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Random Walker Ranking for NCAA Division I-A Football
Random Walker Ranking for NCAA Division I-A Football Thomas Callaghan, Peter J. Mucha, and Mason A. Porter 1. INTRODUCTION. The Bowl Championship Series (BCS) agreement was cre- ated in 1998 to match the top two NCAA Division I-A college football teams in a National Championship game at the end of each season. As part of the agreement, the official BCS Standings are used to pick which two teams most deserve to play in the championship game and which teams should play in the other major bowl games. The BCS Standings have significant financial ramifications, as the National Champi- onship Game and major bowl games yield financial payouts to the conferences of the appearing teams projected to be $14–$17 million per team in 2007 [1]. In addition to this direct financial benefit, BCS bowl appearances likely generate indirect gains from increases in both alumni contributions and student applications. Before the BCS, the matchups in many bowl games were determined according to conference traditions, so matches between the #1 and #2 teams in the nation rarely occurred. At times, this yielded multiple undefeated teams and co-National Champions (most recently, Ne- braska and Michigan in 1997). On other occasions, a single team with an arguably easy schedule could go undefeated and be declared National Champion by polls with- out ever having played a "major" opponent (e.g., BYU in 1984). The BCS system endeavors to address these problems while maintaining the tra- dition of finishing the season with bowl games. Prior to 2004, BCS Standings were determined by a combination of two polls (coaches and sportswriters), selected algo- rithmic rankings, strength of schedule, losses, and bonus points awarded for defeating highly-ranked teams. This system double-counted key contributions, because reason- able polls and rankings already took losses and strength of schedule into account (see, for example, [4]). In 2004, the BCS instituted a formula that simply averages polls and computer rankings, but the system continues to vary from year to year as specific polls and computer systems are added and removed. The fundamental difficulty in accurately ranking or even agreeing on a ranking methodology for college football lies in two factors—the dearth of games played by each team and the large disparities in types and difficulties of individual schedules. For instance, should an undefeated team from a purportedly weaker conference with few tough nonconference opponents be ranked ahead of a "major conference" team that lost a game or two while playing difficult opposition both in its conference and in its nonconference matchups? While each game outcome is an imperfect paired com- parison between two teams, the schedule of 10–13 regular season games (including conference championships) played by each of the 119 Division I-A football teams severely limits the quantity of information relative to, for example, college and profes- sional basketball and baseball schedules. Moreover, most of the Division I-A football teams play the majority of their games within their conferences, and there are signifi- cant variations in the level of play across different conferences that further complicate attempts to select the top two teams from the available information. To make matters worse, it is not even clear what the phrase "top two" should mean: does it refer to the two teams with the best overall seasons or the two playing the best at the end of the season? November 2007] RANDOM WALKER RANKING 761 Despite the obvious difficulties, many systems for ranking college football teams have been promoted by mathematically and computationally inclined fans (see, for example, those listed by Massey [15]). Many of these schemes are relatively com- plicated mathematically, making it virtually impossible for the lay sports enthusiast to understand the ranking methodology and its underlying assumptions. Worse still, essential details of many of the algorithms currently used by the BCS are not even openly declared (the only one completely declared in the public domain is the system by Colley [6], though some others are at least partially explained). Of those that are at least partially public, some include seemingly arbitrary parameters whose effects are difficult to interpret, and others are tweaked periodically to obtain the purportedly most reasonable ranking based on previous results. In this article, we demonstrate that a simply-explained algorithm constructed by crudely mimicking the behavior of voters can provide reasonable rankings. We define a collection of voting automatons (random walkers) each of which declares its prefer- ence for a single team. Each automaton repeatedly selects a game at random from its preferred team's schedule and decides whether to change its preference to the oppo- nent as biased by the game outcome, preferring but not absolutely certain to go with the winner, repeating this process indefinitely. In the simplest implementation of this process, the probability p of choosing the winner is kept constant across voters and games played, with p > 1/2 because on average the winner should be recognized as the better team and p < 1 to allow a given voter to argue that the losing team is still the better team (moreover, the p = 1 limit can be mathematically more complicated in certain scenarios, as discussed in section 3). The voting automatons are nothing more than independent, biased random walkers on a graph connecting the teams (vertices) by their head-to-head games (edges). These "voters" thereby obey idealized behavioral rules dictated by one of the most natural arguments relating the relative ranking of two teams: "my team beat your team." Indeed, the statistics of such biased random walkers can be presented as nothing more than the logical extension of this argument repeated ad infinitum. This algorithm is easy to explain in terms of the "microscopic" behavior of indi- vidual walkers who randomly change their opinion about which team is best (biased by the outcomes of individual games). Of course, this behavior is grossly simplis- tic compared with real-world poll voters. In fact, under the specified range of p, a single walker will never reach a definitive conclusion about which team is the best; rather, it will forever change its allegiance from one team to another, ultimately travers- ing the entire graph. However, the "macroscopic" total number of votes cast for each team by an aggregate of random-walking voters quickly reaches a statistically-steady ranking. The advantage of the algorithm discussed here is that it can be easily understood in terms of single-voter behavior. Additionally, it has a single explicit, precisely-defined parameter with a meaningful interpretation at the single-voter level. We do not claim that this ranking is superior to other algorithms, nor do we review the vast number of ranking systems available, as numerous reviews are already available (see, for ex- ample, [11], [20], [7], [14] for reviews of different ranking methodologies and the list of algorithms and "Bibliography on College Football Ranking Systems" maintained by [22]). We do not even claim that this ranking algorithm is wholly novel; indeed, the resulting linear algebra problem is in the class of "direct methods" discussed by Keener [11] and has many similarities to the linear algebra problem solved by Colley [6]. Rather, we propose this random-walker ranking on the strength of its simple in- terpretation: our intent is to show that this simply-defined ranking yields reasonable results. 762 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 114 The remainder of this article is organized as follows: In section 2, we give the math- ematical definition of the ranking algorithm and examine its statistical properties. In section 3, we investigate the algorithm's asymptotic behavior for extreme values of the probability p. In section 4, we present results for special cases involving round-robin tournaments. We then examine recent historical outcomes from real NCAA Division I-A seasons in section 5. In section 6, we discuss properties of the graphs defined by the games played in a given year according to real NCAA Division I-A schedules, pay- ing special attention to hierarchical structure and the interplay between that structure and random-walker rankings. We conclude in section 7 by discussing some possible generalizations of this ranking algorithm. 