Self-isolation or borders closing: What prevents the spread of the epidemic better?

Pandemic propagation of COVID-19 motivated us to discuss the impact of the human network clustering on epidemic spreading. Today, there are two clustering mechanisms which prevent of uncontrolled disease propagation in a connected network: an "internal" clustering, which mimics self-isolation (SI) in local naturally arranged communities, and an "external" clustering, which looks like a sharp frontiers closing (FC) between cities and countries, and which does not care about the natural connections of network agents. SI networks are "evolutionarily grown" under the condition of maximization of small cliques in the entire network, while FC networks are instantly created. Running the standard SIR model on clustered SI and FC networks, we demonstrate that the evolutionary grown clustered network prevents the spread of an epidemic better than the instantly clustered network with similar parameters. We find that SI networks have the scale-free property for the degree distribution P(k)∼k^{η}, with a small critical exponent -2<η<-1. We argue that the scale-free behavior emerges as a result of the randomness in the initial degree distributions.

Spectral peculiarity and criticality of a human connectome.

We have performed the comparative spectral analysis of structural connectomes for various organisms using open-access data. Our results indicate new peculiar features of connectomes of higher organisms. We found that the spectral density of adjacency matrices of human connectome has maximal deviation from the one of randomized network, compared to other organisms. Considering the network evolution induced by the preference of 3-cycles formation, we discovered that for macaque and human connectomes the evolution with the conservation of local clusterization is crucial, while for primitive organisms the conservation of averaged clusterization is sufficient. Investigating for the first time the level spacing distribution of the spectrum of human connectome Laplacian matrix, we explicitly demonstrate that the spectral statistics corresponds to the critical regime, which is hybrid of Wigner-Dyson and Poisson distributions. This observation provides strong support for debated statement of the brain criticality.

Finite plateau in spectral gap of polychromatic constrained random networks.

We consider critical behavior in the ensemble of polychromatic Erdos-Rényi networks and regular random graphs, where network vertices are painted in different colors. The links can be randomly removed and added to the network subject to the condition of the vertex degree conservation. In these constrained graphs we run the Metropolis procedure, which favors the connected unicolor triads of nodes. Changing the chemical potential, µ, of such triads, for some wide region of µ, we find the formation of a finite plateau in the number of intercolor links, which exactly matches the finite plateau in the network algebraic connectivity (the value of the first nonvanishing eigenvalue of the Laplacian matrix, λ_{2}). We claim that at the plateau the spontaneously broken Z_{2} symmetry is restored by the mechanism of modes collectivization in clusters of different colors. The phenomena of a finite plateau formation holds also for polychromatic networks with M≥2 colors. The behavior of polychromatic networks is analyzed via the spectral properties of their adjacency and Laplacian matrices.

Spontaneous symmetry breaking and phase coexistence in two-color networks.

We consider an equilibrium ensemble of large Erdos-Renyi topological random networks with fixed vertex degree and two types of vertices, black and white, prepared randomly with the bond connection probability p. The network energy is a sum of all unicolor triples (either black or white), weighted with chemical potential of triples µ. Minimizing the system energy, we see for some positive µ the formation of two predominantly unicolor clusters, linked by a string of N_{bw} black-white bonds. We have demonstrated that the system exhibits critical behavior manifested in the emergence of a wide plateau on the N_{bw}(µ) curve, which is relevant to a spinodal decomposition in first-order phase transitions. In terms of a string theory, the plateau formation can be interpreted as an entanglement between baby universes in two-dimensional gravity. We conjecture that the observed classical phenomenon can be considered as a toy model for the chiral condensate formation in quantum chromodynamics.

Eigenvalue tunneling and decay of quenched random network.

We consider the canonical ensemble of N-vertex Erdos-Rényi (ER) random topological graphs with quenched vertex degree, and with fugacity µ for each closed triple of bonds. We claim complete defragmentation of large-N graphs into the collection of [p^{-1}] almost full subgraphs (cliques) above critical fugacity, µ_{c}, where p is the ER bond formation probability. Evolution of the spectral density, ρ(λ), of the adjacency matrix with increasing µ leads to the formation of a multizonal support for µ>µ_{c}. Eigenvalue tunneling from the central zone to the side one means formation of a new clique in the defragmentation process. The adjacency matrix of the network ground state has a block-diagonal form, where the number of vertices in blocks fluctuates around the mean value Np. The spectral density of the whole network in this regime has triangular shape. We interpret the phenomena from the viewpoint of the conventional random matrix model and speculate about possible physical applications.

Islands of stability in motif distributions of random networks.

We consider random nondirected networks subject to dynamics conserving vertex degrees and study, analytically and numerically, equilibrium three-vertex motif distributions in the presence of an external field h coupled to one of the motifs. For small h, the numerics is well described by the "chemical kinetics" for the concentrations of motifs based on the law of mass action. For larger h, a transition into some trapped motif state occurs in Erdos-Rényi networks. We explain the existence of the transition by employing the notion of the entropy of the motif distribution and describe it in terms of a phenomenological Landau-type theory with a nonzero cubic term. A localization transition should always occur if the entropy function is nonconvex. We conjecture that this phenomenon is the origin of the motifs' pattern formation in real evolutionary networks.

Planar diagrams from optimization for concave potentials.

We propose a toy model of a heteropolymer chain capable of forming planar secondary structures typical for RNA molecules. In this model, the sequential intervals between neighboring monomers along a chain are considered as quenched random variables, and energies of nonlocal bonds are assumed to be concave functions of those intervals. A few factors are neglected: the contribution of loop factors to the partition function, the variation in energies of different types of complementary nucleotides, the stacking interactions, and constraints on the minimal size of loops. However, the model captures well the formation of folded structures without pseudoknots in an arbitrary sequence of nucleotides. Using the optimization procedure for a special class of concave-type potentials, borrowed from optimal transport analysis, we derive the local difference equation for the ground state free energy of the chain with the planar (RNA-like) architecture of paired links. We consider various distribution functions of intervals between neighboring monomers (truncated Gaussian and scale free) and demonstrate the existence of a topological crossover from sequential to essentially nested configurations of paired links.

New alphabet-dependent morphological transition in random RNA alignment.

We study the fraction f of nucleotides involved in the formation of a cactuslike secondary structure of random heteropolymer RNA-like molecules. In the low-temperature limit, we study this fraction as a function of the number c of different nucleotide species. We show, that with changing c, the secondary structures of random RNAs undergo a morphological transition: f(c)→1 for c≤c(cr) as the chain length n goes to infinity, signaling the formation of a virtually perfect gapless secondary structure; while f(c)<1 for c>c(cr), which means that a nonperfect structure with gaps is formed. The strict upper and lower bounds 2≤c(cr)≤4 are proven, and the numerical evidence for c(cr) is presented. The relevance of the transition from the evolutional point of view is discussed.