Game Chromatic Number of Shackle Graphs

Coloring vertices on graph is one of the topics of discrete mathematics that are still developing until now. Exploration Coloring vertices develops in the form of a game known as a coloring game. Let G graph. The smallest number k such that the graph G can be colored in a coloring game is called game chromatic number. Notated as χ_g (G). The main objective of this research is to prove game chromatic numbers from graphsThis study examines and proves game chromatic numbers from graphs shack(K_n,v_i,t),shack(S_n,v_i,t), and shack(K_(n,n),v_i,t). The research method used in this research is qualitative. The result show that χ_g (shack(K_n,v_i,t))=n,and χ_g (shack(S_n,v_i,t))=χ_g (shack(K_(n,n),v_i,t))=3.  The game chromatic number of the shackle graph depends on the subgraph and linkage vertices. Therefore, it is necessary to make sure the vertex linkage is colored first

Information Cascades on Arbitrary Topologies

In this paper, we study information cascades on graphs. In this setting, each node in the graph represents a person. One after another, each person has to take a decision based on a private signal as well as the decisions made by earlier neighboring nodes. Such information cascades commonly occur in practice and have been studied in complete graphs where everyone can overhear the decisions of every other player. It is known that information cascades can be fragile and based on very little information, and that they have a high likelihood of being wrong. Generalizing the problem to arbitrary graphs reveals interesting insights. In particular, we show that in a random graph $G(n,q)$, for the right value of $q$, the number of nodes making a wrong decision is logarithmic in $n$. That is, in the limit for large $n$, the fraction of players that make a wrong decision tends to zero. This is intriguing because it contrasts to the two natural corner cases: empty graph (everyone decides independently based on his private signal) and complete graph (all decisions are heard by all nodes). In both of these cases a constant fraction of nodes make a wrong decision in expectation. Thus, our result shows that while both too little and too much information sharing causes nodes to take wrong decisions, for exactly the right amount of information sharing, asymptotically everyone can be right. We further show that this result in random graphs is asymptotically optimal for any topology, even if nodes follow a globally optimal algorithmic strategy. Based on the analysis of random graphs, we explore how topology impacts global performance and construct an optimal deterministic topology among layer graphs

Characterizing Strategic Cascades on Networks

Transmission of disease, spread of information and rumors, adoption of new products, and many other network phenomena can be fruitfully modeled as cascading processes, where actions chosen by nodes influence the subsequent behavior of neighbors in the network graph. Current literature on cascades tends to assume nodes choose myopically based on the state of choices already taken by other nodes. We examine the possibility of strategic choice, where agents representing nodes anticipate the choices of others who have not yet decided, and take into account their own influence on such choices. Our study employs the framework of Chierichetti et al. [2012], who (under assumption of myopic node behavior) investigate the scheduling of node decisions to promote cascades of product adoptions preferred by the scheduler. We show that when nodes behave strategically, outcomes can be extremely different. We exhibit cases where in the strategic setting 100% of agents adopt, but in the myopic setting only an arbitrarily small epsilon % do. Conversely, we present cases where in the strategic setting 0% of agents adopt, but in the myopic setting (100-epsilon)% do, for any constant epsilon > 0. Additionally, we prove some properties of cascade processes with strategic agents, both in general and for particular classes of graphs.Comment: To appear in EC 201

Theoretical Tools for Network Analysis: Game Theory, Graph Centrality, and Statistical Inference.

A computer-driven data explosion has made the difficulty of interpreting large data sets of interconnected entities ever more salient. My work focuses on theoretical tools for summarizing, analyzing, and understanding network data sets, or data sets of things and their pairwise connections. I address four network science issues, improving our ability to analyze networks from a variety of domains. I first show that the sophistication of game-theoretic agent decision making can crucially effect network cascades: differing decision making assumptions can lead to dramatically different cascade outcomes. This highlights the importance of diligence when making assumptions about agent behavior on networks and in general. I next analytically demonstrate a significant irregularity in the popular eigenvector centrality, and propose a new spectral centrality measure, nonbacktracking centrality, showing that it avoids this irregularity. This tool contributes a more robust way of ranking nodes, as well as an additional mathematical understanding of the effects of network localization. I next give a new model for uncertain networks, networks in which one has no access to true network data but instead observes only probabilistic information about edge existence. I give a fast maximum-likelihood algorithm for recovering edges and communities in this model, and show that it outperforms a typical approach of thresholding to an unweighted network. This model gives a better tool for understanding and analyzing real-world uncertain networks such as those arising in the experimental sciences. Lastly, I give a new lens for understanding scientific literature, specifically as a hybrid coauthorship and citation network. I use this for exploratory analysis of the Physical Review journals over a hundred-year period, and I make new observations about the interplay between these two networks and how this relationship has changed over time.PhDComputer Science and EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/133463/1/travisbm_1.pd

How to Schedule a Cascade in an Arbitrary Graph

When individuals in a social network make decisions that depend on what others have done earlier, there is the potential for a cascade to form — a run of behaviors that are highly correlated. In an arbitrary network, the outcome of such a cascade can depend sensitively on the order in which nodes make their decisions, but to do date there has been very little investigation of how this dependence works, or how to choose an order to optimize various parameters of the cascade. Here we formulate the problem of ordering the nodes in a cascade to maximize the expected number of “favorable” decisions — those that support a given option. We provide an algorithm that ensures an expected linear number of favorable decisions in any graph, and we show that the performance bounds for our algorithm are essentially the best achievable assuming P != NP