Survey of Distributed Decision

We survey the recent distributed computing literature on checking whether a given distributed system configuration satisfies a given boolean predicate, i.e., whether the configuration is legal or illegal w.r.t. that predicate. We consider classical distributed computing environments, including mostly synchronous fault-free network computing (LOCAL and CONGEST models), but also asynchronous crash-prone shared-memory computing (WAIT-FREE model), and mobile computing (FSYNC model)

Nested hierarchies in planar graphs

We construct a partial order relation which acts on the set of 3-cliques of a maximal planar graph G and defines a unique hierarchy. We demonstrate that G is the union of a set of special subgraphs, named `bubbles', that are themselves maximal planar graphs. The graph G is retrieved by connecting these bubbles in a tree structure where neighboring bubbles are joined together by a 3-clique. Bubbles naturally provide the subdivision of G into communities and the tree structure defines the hierarchical relations between these communities

Encoding dynamics for multiscale community detection: Markov time sweeping for the Map equation

The detection of community structure in networks is intimately related to finding a concise description of the network in terms of its modules. This notion has been recently exploited by the Map equation formalism (M. Rosvall and C.T. Bergstrom, PNAS, 105(4), pp.1118--1123, 2008) through an information-theoretic description of the process of coding inter- and intra-community transitions of a random walker in the network at stationarity. However, a thorough study of the relationship between the full Markov dynamics and the coding mechanism is still lacking. We show here that the original Map coding scheme, which is both block-averaged and one-step, neglects the internal structure of the communities and introduces an upper scale, the `field-of-view' limit, in the communities it can detect. As a consequence, Map is well tuned to detect clique-like communities but can lead to undesirable overpartitioning when communities are far from clique-like. We show that a signature of this behavior is a large compression gap: the Map description length is far from its ideal limit. To address this issue, we propose a simple dynamic approach that introduces time explicitly into the Map coding through the analysis of the weighted adjacency matrix of the time-dependent multistep transition matrix of the Markov process. The resulting Markov time sweeping induces a dynamical zooming across scales that can reveal (potentially multiscale) community structure above the field-of-view limit, with the relevant partitions indicated by a small compression gap.Comment: 10 pages, 6 figure

Pseudorandom Self-Reductions for NP-Complete Problems

A language L is random-self-reducible if deciding membership in L can be reduced (in polynomial time) to deciding membership in L for uniformly random instances. It is known that several "number theoretic" languages (such as computing the permanent of a matrix) admit random self-reductions. Feigenbaum and Fortnow showed that NP-complete languages are not non-adaptively random-self-reducible unless the polynomial-time hierarchy collapses, giving suggestive evidence that NP may not admit random self-reductions. Hirahara and Santhanam introduced a weakening of random self-reductions that they called pseudorandom self-reductions, in which a language L is reduced to a distribution that is computationally indistinguishable from the uniform distribution. They then showed that the Minimum Circuit Size Problem (MCSP) admits a non-adaptive pseudorandom self-reduction, and suggested that this gave further evidence that distinguished MCSP from standard NP-Complete problems. We show that, in fact, the Clique problem admits a non-adaptive pseudorandom self-reduction, assuming the planted clique conjecture. More generally we show the following. Call a property of graphs ? hereditary if G ? ? implies H ? ? for every induced subgraph of G. We show that for any infinite hereditary property ?, the problem of finding a maximum induced subgraph H ? ? of a given graph G admits a non-adaptive pseudorandom self-reduction

Improved Sum-of-Squares Lower Bounds for Hidden Clique and Hidden Submatrix Problems

