On local weak limit and subgraph counts for sparse random graphs

We use an inequality of Sidorenko to show a general relation between local and global subgraph counts and degree moments for locally weakly convergent sequences of sparse random graphs. This yields an optimal criterion to check when the asymptotic behaviour of graph statistics such as the clustering coefficient and assortativity is determined by the local weak limit. As an application we obtain new facts for several common models of sparse random intersection graphs where the local weak limit, as we see here, is a simple random clique tree corresponding to a certain two-type Galton-Watson branching process

Random graphs with few disjoint cycles

The classical Erd\H{o}s-P\'{o}sa theorem states that for each positive integer k there is an f(k) such that, in each graph G which does not have k+1 disjoint cycles, there is a blocker of size at most f(k); that is, a set B of at most f(k) vertices such that G-B has no cycles. We show that, amongst all such graphs on vertex set {1,..,n}, all but an exponentially small proportion have a blocker of size k. We also give further properties of a random graph sampled uniformly from this class; concerning uniqueness of the blocker, connectivity, chromatic number and clique number. A key step in the proof of the main theorem is to show that there must be a blocker as in the Erd\H{o}s-P\'{o}sa theorem with the extra `redundancy' property that B-v is still a blocker for all but at most k vertices v in B

BIG DATA CONCEPT IN THE FOOD SUPPLY CHAIN: SMALL MARKETS CASE

The strategies of competitive advantage are changing dramatically, because of high technology development. The data size in the world is multiplying rapidly - the amount of information in the world doubles every 12 months. Therefore, the authors analyzed big data in the food supply chain. The food industry‘s supply is complicated, because of various regulations and a demand for high quality products just on time. Various companies are transporting partial freight. Therefore, the visibility, lead-time and cost minimization is essential for them. However, they are unable to use all the gathered information and are not utilizing the potential that is possible. The problem of data analysis is a bigger concern to the smaller markets. Many of the small markets are less developed countries that still is not using big data in their enterprises. In addition, new technologies are developing in the big data industry. Therefore, the gap of technology will increase even more between the large and small markets. The analyzed innovation level and technology usage indicated a need for the food industry to change competiveness strategies. Therefore, the authors developed a competiveness strategy that is oriented to the small market’s food industry.JEL Codes - C80, L6

On small subgraphs in a random intersection digraph

Given a set of vertices V and a set of attributes W let each vertex v ∈ V include an attribute w ∈ W into a set S − (v) with probability p − and let it include w into a set S + (v) with probability p + independently for each w ∈ W . The random binomial intersection digraph on the vertex set V is defined as follows: for each u, v ∈ V the arc uv is present if S − (u) and S + (v) are not disjoint. For any h = 2, 3, . . . we determine the birth threshold of the complete digraph on h vertices and describe the configurations of intersecting sets that realise the threshold