Sampling Algorithms for Butterfly Counting on Temporal Bipartite Graphs

Temporal bipartite graphs are widely used to denote time-evolving relationships between two disjoint sets of nodes, such as customer-product interactions in E-commerce and user-group memberships in social networks. Temporal butterflies, $(2,2)$-bicliques that occur within a short period and in a prescribed order, are essential in modeling the structural and sequential patterns of such graphs. Counting the number of temporal butterflies is thus a fundamental task in analyzing temporal bipartite graphs. However, existing algorithms for butterfly counting on static bipartite graphs and motif counting on temporal unipartite graphs are inefficient for this purpose. In this paper, we present a general framework with three sampling strategies for temporal butterfly counting. Since exact counting can be time-consuming on large graphs, our approach alternatively computes approximate estimates accurately and efficiently. We also provide analytical bounds on the number of samples each strategy requires to obtain estimates with small relative errors and high probability. We finally evaluate our framework on six real-world datasets and demonstrate its superior accuracy and efficiency compared to several baselines. Overall, our proposed framework and sampling strategies provide efficient and accurate approaches to approximating temporal butterfly counts on large-scale temporal bipartite graphs.Comment: 10 pages, 10 figures; under revie

Short directed cycles in bipartite digraphs

The Caccetta-H\"aggkvist conjecture implies that for every integer $k\ge 1$, if $G$ is a bipartite digraph, with $n$ vertices in each part, and every vertex has out-degree more than $n/(k+1)$, then $G$ has a directed cycle of length at most $2k$. If true this is best possible, and we prove this for $k = 1,2,3,4,6$ and all $k\ge 224,539$. More generally, we conjecture that for every integer $k\ge 1$, and every pair of reals $\alpha, \beta> 0$ with $k\alpha +\beta>1$, if $G$ is a bipartite digraph with bipartition $(A,B)$, where every vertex in $A$ has out-degree at least $\beta|B|$, and every vertex in $B$ has out-degree at least $\alpha|A|$, then $G$ has a directed cycle of length at most $2k$. This implies the Caccetta-H\"aggkvist conjecture (set $\beta>0$ and very small), and again is best possible for infinitely many pairs $(\alpha,\beta)$. We prove this for $k = 1,2$, and prove a weaker statement (that $\alpha+\beta>2/(k+1)$ suffices) for $k=3,4$

Counting invertible Schr\"odinger Operators over Finite Fields for Trees, Cycles and Complete Graphs

We count invertible Schr\"odinger operators (perturbations by diagonal matrices of the adjacency matrix) over finite fieldsfor trees, cycles and complete graphs.This is achieved for trees through the definition and use of local invariants (algebraic constructions of perhapsindependent interest).Cycles and complete graphs are treated by ad hoc methods.Comment: Final version to appear in Electronic Journal of Combinatoric

Double-Edge Factor Graphs: Definition, Properties, and Examples

Some of the most interesting quantities associated with a factor graph are its marginals and its partition sum. For factor graphs \emph{without cycles} and moderate message update complexities, the sum-product algorithm (SPA) can be used to efficiently compute these quantities exactly. Moreover, for various classes of factor graphs \emph{with cycles}, the SPA has been successfully applied to efficiently compute good approximations to these quantities. Note that in the case of factor graphs with cycles, the local functions are usually non-negative real-valued functions. In this paper we introduce a class of factor graphs, called double-edge factor graphs (DE-FGs), which allow local functions to be complex-valued and only require them, in some suitable sense, to be positive semi-definite. We discuss various properties of the SPA when running it on DE-FGs and we show promising numerical results for various example DE-FGs, some of which have connections to quantum information processing.Comment: Submitte

The Complexity of Approximately Counting Stable Roommate Assignments

We investigate the complexity of approximately counting stable roommate assignments in two models: (i) the $k$-attribute model, in which the preference lists are determined by dot products of "preference vectors" with "attribute vectors" and (ii) the $k$-Euclidean model, in which the preference lists are determined by the closeness of the "positions" of the people to their "preferred positions". Exactly counting the number of assignments is #P-complete, since Irving and Leather demonstrated #P-completeness for the special case of the stable marriage problem. We show that counting the number of stable roommate assignments in the $k$-attribute model ($k \geq 4$) and the 3-Euclidean model($k \geq 3$) is interreducible, in an approximation-preserving sense, with counting independent sets (of all sizes) (#IS) in a graph, or counting the number of satisfying assignments of a Boolean formula (#SAT). This means that there can be no FPRAS for any of these problems unless NP=RP. As a consequence, we infer that there is no FPRAS for counting stable roommate assignments (#SR) unless NP=RP. Utilizing previous results by the authors, we give an approximation-preserving reduction from counting the number of independent sets in a bipartite graph (#BIS) to counting the number of stable roommate assignments both in the 3-attribute model and in the 2-Euclidean model. #BIS is complete with respect to approximation-preserving reductions in the logically-defined complexity class #RH\Pi_1. Hence, our result shows that an FPRAS for counting stable roommate assignments in the 3-attribute model would give an FPRAS for all of #RH\Pi_1. We also show that the 1-attribute stable roommate problem always has either one or two stable roommate assignments, so the number of assignments can be determined exactly in polynomial time