Outgoing Edges Research Articles

Flow Allocation Games

We study a game-theoretic variant of the maximum circulation problem. In a flow allocation game, we are given a directed flow network. Each node is a rational agent and can strategically allocate any incoming flow to the outgoing edges. Given the strategy choices of all agents, a maximal circulation that adheres to the chosen allocation strategies evolves in the network. Each agent wants to maximize the amount of flow through his or her node. Flow allocation games can be used to express strategic incentives of clearing in financial networks. We provide a cumulative set of results on the existence and computational complexity of pure Nash and strong equilibria as well as tight bounds on the (strong) prices of anarchy and stability. Our results show an interesting dichotomy. Ranking strategies over individual flow units allows us to obtain optimal strong equilibria for many objective functions. In contrast, more intuitive ranking strategies over edges can give rise to unfavorable incentive properties. Funding: This work was supported by Deutsche Forschungsgemeinschaft Research Group ADYN [411362735].

Conservative trees

The conservative number of a graph G is the minimum M such that G admits an orientation and a labeling of its edges with distinct numbers in {1,2,…,M} so that at each vertex of degree at least three, the sum of the labels of the incoming edges minus the sum of the labels of the outgoing edges is zero. A graph is conservative if its conservative number and its size are equal.In this work we determine the conservative number of several classes of trees and study the relation between conservative trees and graceful cycles with chords. We also show that for a given size of its base cycle, shell-type graphs with the maximum number of consecutive 4-faces are graceful.

Recursive State Machine Guided Graph Folding for Context-Free Language Reachability

Context-free language reachability (CFL-reachability) is a fundamental framework for program analysis. A large variety of static analyses can be formulated as CFL-reachability problems, which determines whether specific source-sink pairs in an edge-labeled graph are connected by a reachable path, i.e., a path whose edge labels form a string accepted by the given CFL. Computing CFL-reachability is expensive. The fastest algorithm exhibits a slightly subcubic time complexity with respect to the input graph size. Improving the scalability of CFL-reachability is of practical interest, but reducing the time complexity is inherently difficult. In this paper, we focus on improving the scalability of CFL-reachability from a more practical perspective---reducing the input graph size. Our idea arises from the existence of trivial edges, i.e., edges that do not affect any reachable path in CFL-reachability. We observe that two nodes joined by trivial edges can be folded---by merging the two nodes with all the edges joining them removed---without affecting the CFL-reachability result. By studying the characteristic of the recursive state machines (RSMs), an alternative form of CFLs, we propose an approach to identify foldable node pairs without the need to verify the underlying reachable paths (which is equivalent to solving the CFL-reachability problem). In particular, given a CFL-reachability problem instance with an input graph G and an RSM, based on the correspondence between paths in G and state transitions in RSM, we propose a graph folding principle, which can determine whether two adjacent nodes are foldable by examining only their incoming and outgoing edges. On top of the graph folding principle, we propose an efficient graph folding algorithm GF. The time complexity of GF is linear with respect to the number of nodes in the input graph. Our evaluations on two clients (alias analysis and value-flow analysis) show that GF significantly accelerates RSM/CFL-reachability by reducing the input graph size. On average, for value-flow analysis, GF reduces 60.96% of nodes and 42.67% of edges of the input graphs, obtaining a speedup of 4.65× and a memory usage reduction of 57.35%. For alias analysis, GF reduces 38.93% of nodes and 35.61% of edges of the input graphs, obtaining a speedup of 3.21× and a memory usage reduction of 65.19%.

Numerical relaxation limit and outgoing edges in a central scheme for networked conservation laws

AbstractA recently introduced scheme for networked conservation laws is analyzed in various experiments. The scheme makes use of a novel relaxation approach that governs the coupling conditions of the network and does not require a solution of the Riemann problem at the nodes. We numerically compare the dynamics of the solution obtained by the scheme to solutions obtained using a classical coupling condition. In particular, we investigate the case of two outgoing edges in the Lighthill–Whitham–Richards model of traffic flow and in the Buckley–Leverett model of two phase flow. Moreover, we numerically study the asymptotic preserving property of the scheme by comparing it to its preliminary form before the relaxation limit in a 1‐to‐1 network.

