arXiv subjects

Mark Goldberg

Publications and source records attributed to Mark Goldberg.

4 recordsLinked to original sources

arXiv2026

Tracking and Predicting Evolution of Social Communities

We develop an algorithmic framework for studying the evolution of communities in social networks. We begin with the theoretical foundation, from which we conclude that an evolution is at most as strong as its weakest link. This allows us to formulate an efficient algorithm to identify all evolutionary sequences in a dynamic social network. We use this algorithm to empirically study community evolution in several large social networks, to identify those features of the early stages of a community that indicate whether a community is going to be shortlived or not. Our results show that it is possible to correlate the lifespan of a community to structural parameters of its early evolution; these conclusions are robust across all the social networks we have investigated.

cs.SI↗

arXiv2010

Embedding a Forest in a Graph

For \math{p\ge 1}, we prove that every forest with \math{p} trees whose sizes are $a_1,..., a_p$ can be embedded in any graph containing at least $\sum_{i=1}^p (a_i + 1)$ vertices and having a minimum degree at least $\sum_{i=1}^p a_i$.

math.CO↗

arXiv1995

Path optimization and near-greedy analysis for graph partitioning: an empirical study

This paper presents the results of an experimental study of graph partitioning. We describe a new heuristic technique, path optimization, and its application to two variations of graph partitioning: the max_cut problem and the min_quotient_cut problem. We present the results of computational comparisons between this technique and the Kernighan-Lin algorithm, the simulated annealing algorithm, the FLOW-lagorithm the multilevel algorithm, and teh recent 0.878-approximation algorithm. The experiments were conducted on two classes of graphs that have become standard for such tests: random and random geometric. They show that for both classes of inputs and both variations of the problem, the new heuristic is competitive with the other algorithms and holds an advantage for min_quotient_cut when applied to very large, sparse geometric graphs. In the last part of the paper, we describe an approach to analyzing graph partitioning algorithms from the statistical point of view. Every partitioning of a graph is viewed as a result achieved by a "near gready" partitioning algorithm. The experiments show that for "good" partitionings, the number of non-greedy steps needed to obtain them is quite small; moreover, it is "statistically" smaller for better partitionings. This led us to conjecture that there exists an "optimal" distribution of the non-greedy steps that characterize the classes of graphs that we studied.

math.CO↗

arXiv1994

Inverting sets and the packing problem

Given a set $V$, a subset $S$, and a permutation $π$ of $V$, we say that $π$ permutes $S$ if $π(S) \cap S = \emptyset$. Given a collection $\cS = \{V; S_1,\ldots , S_m\}$, where $S_i \subseteq V ~~(i=1,\ldots ,m)$, we say that $\cS$ is invertible if there is a permutation $π$ of $V$ such that $π(S_i) \subseteq V-S_i$. In this paper, we present necessary and sufficient conditions for the invertibility of a collection and construct a polynomial algorithm which determines whether a given collection is invertible. For an arbitrary collection, we give a lower bound for the maximum number of sets that can be inverted. Finally, we consider the problem of constructing a collection of sets such that no sub-collection of size three is invertible. Our constructions of such collections come from solutions to the packing problem with unbounded block sizes. We prove several new lower and upper bounds for the packing problem and present a new explicit construction of packing.

math.CO↗