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 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 (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 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 sensus games (0.17, P < 0.03). The evidence again suggests that or position, which is the number of color changes per second in an influence seems more determined by the behavioral characteris-

4of5 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1001280107 Judd et al. Michael Kearns’ experiments on coloring games 2006

conflict resolution Classical graph coloring Chromatic graph theory

Coloring graphs in a greedy and selfish way Coloring games on graphs Applications of graph coloring games

• dynamics of social networks

• conflict resolution

• Internet economics

• on-line optimization + scheduling

• • • Hypergraph y

x

z

Hypergraph G = (V,E) V = {voters} E = {chat groups} An interaction voter model on hypergraphs

G = (V,E) V = {voters} E = {chat groups}

Colors for voters: red, blue or * (undecided).

For g ∈E, strategy Xgz: voters in g, interact and change their colors to pattern z.

pg (z) : probability that X gz is chosen.

In each round, one chat group g is randomly chosen. An interaction voter model on hypergraphs

G = (V,E) V = {voters} E = {chat groups}

Colors for voters: red, blue or * (undecided).

For g ∈E, strategy Xgz: voters in g, interact and change their colors to pattern z.

pg (z) : probability that X gz is chosen.

In each round, one chat group g is randomly chosen. What happens after “many” rounds? A general interaction voter model on hypergraphs

Hard!

• (Beyond) NP-hard.

• No good heuristics.

• No good guess for `cut-off’ time.

• Experiments reveal little. Partial results on an interaction voter model on hypergraphs

• If strategies X gz with nontrivial probability p g (z) are consistent (i.e., intersection of two nontrivial color patterns of edges has no conflict), then the process converges in O ( m l og n ) time.

• In general, the coloring does not converge. No equilibrium.

However, if the nontrivial strategies are “memoryless”, then the corresponding random walks on the state graph converges in O ( m l og n ) time and have many strongproperties. Many questions

• Random walks on state graphs might not converge. • Might converge but takes exponential time. • Cut-off time for experiments?

Fig. 5. Progression of vertex colors in four consensus experiments. The x-axis measures experimentKearns duration et. (thusal. image2006 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 colors in approximate unison. In the (successful) leftmost 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 sensus games (0.17, P < 0.03). The evidence again suggests that or position, which is the number of color changes per second in an influence seems more determined by the behavioral characteris-

4of5 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1001280107 Judd et al. An edge flipping process:

Choose an edge {x,y} at random.

Color both x and y

blue with probability p,

red with probability 1-p.

G Repeat. Many Questions:

What does a typical configuration look like?

What is the stationary distribution for color patterns?

How long does it take to converge?

What are the most likely and least likely patterns?

Simulation? Examples? Computation?

All tells little. The state graph H of edge flipping on G:

V(H) : coloring configurations of G,

Each directed edge in E(H) ---- flipping one edge from one coloring of G to another. V(H ) = 2n

G=P 3 H

Loops are omitted. Coloring problems on a graph G G=P3

Random walks on the state graph H.

H The state graph H of edge flipping on G:

V(H) : coloring configurations of G,

Each directed edge in E(H) ---- flipping one edge from one coloring of G to another. V(H ) = 2n

G=P 3 H

Loops are omitted. The random walk of edge flipping on the state graph H converges to a stationary distribution π . (If exists. ) With probability π ( x ), a coloring configuration x is the resulting configuration after t rounds when t is sufficiently large.

G=P 3 H

Loops are omitted. Many amazing properties about random walks on the state graph H of the flipping process.

Eigenvalues are real. Elegant formula using spectral theory on semigroups Eigenvectors? stationary distribution? Not much known.

G=P3 H Spectral graph theory on semigroups

A semigroup is a set with an associative binary operation.

Example: • Intergers • Groups are semigroups. • S = sequences in {1,..., N} = [N] ∧ (x1,...xl )(y1,..., ym ) = (x1,...xl , y1,..., ym )

∧ where ( ) means deletion of any element that has already occurred to its left. Spectral graph theory on semigroups

Cayley graphs G = (V,E)

V ! a group H =A5 u ! v ! u = gv for g"B,a chosen subset of H.

B={(12345),(54321),(12)(34)}!A5

Cayley graphs G = (V,E) V ! a group H u ! v ! u = gv for g"B,a chosen subset of H.

0 adjacency matrix ! 0

irreducible representations

Spectral graph theory on semigroups

A semigroup is a set with an associative binary operation.

Example: • Intergers • Groups are semigroups. • S = sequences in {1,..., N} = [N] ∧ (x1,...xl )(y1,..., ym ) = (x1,...xl , y1,..., ym )

∧ where ( ) means deletion of any element that has already occurred to its left. A semigroup S is a left regular band (LRB) if

2 For all x ∈S, x = x, (idempotent).

For all x , y ∈S, xyx = xy.

Example: S = sequences in {1,..., N} = [N] ∧ (x1,...xl )(y1,..., ym ) = (x1,...xl , y1,..., ym )

∧ where ( ) means deletion of any element that has already occurred to its left.