2. RANKING WITH RANDOM WALKERS. For each team i we denote the num- ber of games it played by n i , its wins by wi , and its losses by li . A tie, which was possible prior to the current NCAA overtime format, is treated as half a win and half a loss, so n i = wi + li always holds. The number of random walkers declaring their preference for team i is denoted vi , and the condition that the total number of voters remain constant is given by i vi = Q. If team i beats team j, then the average rate at which a walker voting for team j changes its allegiance to team i is proportional to p, and the rate at which a walker already voting for team i switches to team j is proportional to 1 − p. For simplicity, we ignore the dates of games, selecting each of the n i games played by team i with equal probability. To avoid rewarding teams for playing more games, the rate at which a given voter considers a given game is taken to be independent of n i . Under this selec- tion of rates, we find it more natural to express the independent random-walker dynam- ics using ordinary differential equations (ODEs) instead of Markov chains, though our entire discussion can certainly be recast in terms of Markov chains, in which case the approach has some similarities with the PageRank citation ranking [3] underlying the Google search engine (see [13]). Formulating the present problem in terms of ODEs has the added advantage of avoiding complications due to cycles of different lengths (such cycles could alternatively be removed through explicit time-averaging). The expected rate of change of the number of votes cast for each team in this random walk is quantified by a homogeneous system of linear differential equations, v = Dv, (1) where v is the T -element vector of the number vi of votes cast for each of the T teams, and the elements of the square matrix D are Dii = − pli − (1 − p)wi , 1 (2 p − 1) Di j = Ni j + Ai j (i = j), (2) 2 2 in which Ni j is the number of games played between teams i and j and Ai j is the number of times team i beats team j minus the number of times i loses to j. That is, if Ni j belongs to {0, 1}, then ⎧ ⎨ +1 if team i beat team j, Ai j = −1 if team i lost to team j, (3) ⎩ 0 if team i tied or did not play team j. When two teams play each other multiple times, we obtain Ai j as summed over those games. Because 1/2 < p < 1, the off-diagonal elements Di j are nonnegative and van- ish if and only if Ni j = 0. November 2007] RANDOM WALKER RANKING 763 Ideally, one would consider the giant connected component in the college football schedule network, including Division I-A as a subgraph, but it was easier to analyze the data (obtained from [9] and [23]) when the information was selectively restricted to the graph of Division I-A teams. Because many Division I-A teams play some non-I-A opponents, we represent all of these connected non-I-A teams collectively as a single node. These teams usually do not fare well against Division I-A competition, so this new "team" achieves a low ranking and does not significantly affect the random walker populations, except to penalize the Division I-A teams they defeat and to maintain the constraint that the total number of votes Q remain constant (i.e., voters do not leave the graph). Equilibrium. The matrix D encompasses all the connections and win-loss outcomes between teams. The steady-state equilibrium of (1,2) satisfies D¯ = 0 v (4) ¯ and gives the expected populations v of the random walkers voting for each team. This information can then be used directly to rank the teams. Despite the simplistic behavior of an individual random walker, the behavior of an aggregate of voters (or equivalently, because of the independence assumption, the long-time average of a single voter) ap- pears to yield reasonably robust orderings of the top teams. Unsurprisingly, the number of votes cast for a given team varies substantially for different values of p; neverthe- less, the relative ranking of the top few teams can remain similar across a wide range of p, as we discuss for recent historical examples in section 5. ¯ The equilibrium point v lies in the null-space of D; that is, it is an eigenvector associated with a zero eigenvalue. We stress that this equilibrium does not require a no-net-flow detailed balance along each edge; rather, we only require zero net flow out of each node. For instance, a schedule of three teams (i, j, and k) in which each plays only two games such that i beats j, j beats k, and k beats i leads to a cyclic flow of votes around the triangle with a statistical equilibrium that arises when each team receives an equal number of votes. An important property of these random walkers is that the matrix D yields a single ¯ attracting equilibrium v for a given p in [1/2, 1), provided the underlying graph repre- senting games played between teams consists of a single connected component. This can be proved by recasting (1)–(4) as an eigenvalue problem and applying the Perron- Frobenius theorem, as described by [11], provided the graph consists of a single con- nected component and p < 1. In the absence of these conditions, the Perron-Frobenius theorem cannot be applied because the resulting matrices are no longer irreducible. Alternatively, given the random-walker interpretation built into the rate matrices we define, one can easily demonstrate that the expected populations achieve a unique at- tracting state: 1. The column sums of D vanish because the sum of the populations remains con- stant: 0 = Q = i vi = i j Di j v j = j ( i Di j )v j for all v, requiring that the inner sum vanishes (i.e., the dynamics confine v j to a hyperplane of codimension 1). 2. The off-diagonal elements of D are nonnegative and—once the graph consists of a single connected component—it can be shown that all off-diagonal elements of the dth power of the short-time approximate mapping matrix (I + tD)d are positive for sufficiently small nonzero t, where I is the identity and d is the diameter of the graph. Because of this property, any vertices with v j = 0 have growing populations. In other words, given nonnegative v j with positive total 764 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 114 population Q, all average flows along the v j = 0 faces enter the hyperquadrant in which v j > 0 for all j. 3. Finally, because (1) is a linear, constant coefficient system, it can be shown that the only possibility consistent with the foregoing observations is that the null- space of D is one-dimensional (there is a single eigendirection associated with ¯ eigenvalue zero), and that there is a unique attracting sink v in the constant- population hyperplane at the point of its transverse intersection with the null eigenvector. This argument breaks down if either the graph is not connected or p = 1, because subgraphs with zero walker population can remain so in such situations. Statistics. Any initial voter distribution eventually randomizes completely, so the ¯ steady-state distribution of the number of votes vi cast for the ith team is binomial ¯ (for Q trials) with probability vi /Q, mean vi , and variance vi (1 − vi /Q). The joint ¯ ¯ ¯ probability density function of two vertices is not perfectly independent, as the sum over all vertices equals the number of random walkers Q. However, it is still obtained from a binomial distribution of Q random trials across the vertices. (The two of inter- ¯ ¯ est have probabilities ri = vi /Q and r j = v j /Q.) We can exploit this fact to measure the confidence in the relative ranking of two teams in terms of the minimum number of voters Q min required to ensure that the expected difference between the number of votes cast for each team is larger than the standard deviation of that difference: ri + r j − (ri − r j )2 ri + r j Q min = = − 1. (5) (ri − r j )2 (ri − r j )2 Because the statistical properties of the random walkers follow directly from the linear algebra problem (1)–(4), there is no need to simulate independent random walkers to obtain rankings. This simplicity disappears if interactions between random walkers are included, as considered briefly in section 7. 3. ASYMPTOTICS AT LARGE AND SMALL p. For a given probability p the expected populations depend in a complex manner on the details of game schedules and outcomes. In an attempt to clarify the effects of selecting a given p, it is instructive to investigate analytically the limiting behaviors near p = 1/2 and p = 1. We demon- strate, in particular, that the main contributions near p = 1/2 include a measurement of schedule strength, whereas behavior near p = 1 is dominated by undefeated teams and by subgraphs of teams that go undefeated against teams outside the subgraph. On any given Saturday. Consider p = 1/2 + ε, where ε 1. The rate matrix (2) becomes 1 Di j = ij + ε Di j , 2 where Δ is the graph Laplacian: i j = 1 (i = j) if nodes i and j are connected (the two teams played each other) and ii = −n i . By (2), D has the same elements as A off the diagonal and the values wi − li on the diagonal. A power series of the equilibrium probabilities, v j = v (0) + ε v (1) + ε 2 v (2) + · · · , then satisfies ¯ ¯j ¯j ¯j ∞ 1 (0) 1 (k) 0= Δ¯ + v εk Δ¯ + D¯ (k−1) , v v (6) 2 k=1 2 November 2007] RANDOM WALKER RANKING 765 subject to the normalization condition ¯ (k) j v j = Qδk0 for Q voters (δk0 = 1 when k = 0, otherwise δk0 = 0). The O(ε 0 ) contribution requires v (0) = Q/T for each j, distributing the Q votes ¯j equally across the T nodes of the graph. With Di j v (0) = 2(wi − li )Q/T , the O(ε 1 ) ¯j condition becomes 4Q ¯ (1) ijvj =− (wi − li ). (7) j T That is, the first correction v (1) at p near 1/2 is a potential that satisfies a discrete ¯j Poisson equation (subject to the constraint ¯ (1) j v j = 0) with charges proportional to the win-loss record of each team. It thus incorporates the record of the jth team and is also