Abstract. We study the out-of-equilibrium dynamics of the spherical ferromagnet after a quench to its critical temperature. We calculate correlation and response functions for spin observables which probe lengthscales much larger than the lattice spacing but smaller than the system size, and find that the asymptotic fluctuation-dissipation ratio (FDR) X ∞ is the same as for local observables. This is consistent with our earlier results for the Ising model in dimension d = 1 and d = 2. We also check that bond observables, both local and long-range, give the same asymptotic FDR. In the second part of the paper the analysis is extended to global observables, which probe correlations among all N spins. Here non-Gaussian fluctuations arising from the spherical constraint need to be accounted for, and we develop a systematic expansion in 1/ √ N to do this. Applying this to the global bond observable, i.e. the energy, we find that non-Gaussian corrections change its FDR to a nontrivial value which we calculate exactly for all dimensions d > 2. Finally, we consider quenches from magnetized initial states. Here even the FDR for the global spin observable, i.e. the magnetization, is nontrivial. It differs from the one for unmagnetized states even in d > 4, signalling the appearance of a distinct dynamical universality class of magnetized critical coarsening. For lower d the FDR is irrational even to first order in 4 − d and d − 2, the latter in contrast to recent results for the n-vector model.

Abstract. Pattern-diluted associative networks were introduced recently as models for the immune system, with nodes representing T-lymphocytes and stored patterns representing signalling protocols between T-and B-lymphocytes. It was shown earlier that in the regime of extreme pattern dilution, a system with N T Tlymphocytes can manage a number N B = O(N δ T ) of B-lymphocytes simultaneously, with δ < 1. Here we study this model in the extensive load regime N B = αN T , with also a high degree of pattern dilution, in agreement with immunological findings. We use graph theory and statistical mechanical analysis based on replica methods to show that in the finite-connectivity regime, where each T-lymphocyte interacts with a finite number of B-lymphocytes as N T → ∞, the T-lymphocytes can coordinate effective immune responses to an extensive number of distinct antigen invasions in parallel. As α increases, the system eventually undergoes a second order transition to a phase with clonal cross-talk interference, where the system's performance degrades gracefully. Mathematically, the model is equivalent to a spin system on a finitely connected graph with many short loops, so one would expect the available analytical methods, which all assume locally tree-like graphs, to fail. Yet it turns out to be solvable. Our results are supported by numerical simulations.

We study the tailoring of structured random graph ensembles to real networks, with the objective of generating precise and practical mathematical tools for quantifying and comparing network topologies macroscopically, beyond the level of degree statistics. Our family of ensembles can produce graphs with any prescribed degree distribution and any degree-degree correlation function, its control parameters can be calculated fully analytically, and as a result we can calculate (asymptotically) formulae for entropies and complexities, and for information-theoretic distances between networks, expressed directly and explicitly in terms of their measured degree distribution and degree correlations.

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.