To better understand the temporal characteristics and the lifetime of fluctuations in stochastic processes in networks, we investigated diffusive persistence in various graphs. Global diffusive persistence is defined as the fraction of nodes for which the diffusive field at a site (or node) has not changed sign up to time t (or, in general, that the node remained active or inactive in discrete models). Here we investigate disordered and random networks and show that the behavior of the persistence depends on the topology of the network. In two-dimensional (2D) disordered networks, we find that above the percolation threshold diffusive persistence scales similarly as in the original 2D regular lattice, according to a power law P(t, L) ti t-theta with an exponent theta ' 0.186, in the limit of large linear system size L. At the percolation threshold, however, the scaling exponent changes to theta ' 0.141, as the result of the interplay of diffusive persistence and the underlying structural transition in the disordered lattice at the percolation threshold. Moreover, studying finite -size effects for 2D lattices at and above the percolation threshold, we find that at the percolation threshold, the long-time asymptotic value obeys a power law P(t, L) ti L-z theta with z ' 2.86 instead of the value of z = 2 normally associated with finite -size effects on 2D regular lattices. In contrast, we observe that in random networks without a local regular structure, such as Erdos-Renyi networks, no simple power -law scaling behavior exists above the percolation threshold.

COVID-19 is not a universal killer. We study the spread of COVID-19 at the county level for the United States up until the 15$^{th}$ of August, 2020. We show that the prevalence of the disease and the death rate are correlated with the local socio-economic conditions often going beyond local population density distributions, especially in rural areas. We correlate the COVID-19 prevalence and death rate with data from the US Census Bureau and point out how the spreading patterns of the disease show asymmetries in urban and rural areas separately and are preferentially affecting the counties where a large fraction of the population is non-white. Our findings can be used for more targeted policy building and deployment of resources for future occurrence of a pandemic due to SARS-CoV-2. Our methodology, based on interpretable machine learning and game theory, can be extended to study the spread of other diseases.

We introduce a continuous-time analog solver for MaxSAT, a quintessential class of NP-hard discrete optimization problems, where the task is to find a truth assignment for a set of Boolean variables satisfying the maximum number of given logical constraints. We show that the scaling of an invariant of the solver's dynamics, the escape rate, as function of the number of unsatisfied clauses can predict the global optimum value, often well before reaching the corresponding state. We demonstrate the performance of the solver on hard MaxSAT competition problems. We then consider the two-color Ramsey number $R(m,m)$ problem, translate it to SAT, and apply our algorithm to the still unknown $R(5,5)$. We find edge colorings without monochromatic 5-cliques for complete graphs up to 42 vertices, while on 43 vertices we find colorings with only two monochromatic 5-cliques, the best coloring found so far, supporting the conjecture that $R(5,5) = 43$.

Many real-life optimization problems can be formulated in Boolean logic as MaxSAT, a class of problems where the task is finding Boolean assignments to variables satisfying the maximum number of logical constraints. Since MaxSAT is NP-hard, no algorithm is known to efficiently solve these problems. Here we present a continuous-time analog solver for MaxSAT and show that the scaling of the escape rate, an invariant of the solver’s dynamics, can predict the maximum number of satisfiable constraints, often well before finding the optimal assignment. Simulating the solver, we illustrate its performance on MaxSAT competition problems, then apply it to two-color Ramsey number R ( m , m ) problems. Although it finds colorings without monochromatic 5-cliques of complete graphs on N ≤ 42 vertices, the best coloring for N = 43 has two monochromatic 5-cliques, supporting the conjecture that R (5, 5) = 43. This approach shows the potential of continuous-time analog dynamical systems as algorithms for discrete optimization.

