Monotone Maps, Sphericity and Bounded Second Eigenvalue

We consider {\em monotone} embeddings of a finite metric space into low dimensional normed space. That is, embeddings that respect the order among the distances in the original space. Our main interest is in embeddings into Euclidean spaces. We observe that any metric on $n$ points can be embedded into $l_2^n$, while, (in a sense to be made precise later), for almost every $n$-point metric space, every monotone map must be into a space of dimension $\Omega(n)$. It becomes natural, then, to seek explicit constructions of metric spaces that cannot be monotonically embedded into spaces of sublinear dimension. To this end, we employ known results on {\em sphericity} of graphs, which suggest one example of such a metric space - that defined by a complete bipartitegraph. We prove that an $\delta n$-regular graph of order $n$, with bounded diameter has sphericity $\Omega(n/(\lambda_2+1))$, where $\lambda_2$ is the second largest eigenvalue of the adjacency matrix of the graph, and 0 < \delta \leq \half is constant. We also show that while random graphs have linear sphericity, there are {\em quasi-random} graphs of logarithmic sphericity. For the above bound to be linear, $\lambda_2$ must be constant. We show that if the second eigenvalue of an $n/2$-regular graph is bounded by a constant, then the graph is close to being complete bipartite. Namely, its adjacency matrix differs from that of a complete bipartite graph in only $o(n^2)$ entries. Furthermore, for any 0 < \delta < \half, and $\lambda_2$, there are only finitely many $\delta n$-regular graphs with second eigenvalue at most $\lambda_2$

Local Access to Random Walks

For a graph G on n vertices, naively sampling the position of a random walk of at time t requires work ?(t). We desire local access algorithms supporting position_G(t) queries, which return the position of a random walk from some fixed start vertex s at time t, where the joint distribution of returned positions is 1/poly(n) close to those of a uniformly random walk in ?? distance. We first give an algorithm for local access to random walks on a given undirected d-regular graph with O?(1/(1-?)?n) runtime per query, where ? is the second-largest eigenvalue of the random walk matrix of the graph in absolute value. Since random d-regular graphs G(n,d) are expanders with high probability, this gives an O?(?n) algorithm for a graph drawn from G(n,d) whp, which improves on the naive method for small numbers of queries. We then prove that no algorithm with subconstant error given probe access to an input d-regular graph can have runtime better than ?(?n/log(n)) per query in expectation when the input graph is drawn from G(n,d), obtaining a nearly matching lower bound. We further show an ?(n^{1/4}) runtime per query lower bound even with an oblivious adversary (i.e. when the query sequence is fixed in advance). We then show that for families of graphs with additional group theoretic structure, dramatically better results can be achieved. We give local access to walks on small-degree abelian Cayley graphs, including cycles and hypercubes, with runtime polylog(n) per query. This also allows for efficient local access to walks on polylog degree expanders. We show that our techniques apply to graphs with high degree by extending or results to graphs constructed using the tensor product (giving fast local access to walks on degree n^? graphs for any ? ? (0,1]) and Cartesian product

Entropy of eigenfunctions on quantum graphs

We consider families of finite quantum graphs of increasing size and we are interested in how eigenfunctions are distributed over the graph. As a measure for the distribution of an eigenfunction on a graph we introduce the entropy, it has the property that a large value of the entropy of an eigenfunction implies that it cannot be localised on a small set on the graph. We then derive lower bounds for the entropy of eigenfunctions which depend on the topology of the graph and the boundary conditions at the vertices. The optimal bounds are obtained for expanders with large girth, the bounds are similar to the ones obtained by Anantharaman et.al. for eigenfunctions on manifolds of negative curvature, and are based on the entropic uncertainty principle. For comparison we compute as well the average behaviour of entropies on Neumann star graphs, where the entropies are much smaller. Finally we compare our lower bounds with numerical results for regular graphs and star graphs with different boundary conditions.Comment: 28 pages, 3 figure

Numerical Investigation of Graph Spectra and Information Interpretability of Eigenvalues

We undertake an extensive numerical investigation of the graph spectra of thousands regular graphs, a set of random Erd\"os-R\'enyi graphs, the two most popular types of complex networks and an evolving genetic network by using novel conceptual and experimental tools. Our objective in so doing is to contribute to an understanding of the meaning of the Eigenvalues of a graph relative to its topological and information-theoretic properties. We introduce a technique for identifying the most informative Eigenvalues of evolving networks by comparing graph spectra behavior to their algorithmic complexity. We suggest that extending techniques can be used to further investigate the behavior of evolving biological networks. In the extended version of this paper we apply these techniques to seven tissue specific regulatory networks as static example and network of a na\"ive pluripotent immune cell in the process of differentiating towards a Th17 cell as evolving example, finding the most and least informative Eigenvalues at every stage.Comment: Forthcoming in 3rd International Work-Conference on Bioinformatics and Biomedical Engineering (IWBBIO), Lecture Notes in Bioinformatics, 201