heavily influenced by the records of the nearest neighbors and other close teams in the graph. In other words, the first correction is strongly influenced by a "strength of schedule" notion. Furthermore, it is only with the second-order term v (2) ¯j that information pertaining to specific games won or lost by a given team begins to be incorporated, as v (1) includes only net records in (7). ¯j Winner takes all. The asymptotic behavior for p = 1 − ε (ε 1) is more compli- cated because the limiting state depends on the number of undefeated teams and other schedule details. The single-equilibrium argument of section 2 breaks down at p = 1 because off-diagonal elements of (I + tD)d are not necessarily positive. There can then be multiple equilibrium states, but only one of these states is achieved in the limit as p → 1. The simplest situation to consider asymptotically occurs when a single undefeated team garners all random walker votes for p = 1 − ε as ε → 0. This condition requires both that there be only a single undefeated and untied team and that there be no sub- graphs of other teams in the network that collectively win all of their games against teams outside that subgraph. The transition rates are then written Di j = Di(0) − ε Di j , j and we again expand v j = v (0) + ε v (1) + ε 2 v (2) + · · · . ¯ ¯j ¯j ¯j Here, D(0) agrees with (N + A)/2 off the diagonal but has the negation of the number of losses by the corresponding teams on the diagonal (recall that a tie counts as half a win and half a loss), and D remains as defined earlier in the perturbation around p = 1/2. The limiting state v (0) = Qδ j u casts all Q votes for the single undefeated team ¯j ("u"). Calculation of the first-order correction then requires solution of D(0) v(1) = b, ¯ where b j = Q D j u , which offers neither simplification nor intuition beyond the original rate equations (1)–(4). 4. ROUND-ROBIN EXAMPLES. The asymptotic analyses of the previous section were necessarily limited by the generality of possible schedule topologies. As a means of developing further intuition, we consider the special cases of round-robin tourna- ments in which every team plays every other team exactly once. On any given Saturday (revisited). Returning to the case p = 1/2 + ε, consider a round-robin tournament of T teams, each of which plays exactly T − 1 games, one against each of the available opponents. Then the graph Laplacian is the matrix with 766 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 114 1s off the diagonal and −(T − 1) for each diagonal entry. Subject to the constraint ¯ (k) j v j = 0 (k > 0), we immediately obtain j ¯ (k) ¯ (k) i j v j = −T vi , so (6) yields 2 vi(k+1) = ¯ D vi(k) . ¯ T The zero-sum constraint is maintained because the column sums of D are identically zero by definition. This simple iterative relation for vi(k+1) can then be summed when ¯ ε < 1/2 (i.e., p < 1) to arrive at −1 2 v= I− ¯ D v (0) . ¯ T We make special note of the first-order correction. In section 3, we saw that strength of schedule and the direct win-loss record both appear at O(ε). Hence, one expects the win-loss record of an individual team to determine its O(ε) contribution fully in the round-robin case, where everyone playing everyone else removes schedule inequities. Indeed, equation (7) gives 4Q vi(1) = ¯ (wi − li ) (8) T2 when applied to round-robin tournaments. Stated differently, the asymptotic rankings near p = 1/2 in a round-robin tournament are linear, and the slopes are set by the win-loss records. Perfectly ordered teams. Because the asymptotic expansion for p = 1 − is com- plicated by the difficult determination of the base state ( = 0), we consider this expansion only for the round-robin case of T "perfectly ordered" teams, which we define as the special case where team i beats team j whenever i < j. In this situa- tion, the D(0) matrix of section 3 is upper triangular, with 1s above the diagonal and Dii = −li = −(i − 1). Because there are no cycles here, it is clear that the single un- defeated team garners all votes in the limit as ε → 0, so vi(0) = Qδi1 . We can then ¯ show that −(T − 1) if i = 1, vi(1) = ¯ T /[li (li + 1)] if i > 1. Mixed ordering. Finally, it is worth asking whether round-robin tournaments can give any indication about useful values of the bias parameter p. Specifically, it is reasonable to expect that teams in a round-robin schedule should be ranked in a manner consistent with their win-loss records, as their schedules contain no inequities. A 6-5 team in a 12-team round robin should presumably be ranked higher than a 5-6 team, even if only marginally so. As we already saw from (8), the O(ε) term in an expansion with p = 1/2 + ε for a round-robin competition does indeed agree with the rank ordering given by win-loss records. Do the terms that are higher order in ε continue to respect the win-loss ordering in a round-robin setting? Equivalently, what happens for values of p further from 1/2? One might reasonably ask whether this win-loss ordering is generically preserved for all p in the range 1/2 < p < 1 (it is not). It is then natural to ask whether the win-loss ordering is always preserved up to "crossing" values bounded away from 1/2. That November 2007] RANDOM WALKER RANKING 767 is, for a given round-robin outcome, we define pc to be the minimum p (≥ 1/2) such that the random-walker ordering crosses from consistent to inconsistent with win-loss records. We then ask whether pc is bounded from below. However, it seems clear from examples of specific outcomes that there is no lower bound for pc except for p = 1/2 (where, by definition, all teams are ranked equally). We obtain pc arbitrarily close to 1/2 through a simple modification to the perfectly- ordered tournament. Starting from a win-loss matrix of T teams where team i beats j if i < j, we modify only the games played by the team with the best losing record. To be precise, for even T we change only games played by team k = (T /2) + 1, with (T /2) − 1 wins and T /2 losses. We now swap the win-loss outcome of every game played by team k except for the game between k and k − 1, which we maintain as a win for team k − 1. These outcome switches modify the win-loss record of every team except for teams k and k − 1; in particular, the latter still has a better record than the former. Nevertheless, we observe numerically that ( pc − 1/2) ∼ T −1 for these explicitly-constructed mixed-ordered round-robin events, with team k ranked above team k − 1 above pc . The realization that there is no lower bound for pc away from 1/2 indicates an ob- vious limitation to this simply-constructed ranking system. Specifically, because the random walkers inherently represent first-place votes, rank-ordered crossings involv- ing teams ranked #T /2 and #(T /2) + 1 are not very surprising. The number of votes cast for each team in the statistical equilibrium is of course important, in that the few votes cast for the lowest-ranked teams by design impact the number of votes cast for each team they play, ensuring that strength of schedule is inherently incorporated. However, the emphasis on highly-ranked teams means that a team can improve its ranking by beating a highly-ranked team more than it might be penalized for losing to a lower-ranked team. These round-robin examples then clearly call into question the accuracy of the precise rankings of those middle teams and thus lead to questions about the scheme as a whole. A system based only on first-place votes should presumably be strongest in its ranking of the top teams, though we have no mathematical proof that these walkers do so in any "optimal" sense. This motivates us to consider how the random-walker rankings fare in comparison with the historical record. 5. RECENT HISTORY. The utility of a ranking algorithm lies in its performance in the face of real data. Accordingly, we investigated college football rankings for each season from 1970 to 2005, restricting our discussion here to 2001 and 2002, as these seasons represent recent extreme situations in attempting to pick the top two teams (see, for example, [1]). The 2003, 2004, and 2005 seasons are discussed on our web page [16], where rankings for future seasons will also be maintained. The only certainty in the 2001 pre-bowl rankings was that Miami belonged in the National Championship Game, because it was the only undefeated team in Division I-A. Indeed, both polls and all eight algorithms used by the BCS that year picked Miami #1 going into the bowl games. That season's controversy concerned Nebraska's selection as the #2 BCS team, narrowly surpassing BCS #3 Colorado despite Col- orado's late-season rout of Nebraska. The fact that BCS #4 Oregon had been ranked #2 in both polls was also mentioned on occasion. After the bowl games, in which Miami defeated Nebraska and Oregon defeated Colorado, it was Oregon's absence from the championship game that became the centerpiece of national controversy. The random walkers select Oregon #2 for p > .5 up to p ≈ .62, above which Nebraska takes #2 in a narrow range up to p ≈ .68. Above that value, the random walkers defy conven- tional wisdom by selecting BCS #6 Tennessee as #2. In fact, they choose