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Graph Coloring Games and Voter Models

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Graph Coloring Games and Voter Models Graph coloring games and voter models Fan Chung Graham University of California, San Diego Joint work with Ron Graham and Alex Tsiatas The Power Grid Graph drawn by Derrick Bezanson 2010 A graph G = (V,E) edge vertex Graph models _____________________________ Vertices Edges cities flights people pairs of friends authors coauthorship telephones phone calls web pages linkings genes regulatory aspects Hypergraph y x z An hyperedge : a subset of nodes Example : { x,y,z} Hypergraph G = (V,E) V = {x, y,z,r,s} E = {{x, y,z},{z,r,s},{y,r}} y r x z s Hypergraph models vertices hyperedges voters chatgroups students classes genes regulatory aspects Graph Theory has 250 years of history. Leonhard Euler The bridges of Königsburg 1707-1783 Is it possible to walk over every bridge once and only once? History on hypergraphs Euler extremal set theory Ramsey 1930 matroid theory Whitney 1935 codes Erdos-Ko-Rado“ 1961 Tutte 1947 Berge 1959 Geometric graphs Algebraic graphs real graphs Massive data Massive graphs • WWW-graphs • Call graphs • Acquaintance graphs • Graphs from any data a.base An induced subgraph of the collaboration graph with authors of Erdös number ! 2. A subgraph of the Hollywood graph. Difficulties in understanding complex graphs • Too large! Only can observe partial subgraphs, but wish to capture the behavior of the whole graph. • Evolving! Wish to know which way to go. • No centralized/distributed rules! Selfish players! Wish to predict the winner any way. Numerous questions arise in dealing with large realistic networks • How are these graphs formed? • What are the basic structures of such xxgraphs? • What principles dictate their behavior? • How are subgraphs related to the large xxhost graph? • What are the main graph invariants xxcapturing the properties of such graphs? New problems and directions • Classical random graph theory Random graphs with any given degrees • Percolation on special graphs Percolation on general graphs • Correlation among vertices Pagerank of a graph • Graph coloring/routing Network games Several examples • Random graphs with any given degrees Diameter of random power law graphs • Diameter of random trees of a given graph • Percolation and giant components in a graph • Correlation between vertices xxxxxxxxxxxxxThe pagerank of a graphs • Graph coloring and network games Games on networks • Routing games Selfish routing, Price of anarchy Braess paradox • Ranking games • Coloring games Consensus games, primary games, Games on networks • Routing games Selfish routing, Price of anarchy Braess paradox • Ranking games • Coloring games Consensus games, primary games, Michael Kearns’ experiments on coloring games 2006 consensus Consensus experiments Fig. 5. Progression of vertex colors in four consensus experiments. The x-axis measures experiment duration (thus image width is proportional to time to completion), and there is a horizontal row for each of the 36 players showing its current color at each moment of the experiment. The first two images are for rewiring parameter q 0, and the last two for q 0.1. Players in the same clique (before any rewiring) are in six consecutive rows, thus explaining the overall ¼ ¼ tendency for such blocks of six to change colorsJudd, in approximate Kearns, unison. In Vorobeychik the (successful) leftmost 2010 experiment, we see a rapid reduction from 9 to 3 and then just 2 colors used by the population; a long stalemate between two halves of the network is gradually broken by yellow cliques adopting brown. The second image shows the only consensus experiment out of 18 total to not successfully complete. Again we see a rapid reduction to 3 and then 2 colors, but this time the dynamics are much richer; shortly before the halfway mark, we see that blue is chosen by the vast majority of the subjects but there remains an orange holdout clique. A short while later, this clique has adopted blue, but not before causing a cascade of orange to ripple through the network, where it then proceeds to take hold and in fact constitute the majority near the end of the time limit. The rightmost two images (where q 0.1) complete significantly faster (consistent ¼ with the reported overall effect of q on consensus games), and show similar overall dynamics but with more mixing and experimentation by subjects. The images also reveal interesting acts of individual behavior, including attempts at signalling and apparently “irrational” experimentation such as playing a color not chosen by anyone else in the network. define the overall neighborhood influence of a player in a given experiment. For this measure we again find that human variation experiment to be the sum of all the influence credits as described, is significant