Vince Grolmusz

SUPERPOLYNOMIAL SIZE SET-SYSTEMS WITH RESTRICTED INTERSECTIONS MOD 6 AND EXPLICIT RAMSEY GRAPHS

Dedicated to the memory of Paul ErdősWe construct a system of subsets of a set of n elements such that the size of each set is divisible by 6 but their pairwise intersections are not divisible by 6. The result generalizes to all non-prime-power moduli m in place of m=6. This result is in sharp contrast with results of Frankl and Wilson (1981) for prime power moduli and gives strong negative answers to questions by Frankl and Wilson (1981) and Babai and Frankl (1992). We use our set-system to give an explicit Ramsey-graph construction, reproducing the logarithmic order of magnitude of the best previously known construction due to Frankl and Wilson (1981). Our construction uses certain mod m polynomials, discovered by Barrington, Beigel and Rudich (1994).

When the Web meets the cell

MOTIVATION Enormous and constantly increasing quantity of biological information is represented in metabolic and in protein interaction network databases. Most of these data are freely accessible through large public depositories. The robust analysis of these resources needs novel technologies, being developed today. RESULTS Here we demonstrate a technique, originating from the PageRank computation for the World Wide Web, for analyzing large interaction networks. The method is fast, scalable and robust, and its capabilities are demonstrated on metabolic network data of the tuberculosis bacterium and the proteomics analysis of the blood of melanoma patients. AVAILABILITY The Perl script for computing the personalized PageRank in protein networks is available for non-profit research applications (together with sample input files) at the address: http://uratim.com/pp.zip.

AmphoraNet: The webserver implementation of the AMPHORA2 metagenomic workflow suite

MOTIVATION Metagenomics went through an astonishing development in the past few years. Today not only gene sequencing experts, but numerous laboratories of other specializations need to analyze DNA sequences gained from clinical or environmental samples. Phylogenetic analysis of the metagenomic data presents significant challenges for the biologist and the bioinformatician. The program suite AMPHORA and its workflow version are examples of publicly available software that yields reliable phylogenetic results for metagenomic data. RESULTS Here we present AmphoraNet, an easy-to-use webserver that is capable of assigning a probability-weighted taxonomic group for each phylogenetic marker gene found in the input metagenomic sample; the webserver is based on the AMPHORA2 workflow. Since a large proportion of molecular biologists uses the BLAST program and its clones on public webservers instead of the locally installed versions, we believe that the occasional user may find it comfortable that, in this version, no time-consuming installation of every component of the AMPHORA2 suite or expertise in Linux environment is required. AVAILABILITY The webserver is freely available at http://amphoranet.pitgroup.org; no registration is required.

A note on the PageRank of undirected graphs

The PageRank is a widely used scoring function of networks in general and of the World Wide Web graph in particular. The PageRank is defined for directed graphs, but in some special cases applications for undirected graphs occur. In the literature it is widely - but not exclusively - noted that the PageRank for undirected graphs is proportional to the degrees of the vertices of the graph. We prove that statement for a particular personalization vector in the definition of the PageRank, and we also show that in general, the PageRank of an undirected graph is not exactly proportional to the degree distribution of the graph: our main theorem gives an upper and a lower bound to the L 1 norm of the difference of the PageRank and the degree distribution vectors. A necessary and sufficient condition is also given for the PageRank for being proportional to the degree. PageRank is usually computed for directed graphs;PageRank of undirected graphs is investigated;Erroneously, some sources remark that the PR of undirected graphs are proportional to the degrees;We show that this happens very rarely;A necessary and sufficient condition is given for the PR for being proportional to the degree.

Constructing set systems with prescribed intersection sizes

Let f be an n-variable polynomial with positive integer coefficients, and let H = {H1, H2,..., Hm} be a set system on the n-element universe. We define a set system f(H) = {G1, G2,..., Gm} and prove that f(Hi1 ∩ Hi2 ∩... ∩ Hik) = |Gi1 ∩ Gi2 ∩ ... ∩ Gik|, for any 1 ≤ k ≤ m, where f(Hi1 ∩ Hi2 ∩... ∩Hik) denotes the value of f on the characteristic vector of Hi1 ∩ Hi2 ∩... ∩ Hik. The construction of f(H) is a straightforward polynomial-time algorithm from the set system H and the polynomial f. In this paper we use this algorithm for constructing set systems with prescribed intersection sizes modulo an integer. As a by-product of our method, some upper bounds on the number of sets in set systems with prescribed intersection sizes are extended.

On k-wise set-intersections and k-wise Hamming-distances

We prove a version of the Ray-Chaudhuri?Wilson and Frankl?Wilson theorems for k-wise intersections and also generalize a classical code-theoretic result of Delsarte for k-wise Hamming distances. A set of code-words a1, a2,?,ak of length n have k-wise Hamming-distance ?, if there are exactly ? such coordinates, where not all of their coordinates coincide (alternatively, exactly n?? of their coordinates are the same). We show a Delsarte-like upper bound: codes with few k-wise Hamming-distances must contain few code-words.