Nonabelian flows in networks

AbstractIn this work we consider a generalization of graph flows. A graph flow is, in its simplest formulation, a labelling of the directed edges with real numbers subject to various constraints. A common constraint is conservation in a vertex, meaning that the sum of the labels on the incoming edges of this vertex equals the sum of those on the outgoing edges. One easy fact is that if a flow is conserving in all but one vertex, then it is also conserving in the remaining one. In our generalization we do not label the edges with real numbers, but with elements from an arbitrary group, where this fact becomes false in general. As we will show, graphs with the property that conservation of a flow in all but one vertex implies conservation in all vertices are precisely the planar graphs.

Enhanced electroencephalography effective connectivity in frontal low-gamma band correlates of emotional regulation after mindfulness training.

Practicing mindfulness, focusing attention on the internal and external experiences occurring in the present moment with open and nonjudgement stance, can lead to the development of emotional regulation skills. Yet, the effective connectivity of brain regions during mindfulness has been largely unexplored. Studies have shown that mindfulness practice promotes functional connectivity in practitioners, potentially due to improved emotional regulation abilities and increased connectivity in the lateral prefrontal areas. To examine the changes in effective connectivity due to mindfulness training, we analyzed electroencephalogram (EEG) signals taken before and after mindfulness training, focusing on training-related effective connectivity changes in the frontal area. The mindfulness training group participated in an 8-week mindfulness-based stress reduction (MBSR) program. The control group did not take part. Regardless of the specific mindfulness practice used, low-gamma band effective connectivity increased globally after the mindfulness training. High-beta band effective connectivity increased globally only during Breathing. Moreover, relatively higher outgoing effective connectivity strength was seen during Resting and Breathing and Body-scan. By analyzing the changes in outgoing and incoming connectivity edges, both F7 and F8 exhibited strong parietal connectivity during Resting and Breathing. Multiple regression analysis revealed that the changes in effective connectivity of the right lateral prefrontal area predicted mindfulness and emotional regulation abilities. These results partially support the theory that the lateral prefrontal areas have top-down modulatory control, as these areas have high outflow effective connectivity, implying that mindfulness training cultivates better emotional regulation.

Some Properties of Stochastic Matrices and Non-Homogeneous Markov Chains Generated by Nonlinearities in the Resource Network Model

The resource network is a non-linear threshold model where vertices exchange resource in infinite discrete time. The model is represented by a directed weighted graph. At each time step, all vertices send their resources along all output edges following one of two rules. For each vertex, the threshold value for changing the operation rule is equal to the total weight of its outgoing edges. If all vertices have resources less than their thresholds, the network is completely described by a homogeneous Markov chain. If at least one of the vertices has a resource above the threshold, the network is described by a non-homogeneous Markov chain. The purpose of this article is to describe and investigate non-homogeneous Markov chains generated by the resource network model. It is proven that they are strongly ergodic. In addition, stochastic matrices of a special form were studied. A number of new properties were revealed for them. The results obtained were generalized to arbitrary stochastic matrices.

A Weighted and Normalized Gould-Fernandez brokerage measure.

The Gould and Fernandez local brokerage measure defines brokering roles based on the group membership of the nodes from the incoming and outgoing edges. This paper extends on this brokerage measure to account for weighted edges and introduces the Weighted-Normalized Gould-Fernandez measure (WNGF). The value added of this new measure is demonstrated empirically with both a macro level trade network and a micro level organization network. The measure is first applied to the EUREGIO inter-regional trade dataset and then to an organizational network in a research and development (R&D) group. The results gained from the WNGF measure are compared to those from two dichotomized networks: a threshold and a multiscale backbone network. The results show that the WNGF generates valid results, consistent with those of the dichotomized network. In addition, it provides the following advantages: (i) it ensures information retention; (ii) since no alterations and decisions have to be made on how to dichotomize the network, the WNGF frees the user from the burden of making assumptions; (iii) it provides a nuanced understanding of each node's brokerage role. These advantages are of special importance when the role of less connected nodes is considered. The two empirical networks used here are for illustrative purposes. Possible applications of WNGF span beyond regional and organizational studies, and into all those contexts where retaining weights is important, for example by accounting for persisting or repeating edges compared to one-time interactions. WNGF can also be used to further analyze networks that measure how often people meet, talk, text, like, or retweet. WNGF makes a relevant methodological contribution as it offers a way to analyze brokerage in weighted, directed, and even complete graphs without information loss that can be used across disciplines and different type of networks.