Tennessee as #2 over the widest range of probabilities (see Figure 1a). This ranking is explained 768 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 114 2001 pre-bowl 2002 pre-bowl 12 Miami 11−0 10 Miami 12−0 Tennessee 10−2 Ohio St. 13−0 10 Nebraska 11−1 USC 10−2 Oregon 10−1 8 Georgia 12−1 Colorado 10−2 8 Florida 9−2 Oklahoma 11−2 Stanford 9−2 6 Iowa 11−1 Rank Rank LSU 9−3 Notre Dame 10−2 6 4 4 2 2 0 0 0.5 0.6 0.7 0.8 0.9 1 0.5 0.6 0.7 0.8 0.9 1 p p (a) (b) Figure 1. Pre-bowl rank orderings of teams by the expected populations of random walkers at different values of the probability p of voting for the winner of a given game for (a) 2001 and (b) 2002. in part by the fact that the simplest random-walking voter algorithm presented here does not distinguish games based on the date played. Tennessee had been the most likely team to be picked to face Miami until they lost the Southeastern Conference (SEC) championship game on the final day before the BCS selections. Even with the loss (because the date of that loss does not affect things here), the random walkers congregate near Tennessee, in part because the SEC as a whole is highly ranked that season. Indeed, in the limit as p → 1, Florida is ranked #3 and LSU is ranked #4, with Oregon falling to #5 and both Nebraska and Colorado falling completely out of the top ten. This SEC near-dominance is generated by a number of nearly-undefeated sub- graphs: Tennessee (10-2) lost to Georgia and lost to LSU the second time they played (in the SEC championship), Florida (9-2) lost to Auburn and Tennessee, and LSU (9- 3) lost to Mississippi, Florida, and to Tennessee in their first meeting. Not counting games against each other, this trio of closely-linked teams had a collective 24-3 record prior to the bowl games, with all three losses against SEC conference opponents, two of whom were played multiple times by this group (Florida beat Georgia and LSU beat Auburn). Those three losses came against teams who also had all but one of their losses in the SEC (Auburn lost to Syracuse). Florida falls to #8 at smaller values of p (closer to 1/2), and LSU is not even in the top ten for p below ≈ .77. Nevertheless, they continue to help Tennessee's ranking. In contrast, the BCS system worked virtually without controversy in selecting teams for the National Championship Game at the end of the 2002 season, as there were pre- cisely two undefeated teams, both from major conferences. Both polls picked Miami and Ohio State as the top two teams, in agreement with six of the seven ranking algo- rithms used that year. The only nonconformist was the New York Times ranking system, which picked Miami and USC as the top two teams—the latter presumably in part due to its difficult schedule. For p ≈ 1/2, the random walkers also rank USC in the top two based on the strength of its schedule (see Figure 1b), but they agree across most values of p that the top two teams are Miami and Ohio State. One can explore the differences between the 2001 and 2002 pre-bowl results further by exploiting the statistical properties of expected vote totals to express a measure of confidence in the resulting rankings. The Q min required to distinguish successfully between the top rank-ordered pairs of teams, defined by (5) in section 2, is plotted for the 2001 and 2002 pre-bowl rankings in Figure 2. In particular, we note that the relative numbers of voters required to distinguish #1 from #2 in 2001 and #2 from #3 in November 2007] RANDOM WALKER RANKING 769 0 2001 pre-bowl 0 2002 pre-bowl 10 10 1 v. 2 1 v. 2 2 v. 3 2 v. 3 3 v. 4 3 v. 4 4 v. 5 4 v. 5 1/Q min 1/Q min −5 −5 10 10 −10 −10 10 10 0.5 0.6 0.7 0.8 0.9 1 0.5 0.6 0.7 0.8 0.9 1 p p (a) (b) Figure 2. Degree of confidence in the ordering of pairs of teams in the (a) 2001 and (b) 2002 pre-bowl rankings, as quantified by 1/Q min , where Q min denotes the minimum number of independent random walkers necessary to ensure that the standard deviation of the difference between the expected populations of two teams is smaller than the expected difference (see equation (5)). 2002 are significantly smaller (indicating a higher degree of confidence in the chosen ordering) than, in particular, the distinctions between #2 and #3 in 2001 or #1 and #2 in 2002. One should, of course, be careful in applying this measure of confidence too broadly, because the percentage of total votes that remain available to distinguish #4 from #5 will in general be smaller than those available to distinguish #1 from #2. Accordingly, direct comparisons of the same rank distinction across different years are more reasonable. For example, observe the typically lower degree of confidence in distinguishing #3 from #4 and #4 from #5 in 2001 (when Nebraska, Colorado, Oregon, and Tennessee were all trying to lay claim to the #2 spot) compared with the situation in 2002. −1 10 Population of Losing Team −2 10 −3 10 −3 −2 −1 10 10 10 Population of Winning Team Figure 3. Winning and losing teams plotted according to the 2001 pre-bowl rankings of each team at p = 0.8, given by the expected fraction of random walkers populating ("voting for") each team. After seeing these results, we should discuss at least briefly which values of p might yield good rankings. Perhaps neither the dominance of strength of schedule near p = 1/2 (as illustrated, for example, by USC in 2001) nor the emphasis on undefeated 770 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 114 seasons near p = 1 (e.g., BYU in 1984) are appropriate. Rather, we might naively argue that a balanced value of p is preferred. Alternatively, we could consider various objectives for optimizing p. For instance, Figure 3 shows the organization of the win-loss outcome of every game in the 2001 pre-bowl schedule according to the random walker populations determined with p = 0.8. The votes are not (of course) strictly ordered according to the winner and loser of each individual game, but the winners have more votes on average. Instances where a higher-ranked team has lost to a lower-ranked team are frequently referred to as "rank- ing violations" and can be used to construct one measure of the quality of different ranking systems (see the comparisons page maintained by Massey [15]). Additionally, the goal of minimizing the number of such violations and determining rank order- ings from the multitude of such nonunique minimizations can itself be used to define ranking algorithms [5], [18]. An obvious optimization procedure here is to select p to minimize the number of ranking violations. While we observe that this optimal p varies from year to year, it frequently takes a value near the center of the allowed interval. Other measures of the quality of the ranking may be naturally inspired by Figure 3; for example, we might minimize the violations in another norm, such as one giving their distance from nonviolation. Another approach based on logistic regres- sion of a generalized version of the present ranking system has been implemented by Kvam and Sokol [12] for college basketball, where the best p transition probabilities include information about point spread and home-court advantage. Finally, we might wonder whether some optimization would connect this class of "direct methods" [11] to the more statistically-sophisticated maximum likelihood methods originally used for paired comparisons in the pioneering works of Zermelo [24] or Bradley and Terry [2]. Rather than continuing these lines of inquiry here, we instead focus our remaining attention on trying to understand the connections between our biased random walkers and the properties of the underlying networks. 6. NETWORK STRUCTURE. The voting automatons randomly walk on a graph consisting of the Division I-A football teams (vertices) connected by the games played between the teams (edges). We note that this is not the only ranking system motivated by direct interest in the underlying network; indeed, while the random walkers here provide rankings through their dynamics on the network, Park and Newman [19] have developed a ranking system determined completely by the directed graph defined by wins. Because the random walkers propagate in a strongly heterogeneous network topology, it is important to try to understand how that topology affects the walkers' behavior and the resulting rankings. Each NCAA Division I-A football season consists of 650–750 games between about 115 teams (119 in 2005). In every season since 1990, this graph becomes a single connected component by the third or fourth week of the season (without counting faux connections via the single "non-I-A" node that we include for simplicity). The relative quickness in achieving a single connected component (after only about two hundred games) results in part because schools typically play many of their nonconference games at the beginning of a season. The degree of each vertex—that is, the number of games played by each team—varies in a narrow range. Each vertex has between ten and thirteen edges prior to the bowl games and between ten and fourteen connections after the bowl games. The diameter of the graph, determined by counting the number of edges along the longest geodesic path, is 4 in every post-bowl