arXiv subjects

Fan Chung

Publications and source records attributed to Fan Chung.

21 records · Page 2Linked to original sources

arXiv2005

De Bruijn Covering Codes for Rooted Hypergraphs

What is the length of the shortest sequence $S$ of reals so that the set of consecutive $n$-words in $S$ form a covering code for permutations on $\{1,2, >..., n\}$ of radius $R$ ? (The distance between two $n$-words is the number of transpositions needed to have the same order type.) The above problem can be viewed as a special case of finding a De Bruijn covering code for a rooted hypergraph. Each edge of a rooted hypergraph contains a special vertex, called the {\it root} of the edge, and each vertex is the root of a unique edge, called its {\it ball}. A De Bruijn covering code is a subset of the roots such that every vertex is in some edge containing a chosen root. Under some mild conditions, we obtain an upper bound for the shortest length of a De Bruijn covering code of a rooted hypergraph, a bound which is within a factor of $\log n$ of the lower bound.

math.CO↗

arXiv2003

De Bruijn Cycles for Covering Codes

A de Bruijn covering code is a q-ary string S so that every q-ary string is at most R symbol changes from some n-word appearing consecutively in S. We introduce these codes and prove that they can have length close to the smallest possible covering code. The proof employs tools from field theory, probability, and linear algebra. We also prove a number of ``spectral'' results on de Bruijn covering codes. Included is a table of the best known bounds on the lengths of small binary de Bruijn covering codes, up to R=11 and n=13, followed by several open questions in this area.

math.CO↗

arXiv2002

Duplication Models for Biological Networks

Are biological networks different from other large complex networks? Both large biological and non-biological networks exhibit power-law graphs (number of nodes with degree k, N(k) ~ k-b) yet the exponents, b, fall into different ranges. This may be because duplication of the information in the genome is a dominant evolutionary force in shaping biological networks (like gene regulatory networks and protein-protein interaction networks), and is fundamentally different from the mechanisms thought to dominate the growth of most non-biological networks (such as the internet [1-4]). The preferential choice models non-biological networks like web graphs can only produce power-law graphs with exponents greater than 2 [1-4,8]. We use combinatorial probabilistic methods to examine the evolution of graphs by duplication processes and derive exact analytical relationships between the exponent of the power law and the parameters of the model. Both full duplication of nodes (with all their connections) as well as partial duplication (with only some connections) are analyzed. We demonstrate that partial duplication can produce power-law graphs with exponents less than 2, consistent with current data on biological networks. The power-law exponent for large graphs depends only on the growth process, not on the starting graph.

cond-mat↗