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Register a new userThermodynamics of spin systems on small-world hypergraphs
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We study the thermodynamic properties of spin systems on small-world hypergraphs, obtained by superimposing sparse Poisson random graphs with p-spin interactions onto a one-dimensional Ising chain with nearest-neighbor interactions. We use replica-symmetric transfer-matrix techniques to derive a set of fixed-point equations describing the relevant order parameters and free energy, and solve them employing population dynamics. In the special case where the number of connections per site is of the order of the system size we are able to solve the model analytically. In the more general case where the number of connections is finite we determine the static and dynamic ferromagnetic-paramagnetic transitions using population dynamics. The results are tested against Monte-Carlo simulations.
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It was recently claimed that on d-dimensional small-world networks with a density p of shortcuts, the typical separation s(p) ~ p^{-1/d} between shortcut-ends is a characteristic length for shortest-paths{cond-mat/9904419}. This contradicts an earlier argument suggesting that no finite characteristic length can be defined for bilocal observables on these systems {cont-mat/9903426}. We show analytically, and confirm by numerical simulation, that shortest-path lengths ell(r) behave as ell(r) ~ r for r < r_c, and as ell(r) ~ r_c for r > r_c, where r is the Euclidean separation between two points and r_c(p,L) = p^{-1/d} log(L^dp) is a characteristic length. This shows that the mean separation s between shortcut-ends is not a relevant length-scale for shortest-paths. The true characteristic length r_c(p,L) diverges with system size L no matter the value of p. Therefore no finite characteristic length can be defined for small-world networks in the thermodynamic limit.
We investigate the critical properties of the Ising model in two dimensions on {it directed} small-world lattice with quenched connectivity disorder. The disordered system is simulated by applying the Monte Carlo update heat bath algorithm. We calculate the critical temperature, as well as the critical exponents $gamma/ u$, $beta/ u$, and $1/ u$ for several values of the rewiring probability $p$. We find that this disorder system does not belong to the same universality class as the regular two-dimensional ferromagnetic model. The Ising model on {it directed} small-world lattices presents in fact a second-order phase transition with new critical exponents which do not dependent of $p$, but are identical to the exponents of the Ising model and the spin-1 Blume-Capel model on {it directed} small-world network.
The vertex-cover problem on the Hanoi networks HN3 and HN5 is analyzed with an exact renormalization group and parallel-tempering Monte Carlo simulations. The grand canonical partition function of the equivalent hard-core repulsive lattice-gas problem is recast first as an Ising-like canonical partition function, which allows for a closed set of renormalization group equations. The flow of these equations is analyzed for the limit of infinite chemical potential, at which the vertex-cover problem is attained. The relevant fixed point and its neighborhood are analyzed, and non-trivial results are obtained both, for the coverage as well as for the ground state entropy density, which indicates the complex structure of the solution space. Using special hierarchy-dependent operators in the renormalization group and Monte-Carlo simulations, structural details of optimal configurations are revealed. These studies indicate that the optimal coverages (or packings) are not related by a simple symmetry. Using a clustering analysis of the solutions obtained in the Monte Carlo simulations, a complex solution space structure is revealed for each system size. Nevertheless, in the thermodynamic limit, the solution landscape is dominated by one huge set of very similar solutions.
We calculate the number of metastable configurations of Ising small-world networks which are constructed upon superimposing sparse Poisson random graphs onto a one-dimensional chain. Our solution is based on replicated transfer-matrix techniques. We examine the denegeracy of the ground state and we find a jump in the entropy of metastable configurations exactly at the crossover between the small-world and the Poisson random graph structures. We also examine the difference in entropy between metastable and all possible configurations, for both ferromagnetic and bond-disordered long-range couplings.
Mapping a complex network to an atomic cluster, the Anderson localization theory is used to obtain the load distribution on a complex network. Based upon an intelligence-limited model we consider the load distribution and the congestion and cascade failures due to attacks and occasional damages. It is found that the eigenvector centrality (EC) is an effective measure to find key nodes for traffic flow processes. The influence of structure of a WS small-world network is investigated in detail.
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