Efficient solutions to NP-complete problems would significantly benefit both science and industry. However, such problems are intractable on digital computers based on the von Neumann architecture, thus creating the need for alternative solutions to tackle such problems. Recently, a deterministic, continuous-time dynamical system (CTDS) was proposed (Nat.Phys. {\bf 7}(12), 966 (2011)) to solve a representative NP-complete problem, Boolean Satisfiability (SAT). This solver shows polynomial analog time-complexity on even the hardest benchmark $k$-SAT ($k \geq 3$) formulas, but at an energy cost through exponentially driven auxiliary variables. This paper presents a novel analog hardware SAT solver, AC-SAT, implementing the CTDS via incorporating novel, analog circuit design ideas. AC-SAT is intended to be used as a co-processor and is programmable for handling different problem specifications. It is especially effective for solving hard $k$-SAT problem instances that are challenging for algorithms running on digital machines. Furthermore, with its modular design, AC-SAT can readily be extended to solve larger size problems, while the size of the circuit grows linearly with the product of the number of variables and number of clauses. The circuit is designed and simulated based on a 32nm CMOS technology. SPICE simulation results show speedup factors of $\sim$10$^4$ on even the hardest 3-SAT problems, when compared with a state-of-the-art SAT solver on digital computers. As an example, for hard problems with $N=50$ variables and $M=212$ clauses, solutions are found within from a few $ns$ to a few hundred $ns$.

Understanding the rat neurochemical connectome is fundamental for exploring neuronal information processing. By using advanced data mining, supervised machine learning, and network analysis, this study integrates over 5 decades of neuroanatomical investigations into a multiscale, multilayer neurochemical connectome of the rat brain. This neurochemical connectivity database (ChemNetDB) is supported by comprehensive systematically-determined receptor distribution maps. The rat connectome has an onion-type structural organization and shares a number of structural features with mesoscale connectomes of mouse and macaque. Furthermore, we demonstrate that extremal values of graph theoretical measures (e.g., degree and betweenness) are associated with evolutionary-conserved deep brain structures such as amygdala, bed nucleus of the stria terminalis, dorsal raphe, and lateral hypothalamus, which regulate primitive, yet fundamental functions, such as circadian rhythms, reward, aggression, anxiety, and fear. The ChemNetDB is a freely available resource for systems analysis of motor, sensory, emotional, and cognitive information processing.

Transient chaos is a ubiquitous phenomenon characterizing the dynamics of phase-space trajectories evolving towards a steady-state attractor in physical systems as diverse as fluids, chemical reactions, and condensed matter systems. Here we show that transient chaos also appears in the dynamics of certain efficient algorithms searching for solutions of constraint satisfaction problems that include scheduling, circuit design, routing, database problems, and even Sudoku. In particular, we present a study of the emergence of hardness in Boolean satisfiability (k-SAT), a canonical class of constraint satisfaction problems, by using an analog deterministic algorithm based on a system of ordinary differential equations. Problem hardness is defined through the escape rate κ, an invariant measure of transient chaos of the dynamical system corresponding to the analog algorithm, and it expresses the rate at which the trajectory approaches a solution. We show that for a given density of constraints and fixed number of Boolean variables N, the hardness of formulas in random k-SAT ensembles has a wide variation, approximable by a lognormal distribution. We also show that when increasing the density of constraints α, hardness appears through a second-order phase transition at α_{χ} in the random 3-SAT ensemble where dynamical trajectories become transiently chaotic. A similar behavior is found in 4-SAT as well, however, such a transition does not occur for 2-SAT. This behavior also implies a novel type of transient chaos in which the escape rate has an exponential-algebraic dependence on the critical parameter κ∼N^{B|α-α_{χ}|^{1-γ}} with 0<γ<1. We demonstrate that the transition is generated by the appearance of metastable basins in the solution space as the density of constraints α is increased.