Separating the Communication Complexities of MOD m and MOD p Circuits

We prove in this paper that it is much harder to evaluate depth-2, size-N circuits with MOD m gates than with MOD p gates by k-party communication protocols: we show a k-party protocol which communicates O(1) bits to evaluate circuits with MOD p gates, while evaluating circuits with MOD m gates needs ?(N) bits, where p denotes a prime and m denotes a composite, non-prime power number. As a corollary, for all m, we show a function, computable with a depth-2 circuit with MOD m gates, but not with any depth-2 circuit with MOD p gates. Obviously, the k?-party protocols are not weaker than the k-party protocols, for k? >k. Our results imply that if there is a prime p between k and k?: k < p ? k?, then there exists a function which can be computed by a k?-party protocol with a constant number of communicated bits, while any k-party protocol needs linearly many bits of communication. This result gives a hierarchy theorem for multi-party protocols.

Equal Opportunity for Low-Degree Network Nodes: A PageRank-Based Method for Protein Target Identification in Metabolic Graphs

Biological network data, such as metabolic-, signaling- or physical interaction graphs of proteins are increasingly available in public repositories for important species. Tools for the quantitative analysis of these networks are being developed today. Protein network-based drug target identification methods usually return protein hubs with large degrees in the networks as potentially important targets. Some known, important protein targets, however, are not hubs at all, and perturbing protein hubs in these networks may have several unwanted physiological effects, due to their interaction with numerous partners. Here, we show a novel method applicable in networks with directed edges (such as metabolic networks) that compensates for the low degree (non-hub) vertices in the network, and identifies important nodes, regardless of their hub properties. Our method computes the PageRank for the nodes of the network, and divides the PageRank by the in-degree (i.e., the number of incoming edges) of the node. This quotient is the same in all nodes in an undirected graph (even for large- and low-degree nodes, that is, for hubs and non-hubs as well), but may differ significantly from node to node in directed graphs. We suggest to assign importance to non-hub nodes with large PageRank/in-degree quotient. Consequently, our method gives high scores to nodes with large PageRank, relative to their degrees: therefore non-hub important nodes can easily be identified in large networks. We demonstrate that these relatively high PageRank scores have biological relevance: the method correctly finds numerous already validated drug targets in distinct organisms (Mycobacterium tuberculosis, Plasmodium falciparum and MRSA Staphylococcus aureus), and consequently, it may suggest new possible protein targets as well. Additionally, our scoring method was not chosen arbitrarily: its value for all nodes of all undirected graphs is constant; therefore its high value captures importance in the directed edge structure of the graph.

Lower Bounds for (MOD p - MOD m ) Circuits

Modular gates are known to be immune for the random restriction techniques of Ajtai (1983), Furst, Saxe, and Sipser (1984), Yao (1985), and Ha stad (1986). We demonstrate here a random clustering technique which overcomes this difficulty and is capable of proving generalizations of several known modular circuit lower bounds of Barrington, Straubing, and Th{erien (1990), Krause and Pudl{ak (1994), and others, characterizing symmetric functions computable by small (MODp, ANDt, MODm) circuits. Applying a degree-decreasing technique together with random restriction methods for the AND gates at the bottom level, we also prove a hard special case of the constant degree hypothesis of Barrington, Straubing, and Th{erien (1990) and other related lower bounds for certain (MODp, MODm, AND) circuits. Most of the previous lower bounds on circuits with modular gates used special definitions of the modular gates (i.e., the gate outputs one if the sum of its inputs is divisible by m or is not divisible by m) and were not valid for more general MODm gates. Our methods are applicable, and our lower bounds are valid for the most general modular gates as well.

Graph Theoretical Analysis Reveals: Women's Brains Are Better Connected than Men's.

Deep graph-theoretic ideas in the context with the graph of the World Wide Web led to the definition of Google’s PageRank and the subsequent rise of the most popular search engine to date. Brain graphs, or connectomes, are being widely explored today. We believe that non-trivial graph theoretic concepts, similarly as it happened in the case of the World Wide Web, will lead to discoveries enlightening the structural and also the functional details of the animal and human brains. When scientists examine large networks of tens or hundreds of millions of vertices, only fast algorithms can be applied because of the size constraints. In the case of diffusion MRI-based structural human brain imaging, the effective vertex number of the connectomes, or brain graphs derived from the data is on the scale of several hundred today. That size facilitates applying strict mathematical graph algorithms even for some hard-to-compute (or NP-hard) quantities like vertex cover or balanced minimum cut. In the present work we have examined brain graphs, computed from the data of the Human Connectome Project, recorded from male and female subjects between ages 22 and 35. Significant differences were found between the male and female structural brain graphs: we show that the average female connectome has more edges, is a better expander graph, has larger minimal bisection width, and has more spanning trees than the average male connectome. Since the average female brain weighs less than the brain of males, these properties show that the female brain has better graph theoretical properties, in a sense, than the brain of males. It is known that the female brain has a smaller gray matter/white matter ratio than males, that is, a larger white matter/gray matter ratio than the brain of males; this observation is in line with our findings concerning the number of edges, since the white matter consists of myelinated axons, which, in turn, roughly correspond to the connections in the brain graph. We have also found that the minimum bisection width, normalized with the edge number, is also significantly larger in the right and the left hemispheres in females: therefore, the differing bisection widths are independent from the difference in the number of edges.