This limitation can be overcome by relaxing the AP hard constraints. A new parameter controls the importance of the constraints compared to the aim of maximizing the overall similarity, and allows to interpolate between the simple case where each data point selects its closest neighbor as an exemplar and the original AP. The resulting soft-constraint affinity propagation (SCAP) becomes more informative, accurate and leads to more stable clustering. Even though a new a priori free parameter is introduced, the overall dependence of the algorithm on external tuning is reduced, as robustness is increased and an optimal strategy for parameter selection emerges more naturally. SCAP is tested on biological benchmark data, including in particular microarray data related to various cancer types. We show that the algorithm efficiently unveils the hierarchical cluster structure present in the data sets. Further on, it allows to extract sparse gene expression signatures for each cluster.

We consider the K-satisfiability problem on a regular d-ary rooted tree. For this model, we demonstrate how we can calculate in closed form, the moments of the total number of solutions as a function of d and K, where the average is over all realizations, for a fixed assignment of the surface variables. We find that different moments pick out different 'critical' values of d, below which they diverge as the total number of variables on the tree → ∞ and above which they decay. We show that K-SAT on the random graph also behaves similarly. We also calculate exactly the fraction of instances that have solutions for all K.On the tree, this quantity decays to 0 (as the number of variables increases) for any d > 1.However the recursion relations for this quantity have a non-trivial fixed-point solution which indicates the existence of a different transition in the interior of an infinite rooted tree.

In this paper we solve the Blume-Capel model on a complete graph in the presence of random crystal field with a distribution, P (∆ i ) = pδ(∆ i − ∆) + (1 − p)δ(∆ i + ∆), using large deviation techniques. We find that the first order transition of the pure system is destroyed for 0.046 < p < 0.954 for all values of the crystal field, ∆. The system has a line of continuous transition for this range of p from −∞ < ∆ < ∞. For values of p outside this interval, the phase diagram of the system is similar to the pure model, with a tricritical point separating the line of first order and continuous transitions. We find that in this regime, the order vanishes for large ∆ for p < 0.046(and for large −∆ for p > 0.954) even at zero temperature.

We obtain the phase diagram for the Blume-Capel model with bimodal distribution for random crystal fields, in the space of three fields: temperature(T ), crystal field(∆) and magnetic field (H). We find that three critical lines meet at a tricritical point, but only for weak disorder. As disorder strength increases there is no tricritical point in the phase diagram. We instead find a bicritical end point, where only two of the critical lines meet on a first order surface in the H = 0 plane. For intermediate strengths of disorder, the phase diagram has critical end points along with the bicritical end point. One needs to look at the phase diagram in the space of three fields to identify various such multicritical points.

We study the efficiency of the incomplete enumeration algorithm for linear and branched polymers. There is a qualitative difference in the efficiency in these two cases. The average time to generate an independent sample of configuration of polymer with n monomers varies as n 2 for linear polymers for large n, but as exp(cn α) for branched (undirected and directed) polymers, where 0 < α < 1. On the binary tree, our numerical studies for n of order 10 4 gives α = 0.333 ± 0.005. We argue that α = 1/3 exactly in this case.

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We study the random field p-spin model with Ising spins on a fully connected graph using the theory of large deviations in this paper. This is a good model to study the effect of quenched random field on systems which have a sharp first order transition in the pure state. For p = 2, the phase-diagram of the model, for bimodal distribution of the random field, has been well studied and is known to undergo a continuous transition for lower values of the random field (h) and a first order transition beyond a threshold, h tp (≈ 0.439). We find the phase diagram of the model, for all p ≥ 2, with bimodal random field distribution, using large deviation techniques. We also look at the fluctuations in the system by calculating the magnetic susceptibility. For p = 2, beyond the tricritical point in the regime of first order transition, we find that for h tp < h < 0.447, magnetic susceptibility increases rapidly (even though it never diverges) as one approaches the transition from the high temperature side. On the other hand, for 0.447 < h ≤ 0.5, the high temperature behaviour is well described by the Curie-Weiss law. For all p ≥ 2, we find that for larger magnitudes of the random field (h > h o = 1/p!), the system does not show ferromagnetic order even at zero temperature. We find that the magnetic susceptibility for p ≥ 3 is discontinuous at the transition point for h < h o .