatorics of cycles in random permutations of $N$ objects. In this context our results recover and extend known results. In contrast, the statistical properties of cycles in undirected 2-RRGs have not been studied before.

cond-mat.stat-mech↗

arXiv2022

The mean and variance of the distribution of shortest path lengths of random regular graphs

The distribution of shortest path lengths (DSPL) of random networks provides useful information on their large scale structure. In the special case of random regular graphs (RRGs), which consist of $N$ nodes of degree $c \ge 3$, the DSPL, denoted by $P(L=\ell)$, follows a discrete Gompertz distribution. Using the discrete Laplace transform we derive a closed-form expression for the moment generating function of the DSPL of RRGs. From the moment generating function we obtain closed-form expressions for the mean and variance of the DSPL. More specifically, we find that the mean distance between pairs of distinct nodes is given by $\langle L \rangle = \frac{\ln N}{\ln (c-1)} + \frac{1}{2} - \frac{ \ln c - \ln (c-2) +γ}{\ln (c-1)} + \mathcal{O} \left( \frac{\ln N}{N} \right)$, where $γ$ is the Euler-Mascheroni constant. While the leading term is known, this result includes a novel correction term, which yields very good agreement with the results obtained from direct numerical evaluation of $\langle L \rangle$ via the tail-sum formula and with the results obtained from computer simulations. However, it does not account for an oscillatory behavior of $\langle L \rangle$ as a function of $c$ or $N$. These oscillations are negligible in sparse networks but detectable in dense networks. We also derive an expression for the variance ${\rm Var}(L)$ of the DSPL, which captures the overall dependence of the variance on $c$ but does not account for the oscillations. The oscillations are due to the discrete nature of the shell structure around a random node. They reflect the profile of the filling of new shells as $N$ is increased. The results for the mean and variance are compared to the corresponding results obtained in other types of random networks. The relation between the mean distance and the diameter is discussed.

cond-mat.stat-mech↗

arXiv2022

Uncovering the non-equilibrium stationary properties in sparse Boolean networks

Dynamic processes of interacting units on a network are out of equilibrium in general. In the case of a directed tree, the dynamic cavity method provides an efficient tool that characterises the dynamic trajectory of the process for the linear threshold model. However, because of the computational complexity of the method, the analysis has been limited to systems where the largest number of neighbours is small. We devise an efficient implementation of the dynamic cavity method which substantially reduces the computational complexity of the method for systems with discrete couplings. Our approach opens up the possibility to investigate the dynamic properties of networks with fat-tailed degree distribution. We exploit this new implementation to study properties of the non-equilibrium steady-state. We extend the dynamical cavity approach to calculate the pairwise correlations induced by different motifs in the network. Our results suggest that just two basic motifs of the network are able to accurately describe the entire statistics of observed correlations. Finally, we investigate models defined on networks containing bi-directional interactions. We observe that the stationary state associated with networks with symmetric or anti-symmetric interactions is biased towards the active or inactive state respectively, even if independent interaction entries are drawn from a symmetric distribution. This phenomenon, which can be regarded as a form of spontaneous symmetry-breaking, is peculiar to systems formulated in terms of Boolean variables, as opposed to Ising spins.

cond-mat.dis-nn↗

arXiv2021

Overcoming the complexity barrier of the dynamic message-passing method in networks with fat-tailed degree distributions

The dynamic cavity method provides the most efficient way to evaluate probabilities of dynamic trajectories in systems of stochastic units with unidirectional sparse interactions. It is closely related to sum-product algorithms widely used to compute marginal functions from complicated global functions of many variables, with applications in disordered systems, combinatorial optimization and computer science. However, the complexity of the cavity approach grows exponentially with the in-degrees of the interacting units, which creates a de-facto barrier for the successful analysis of systems with fat-tailed in-degree distributions. In this manuscript, we present a dynamic programming algorithm that overcomes this barrier by reducing the computational complexity in the in-degrees from exponential to quadratic, whenever couplings are chosen randomly from (or can be approximated in terms of) discrete, possibly unit-dependent, sets of equidistant values. As a case study, we analyse the dynamics of a random Boolean network with a fat-tailed degree distribution and fully asymmetric binary $\pm J$ couplings, and we use the power of the algorithm to unlock the noise dependent heterogeneity of stationary node activation patterns in such a system.

cond-mat.dis-nn↗

arXiv2021

Cavity and replica methods for the spectral density of sparse symmetric random matrices