by Melinda Varga There are many problems, which cannot be solved with today’s digital computers. One of the most studied such problem is Boolean satisfiability (k-SAT), which asks to find the truth-values for a set of Boolean variables in a way to satisfy a given number of constraints. This problem appears in many real-world applications, and it has a key role in the theory of computational complexity and in particular NP-completeness: if one would find an efficient (polynomial-time) algorithm to solve k-SAT (for k ≥ 3), then we would be able to generate solutions efficiently to all problems from the NP class (Cook-Levin theorem), i.e., to a very large number of hard problems. The Thesis focuses on the k-SAT problem and presents a novel approach to it, using a deterministic continuous-time dynamical system. This dynamical system solves the problem efficiently (in polynomial continuous-time) at the expense of exponential fluctuations in its energy function, while it also shows that problem hardness is translated into a transiently chaotic behavior of the analog trajectories by this system. We use the escape rate, an invariant measure of transient chaos, to show that hardness appears through a second-order phase transition in the random 3-SAT ensemble and a similar behavior is found in 4-SAT as well, however, such transition does not occur for 2-SAT (which is in class P, hence easily solvable by a polynomial-time algorithm). Since the solver (i.e., the dynamical system expressed as ordinary differential

Multilevel societies, containing hierarchically nested social levels, are remarkable social structures whose origins are unclear. The social relationships of sperm whales are organized in a multilevel society with an upper level composed of clans of individuals communicating using similar patterns of clicks (codas). Using agent-based models informed by an 18-year empirical study, we show that clans are unlikely products of stochastic processes (genetic or cultural drift) but likely originate from cultural transmission via biased social learning of codas. Distinct clusters of individuals with similar acoustic repertoires, mirroring the empirical clans, emerge when whales learn preferentially the most common codas (conformism) from behaviourally similar individuals (homophily). Cultural transmission seems key in the partitioning of sperm whales into sympatric clans. These findings suggest that processes similar to those that generate complex human cultures could not only be at play in non-human societies but also create multilevel social structures in the wild.

Evolutionary 2×2 games are studied with players located on a square lattice. During the evolution the randomly chosen neighboring players try to maximize their collective income by adopting a random strategy pair with a probability dependent on the difference of their summed payoffs between the final and initial states assuming quenched strategies in their neighborhood. In the case of the anticoordination game this system behaves like an antiferromagnetic kinetic Ising model. Within a wide region of social dilemmas this dynamical rule supports the formation of similar spatial arrangement of the cooperators and defectors ensuring the optimum total payoff if the temptation to choose defection exceeds a threshold value dependent on the sucker's payoff. The comparison of the results with those achieved for pairwise imitation and myopic strategy updates has indicated the relevant advantage of pairwise collective strategy update in the maintenance of cooperation.

Random networks with co-existing positive and negative links are studied from the viewpoint of the NP hard correlation clustering problem. The task is to produce a clustering of the vertices which maximizes the number of positive edges within clusters and the number of negative edges between clusters. Simulated annealing, Monte Carlo renormalization and molecular dynamics optimization are used to find the optimal cluster structure. Recently, this problem was studied for globally coupled systems and an interesting phase-transition-like phenomenon was predicted: in the thermodynamic limit the relative size of the largest cluster, r, exhibits a step-like behavior as a function of the density of positive links q (r = 0 if q < 1/2 and r = 1 if q > 1/2). Here we prove that when considering random networks with a constant bond density, the same phase transition is expected. A totally different result emerges however, when networks with a fixed average number of connections per node are considered. In such cases a nontrivial spin-glass-type behavior is found, where the location of the critical point shifts toward q > 1/2 values. The results also suggest that instead of the simple step-like behavior, the r(q) curve has a more complex shape, which depends on the specific topology of the considered network.

The development of anti-factor VIII/IX antibodies (inhibitor-induction) and the transmission of viral infections are the most significant complications of haemophilia treatment. The Humafactor-8 and Humafactor-9 are high-purity pasteurized factor VIII and IX concentrates, which are produced from pooled plasma of Hungarian donors by ion-exchange chromatography. The clinical study has been accomplished in two steps: first we have demonstrated the biological efficacy of the concentrates in a phase IV trial. After that we followed 13 patients with severe haemophilia for 6 months in respect of virus-safety and inhibitor-induction. According to our results the recently developed domestic FVIII/FIX concentrates display appropriate biological activities and they are safe as blood-borne virus-transmission and immunogenicity are concerned.