Memory has a great impact on the evolution of every process related to human societies. Among them, the evolution of an epidemic is directly related to the individuals' experiences. Indeed, any real epidemic process is clearly sustained by a non-Markovian dynamics: memory effects play an essential role in the spreading of diseases. Including memory effects in the susceptible-infected-recovered (SIR) epidemic model seems very appropriate for such an investigation. Thus, the memory prone SIR model dynamics is investigated using fractional derivatives. The decay of long-range memory, taken as a power-law function, is directly controlled by the order of the fractional derivatives in the corresponding nonlinear fractional differential evolution equations. Here we assume "fully mixed" approximation and show that the epidemic threshold is shifted to higher values than those for the memoryless system, depending on this memory "length" decay exponent. We also consider the SIR model on structured networks and study the effect of topology on threshold points in a non-Markovian dynamics. Furthermore, the lack of access to the precise information about the initial conditions or the past events plays a very relevant role in the correct estimation or prediction of the epidemic evolution. Such a "constraint" is analyzed and discussed.

We generalize the theory of k-core percolation on complex networks to k-core percolation on multiplex networks, where k≡(k(1),k(2),...,k(M)). Multiplex networks can be defined as networks with vertices of one kind but M different types of edges, representing different types of interactions. For such networks, the k-core is defined as the largest subgraph in which each vertex has at least k(i) edges of each type, i=1,2,...,M. We derive self-consistency equations to obtain the birth points of the k-cores and their relative sizes for uncorrelated multiplex networks with an arbitrary degree distribution. To clarify our general results, we consider in detail multiplex networks with edges of two types and solve the equations in the particular case of Erdős-Rényi and scale-free multiplex networks. We find hybrid phase transitions at the emergence points of k-cores except the (1,1)-core for which the transition is continuous. We apply the k-core decomposition algorithm to air-transportation multiplex networks, composed of two layers, and obtain the size of (k(1),k(2))-cores.

We check the universality properties of the two-dimensional Abelian sandpile model by computing some of its properties on the honeycomb lattice. Exact expressions for unit height correlation functions in presence of boundaries and for different boundary conditions are derived. Also, we study the statistics of the boundaries of avalanche waves by using the theory of SLE and suggest that these curves are conformally invariant and described by SLE 2 .

The spread of one disease, in some cases, can stimulate the spreading of another infectious disease. Here, we treat analytically a symmetric co-infection model for spreading of two diseases on a twolayer multiplex network. We allow layer overlapping, but we assume that each layer is random and locally loop-less. Infection with one of the diseases increases the probability of getting infected with the other. Using the generating function method, we calculate exactly the fraction of individuals infected with both diseases (so-called co-infected clusters) in the stationary state, as well as the epidemic spreading thresholds and the phase diagram of the model. With increasing cooperation, we observe a tricritical point and the type of transition changes from continuous to hybrid. Finally, we compare the co-infected clusters in the case of co-operating diseases with the so-called "viable" clusters in networks with dependencies.

We introduce a k-leaf removal algorithm as a generalization of the so-called leaf removal algorithm. In this pruning algorithm, vertices of degree smaller than k, together with their first nearest neighbors and all incident edges are progressively removed from a random network. As the result of this pruning the network is reduced to a subgraph which we call the Generalized k-core (Gkcore). Performing this pruning for the sequence of natural numbers k, we decompose the network into a hierarchy of progressively nested Gk-cores. We present an analytical framework for description of Gk-core percolation for undirected uncorrelated networks with arbitrary degree distributions (configuration model). To confirm our results, we also derive rate equations for the k-leaf removal algorithm which enable us to obtain the structural characteristics of the Gk-cores in another way. Also we apply our algorithm to a number of real-world networks and perform the Gk-core decomposition for them.