(12)(13) = (123) (12)(13)(23) = (123)(23) = (123) = (12)(132) History of LRB

1940 Klein-Barem defined “LRB”. .. 1947 Schutzenberger also introduced “LRB”.

1991 Phatarfod, Tsetlin library, self-organizing search.

1996 Fill, Move-to-front search.

1998 Brown + Diaconis, card shuffling, random walks

1999 Bidigare + Hanlon + Rockmore, generalized Tsetlin library, hyperplane random walks. 2000 Brown, semigroup Markov chains {w } For a probability distribution x on S, ∑wx = 1, x∈S we define a random walk with transition probability matrix

P(s,t) = ∑ wx x∈S xs=t The eigenvalues of a random walk on chambers of semigroup S: For each flat X ∈ L , the lattice constructed from S,

λX = ∑ wx supp x⊆ X

with multiplicity ∑ mX = cX Y_≻ X {w } For a probability distribution x on S, ∑wx = 1, x∈S we define a random walk with transition probability matrix

P(s,t) = ∑ wx x∈S xs=t The eigenvalues of a random walk on chambers of semigroup S: ? ? For each flat X ∈ L , the lattice constructed from S, ? λX = ∑ wx ? supp x⊆ X ? ? with multiplicity ∑ mX = cX ? Y_≻ X Definitions: From a semigroup S, we can construct:

A partial order on S x ≤ y ⇔ xy = y

Example: (12) ≤ (123) (13) ≤ (123)

A chamber is a maximal element in the poset S.

Example: chambers = strings of length n Definitions: From a semigroup S, we can construct: A partial order on S x ≤ y ⇔ xy = y A chamber is a maximal element in the poset S.

A semilattice L, can be constructed from S: Define y ≺_ x ⇔ xy = x. supp x = the equivalence class containing x under ≺_ supp xy = supp x ∨ supp y For a flat Y ∈L, S = S = {z ∈S : z ≻ y}for y ∈Y. ≻Y ≻ y c = S . Y ≻ y Edge flipping semigroup

G : a graph on n vertices and m edges.

A deck of cards: each card ← → an action c (on a state σ )

Example: σ -- pick edge {3,4} and color both 3,4 blue.

Define the product of two cards:

c1c2σ = c1(c2σ ) Edge flipping semigroup

G : a graph on n vertices and m edges.

A deck of cards: each card ← → an action c (on a state σ )

Example: σ -- pick edge {3,4} and color both 3,4 blue.

Define the product of two cards:

c1c2σ = c1(c2σ ) Associative

(c1c2 )c3 = c1(c2c3 ) Edge flipping semigroup

G : a graph on n vertices and m edges.

A deck of cards: each card ← → an action c (on a state σ )

Example: σ -- pick edge {3,4} and color both 3,4 blue.

Define c1c2σ = c1(c2σ ) 2 cards are for each edge.

The edge flipping semigroup is generated by 2m cards and is a LRB. {w } For a probability distribution x on S, ∑wx = 1, x∈S we define a random walk with transition probability matrix

P(s,t) = ∑ wx x∈S xs=t The eigenvalues of a random walk on chambers of semigroup S: For each flat X ∈ L , the lattice constructed from S,

λX = ∑ wx supp x⊆ X

with multiplicity ∑ mX = cX Y_≻ X abcde 1 (x 1)

3 1 3 (x 1) 4 (x 1) abcd abde 2 bcde 4 (x 1)

1 1 1 (x 1) (x2) 2 (x2) abc bcd 2 cde 2

1 1 1 1 (x3) (x2) (x2) (x3) 4 ab bc 4 cd 4 de 4

∅ 0(x13) Eigenvalues with multiplicity for the state graph P5 • An alternative description for the spectra of random walks on the state graph of

• Formula for eigenvalues of the edge flipping semigroup. Theorem: For a graph G on n vertices and m edges, for each card x, let w = 1, wx = Pr(σ → xσ ), ∑ x x∈S The associated random walk of edge flipping on the

state graph of G has eigenvalues λ T :

For each subset T ⊆ V (G),

with multiplicity 1. λT = ∑ wx supp x⊆T

For the case of w = 1 / (2m), c e(T ) λ = . T m abcde 1

abcd 3/4 abce 1/2 abde 1/2 acde 1/2 bcde 3/4

abc 1/2 abd 1/4 abe 1/4 acd 1/4 ace 0 ade 1/4 bcd 1/2 bce 1/4 bde 1/4 cde 1/2

ab 1/4 ac 0 ad 0 ae 0 bc 1/4 bd 0 be 0 cd 1/4 ce 0 de 1/4

a 0 b 0 c 0 d 0 e 0

∅ 0

Eigenvalues with multiplicity for the state graph P5 abcde 1 (x 1)

3 1 3 (x 1) 4 (x 1) abcd abde 2 bcde 4 (x 1)

