Multiplex networks (a system of multiple networks that have different types of links but share a common set of nodes) arise naturally in a wide spectrum of fields. Theoretical studies show that in such multiplex networks, correlated edge dynamics between the layers can have a profound effect on dynamical processes. However, how to extract the correlations from real-world systems is an outstanding challenge. Here we introduce the Multiplex Markov chain to quantify correlations in edge dynamics found in longitudinal data of multiplex networks. By comparing the results obtained from the multiplex perspective to a null model which assumes layers in a network are independent, we can identify real correlations as distinct from simultaneous changes that occur due to random chance. We use this approach on two different data sets: the network of trade and alliances between nation states, and the email and co-commit networks between developers of open source software. We establish the existence of “dynamical spillover” showing the correlated formation (or deletion) of edges of different types as the system evolves. The details of the dynamics over time provide insight into potential causal pathways.
Collective organization in matter plays a significant role in its expressed physical properties. Typically, it is detected via an order parameter, appropriately defined for each given system's observed emergent patterns. Recent developments in information theory, however, suggest quantifying collective organization in a system-and phenomenon-agnostic way: decomposing the system's thermodynamic entropy density into a localized entropy, that is solely contained in the dynamics at a single location, and a bound entropy, that is stored in space as domains, clusters, excitations, or other emergent structures. As a concrete demonstration, we compute this decomposition and related quantities explicitly for the nearest-neighbor Ising model on the 1D chain, on the Bethe lattice with coordination number k = 3, and on the 2D square lattice, illustrating its generality and the functional insights it gives near and away from phase transitions. In particular, we consider the roles that different spin motifs play (in cluster bulk, cluster edges, and the like) and how these affect the dependencies between spins.
Entanglement within qubits are studied for the subspace of definite particle
states or definite number of up spins. A transition from an algebraic decay of
entanglement within two qubits with the total number $N$ of qubits, to an
exponential one when the number of particles is increased from two to three is
studied in detail. In particular the probability that the concurrence is
non-zero is calculated using statistical methods and shown to agree with
numerical simulations. Further entanglement within a block of $m$ qubits is
studied using the log-negativity measure which indicates that a transition from
algebraic to exponential decay occurs when the number of particles exceeds $m$.
Several algebraic exponents for the decay of the log-negativity are
analytically calculated. The transition is shown to be possibly connected with
the changes in the density of states of the reduced density matrix, which has a
divergence at the zero eigenvalue when the entanglement decays algebraically.Comment: Substantially added content (now 24 pages, 5 figures) with a
discussion of the possible mechanism for the transition. One additional
author in this version that is accepted for publication in Phys. Rev.
The "power of choice" has been shown to radically alter the behavior of a number of randomized algorithms. Here we explore the effects of choice on models of tree and network growth. In our models each new node has k randomly chosen contacts, where k > 1 is a constant. It then attaches to whichever one of these contacts is most desirable in some sense, such as its distance from the root or its degree. Even when the new node has just two choices, i.e., when k = 2, the resulting network can be very different from a random graph or tree. For instance, if the new node attaches to the contact which is closest to the root of the tree, the distribution of depths changes from Poisson to a traveling wave solution. If the new node attaches to the contact with the smallest degree, the degree distribution is closer to uniform than in a random graph, so that with high probability there are no nodes in the network with degree greater than O(log log N). Finally, if the new node attaches to the contact with the largest degree, we find that the degree distribution is a power law with exponent −1 up to degrees roughly equal to k, with an exponential cutoff beyond that; thus, in this case, we need k 1 to see a power law over a wide range of degrees.
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