We review the problem of how to compute the spectral density of sparse symmetric random matrices, i.e. weighted adjacency matrices of undirected graphs. Starting from the Edwards-Jones formula, we illustrate the milestones of this line of research, including the pioneering work of Bray and Rodgers using replicas. We focus first on the cavity method, showing that it quickly provides the correct recursion equations both for single instances and at the ensemble level. We also describe an alternative replica solution that proves to be equivalent to the cavity method. Both the cavity and the replica derivations allow us to obtain the spectral density via the solution of an integral equation for an auxiliary probability density function. We show that this equation can be solved using a stochastic population dynamics algorithm, and we provide its implementation. In this formalism, the spectral density is naturally written in terms of a superposition of local contributions from nodes of given degree, whose role is thoroughly elucidated. This paper does not contain original material, but rather gives a pedagogical overview of the topic. It is indeed addressed to students and researchers who consider entering the field. Both the theoretical tools and the numerical algorithms are discussed in detail, highlighting conceptual subtleties and practical aspects.

cond-mat.stat-mech↗

arXiv2021

Second largest Eigenpair Statistics for Sparse Graphs

We develop a formalism to compute the statistics of the second largest eigenpair of weighted sparse graphs with $N\gg 1$ nodes, finite mean connectivity and bounded maximal degree, in cases where the top eigenpair statistics is known. The problem can be cast in terms of optimisation of a quadratic form on the sphere with a fictitious temperature, after a suitable deflation of the original matrix model. We use the cavity and replica methods to find the solution in terms of self-consistent equations for auxiliary probability density functions, which can be solved by an improved population dynamics algorithm enforcing eigenvector orthogonality on-the-fly. The analytical results are in perfect agreement with numerical diagonalisation of large (weighted) adjacency matrices, focussing on the cases of random regular and Erdős-Rényi graphs. We further analyse the case of sparse Markov transition matrices for unbiased random walks, whose second largest eigenpair describes the non-equilibrium mode with the largest relaxation time. We also show that the population dynamics algorithm with population size $N_P$ does not actually capture the thermodynamic limit $N\to\infty$ as commonly assumed: the accuracy of the population dynamics algorithm has a strongly non-monotonic behaviour as a function of $N_P$, thus implying that an optimal size $N_P^\star=N_P^\star(N)$ must be chosen to best reproduce the results from numerical diagonalisation of graphs of finite size $N$.

cond-mat.stat-mech↗

arXiv2020

Opinion dynamics with emergent collective memory: the impact of a long and heterogeneous news history

In modern society people are being exposed to numerous information, with some of them being frequently repeated or more disruptive than others. In this paper we use a model of opinion dynamics to study how this news impact the society. In particular, our study aims to explain how the exposure of the society to certain events deeply change people's perception of the present and future. The evolution of opinions which we consider is influenced both by external information and the pressure of the society. The latter includes imitation, differentiation, homophily and its opposite, xenophobia. The combination of these ingredients gives rise to a collective memory effect, which is triggered by external information. In this paper we focus our attention on how this memory arises when the order of appearance of external news is random. We will show which characteristics a piece of news needs to have in order to be embedded in the society's memory. We will also provide an analytical way to measure how many information a society can remember when an extensive number of news items is presented. Finally we will show that, when a certain piece of news is present in the society's history, even a distorted version of it is sufficient to trigger the memory of the originally stored information.

physics.soc-ph↗

arXiv2020

Statistical analysis of edges and bredges in configuration model networks

A bredge (bridge-edge) is an edge whose deletion would split the network component on which it resides into two components. Bredges are vulnerable links that play an important role in network collapse processes, which may result from node or link failures, attacks or epidemics. Therefore, the abundance and properties of bredges affect the resilience of the network. We present analytical results for the statistical properties of bredges in configuration model networks. Using a generating function approach based on the cavity method, we calculate the probability $\hat P(e\in{\rm B})$ that a random edge e in a configuration model network with degree distribution P(k) is a bredge (B). We also calculate the joint degree distribution $\hat P(k,k'|{\rm B})$ of the end-nodes of a random bredge. We examine the distinct properties of bredges on the giant component (GC) and on the finite tree components (FC) of the network. On the finite components all the edges are bredges and there are no degree-degree correlations. We calculate the probability $\hat P(e\in{\rm B}|{\rm GC})$ that a random edge on the giant component is a bredge. We also calculate the joint degree distribution $\hat P(k,k'|{\rm B},{\rm GC})$ of the end-nodes of bredges and the joint degree distribution $\hat P(k,k'|{\rm NB},{\rm GC})$ of the end-nodes of non-bredge (NB) edges on the giant component. Surprisingly, it is found that the degrees k and k' of the end-nodes of bredges are correlated, while the degrees of the end-nodes of NB edges are uncorrelated. We thus conclude that all the degree-degree correlations on the giant component are concentrated on the bredges. We calculate the covariance of end-nodes of bredges and show it is negative, namely bredges tend to connect high degree nodes to low degree nodes. The implications of the results are discussed in the context of common attack scenarios and dismantling processes.