under a standard ANOVA test, and furthermore that divided by its degree and number of color changes; thus the the change rate of a human subject is strongly negatively corre- measure is already normalized to compensate for either a large lated with both their neighborhood and outcome influence in number of neighbors or players who change color frequently. both game types (correlations ranging from −0.40 to −0.51 de- Our second measure of influence is called outcome influence, pending on the influence and game type, all with P < 0.02). and is simply the amount of time elapsed between a player’s final We thus conclude that more stable players exhibit greater influ- color change and the end of the experiment. We thus attribute the ence. Note that this finding is not obvious a priori, since neigh- most credit to those players who anticipate the eventual global borhood influence is already normalized for change rate, and for solution, and thus may have participated in its creation. Under outcome influence the least stable players may still settle on their this definition, the outcome influence of the last player to com- final color relatively early. plete a global solution is zero. Note that neighborhood influence We also examined the notion of stubbornness, defined as the depends intimately on network structure, while outcome influ- proportion of time that a player or vertex chooses a “noncompli- ence does not do so in any obvious or direct way. ant” color. In consensus, a color is noncompliant if there is another played by more neighbors. In coloring, a color is noncom- Influence Variation by Subject and Position. Armed with these two pliant if there is another played by fewer neighbors. As with notions of influence, our first interest is to examine whether the change rate, we find stubbornness variation among subjects to influence of an agent can be understood as arising primarily from be significant, and stubbornness is strongly negatively correlated their position in the network, or is to a significant extent a prop- with both influence types (−0.45 for neighborhood influence, erty of the actual human subject. In our experimental methodol- −0.50 for outcome influence, both with P < 0.02) in consensus ogy, human subjects are randomly assigned to vertices at the games (but not in coloring games). Thus it appears that rather beginning of each experiment. This policy helps to separate than causing others to react to or coalesce around their noncom- the effect of the human from the effect of the vertex (network pliant choices, stubborn players reduce their impact on their position). neighbors and the outcome of experiments. For each of our two notions of influence, we use a standard Despite the fact that human variation in influence cannot be ANOVA test to detect whether variability of average influence explained by network position alone, and that there appear to be among the 36 human subjects exceeds the variability of influence a number of other behavioral traits that strongly correlate with due to experimental conditions, such as random assignments of influence, it remains reasonable to ask what purely structural subjects to network vertices. properties of network position still have some explanatory value. For neighborhood influence, we find that human subject varia- The degree of a vertex is only weakly correlated with influence in bility is significant in coloring games (with P < 0.001), but not consensus games (0.21 for neighborhood influence, 0.10 for out- consensus games; while for outcome influence, human variation come influence, both with P < 0.03). Also, degree is not signifi- is significant with P < 0.05 in both game types. We conclude that cantly correlated with either type of influence in coloring games. overall, influence is not a product of just an agent’s network posi- Vertex centrality (average inverse shortest path distance to all tion, but arises from their intrinsic human behavioral patterns other vertices) is significantly correlated only with outcome influ- as well. ence in consensus games (0.26 with P < 0.03). The clustering coefficient of a vertex, which can be viewed as a measure of con- Correlates of Influence. Our two measures of influence quantify the nectivity in a vertex’s neighborhood, is significantly correlated (apparent) effects of a player or vertex on other players or ver- only with outcome influence (0.086, P < 0.05 in coloring, tices. We now examine a number of correlates of influence that −0.22, P < 0.001 in consensus). Similarly, being one of the “con- are determined only by a player’s own actions, or their apparent nector” vertices between cliques in the baseline network before reaction to their neighbors. rewiring is correlated only with neighborhood influence in con- For instance, consider the change rate or instability of a player
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