graph since 1970. We also considered the local clustering coefficient Ci for the ith team in the network (see [21], [17]), given by November 2007] RANDOM WALKER RANKING 771 number of triangles connected to vertex i Ci = . (9) number of triples centered on vertex i Unsurprisingly, clustering coefficients provide one means of identifying the strong heterogeneity of the network. Each conference typically has a different average local clustering coefficient, which also varies from the coefficients computed for indepen- dent teams such as Notre Dame and Navy. Several other network properties can also be calculated (including average path lengths and various notions of centrality and connectedness; see, for example, [17]), but such computations do not necessarily help explain the random-walker statistics. An important exception is the graph's "community structure" [8], which indicates the hier- archies present in the network and is useful for understanding the nature of the confer- ence scheduling and the resulting effect on the random-walker statistics. We computed community structure using the notion of "edge betweenness," defined as the number of geodesics that traverse each edge, using the algorithm given by [8]. Briefly, the edge with the highest betweenness is removed from the graph, and the betweenness is re- calculated for this modified graph to determine the next edge to remove; the process is then repeated until no edges remain. The removal of some of these edges breaks a connected component of the graph into two parts, grouping the network into a hierar- chy of communities as the algorithm is iterated. Girvan and Newman [8] demonstrated that their algorithm closely reproduced the predefined conference structure of the 2000 football schedule, and we found similar results for other years. Using the 2001 season as an example, Figure 4 portrays the college football com- munity structure. The conferences and their subdivisions are reasonably reconstructed based on their relative strengths of community, and the tree ("dendrogram") in the figure indicates the relative closeness of different conferences and of the independent teams. Note, for example, the close connections between Notre Dame, Navy, and the Big East (from Virginia Tech counterclockwise to Miami (Florida)). This community structure is intimately linked to the dynamics of the random- walking voters, as the specific pairings of interconference games and the outcomes of those games strongly influence the flow of voters into and out of the given conferences and more general structures such as the divisions inside large conferences and larger hierarchical groups of conferences. The 2001 pre-bowl community structure in Figure 4 is gray-scaled according to the average number of voters per team at each level of the hierarchy (for p = 0.7). Such a plot indicates the relatively high vote counts given to the SEC, Pacific-10 (Pac-10), and Big 12; it also shows the significantly smaller average vote counts for Big East teams, despite Miami's first-place standing. We further quantify the importance of the relationships between conferences by di- rectly measuring the effect of reversing the outcomes of individual games. Keeping the games in the order in which their respective edges were removed in developing the community structure, we measure the difference between the original voter pop- ulations and the new populations calculated with the win-loss outcome of that single game reversed. The dominant effect of such reversals is to change the rankings of the two teams involved in that game, so we calculate the change to the global distribu- ¯ tion of votes with the quantity i = j,k |vi |2 , where the edge removed corresponds to a game between teams j and k. Plotting this quantity (see Figure 5) versus the order of the edge removed reveals a sharp transition in average magnitude between the first approximately two hundred edges and those that follow. This corresponds roughly to the number of edges removed in the community structure determination at which the hierarchy breaks up into the different conferences; these first roughly two hundred 772 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 114 Cle ke Fore na State Weste n Michig an Bow Northern Illinois Ea s t Cen tate ling er li Geomson st rn Mic tral en St hio Caro Te xas Caro h T ay s A ch a Ball State Maryland Florida S B xa Te lin Te rth a Tec Gre Virginia Mic Toledo Duke North higan O ex lor &M 0.021086 No rgi hig te Bu kron ll Wa e M on an O lo an at Ma K hio t a A ha t ia ne S ffa St om a M C ig m c ta rs en ) ah om ic a O kla as i ( ti te te h kl h M do ta O cu No in ra S ne lo sas tate rth s o w I ot C an a S ka Mi es owa a K w ras ri ch ter Io eb ou i Wi Illin gann N iss as sco oi M ans tate ate ns s K ah S o St Pen Indian in Ut lorad n Ohio State a Co h ung St Uta ham Yo Purdate Brig ming Notre Dame ue Wyo rce Air Fo Las Vegas Navy Virginia Tech Nevada− New Mexico West Virginia San Diego State Pittsburgh Tulsa Texas−El Pa Boston College Southe so Rutgers Fresn rn Methodist Temple Nev o State use Haw ada Syrac ida) i r (Flo zona San aii Miam Ari rnia Lo Jos ifo d R uisi e Sta Cal anfor te Boice ana T te St Sta on A ise ech na eg te L lab St izo Or Sta ia M oui ama ate Ar n Ci em svi −B on or LA nc ph lle irm eg lif C on H as ane rn in is ing Or Ca U ngt ou t C ip u m e E l na iss A laba tat ha Lo Mis pi S bur a Tu out y n st ar er hi ti m S rm Divi ristia uis si ta n A nS on ol th as ian ssi te A n− Ch Sou Vand Sta pi No xas l Flor a ou to th C erb te Te ntra lorid W p Ken rolina t S he Ce th F ng il Sou y State o State Tenn tucky Tro Mexic in Geor e New iana−Lafa hi gia Louis as State Florida Arkans e State Arkansas North Texa a Idaho esse Louisiana−Monroe as Mi a p = 0.7 W a ss sio n ss iss nI Mi Middle Tennesse ip ida A pi s yette Figure 4. Graphical depiction of the community structure of the 2001 pre-bowl network, with gray-scale coding of the average votes per team for each community at p = 0.7. edges removed are predominantly interconference games, whereas those that follow are intraconference games. 7. DISCUSSION. We have developed a simply-defined ranking algorithm in which random walkers adjust their "votes" for the best team on the network of teams con- nected by their head-to-head games, with a single explicit parameter describing the preference towards selecting the winner of a given game. We investigated the asymp- totic behavior of this algorithm in the two extremes of the probability parameter, pay- ing special attention to round-robin tournaments, and explored the results of the 2001 and 2002 NCAA Division I-A college football seasons in detail. More recent seasons are discussed on our website [16]. Finally, we have connected the rankings to the un- derlying network of games played by Division I-A teams, quantitatively demonstrating the importance of interconference game outcomes and the relation to the community structure of the graph. Of course, many generalizations of the random-walker ranking algorithm described here are possible, ranging from trivial redefinitions of the rate matrix D to fundamental November 2007] RANDOM WALKER RANKING 773 −5 10 Effect of reversal −6 10 −7 10 0 200 400 600 Order of edge removal Figure 5. The effect of reversing the outcome of a single game played between teams j and k, quantified ¯ by i = j,k |vi |2 (for p = 0.7) and plotted versus the order of that game according to edge removal in the community structure determination. changes in the mathematical tools required to investigate the random-walking voters. The simplest generalizations are those that modify the rate matrix D without changing the independence of the random walkers themselves. For instance, one can readily incorporate margin of victory, home-field advantage, and game date into the definition of D by replacing the constant probability p of voting for the winner with a function that includes these components (see, for example, [12]). The resulting transition rates of walking each direction on a single edge are still defined by the outcome of the game represented by that edge, and the associated linear algebra problem determines the probability distribution of each state. It would be natural to make the probability of going towards the winner along a given edge higher for a larger margin of victory or for a game won on the road. However, incorporating any such qualitative assertions into quantitative variations in the transition rates requires adding more parameters to the ranking algorithm and determining reasonable values of those parameters, in stark contrast to the minimalist philosophy espoused here. Alternatively, each random walker can be assigned two votes instead of one. This change is particularly sensible if the point is to select the two teams to play head-to- head in the National Championship Game. We considered such rankings generated from random walkers, each of which holds two equal votes (as opposed to a #1 vote and a separate #2 vote), using the same probability parameter p, subject to the addi- tional constraint that a given voter must cast the votes for two different teams. This two-vote constrained random walk is most easily