1 1 1 (x 1) (x2) 2 (x2) abc bcd 2 cde 2

1 1 1 1 (x3) (x2) (x2) (x3) 4 ab bc 4 cd 4 de 4

∅ 0(x13) Eigenvalues with multiplicity for the state graph P5 abcde 1

abcd 3/4 abce 1/2 abde 1/2 acde 1/2 bcde 3/4

abc 1/2 abd 1/4 abe 1/4 acd 1/4 ace 0 ade 1/4 bcd 1/2 bce 1/4 bde 1/4 cde 1/2

ab 1/4 ac 0 ad 0 ae 0 bc 1/4 bd 0 be 0 cd 1/4 ce 0 de 1/4

a 0 b 0 c 0 d 0 e 0

∅ 0

Eigenvalues with multiplicity for the state graph P5 For the case of having the same weight for each card, the eigenvalues are of the form: j for 0 ≤ j ≤ m −1. m Using eigenvalues to bound the rate of convergence:

Pt − π ≤ m λ t TV ∑ F F F∈L* 1 ≤ n2 (1− )t ≤ e−c m if t ≥ 2m logn + cn. What are the most likely and least likely color patterns?

Stationary distribution of the state graph ? G = Cn , n = a +b. C(a,b) : probability the first a vertices are red and the rest are blue. p(1− p) C(n,0) ∼ γ n , γ = p = 1 arctan( p / (1− p)) for 2 2 γ = π ≈ 0.6336197... G = Cn , n = a +b. C(a,b) : probability the first a vertices are red and the rest are blue. p(1− p) C(n,0) ∼ γ n , γ = arctan( p / (1− p))

1 n C(n − 1,1) ∼ p (− p + γ )γ ,

1 n C(n − 2,2) ∼ p (1+ p − 2γ )γ ,

1 2 n−1 C(n − 3,3) ∼ p (1− p − (2 + p)γ + 3γ )γ ,

⎛ (1− p)(3 − 2 p) 2 3 ⎞ n 2 C(n − 4,4) ∼ 1 − 2(1− p)γ + (3 + p)γ − 4γ γ − . p ⎝⎜ 3 ⎠⎟

Relative order of C(n − k,k),0 ≤ k ≤ 4. normalized in comparison with C(n,0). Relative order of C(n − k,k),0 ≤ k ≤ 4. Edge-flipping on a graph

generalized

flipping vertices,

flipping subsets of vertices/edges on graphs and hypergraphs

voter models memoryless games ... Generalizations of edge flipping

• Graphs Hypergraphs

• 2 colors r colors

• memoryless partial memoryless

• Approximate general voter games by partial memoryless models Nodes are colored in blue, red or .

y Memoryless :

...Xg,z ...Xg,z ... = ...Xg,z ... x

z Theorem: Suppose a voting game on hypergraph G in r

colors has memoryless strategies X gz for edge g and color pattern z on g.

The the state graph of G has eigenvalues λ T :

For each subset T ⊆ V (G),

n−|T | λT = ∑ pg,z with multiplicity ( r − 1 ) g,z g⊆T

For the case of r = 3, pg,z = 1 / (3m), e(T ) λ = with multiplicity T m For the graph coloring game,

Pt − π ≤ λ t (r −1)n− T TV ∑ T T ⊆H

n t For r = 3, t ⎛ k ⎞ ⎛ n⎞ n−k P − π ≤ 1− 2 TV ∑⎜ ⎟ ⎜ ⎟ k=1 ⎝ m⎠ ⎝ k⎠

t 2 ⎛ 1 ⎞ ≤ n 1− ⎝⎜ m⎠⎟ ≤ e−c

if t ≥ 2mlogn + cn. Theorem: For memoryless games, the stationary distribution of the state graph is not easy to determine, but

the fast mixing of random walks leads to sharp estimate for an event A Theorem: For memoryless games, the stationary distribution of the state graph is not easy to determine, but

the fast mixing of random walks leads to sharp estimate for an event A (log1 / ε)m logn in rounds within an π(A) error bound ε. A typical result for memoryless voter game.

Theorem: Chung+Tsiatas 2014 If the interaction strategies are memoryless, the probability that the reds win by 5% can be approximated within an error bound ε in c ( l og( 1 / ) ) m l og n rounds, ε y where n is the number of voters, and m is the number of chat groups. x

z Partial memoryless random walks:

P = βM + (1− β)P' for 0 ≤ β < 1.

memoryless

Can be used to approximate general interaction games An attempt to rigorously analyze general interaction games

• Compare/approximate with a partial memoryless model.

• Construct M by using collected data. P = βM + (1− β)P'

memoryless

• Choose β based on the rate of convergence of M. More questions

• Modeling and analyzing interactive computing

• Bounded memory versus memoryless.

• Bounded look ahead.

• On-line versus off-line analysis. 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 The Power Grid Graph drawn by Derrick Bezanson 2010 Quasirandom graphs and hypergraphs

Fan Chung University of California, San Diego

The Power Grid Graph drawn by Derrick Bezanson 2010