Solvable Models of Random Hetero-Polymers at Finite Density: I. Statics

We introduce $\infty$-dimensional versions of three common models of random hetero-polymers, in which both the polymer density and the density of the polymer-solvent mixture are finite. These solvable models give valuable insight into the problems related to the (quenched) average over the randomness in statistical mechanical models of proteins, without having to deal with the hard geometrical constraints occurring in finite dimensional models. Our exact solution, which is specific to the $\infty$-dimensional case, is compared to the results obtained by a saddle-point analysis and by the grand ensemble approach, both of which canalso be applied to models of finite dimension. We find, somewhat surprisingly, that the saddle-point analysis can lead to qualitatively incorrect results.Comment: 16 pages, 17 figure

Distributed sequential task allocation in foraging swarms

When designing a practical swarm robotics system, self-organized task allocation is key to make best use of resources. Current research in this area focuses on task allocation which is either distributed (tasks must be performed at different locations) or sequential (tasks are complex and must be split into simpler sub-tasks and processed in order). In practice, however, swarms will need to deal with tasks which are both distributed and sequential. In this paper, a classic foraging problem is extended to incorporate both distributed and sequential tasks. The problem is analysed theoretically, absolute limits on performance are derived, and a set of conditions for a successful algorithm are established. It is shown empirically that an algorithm which meets these conditions, by causing emergent cooperation between robots can achieve consistently high performance under a wide range of settings without the need for communication

The influence of memory in a threshold model for distributed task assignment

A nature inspired decentralised multi-agent algorithm is proposed to solve a problem of distributed task selection in which cities produce and store batches of different mail types. Agents must collect and process the mail batches, without a priori knowledge of the available mail at the cities or inter-agent communication. In order to process a different mail type than the previous one, agents must undergo a change-over during which it remains inactive. We propose a threshold based algorithm in order to maximise the overall efficiency (the average amount of mail collected). We show that memory, i.e. the possibility for agents to develop preferences for certain cities, not only leads to emergent cooperation between agents, but also to a significant increase in efficiency (above the theoretical upper limit for any memoryless algorithm), and we systematically investigate the influence of the various model parameters. Finally, we demonstrate the flexibility of the algorithm to changes in circumstances, and its excellent scalability

Heterogeneity in outcomes of repeated instances of percolation experiments

We investigate the heterogeneity of outcomes of repeated instances of percolation experiments in complex networks using a message-passing approach to evaluate heterogeneous, node-dependent probabilities of belonging to the giant or percolating cluster, i.e., the set of mutually connected nodes whose size scales linearly with the size of the system. We evaluate these both for large finite single instances and for synthetic networks in the configuration model class in the thermodynamic limit. For the latter, we consider both Erdos-Rényi and scale-free networks as examples of networks with narrow and broad degree distributions, respectively. For real-world networks we use an undirected version of a Gnutella peer-to-peer file-sharing network with N=62568 nodes as an example. We derive the theory for multiple instances of both uncorrelated and correlated percolation processes. For the uncorrelated case, we also obtain a closed-form approximation for the large mean degree limit of Erdos-Rényi networks

Genetics and competing strategies in a threshold model for mail processing

Multi-agent algorithms inspired by the division of labour in social insects are applied to a problem of distributed mail retrieval in which agents must visit mail producing cities and choose between mail types under certain constraints.The efficiency (i.e. the average amount of mail retrieved per time step), and the flexibility (i.e. the capability of the agents to react to changes in the environment) are investigated both in static and dynamic environments. New rules for mail selection and specialisation are introduced and are shown to exhibit improved efficiency and flexibility compared to existing ones. We employ a genetic algorithm which allows the various rules to evolve and compete. Apart from obtaining optimised parameters for the various rules for any environment, we also observe extinction and speciation. From a more theoretical point of view, in order to avoid finite size effects, most results are obtained for large population sizes. However, we do analyse the influence of population size on the performance. Furthermore, we critically analyse the causes of efficiency loss, derive the exact dynamics of the model in the large system limit under certain conditions, derive theoretical upper bounds for the efficiency, and compare these with the experimental results

Determination of Interaction Potentials of Amino Acids from Native Protein Structures: Test on Simple Lattice Models

We propose a novel method for the determination of the effective interaction potential between the amino acids of a protein. The strategy is based on the combination of a new optimization procedure and a geometrical argument, which also uncovers the shortcomings of any optimization procedure. The strategy can be applied on any data set of native structures such as those available from the Protein Data Bank (PDB). In this work, however, we explain and test our approach on simple lattice models, where the true interactions are known a priori. Excellent agreement is obtained between the extracted and the true potentials even for modest numbers of protein structures in the PDB. Comparisons with other methods are also discussed.Comment: 24 pages, 4 figure

Solvable model for distribution networks on random graphs

We propose a simple model that captures the salient properties of distribution networks, and study the possible occurrence of blackouts, i.e., sudden failings of large portions of such networks. The model is defined on a random graph of finite connectivity. The nodes of the graph represent hubs of the network, while the edges of the graph represent the links of the distribution network. Both, the nodes and the edges carry dynamical two state variables representing the functioning or dysfunctional state of the node or link in question. We describe a dynamical process in which the breakdown of a link or node is triggered when the level of maintenance it receives falls below a given threshold. This form of dynamics can lead to situations of catastrophic breakdown, if levels of maintenance are themselves dependent on the functioning of the net, once maintenance levels locally fall below a critical threshold due to fluctuations. We formulate conditions under which such systems can be analyzed in terms of thermodynamic equilibrium techniques, and under these conditions derive a phase diagram characterizing the collective behavior of the system, given its model parameters. The phase diagram is confirmed qualitatively and quantitatively by simulations on explicit realizations of the graph, thus confirming the validity of our approach. © 2007 The American Physical Society

Magnetization enumerator for LDPC codes - a statistical physics approach

We propose a method based on the magnetization enumerator to determine the critical noise level for Gallager type low density parity check error correcting codes (LDPC). Our method provides an appealingly simple interpretation to the relation between different decoding schemes, and provides more optimistic critical noise levels than those reported in the information theory literature

Dynamics and topographic organization of recursive self-organizing maps

Recently there has been an outburst of interest in extending topographic maps of vectorial data to more general data structures, such as sequences or trees. However, there is no general consensus as to how best to process sequences using topographicmaps, and this topic remains an active focus of neurocomputational research. The representational capabilities and internal representations of the models are not well understood. Here, we rigorously analyze a generalization of the self-organizingmap (SOM) for processing sequential data, recursive SOM (RecSOM) (Voegtlin, 2002), as a nonautonomous dynamical system consisting of a set of fixed input maps. We argue that contractive fixed-input maps are likely to produce Markovian organizations of receptive fields on the RecSOM map. We derive bounds on parameter β (weighting the importance of importing past information when processing sequences) under which contractiveness of the fixed-input maps is guaranteed. Some generalizations of SOM contain a dynamic module responsible for processing temporal contexts as an integral part of the model. We show that Markovian topographic maps of sequential data can be produced using a simple fixed (nonadaptable) dynamic module externally feeding a standard topographic model designed to process static vectorial data of fixed dimensionality (e.g., SOM). However, by allowing trainable feedback connections, one can obtain Markovian maps with superior memory depth and topography preservation. We elaborate on the importance of non-Markovian organizations in topographic maps of sequential data. © 2006 Massachusetts Institute of Technology

Average and reliability error exponents in low-density parity-check codes

We present a theoretical method for a direct evaluation of the average and reliability error exponents in low-density parity-check error-correcting codes using methods of statistical physics. Results for the binary symmetric channel are presented for codes of both finite and infinite connectivity