understood in terms of independent random walks on the (significantly larger!) network in which each vertex represents a possible pair of votes and the edges between vertices include games played between teams representing one of the two votes, with the other vote held fixed. This again immediately reduces to a linear algebra problem for the expected percentage of votes garnered by each team, although it has much higher rank: T (T − 1)/2 for T teams. Clearly, further increases in the number of votes given to each random walker would quickly make the state space so large that solving the exact linear algebra problem would no longer be computationally feasible. As examples, we considered this two-vote generalization for pre-bowl rankings for both 2001 and 2002. Our results were unremarkably similar to our single-vote rank- 774 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 114 ings. In particular, in comparison with the single-vote 2001 case, Miami can (of course) obtain only half the votes in the limit as p → 1 because of the constraint that each walker casts the two votes for two different teams, leaving fifty percent of the votes available to select a #2 team. That fifty percent becomes widely divided among the available candidates, with Tennessee and Oregon getting the largest shares at roughly five percent each. Meanwhile, this major change in the expected populations causes only relatively small changes in the rank orderings, except at extreme values of p, where Tennessee and the rest of the SEC do not fare quite as well here near p = 1 as they do in the single-vote scheme (though Tennessee still maintains the #2 spot at most values of p). At p = 0.75, the single-vote and two-vote systems agree on the same top sixty teams in the 2001 pre-bowl rankings, save for swaps in the orderings of #19-20 and #48-49; at p = 0.9, the top twenty-five are identical; at p = 0.95, however, there are switches in the orderings of #3-4, #5-6, and #9-10 (SEC teams Florida, LSU, and Georgia all do better in the single-vote algorithm), while the rest of the top twenty-five remains identical. Not unexpectedly, the 2002 two-vote rankings are very similar to the single-vote values, with Miami and Ohio State splitting the votes in the limit as p → 1. Even at p = 0.95, the top eighteen teams are identical except for a reversal in the ordering of #6-7. One can also consider generalizations that destroy the independence of individ- ual random walkers. For example, voter decisions could be influenced by the number of other walkers voting for each team in a head-to-head game. Whether they are in- clined to follow the crowd or to try to be nonconformist, such dependence between the random walkers makes the calculation of their aggregate behavior nonlinear and removes most of our knowledge about the probability distributions of votes per team. Another interesting generalization would be to weigh more strongly the effects of up- sets by increasing the flow of votes towards a lower-ranked team that beats a team with more votes. Of course, this flow increase might reverse the ordering of the two teams, thereby removing the upset character and reducing the flow towards the winning team of that game, so there may not even be a statistically-steady ordering of the two teams. A similar complication occurs if one breaks the independence of the walkers by adding a bonus for beating a team ranked in (for example) the top ten that reduces the prob- ability of voting for the loser. This reduced flow towards the losing team could then knock them out of the top ten, thereby causing the bonus to disappear, and allowing the team to rise back into the top ten, and so on. Obviously, the study of any of these interacting random walkers is significantly more difficult than the independent walkers considered in this article. As a last consideration, we return to our discussion in section 4 of crossing proba- bilities pc for round-robin competition and the observation that pc can be very close to 1/2 when a team with a worse record won its games against quality opponents, such as the specific mixed ordering we considered. The random walkers then reward a team more for its high-quality wins than they penalize it for its low-quality losses. This issue can be easily corrected by introducing a new ranking according to the dif- ference of expected vote populations obtained with parameter p ("first-place votes" rewarding high-quality wins) and those obtained with 1 − p ("last-place votes" pe- nalizing low-quality losses). For the specific mixed ordering considered in section 4 this "first-last" generalization gives pc .77 for large T , while Monte Carlo searches preliminarily suggest that pc may indeed be bounded away from 1/2 for these first- last random walker rankings for round-robin tournaments. While this improvement is mathematically more satisfying than the situation for the original "first-only" rankings, the first-only and first-last rankings nevertheless generally agree on the top teams for real football seasons. For instance, in the recently-concluded 2005 season these two November 2007] RANDOM WALKER RANKING 775 pre-bowl rankings at p = 0.75 agree on the ordering of the top twelve teams except for a #10-11 swap, with the revised ("first-last") system giving a slightly smaller num- ber of ranking violations than the first-only system (complete rankings are available from [16]). After eight seasons, the Bowl Championship Series remains as controversial as ever. Even when the system yields an uncontroversial National Championship Game be- cause two teams clearly separate themselves from the field (as in 2002 and 2005), the BCS is still unable to escape controversy. The treatment of the so-called mid-major (or "non-BCS") conferences remains an important issue (see, for example, [4]), lead- ing to a December 2005 Congressional hearing in a House Energy and Commerce subcommittee [10]. We remain committed to the proposition that the use of algorith- mic rankings for determining the college football postseason will only become widely accepted when those rankings have been reasonably explained to the public. In that context, the random walker rankings (and their first-last generalization) provide rea- sonable ways to rank teams algorithmically with methods that can be easily explained and broadly understood. ACKNOWLEDGMENTS. We are grateful to both Kenneth Massey and Mark Newman for several discus- sions over the course of this project, to Michael Abraham for developing some of the codes used in our analysis, and to the anonymous referees for their helpful criticisms. This work was supported in part by an NSF VIGRE grant (DMS-0135290) to the Georgia Institute of Technology, where this work began. REFERENCES 1. Bowl Championship Series, http://www.bcsfootball.org, updated 2006. 2. R. A. Bradley and M. E. Terry, Rank analysis of incomplete block designs. I. The method of paired comparisons, Biometrika 39 (1952) 324–345. 3. S. Brin, L. Page, R. Motwami, and T. Winograd, The PageRank citation ranking: Bringing order to the web, Tech. Report 1999-0120, Computer Science, Stanford University, Stanford, CA, 1999. 4. T. Callaghan, P. J. Mucha, and M. A. Porter, The Bowl Championship Series: A mathematical review, Notices Amer. Math. Soc. 51 (2004) 887–893. 5. J. Coleman, Minimizing game score violations in college football rankings, Interfaces 35 (2005) 483– 496. 6. W. N. Colley, ColleyMatrix: Colley's bias free matrix rankings, updated 2006, http://www. colleyrankings.com/. 7. G. R. Connor and C. P. Grant, An extension of Zermelo's model for ranking by paired comparisons, Eur. J. App. Math. 11 (2000) 225–247. 8. M. Girvan and M. E. J. Newman, Community structure in social and biological networks, Proc. Nat. Acad. Sci 99 (2002) 7821–7826. 9. J. Howell, James Howell's college football scores, downloaded 2003, http://homepages.cae.wisc. edu/~dwilson/rsfc/history/howell/. 10. S. Jenkins, BCS schemers are looking out for no. 1, Washington Post (10 December, 2005), E01. 11. J. P. Keener, The Perron-Frobenius theorem and the ranking of football teams, SIAM Rev. 35 (1993) 80–93. 12. P. Kvam and J.S. Sokol, A Logistic regression/Markov chain model for college basketball rankings, Naval Research Logistics 53 (2006) 788–803. 13. A. N. Langville and C. D. Meyer, A survey of eigenvector methods for web information retrieval, SIAM Review 47 (2005) 135–161. 14. J. Martinich, College football rankings: Do the computers know best?, Interfaces 32 (2002) 85–94. 15. K. Massey, College football ranking comparison, updated 2006, http://www.masseyratings.com/ cf/compare.htm. 16. P. J. Mucha, T. Callaghan, and M. A. Porter, Random walker rankings for NCAA Division I-A football, updated 2006, http://rankings.amath.unc.edu/. 17. M. E. J. Newman, The structure and function of complex networks, SIAM Rev. 45 (2003) 167–256. 18. J. Park, On minimum violations ranking in paired comparisons, (2005), http://www.arxiv.org/abs/ physics/0510242. 776 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 114 19. J. Park and M. E. J. Newman, A network-based ranking system for US college football, J. Stat. Mech. (2005) P10014. 20. R. T. Stefani, Survey of the major world sports rating systems, J. App. Stats. 