Latency and Alphabet Size in the Context of Multicast Network Coding

We study the relation between <i>latency</i> and <i>alphabet size</i> in the context of Multicast Network Coding. Given a graph <inline-formula> <tex-math notation="LaTeX">$G = (V, E)$ </tex-math></inline-formula> representing a communication network, a subset <inline-formula> <tex-math notation="LaTeX">$S \subseteq V$ </tex-math></inline-formula> of sources, each of which initially holds a set of information messages, and a set <inline-formula> <tex-math notation="LaTeX">$T \subseteq V$ </tex-math></inline-formula> of terminals; we consider the problem in which one wishes to design a communication scheme that eventually allows all terminals to obtain all the messages held by the sources. In this study we assume that communication is performed in rounds, where in each round each network node may transmit a single (possibly encoded) information packet on any of its outgoing edges. The objective is to minimize the communication latency, i.e., number of communication rounds needed until all terminals have all the messages of the source nodes. For sufficiently large alphabet sizes (i.e., large block length, packet sizes), it is known that traditional linear multicast network coding techniques (such as random linear network coding) minimize latency. In this work we seek to study the task of minimizing latency in the setting of limited alphabet sizes (i.e., finite block length), and alternatively, the task of minimizing the alphabet size in the setting of bounded latency. We focus on the establishing the computation complexity of the problem and present several intractability results. In particular, through reductive arguments, we prove that it is NP-hard to (i) approximate (and in particular to determine) the minimum alphabet size given a latency constraint; (ii) approximate (and in particular to determine) the minimum latency of communication schemes in the setting of limited alphabet sizes.

Practical Fixed-Parameter Algorithms for Defending Active Directory Style Attack Graphs

Active Directory is the default security management system for Windows domain networks. We study the shortest path edge interdiction problem for defending Active Directory style attack graphs. The problem is formulated as a Stackelberg game between one defender and one attacker. The attack graph contains one destination node and multiple entry nodes. The attacker's entry node is chosen by nature. The defender chooses to block a set of edges limited by his budget. The attacker then picks the shortest unblocked attack path. The defender aims to maximize the expected shortest path length for the attacker, where the expectation is taken over entry nodes. We observe that practical Active Directory attack graphs have small maximum attack path length and are structurally close to trees. We first show that even if the maximum attack path length is a constant, the problem is still w[1]-hard with respect to the defender's budget. Having a small maximum attack path length and a small budget is not enough to design fixed-parameter algorithms. If we further assume that the number of entry nodes is small, then we derive a fixed-parameter tractable algorithm. We then propose two other fixed-parameter algorithms by exploiting the tree-like features. One is based on tree decomposition and requires a small tree width. The other assumes a small number of splitting nodes (nodes with multiple out-going edges). Finally, the last algorithm is converted into a graph convolutional neural network based heuristic, which scales to larger graphs with more splitting nodes.

Planar L-Drawings of Bimodal Graphs

In a planar L-drawing of a directed graph (digraph) each edge $e$ is represented as a polyline composed of a vertical segment starting at the tail of $e$ and a horizontal segment ending at the head of $e$. Distinct edges may overlap, but not cross. Our main focus is on bimodal graphs, i.e., digraphs admitting a planar embedding in which the incoming and outgoing edges around each vertex are contiguous. We show that every plane bimodal graph without 2-cycles admits a planar L-drawing. This includes the class of upward-plane graphs. Bimodal graphs with 2-cycles admit a planar L-drawing if the underlying undirected graph with merged 2-cycles is a planar 3-tree. Finally, outerplanar digraphs admit a planar L-drawing - although they do not always have a bimodal embedding - but not necessarily with an outerplanar embedding.

Notes on weak-odd edge colorings of digraphs

A weak-odd edge coloring of a general digraph D is a (not necessarily proper) coloring of its edges such that for each vertex v ∈ V ( D ) at least one color c satisfies the following conditions: if d D − ( v ) &gt; 0 then c appears an odd number of times on the incoming edges at v ; and if d D + ( v ) &gt; 0 then c appears an odd number of times on the outgoing edges at v . The minimum number of colors sufficient for a weak-odd edge coloring of D is the weak-odd chromatic index, denoted χ ′ wo ( D ) . It is known that χ ′ wo ( D ) ≤ 3 for every digraph D , and that the bound is sharp. In this article we show that the weak-odd chromatic index can be determined in polynomial time. Restricting to edge colorings of D with at most two colors, the minimum number of vertices v ∈ V ( D ) for which no color c satisfies the above conditions is the defect of D , denoted def( D ) . Surprisingly, it turns out that the problem of determining the defect of digraphs is (polynomially) equivalent to the problem of finding the matching number of simple graphs. Moreover, we characterize the classes of associated digraphs and tournaments in terms of the weak-odd chromatic index and the defect.