Given a large data matrix $A\in\mathbb{R}^{n\times n}$, we consider the problem of determining whether its entries are i.i.d. with some known marginal distribution $A_{ij}\sim P_0$, or instead $A$ contains a principal submatrix $A_{{\sf Q},{\sf Q}}$ whose entries have marginal distribution $A_{ij}\sim P_1\neq P_0$. As a special case, the hidden (or planted) clique problem requires to find a planted clique in an otherwise uniformly random graph. Assuming unbounded computational resources, this hypothesis testing problem is statistically solvable provided $|{\sf Q}|\ge C \log n$ for a suitable constant $C$. However, despite substantial effort, no polynomial time algorithm is known that succeeds with high probability when $|{\sf Q}| = o(\sqrt{n})$. Recently Meka and Wigderson \cite{meka2013association}, proposed a method to establish lower bounds within the Sum of Squares (SOS) semidefinite hierarchy. Here we consider the degree-$4$ SOS relaxation, and study the construction of \cite{meka2013association} to prove that SOS fails unless $k\ge C\, n^{1/3}/\log n$. An argument presented by Barak implies that this lower bound cannot be substantially improved unless the witness construction is changed in the proof. Our proof uses the moments method to bound the spectrum of a certain random association scheme, i.e. a symmetric random matrix whose rows and columns are indexed by the edges of an Erd\"os-Renyi random graph.Comment: 40 pages, 1 table, conferenc

Directed network modules

A search technique locating network modules, i.e., internally densely connected groups of nodes in directed networks is introduced by extending the Clique Percolation Method originally proposed for undirected networks. After giving a suitable definition for directed modules we investigate their percolation transition in the Erdos-Renyi graph both analytically and numerically. We also analyse four real-world directed networks, including Google's own webpages, an email network, a word association graph and the transcriptional regulatory network of the yeast Saccharomyces cerevisiae. The obtained directed modules are validated by additional information available for the nodes. We find that directed modules of real-world graphs inherently overlap and the investigated networks can be classified into two major groups in terms of the overlaps between the modules. Accordingly, in the word-association network and among Google's webpages the overlaps are likely to contain in-hubs, whereas the modules in the email and transcriptional regulatory networks tend to overlap via out-hubs.Comment: 21 pages, 10 figures, version 2: added two paragaph

Élőlények kollektív viselkedésének statisztikus fizikája = Statistical physics of the collective behaviour of organisms

Experiments: We have carried out quantitative experiments on the collective motion of cells as a function of their density. A sharp transition could be observed from the random motility in sparse cultures to the flocking of dense islands of cells. Using ultra light GPS devices developed by us, we have determined the existing hierarchical relations within a flock of 10 homing pigeons. Modelling: From the simulations of our new model of flocking we concluded that the information exchange between particles was maximal at the critical point, in which the interplay of such factors as the level of noise, the tendency to follow the direction and the acceleration of others results in large fluctuations. Analysis: We have proposed a novel link-density based approach to finding overlapping communities in large networks. The algorithm used for the implementation of this technique is very efficient for most real networks, and provides full statistics quickly. Correspondingly, we have developed a by now popular, user-friendly, freely downloadable software for finding overlapping communities. Extending our method to the time-dependent regime, we found that large groups in evolving networks persist for longer if they are capable of dynamically altering their membership, thus, an ability to change the group composition results in better adaptability. We also showed that knowledge of the time commitment of members to a given community can be used for estimating the community's lifetime. Experiments: We have carried out quantitative experiments on the collective motion of cells as a function of their density. A sharp transition could be observed from the random motility in sparse cultures to the flocking of dense islands of cells. Using ultra light GPS devices developed by us, we have determined the existing hierarchical relations within a flock of 10 homing pigeons. Modelling: From the simulations of our new model of flocking we concluded that the information exchange between particles was maximal at the critical point, in which the interplay of such factors as the level of noise, the tendency to follow the direction and the acceleration of others results in large fluctuations. Analysis: We have proposed a novel link-density based approach to finding overlapping communities in large networks. The algorithm used for the implementation of this technique is very efficient for most real networks, and provides full statistics quickly. Correspondingly, we have developed a by now popular, user-friendly, freely downloadable software for finding overlapping communities. Extending our method to the time-dependent regime, we found that large groups in evolving networks persist for longer if they are capable of dynamically altering their membership, thus, an ability to change the group composition results in better adaptability. We also showed that knowledge of the time commitment of members to a given community can be used for estimating the community's lifetime