24 (1997) 635–646. 21. D. J. Watts and S. H. Strogatz, Collective dynamics of "small-world" networks, Nature 393 (1998) 440– 442. 22. D. L. Wilson, http://homepages.cae.wisc.edu/~dwilson/rsfc/rate/, updated 2006. 23. P. R. Wolfe, http://prwolfe.bol.ucla.edu/cfootball/, downloaded 2003–2005. 24. E. Zermelo, Die Berechnung der Turnier-Ergebnisse als ein Maximum-problem der Wahrscheinlichkeit- srechnung, Math. Z. 29 (1926) 436–460. THOMAS CALLAGHAN is a graduate student in the Institute for Computational and Mathematical Engi- neering at Stanford University. He is a 2005 graduate of the Georgia Institute of Technology, where he started this work as an REU project at a time when all three authors were in the School of Mathematics there. He won his girlfriend's heart after she used this ranking system to win a college football bowl game pool among her coworkers. A fan of many sports, he is an avid ultimate frisbee player, when he is not too busy with research or autographing the November 2004 ESPN: The Magazine in which this work was featured. He is the only coauthor here with a degree from a Division I-A football school. Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA 94305-4042 USA email@example.com PETER MUCHA is an associate professor in the Department of Mathematics and the Institute for Advanced Materials at the University of North Carolina at Chapel Hill. Previously, he was an assistant professor in the School of Mathematics at Georgia Tech. He received a Ph.D. from Princeton University in 1998. While he greatly enjoys college football, as a graduate of Cornell University he is naturally a much bigger fan of college hockey, which manages to seed its annual sixteen-team postseason tournament by a committee decision that is almost wholly dictated algorithmically (see, for example, http://www.uscho.com). Department of Mathematics & Institute for Advanced Materials, University of North Carolina, Chapel Hill, NC 27599-3250 USA firstname.lastname@example.com MASON PORTER recently moved across the pond to University of Oxford, taking up a position as a univer- sity lecturer in the Centre for Industrial and Applied Mathematics and a tutorial fellow of Somerville College. From June 2005 through September 2007, he was a postdoctoral scholar in the Center for the Physics of In- formation and the Department of Physics at California Institute of Technology. Prior to his 2005 return to his beloved alma mater (B.S., Applied Mathematics, 1998), which also brought him closer to his equally beloved Los Angeles Dodgers, he received a Ph.D. from the Center for Applied Mathematics at Cornell University in 2002 and was a VIGRE Visiting Assistant Professor in the School of Mathematics at Georgia Tech. He notes that Caltech's football team has gone undefeated since 1993 (when it was disbanded) and most of its existing teams tend to win very few games. While this article was in press, Caltech's men's basketball team won its first game in over 10 years and its women's basketball team won its first two games ever. The women's table tennis team, of which Mason was named the official mascot in 2007, was also very successful recently. Mason promises to bring his immense passion for baseball to Oxford. The times, they are achangin'. Centre for Industrial and Applied Mathematics, The University of Oxford, 24-29 St. Giles', Oxford OX1 3LB, United Kingdom firstname.lastname@example.com November 2007] RANDOM WALKER RANKING 777
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Why We Build: Power and Desire in ArchitectureHardback
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- Publisher: Harper Design
- Format: Hardback | 392 pages
- Dimensions: 160mm x 231mm x 33mm | 567g
- Publication date: 20 August 2013
- Publication City/Country: New York
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In an era of brash, expensive, provocative new buildings, a prominent critic argues that emotions--hope, power, sex, our changing relationship to the idea of home--are the most powerful force behind architecture, yesterday and (especially) today.We are living in one of the most dramatic periods in modern architectural history: a time when cityscapes are being redrawn on a yearly basis, architects are testing the very idea of what a building is, and whole cities are being invented overnight, both here in the United States and in exotic locations around the world.In this bold and wide-ranging new work, Rowan Moore--former director of the Architecture Foundation, now a leading architecture critic--explores the reasons behind these changes in our built environment, and how they in turn are changing the way we live in the world. Taking as his starting point dramatic examples such as the High Line in New York City and the outrageous island experiment of Dubai, Moore then reaches far and wide: back in time to explore the Covent Garden brothels of eighteenth-century London and the fetishistic minimalism of Adolf Loos; across the world to assess a software magnate's grandiose mansion in Atlanta and Daniel Libeskind's failed design for the World Trade Center site; and finally to the deeply naturalistic work of Lina Bo Bardi, whom he celebrates as the most underrated architect of the modern era.Provocative and personal, iconoclastic and transforming, Why We Build is that rarest of things: a book about architecture that is also, on every page, a book about people--those chosen few who design buildings, and the rest of us, who use them every day.
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"Supremely ambitious...[Moore] writes with economy, clarity and wit. The prospect of 400 pages in his presence is not an unhappy one."--"Building Design"


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- A Place Depicted Melbourne, Australia
Glass Negative - Building Exterior, by A.J. Campbell, Melbourne, Victoria, circa 1900 Reg. No: MM 47402
- Description Of Content:
- Rear of chapel or church.
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- Transfer from Museum Victoria Natural Science
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|Themes this item is part of:
|A.J. Campbell Collection, Public Life & Institutions Collection
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|Tertiary Institutions
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|buildings & grounds
|Glass Negative: Black & White
|Place Depicted:
|Melbourne, Victoria, Australia
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Overview: Part of a good risk management plan is electronic device security. This is the technology age, and no matter what the business, from small store to a large multi-national corporation, technology is being utilized. With the growth in technology, though, comes an increase in security risks that employers must guard against. When securing electronic devices, employers should protect against employee abuse and protect employees from thieves.
The more technology an employee is given - laptops, phones, tablets, etc. - the more access he or she will have to an employer's business information - this might mean trade secrets, customer data or any other sensitive information that could be used against the employer at the termination of the employee. Having safeguards to protect against such concerns, such as ways to limit data access or creating enforceable employee contracts protecting the information, will help prevent future trouble.
Outside of the concerns about employer information, HR also needs to be concerned with employee information. Securing electronic systems that store personal information about employees or teaching employees how to secure their own information on employer-issued devices will help stop embarrassing security breaches and make sure that employees feel safe from concerns such as identity theft.
Trends: Bring your own device (BYOD) policies are a recent trend where employees access employer networks and servers with their personal technology. This can cause problems for employers who have limited ability to secure the devices and their own information. However, it can also mean significant saving for employees who do not have to pay for electronic devices at the same extent that they would without a BYOD policy.
Author: Ashley Shaw, JD, Legal Editor
FAQs have been added to employers in managing the emerging area of wearable technology in the workplace
California employers seeking to inform employees of the company's position on accepting personal calls and using personal electronic devices during work time should consider including this model policy statement in their handbook.
California employers seeking to communicate to employees that their personal use of company telephones should be very limited and excessive use may be grounds for discipline should consider including this model policy statement in their handbook.
Connecticut employers that engage in electronic surveillance to monitor activity in the workplace and seek to inform employees that they may be subject to electronic monitoring should consider including this model policy statement in their handbook.
HR guidance on the importance of securing employee electronic devices.