cond-mat.dis-nn↗

arXiv2020

Opinion dynamics with memory: how a society is shaped by its own past

In order to understand the development of common orientation of opinions in the modern world we propose a model of a society described as a large collection of agents that exchange their expressed opinions under the influence of their mutual interactions and external events. In particular we introduce an interaction bias which creates a collective memory effect such that the society is able to store and recall information coming from several external signals. Our model shows how the inner structure of the society and its future reactions can be shaped by its own history. We will provide an analytical explanation of how this might occur and we will show the emergent similarity between the reaction of a society modelled in this way and the Hopfield mechanism for information retrieval.

physics.soc-ph↗

arXiv2020

Percolation on the gene regulatory network

We consider a simplified model for gene regulation, where gene expression is regulated by transcription factors (TFs), which are single proteins or protein complexes. Proteins are in turn synthesised from expressed genes, creating a feedback loop of regulation. This leads to a directed bipartite network in which a link from a gene to a TF exists if the gene codes for a protein contributing to the TF, and a link from a TF to a gene exists if the TF regulates the expression of the gene. Both genes and TFs are modelled as binary variables, which indicate, respectively, whether a gene is expressed or not, and a TF is synthesised or not. We consider the scenario where for a TF to be synthesised, all of its contributing genes must be expressed. This results in an ``AND'' gate logic for the dynamics of TFs. By adapting percolation theory to directed bipartite graphs, evolving according to the AND logic dynamics, we are able to determine the necessary conditions, in the network parameter space, under which bipartite networks can support a multiplicity of stable gene expression patterns, under noisy conditions, as required in stable cell types. In particular, the analysis reveals the possibility of a bi-stability region, where the extensive percolating cluster is or is not resilient to perturbations. This is remarkably different from the transition observed in standard percolation theory. Finally, we consider perturbations involving single node removal that mimic gene knockout experiments. Results reveal the strong dependence of the gene knockout cascade on the logic implemented in the underlying network dynamics, highlighting in particular that avalanche sizes cannot be easily related to gene-gene interaction networks.

q-bio.MN↗

arXiv2019

Glassy dynamics on networks: local spectra and return probabilities

The slow relaxation and aging of glassy systems can be modelled as a Markov process on a simplified rough energy landscape: energy minima where the system tends to get trapped are taken as nodes of a random network, and the dynamics are governed by the transition rates among these. In this work we consider the case of purely activated dynamics, where the transition rates only depend on the depth of the departing trap. The random connectivity and the disorder in the trap depths make it impossible to solve the model analytically, so we base our analysis on the spectrum of eigenvalues $λ$ of the master operator. We compute the local density of states $ρ(λ|τ)$ for traps with a fixed lifetime $τ$ by means of the cavity method. This exhibits a power law behaviour $ρ(λ|τ)\simτ|λ|^T$ in the regime of small relaxation rates $|λ|$, which we rationalize using a simple analytical approximation. In the time domain, we find that the probabilities of return to a starting node have a power law-tail that is determined by the distribution of excursion times $F(t)\sim t^{-(T+1)}$. We show that these results arise only by the combination of finite configuration space connectivity and glassy disorder, and interpret them in a simple physical picture dominated by jumps to deep neighbouring traps.

cond-mat.dis-nn↗

arXiv2019

Generating random networks that consist of a single connected component with a given degree distribution

We present a method for the construction of ensembles of random networks that consist of a single connected component with a given degree distribution. This approach extends the construction toolbox of random networks beyond the configuration model framework, in which one controls the degree distribution but not the number of components and their sizes. Unlike configuration model networks, which are completely uncorrelated, the resulting single-component networks exhibit degree-degree correlations. Moreover, they are found to be disassortative, namely high-degree nodes tend to connect to low-degree nodes and vice versa. We demonstrate the method for single-component networks with ternary, exponential and power-law degree distributions.

