Clustering properties of a generalised critical Euclidean network

Many real-world networks exhibit scale-free feature, have a small diameter and a high clustering tendency. We have studied the properties of a growing network, which has all these features, in which an incoming node is connected to its $i$th predecessor of degree $k_i$ with a link of length $\ell$ using a probability proportional to $k^\beta_i \ell^{\alpha}$. For $\alpha > -0.5$, the network is scale free at $\beta = 1$ with the degree distribution $P(k) \propto k^{-\gamma}$ and $\gamma = 3.0$ as in the Barab\'asi-Albert model ($\alpha =0, \beta =1$). We find a phase boundary in the $\alpha-\beta$ plane along which the network is scale-free. Interestingly, we find scale-free behaviour even for $\beta > 1$ for $\alpha < -0.5$ where the existence of a new universality class is indicated from the behaviour of the degree distribution and the clustering coefficients. The network has a small diameter in the entire scale-free region. The clustering coefficients emulate the behaviour of most real networks for increasing negative values of $\alpha$ on the phase boundary.Comment: 4 pages REVTEX, 4 figure

Evolving Networks with Multi-species Nodes and Spread in the Number of Initial Links

We consider models for growing networks incorporating two effects not previously considered: (i) different species of nodes, with each species having different properties (such as different attachment probabilities to other node species); and (ii) when a new node is born, its number of links to old nodes is random with a given probability distribution. Our numerical simulations show good agreement with analytic solutions. As an application of our model, we investigate the movie-actor network with movies considered as nodes and actors as links.Comment: 5 pages, 5 figures, submitted to PR

Stability of shortest paths in complex networks with random edge weights

We study shortest paths and spanning trees of complex networks with random edge weights. Edges which do not belong to the spanning tree are inactive in a transport process within the network. The introduction of quenched disorder modifies the spanning tree such that some edges are activated and the network diameter is increased. With analytic random-walk mappings and numerical analysis, we find that the spanning tree is unstable to the introduction of disorder and displays a phase-transition-like behavior at zero disorder strength $\epsilon=0$. In the infinite network-size limit ($N\to \infty$), we obtain a continuous transition with the density of activated edges $\Phi$ growing like $\Phi \sim \epsilon^1$ and with the diameter-expansion coefficient $\Upsilon$ growing like $\Upsilon\sim \epsilon^2$ in the regular network, and first-order transitions with discontinuous jumps in $\Phi$ and $\Upsilon$ at $\epsilon=0$ for the small-world (SW) network and the Barab\'asi-Albert scale-free (SF) network. The asymptotic scaling behavior sets in when $N\gg N_c$, where the crossover size scales as $N_c\sim \epsilon^{-2}$ for the regular network, $N_c \sim \exp[\alpha \epsilon^{-2}]$ for the SW network, and $N_c \sim \exp[\alpha |\ln \epsilon| \epsilon^{-2}]$ for the SF network. In a transient regime with $N\ll N_c$, there is an infinite-order transition with $\Phi\sim \Upsilon \sim \exp[-\alpha / (\epsilon^2 \ln N)]$ for the SW network and $\sim \exp[ -\alpha / (\epsilon^2 \ln N/\ln\ln N)]$ for the SF network. It shows that the transport pattern is practically most stable in the SF network.Comment: 9 pages, 7 figur

Quasistatic Scale-free Networks

A network is formed using the $N$ sites of an one-dimensional lattice in the shape of a ring as nodes and each node with the initial degree $k_{in}=2$. $N$ links are then introduced to this network, each link starts from a distinct node, the other end being connected to any other node with degree $k$ randomly selected with an attachment probability proportional to $k^{\alpha}$. Tuning the control parameter $\alpha$ we observe a transition where the average degree of the largest node $$ changes its variation from $N^0$ to $N$ at a specific transition point of $\alpha_c$. The network is scale-free i.e., the nodal degree distribution has a power law decay for $\alpha \ge \alpha_c$.Comment: 4 pages, 5 figure

Modulated Scale-free Network in the Euclidean Space

A random network is grown by introducing at unit rate randomly selected nodes on the Euclidean space. A node is randomly connected to its $i$-th predecessor of degree $k_i$ with a directed link of length $\ell$ using a probability proportional to $k_i \ell^{\alpha}$. Our numerical study indicates that the network is Scale-free for all values of $\alpha > \alpha_c$ and the degree distribution decays stretched exponentially for the other values of $\alpha$. The link length distribution follows a power law: $D(\ell) \sim \ell^{\delta}$ where $\delta$ is calculated exactly for the whole range of values of $\alpha$.Comment: 4 pages, 4 figures. To be published in Physical Review

Constrained Markovian dynamics of random graphs

We introduce a statistical mechanics formalism for the study of constrained graph evolution as a Markovian stochastic process, in analogy with that available for spin systems, deriving its basic properties and highlighting the role of the `mobility' (the number of allowed moves for any given graph). As an application of the general theory we analyze the properties of degree-preserving Markov chains based on elementary edge switchings. We give an exact yet simple formula for the mobility in terms of the graph's adjacency matrix and its spectrum. This formula allows us to define acceptance probabilities for edge switchings, such that the Markov chains become controlled Glauber-type detailed balance processes, designed to evolve to any required invariant measure (representing the asymptotic frequencies with which the allowed graphs are visited during the process). As a corollary we also derive a condition in terms of simple degree statistics, sufficient to guarantee that, in the limit where the number of nodes diverges, even for state-independent acceptance probabilities of proposed moves the invariant measure of the process will be uniform. We test our theory on synthetic graphs and on realistic larger graphs as studied in cellular biology.Comment: 28 pages, 6 figure

Measurement of D0-D0 mixing and search for CP violation in D0→K+K-,π+π- decays with the full Belle data set

We report an improved measurement of D0 – D‾0 mixing and a search for CP violation in D0 decays to CP -even final states K+K− and π+π− . The measurement is based on the final Belle data sample of 976 fb −1 . The results are yCP=(1.11±0.22±0.09)% and AΓ=(−0.03±0.20±0.07)% , where the first uncertainty is statistical and the second is systematic