Random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology. In recent years, relevant mathematical results have been obtained in this field, and new ideas have been introduced, which can be fruitfully extended to different areas and disciplines. Here we aim at giving a brief but comprehensive perspective of these progresses, with a particular emphasis on physical aspects.

The thermodynamic properties of non interacting bosons on a complex network can be strongly affected by topological inhomogeneities. The latter give rise to anomalies in the density of states that can induce Bose-Einstein condensation in low dimensional systems also in absence of external confining potentials. The anomalies consist in energy regions composed of an infinite number of states with vanishing weight in the thermodynamic limit. We present a rigorous result providing the general conditions for the occurrence of Bose-Einstein condensation on complex networks in presence of anomalous spectral regions in the density of states. We present results on spectral properties for a wide class of graphs where the theorem applies. We study in detail an explicit geometrical realization, the comb lattice, which embodies all the relevant features of this effect and which can be experimentally implemented as an array of Josephson Junctions.

We show that spatial Bose-Einstein condensation of non-interacting bosons occurs in dimension d < 2 over discrete structures with inhomogeneous topology and with no need of external confining potentials. Josephson junction arrays provide a physical realization of this mechanism. The topological origin of the phenomenon may open the way to the engineering of quantum devices based on Bose-Einstein condensation. The comb array, which embodies all the relevant features of this effect, is studied in detail.PACS numbers: 03.75. Fi, 85.25.Cp, The recent impressive experimental demonstration of Bose-Einstein Condensation (BEC) [1] has stimulated a new wealth of theoretical work aimed to better understanding its basic mechanisms [2] and, possibly, to exploit its consequences for the engineering of quantum devices.It is well known [3] that for an ideal gas of Bose particles BEC does not occur in dimension d ≤ 2, and an ′′ ad hoc ′′ external confining potential is needed to reach the required density of states. The same is true for free bosons living on regular periodic lattices, while the result cannot be extended to more general discrete structures lacking translational invariance.In the following we shall prove that even for d < 2 [4] non-interacting bosons may lead to Bose-Einstein condensation into a single non-degenerate state, provided one resorts to a suitable discrete non-homogeneous support structure: indeed, when the bosonic kinetic degrees of freedom do not depend on metric features only, the particles may feel a sort of effective interaction due to topology. The proposed mechanism for BEC in lower dimensional systems is then a pure effect of the structure of the ambient space and avoids as well the need of resorting to external random potentials as the ones investigated by Huang in [2]; this is a very desirable feature in view of engineering real quantum devices.In practice, the behavior of free bosons over generic discrete structures is made experimentally accessible through the realization of suitable arrays of Josephson junctions. The latter are devices that can be engineered in such a way as to realize a variety of non-homogeneous patterns. We shall show indeed that classical Josephson junction arrays arranged in a non-homogeneous geometry -not even necessarily planar -provide an example of the proposed mechanism for BEC, leading to a single state spatial condensation.Theoretical studies of Josephson junction arrays are based on the short-range Bose-Hubbard model, since the phase diagram of Josephson junction arrays may be derived [5] from an Hamiltonian describing bosons with repulsive interactions over a lattice. In d = 1 the phase diagram has been studied by analytical [6] and quantum Monte Carlo methods [7]; experimentally, Josephson junction arrays are used to study interacting bosons in one dimension. For a generic array the corresponding Hamiltonian is given bywhere A ij is the adjacency matrix: A ij = 1 if the sites i and j are nearest neighbors and A ij = 0 otherwise; a † i creates a boson at site...

We investigated the efficiency of attack strategies to network nodes when targeting several complex model and real-world networks. We tested 5 attack strategies, 3 of which were introduced in this work for the first time, to attack 3 model networks (Erdos and Renyi, Barabasi and Albert preferential attachment network, and scale-free network configuration models) and 3 real networks (Gnutella peer-to-peer network, email network of the University of Rovira i Virgili, and immunoglobulin interaction network). Nodes were removed sequentially according to the importance criterion defined by the attack strategy, and we used the size of the largest connected component (LCC) as a measure of network damage. We found that the efficiency of attack strategies (fraction of nodes to be deleted for a given reduction of LCC size) depends on the topology of the network, although attacks based on either the number of connections of a node or betweenness centrality were often the most efficient strategies. Sequential deletion of nodes in decreasing order of betweenness centrality was the most efficient attack strategy when targeting real-world networks. The relative efficiency of attack strategies often changed during the sequential removal of nodes, especially for networks with power-law degree distribution. © 2014 Elsevier B.V. All rights reserved