cond-mat.dis-nn↗

arXiv2018

Statistical analysis of articulation points in configuration model networks

An articulation point (AP) in a network is a node whose deletion would split the network component on which it resides into two or more components. APs are vulnerable spots that play an important role in network collapse processes, which may result from node failures, attacks or epidemics. Therefore, the abundance and properties of APs affect the resilience of the network to these collapse scenarios. We present analytical results for the statistical properties of APs in configuration model networks. In order to quantify their abundance, we calculate the probability $P(i \in {\rm AP})$, that a random node, i, in a configuration model network with P(K=k), is an AP. We also obtain the conditional probability $P(i \in {\rm AP}|k)$ that a random node of degree k is an AP, and find that high degree nodes are more likely to be APs than low degree nodes. Using Bayes' theorem, we obtain the conditional degree distribution, $P(K=k|{\rm AP})$, over the set of APs and compare it to P(K=k). We propose a new centrality measure based on APs: each node can be characterized by its articulation rank, r, which is the number of components that would be added to the network upon deletion of that node. For nodes which are not APs the articulation rank is $r=0$, while for APs $r \ge 1$. We obtain a closed form expression for the distribution of articulation ranks, P(R=r). Configuration model networks often exhibit a coexistence between a giant component and finite components. To examine the distinct properties of APs on the giant and on the finite components, we calculate the probabilities presented above separately for the giant and the finite components. We apply these results to ensembles of configuration model networks with a Poisson, exponential and power-law degree distributions. The implications of these results are discussed in the context of common attack scenarios and network dismantling processes.

cond-mat.dis-nn↗

arXiv2018

Spectral properties of the trap model on sparse networks

One of the simplest models for the slow relaxation and aging of glasses is the trap model by Bouchaud and others, which represents a system as a point in configuration-space hopping between local energy minima. The time evolution depends on the transition rates and the network of allowed jumps between the minima. We consider the case of sparse configuration-space connectivity given by a random graph, and study the spectral properties of the resulting master operator. We develop a general approach using the cavity method that gives access to the density of states in large systems, as well as localisation properties of the eigenvectors, which are important for the dynamics. We illustrate how, for a system with sparse connectivity and finite temperature, the density of states and the average inverse participation ratio have attributes that arise from a non-trivial combination of the corresponding mean field (fully connected) and random walk (infinite temperature) limits. In particular, we find a range of eigenvalues for which the density of states is of mean-field form but localisation properties are not, and speculate that the corresponding eigenvectors may be concentrated on extensively many clusters of network sites.

cond-mat.dis-nn↗

arXiv2018

Revealing the Micro-Structure of the Giant Component in Random Graph Ensembles

The micro-structure of the giant component of the Erd{\H o}s-Rényi network and other configuration model networks is analyzed using generating function methods. While configuration model networks are uncorrelated, the giant component exhibits a degree distribution which is different from the overall degree distribution of the network and includes degree-degree correlations of all orders. We present exact analytical results for the degree distributions as well as higher order degree-degree correlations on the giant components of configuration model networks. We show that the degree-degree correlations are essential for the integrity of the giant component, in the sense that the degree distribution alone cannot guarantee that it will consist of a single connected component. To demonstrate the importance and broad applicability of these results, we apply them to the study of the distribution of shortest path lengths on the giant component, percolation on the giant component and the spectra of sparse matrices defined on the giant component. We show that by using the degree distribution on the giant component, one obtains high quality results for these properties, which can be further improved by taking the degree-degree correlations into account. This suggests that many existing methods, currently used for the analysis of the whole network, can be adapted in a straightforward fashion to yield results conditioned on the giant component.

cond-mat.stat-mech↗

arXiv2016

Distance distribution in configuration model networks

We present analytical results for the distribution of shortest path lengths between random pairs of nodes in configuration model networks. The results, which are based on recursion equations, are shown to be in good agreement with numerical simulations for networks with degenerate, binomial and power-law degree distributions. The mean, mode and variance of the distribution of shortest path lengths are also evaluated. These results provide expressions for central measures and dispersion measures of the distribution of shortest path lengths in terms of moments of the degree distribution, illuminating the connection between the two distributions.

cond-mat.dis-nn↗