Graph planning with expected finite horizon

Fixed-horizon planning considers a weighted graph and asks to construct a path that maximizes the sum of weights for a given time horizon T. However, in many scenarios, the time horizon is not fixed, but the stopping time is chosen according to some distribution such that the expected stopping time is T. If the stopping-time distribution is not known, then to ensure robustness, the distribution is chosen by an adversary as the worst-case scenario. A stationary plan for every vertex always chooses the same outgoing edge. For fixed horizon or fixed stopping-time distribution, stationary plans are not sufficient for optimality. Quite surprisingly we show that when an adversary chooses the stopping-time distribution with expected stopping-time T, then stationary plans are sufficient. While computing optimal stationary plans for fixed horizon is NP-complete, we show that computing optimal stationary plans under adversarial stopping-time distribution can be achieved in polynomial time.

Evolving Bipartite Model Reveals the Bounded Weights in Mobile Social Networks

Many realistic mobile social networks can be characterized by evolving bipartite graphs [1], [2], in which dynamically added elements are divided into two entities and connected by links between these two entities, such as users and items in recommendation networks, followees and followers in Twitter networks, authors and scientific topics in scholarly networks, etc. However, given the fact that connections between two entities are often weighted, how to mathematically model such weighted evolving bipartite relationships, along with quantitative characterizations, remains unexplored. Motivated by this, we develop a novel Evolving Bipartite Model (EBM), which, based on empirically validated power-law distribution on multiple realistic mobile social networks, discloses that the distribution of total weights of incoming and outgoing edges in networks is determined by the weighting scale and bounded by certain ceilings and floors. Based on these theoretical results, for evolving bipartite networks whose degree follows power-law distribution, their overall weights of vertices can be predicted by EBM. To illustrate, in recommendation networks, the evaluation of items, i.e., total rating scores, can be estimated through the given bounds; in Twitter social networks, the influence of users can be roughly measured; in scholarly networks

A convection–diffusion model on a star-shaped graph

In this paper we study a convection–diffusion equation on a star-shaped graph composed by n incoming edges and m outgoing edges with a nonlinearity $$f\in C^1({\mathbb {R}})$$ satisfying some additional general conditions. First, we prove the global well-posedness of the solutions of the system under consideration. Next, in the particular case that the nonlinear convection is given by $$\partial _x(f(u(t, x))$$ with $$f(s)=-a|s|^{q-1}s$$ with $$q\ge 2$$ and $$a\in {\mathbb {R}}$$ verifying $$(n-m)a\ge 0$$ , we analyze the long time behavior of the solutions. For $$q> 2$$ we find that the asymptotic behavior of the solutions is given by some self-similar profiles of the heat equation on the considered structure. In the case $$q=2$$ , the nonnegative/nonpositive solutions converge to the self-similar profiles of Burgers’ equation. Explicit representations of the limit profiles are obtained.

A theoretical framework to accelerate scheduling improvement heuristics using a new longest path algorithm in perturbed DAGs

ABSTRACT Job-shop scheduling problems are complex and still well-studied manufacturing problems. Improvement heuristic algorithms have been proposed to solve the scheduling problems using makespan as their performance measure. All these heuristics iteratively perturb trial schedules by selecting a new schedule from a set of nearby schedules (neighbourhood); then, recalculate and compare the makespan until a sufficient schedule is determined. Unlike previous studies, we did not generate a new heuristic or a novel neighbourhood calculation. Instead, we proposed a theoretical framework, Algorithm to Visit Affected Node (AVAN), which can be incorporated in qualified heuristics while using their current neighbourhood structure to accelerate the recalculation of the makespan in each iteration. We modelled the system by Directed Acyclic Graph (DAG) where the length of the longest path equals the makespan. The scheduling perturbations are represented by adding and deleting edges. AVAN investigates the configuration of scheduling perturbations (added/deleted edges) to find an appropriate starting point to traverse the graph. AVAN is mathematically more efficient than previous longest path algorithms for perturbed DAG. Its time complexity is , where is the number of affected nodes and is the number of incoming and outgoing edges of the affected nodes.

