Arc-disjoint in- and out-branchings rooted at the same vertex in compositions of digraphs

A digraph $D=(V, A)$ has a good pair at a vertex $r$ if $D$ has a pair of arc-disjoint in- and out-branchings rooted at $r$. Let $T$ be a digraph with $t$ vertices $u_1,\dots , u_t$ and let $H_1,\dots H_t$ be digraphs such that $H_i$ has vertices $u_{i,j_i},\ 1\le j_i\le n_i.$ Then the composition $Q=T[H_1,\dots , H_t]$ is a digraph with vertex set $\{u_{i,j_i}\mid 1\le i\le t, 1\le j_i\le n_i\}$ and arc set $$A(Q)=\cup^t_{i=1}A(H_i)\cup \{u_{ij_i}u_{pq_p}\mid u_iu_p\in A(T), 1\le j_i\le n_i, 1\le q_p\le n_p\}.$$ When $T$ is arbitrary, we obtain the following result: every strong digraph composition $Q$ in which $n_i\ge 2$ for every $1\leq i\leq t$, has a good pair at every vertex of $Q.$ The condition of $n_i\ge 2$ in this result cannot be relaxed. When $T$ is semicomplete, we characterize semicomplete compositions with a good pair, which generalizes the corresponding characterization by Bang-Jensen and Huang (J. Graph Theory, 1995) for quasi-transitive digraphs. As a result, we can decide in polynomial time whether a given semicomplete composition has a good pair rooted at a given vertex

Proximity and Remoteness in Directed and Undirected Graphs

Let $D$ be a strongly connected digraph. The average distance $\bar{\sigma}(v)$ of a vertex $v$ of $D$ is the arithmetic mean of the distances from $v$ to all other vertices of $D$. The remoteness $\rho(D)$ and proximity $\pi(D)$ of $D$ are the maximum and the minimum of the average distances of the vertices of $D$, respectively. We obtain sharp upper and lower bounds on $\pi(D)$ and $\rho(D)$ as a function of the order $n$ of $D$ and describe the extreme digraphs for all the bounds. We also obtain such bounds for strong tournaments. We show that for a strong tournament $T$, we have $\pi(T)=\rho(T)$ if and only if $T$ is regular. Due to this result, one may conjecture that every strong digraph $D$ with $\pi(D)=\rho(D)$ is regular. We present an infinite family of non-regular strong digraphs $D$ such that $\pi(D)=\rho(D).$ We describe such a family for undirected graphs as well

Arc-disjoint strong spanning subdigraphs in compositions and products of digraphs

A digraph $D=(V,A)$ has a good decomposition if $A$ has two disjoint sets $A_1$ and $A_2$ such that both $(V,A_1)$ and $(V,A_2)$ are strong. Let $T$ be a digraph with $t$ vertices $u_1,\dots , u_t$ and let $H_1,\dots H_t$ be digraphs such that $H_i$ has vertices $u_{i,j_i},\ 1\le j_i\le n_i.$ Then the composition $Q=T[H_1,\dots , H_t]$ is a digraph with vertex set $\{u_{i,j_i}\mid 1\le i\le t, 1\le j_i\le n_i\}$ and arc set $$A(Q)=\cup^t_{i=1}A(H_i)\cup \{u_{ij_i}u_{pq_p}\mid u_iu_p\in A(T), 1\le j_i\le n_i, 1\le q_p\le n_p\}.$$ For digraph compositions $Q=T[H_1,\dots H_t]$, we obtain sufficient conditions for $Q$ to have a good decomposition and a characterization of $Q$ with a good decomposition when $T$ is a strong semicomplete digraph and each $H_i$ is an arbitrary digraph with at least two vertices. For digraph products, we prove the following: (a) if $k\geq 2$ is an integer and $G$ is a strong digraph which has a collection of arc-disjoint cycles covering all vertices, then the Cartesian product digraph $G^{\square k}$ (the $k$th powers with respect to Cartesian product) has a good decomposition; (b) for any strong digraphs $G, H$, the strong product $G\boxtimes H$ has a good decomposition

Stochastic Step-wise Feature Selection for Exponential Random Graph Models (ERGMs)

Statistical analysis of social networks provides valuable insights into complex network interactions across various scientific disciplines. However, accurate modeling of networks remains challenging due to the heavy computational burden and the need to account for observed network dependencies. Exponential Random Graph Models (ERGMs) have emerged as a promising technique used in social network modeling to capture network dependencies by incorporating endogenous variables. Nevertheless, using ERGMs poses multiple challenges, including the occurrence of ERGM degeneracy, which generates unrealistic and meaningless network structures. To address these challenges and enhance the modeling of collaboration networks, we propose and test a novel approach that focuses on endogenous variable selection within ERGMs. Our method aims to overcome the computational burden and improve the accommodation of observed network dependencies, thereby facilitating more accurate and meaningful interpretations of network phenomena in various scientific fields. We conduct empirical testing and rigorous analysis to contribute to the advancement of statistical techniques and offer practical insights for network analysis.Comment: 23 pages, 6 tables and 18 figure

Double Oracle Algorithm for Game-Theoretic Robot Allocation on Graphs

We study the problem of game-theoretic robot allocation where two players strategically allocate robots to compete for multiple sites of interest. Robots possess offensive or defensive capabilities to interfere and weaken their opponents to take over a competing site. This problem belongs to the conventional Colonel Blotto Game. Considering the robots' heterogeneous capabilities and environmental factors, we generalize the conventional Blotto game by incorporating heterogeneous robot types and graph constraints that capture the robot transitions between sites. Then we employ the Double Oracle Algorithm (DOA) to solve for the Nash equilibrium of the generalized Blotto game. Particularly, for cyclic-dominance-heterogeneous (CDH) robots that inhibit each other, we define a new transformation rule between any two robot types. Building on the transformation, we design a novel utility function to measure the game's outcome quantitatively. Moreover, we rigorously prove the correctness of the designed utility function. Finally, we conduct extensive simulations to demonstrate the effectiveness of DOA on computing Nash equilibrium for homogeneous, linear heterogeneous, and CDH robot allocation on graphs

Boundary Layers on Sobolev–Besov Spaces and Poisson's Equation for the Laplacian in Lipschitz Domains

AbstractWe study inhomogeneous boundary value problems for the Laplacian in arbitrary Lipschitz domains with data in Sobolev–Besov spaces. As such, this is a natural continuation of work in [Jerison and Kenig,J. Funct. Anal.(1995), 16–219] where the inhomogeneous Dirichlet problem is treated via harmonic measure techniques. The novelty of our approach resides in the systematic use of boundary integral methods. In this regard, the key results are establishing the invertibility of the classical layer potential operators on scales of Sobolev–Besov spaces on Lipschitz boundaries for optimal ranges of indices. Applications toLp-based Helmholtz type decompositions of vector fields in Lipschitz domains are also presented