The effect of avoiding known infected neighbors on the persistence of a recurring infection process

We study a generalization of the classical contact process (SIS epidemic model) on a directed graph G. Our model is a continuous-time interacting particle system in which at every time, each vertex is either healthy or infected, and each oriented edge is either active or inactive. Infected vertices become healthy at rate 1 and pass the infection along each active outgoing edge at rate λ. At rate α, healthy individuals deactivate each incoming edge from their infected neighbors, and an inactive edge becomes active again as soon as its tail vertex becomes healthy. When α=0, this model is the same as the classical contact process on a static graph. We study the persistence time of this epidemic model on the lattice Z, the n-cycle Zn, and the n-star graph. We show that on Z, for every α>0, there is a phase transition in λ between almost sure extinction and positive probability of indefinite survival; on Zn we show that there is a phase transition between poly-logarithmic and exponential survival time as the size of the graph increases. On the star graph, we show that the survival time is nΔ+o(1) for an explicit function Δ(α,λ) whenever α>0 and λ>0. In the cases of Z and Zn, our results qualitatively match what has been shown for the classical contact process, while in the case of the star graph, the classical contact process exhibits exponential survival for all λ>0, which is qualitatively different from our result. This model presents a challenge because, unlike the classical contact process, it has not been shown to be monotonic in the infection parameter λ or the initial infected set.

Resolvable Networks—A Graphical Tool for Representing and Solving SAT

In this paper, we introduce the notion of resolvable networks. A resolvable network is a digraph of subnetworks, where subnetworks may overlap, and the inner structure of subnetworks are not interesting from the viewpoint of the network. There are two special subnetworks, Source and Sink, with the following properties: there is no incoming edge to Source, and there is no outgoing edge from Sink. Any resolvable network can be represented by a satisfiability problem in Boolean logic (shortly, SAT problem), and any SAT problem can be represented by a resolvable network. Because of that, the resolution operation is valid also for resolvable networks. We can use resolution to find out or refine the inner structure of subnetworks. We give also a pessimistic and an optimistic interpretation of subnetworks. In the pessimistic case, we assume that inside a subnetwork, all communication possibilities are represented as part of the resolvable network. In the optimistic case, we assume that each subnetwork is strongly connected. We show that any SAT problem can be visualized using the pessimistic interpretation. We show that transitivity is very limited in the pessimistic interpretation, and in this case, transitivity corresponds to resolution of clauses. In the optimistic interpretation of subnetworks, we have transitivity without any further condition, but not all SAT problems can be represented in this case; however, any such network can be represented as a SAT problem. The newly introduced graphical concept allows to use terminology and tools from directed graphs in the field of SAT and also to give graphical representations of various concepts of satisfiability problems. A resolvable network is also a suitable data structure to study, for example, wireless sensor networks. The visualization power of resolvable networks is demonstrated on some pigeon hole SAT problems. Another important application field could be modeling the communication network of an information bank. Here, a subnetwork represents a dataset of a user which is secured by a proxy. Any communication should be done through the proxy, and this constraint can be checked using our model.

XENet: Using a new graph convolution to accelerate the timeline for protein design on quantum computers.

Graph representations are traditionally used to represent protein structures in sequence design protocols in which the protein backbone conformation is known. This infrequently extends to machine learning projects: existing graph convolution algorithms have shortcomings when representing protein environments. One reason for this is the lack of emphasis on edge attributes during massage-passing operations. Another reason is the traditionally shallow nature of graph neural network architectures. Here we introduce an improved message-passing operation that is better equipped to model local kinematics problems such as protein design. Our approach, XENet, pays special attention to both incoming and outgoing edge attributes. We compare XENet against existing graph convolutions in an attempt to decrease rotamer sample counts in Rosetta's rotamer substitution protocol, used for protein side-chain optimization and sequence design. This use case is motivating because it both reduces the size of the search space for classical side-chain optimization algorithms, and allows larger protein design problems to be solved with quantum algorithms on near-term quantum computers with limited qubit counts. XENet outperformed competing models while also displaying a greater tolerance for deeper architectures. We found that XENet was able to decrease rotamer counts by 40% without loss in quality. This decreased the memory consumption for classical pre-computation of rotamer energies in our use case by more than a factor of 3, the qubit consumption for an existing sequence design quantum algorithm by 40%, and the size of the solution space by a factor of 165. Additionally, XENet displayed an ability to handle deeper architectures than competing convolutions.

On integer balancing of directed graphs

A weighted digraph is balanced if the sums of the weights of the incoming and of the outgoing edges are equal at each vertex. We show that if these sums are integers, then for any edge weights satisfying the balance conditions, there exist integer weights obtained by rounding the original weights up or down that preserve both the balance condition and the sum of all edge weights.