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Ostrom Drugs and GiftRead MoreNamed Best Gift/Book Store in the Bothell Reporter's Best of Northshore 2010 Readers' Choice Awards, Ostroms Drug & Gift has presented patrons with greeting cards, novelty gifts, and pharmaceuticals for half a century. Owned and operated by three generations of the Ramsey family, the store has amassed a loyal staff, with each employee averaging 10 years. In addition to greeting their customers with a friendly face and an enthusiastic chest bump during regular business hours, the staff's community-minded pharmacists make emergency house calls.6414 NE Bothell WayKenmore, WA US
Solarium SkylightRead MoreYou don't need a stopwatch to know that Jiffy Lube's services are fast, but if you had one, the Signature Service Oil Change would clock in at the 20-minute mark. And you'd be getting more than just an oil change. Jiffy Lube technicians also top off fluids, inflate tires, inspect key components, and diagnose issues before they happen—all in less time than it takes to replace a shattered windshield with Saran-Wrap. This blend of speed, quality, and preventative maintenance is at the core of Jiffy Lube's mission to maximize a vehicle's lifespan and keep it out of the repair shop. More than 20,000 of its technicians have completed an extensive training program certified by the National Institute for Automotive Service Excellence (ASE). These experts can perform many of the manufacturer-recommended services offered at dealerships, including air-filtration and electrical services—only without the need for appointments or long wait times. An impressive feat, considering that Jiffy Lube serves roughly 24 million customers each year.7204 NE 175th StKenmore, WA US
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This band's aim is not exactly true. The songs sound like high school gossip, which is not ever cute. At least vocalist Shaant's singing is snotty instead of whiney, keeping Cute Is What We Aim For far away from one stereotypically emo element. "Drama doesn't follow me", Shaant admits during "There's a Class for This". "It rides on my back", he continues. Over-the-top drama is sold with stripped-down, pop-rock hooks that mostly prevent overbearing emotional mood swings. Group vocals, as found on "Risqué" and a few other places, highlight Cute Is What We Aim For hooks. But this group is too precocious for its own good. Song titles alone, like "I Put the 'Metro' in Metronome" and "Sweat the Battle Before the Battle Sweats You", try too hard to impress. This group has much to say, but lacks the right words to say it all. It shouldn't be so cute.
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Colette Carr "Static" (video) (Premiere) // Music
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"Australian sextet Alpine's newest album is a fantastic expansion of their joyous pop sound, but two members give us five records apiece that helped define their unique musical identities.READ the article


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Purported information shows areas of emphasis for next OS
Slides purporting to show Windows 8 features have emerged on the Web.
The Microsoft Kitchen blog published the slides on Monday, citing an Italian blog source. Some of the slides are dated from April and are stamped "Microsoft confidential – NDA Conversation." The slides appear to be a presentation of how Microsoft's rumored operating system under development – "Windows 8" – can help PC maker HP support its customers.
Microsoft, which released Windows 7 in October, hasn't described a successor OS, although new releases happen about every three years. The "Windows 8" project is still a rumor, although CNet author Ina Fried wrote that the name was confirmed in slides at the Microsoft Professional Developer Conference in November, along with a 2012 release date. Such a date seems somewhat consistent with a July 1, 2011 purported release-to-manufacturing date, also not confirmed.
A Microsoft spokesperson didn't confirm the existence of a Windows 8 project at Microsoft. She said in a Monday phone call that Microsoft was looking into whether the slides really came from Microsoft. "If it is real, it could be stuff that's old," she said.
Certainly, the slide materials tend to cover old ground or areas Microsoft has already dealth with in Windows 7. For instance, the takeaway slides suggest that energy efficiency will be an emphasis of Windows 8, along with "instant sleep." The slides also predict that Windows 8 users will see faster wake-up times: "Turn On duration is about half of today's cold boot with "Logoff + Hibernate," one slide states.
A future scenario is presented with regard to identity management. Users will able to establish their identity through the PC's camera and facial recognition technology, according to one slide. In addition, identity will evolve from being "machine centric to user centric" by housing identity information in the Internet cloud.
The slides state that a 3D user interface for PCs is "on the horizon." However, Microsoft announced in late May that it partnered with chipmaker NVidia on such technology.
There's also a slide suggesting that users of Apple's OS consider the interface to be of "high quality" by users, which enforces Apple's brand strength. If the slides are real, that seems a peculiar addition to the slide deck. Why would Microsoft make such a concession about Apple's OS brand strength in slides aimed at Microsoft's No. 1 PC equipment manufacturing partner?
A number of Web sites track alleged Windows 8 "leaks." In the case of a rumored 128-bit Windows 8, the Web site alleging it has since disappeared.
About the Author
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Three-time Federal 100 winner is leaving government in July.
Members of Congress heaped hard questions on federal IT leaders at a June 16 hearing and suggested strongly that somebody needs to be fired.
Senate bill carves out exemption for supercomputing and other lab IT; an earlier version of the measure would have exempted the entire Department of Energy.
OPM is partnering with CSID to try to manage the fallout from a massive breach of some 4 million federal personnel records.
Steve Kelman applauds the OMB's move to make career employees "goal leaders" for performance management initiatives -- but wishes it had come far earlier in the administration.
In an exclusive interview, Jimaye Sones, who was Defense Information Systems Agency comptroller from 2005 to 2013, says he was reassigned after revealing questionable accounting practices at the agency.
Through computer forensics training and internships, veterans are helping Immigration and Customs Enforcement tackle a backlog in child exploitation cases.
After two months on the job studying DOD's cybersecurity and IT needs, Secretary Ashton Carter is set to unveil a new Pentagon cyber strategy in Silicon Valley.
The failure of massive federal IT projects can usually be traced to poor management. The solution is as multilayered as the projects themselves.
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2013 April 24 - 28 CSNS US Coin Signature Auction - Chicago #1184
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1844 Seated Liberty Dollar, PR651844 $1 PR65 NGC. Ex: Pittman-Kaufman. The 1844 proof Seated Liberty dollar is a great rarity, and few examples can match the quality and appeal of this delightful Gem. Q. David Bowers, in his Silver and Trade Dollar Encyclopedia (1993), estimates the original mintage to be between 20 and 30 pieces, and the number of survivors to be between seven and 14 coins. David Akers, on the other hand, cataloging John Jay Pittman's 1844 proof set, from which this specimen came, says: "I would place the number at the high end of that range, possibly 10-12."
Second-Finest Certified
Ex: Pittman-Kaufman
Second-Finest Certified
Ex: Pittman-Kaufman
Spectacular cobalt-blue, violet, and golden-tan iridescence in the mirrored obverse fields accentuates the satiny gold and violet obverse motif on this impressive Gem. The reverse, on the other hand, features light silver-tan color resting on reflective fields, and bright silvery design elements. The devices are sharply detailed, with razor sharp definition throughout. Both surfaces are well-preserved, but high magnification reveals some faint hairlines in the fields. A minute, unobtrusive mark on Liberty's cheek and another above the eagle's right (left facing) wing might help to identify the coin for future catalogers. This piece is certain to draw the attention of connoisseurs of Seated Liberty proof coinage, and we expect intense competition when this lot is called. Census: 1 in 65, 0 finer (3/13).
Ex: Possibly Dr. J. Hewitt Judd; Numismatic Gallery; purchased as part of a complete 1844 silver and minor proof set by John Jay Pittman on 4/5/1948 for $400; John Jay Pittman Collection, Part I (David Akers, 10/1997), lot 833; Philip Kaufman; Summer FUN Signature (Heritage, 6/2008), lot 1247; Internet Coin Auction (Heritage, 8/2008), lot 23479, realized $83,375; Los Angeles Signature (Heritage, 7/2009), lot 1149, realized $ 69,000.
From The Greensboro Collection, Part III.(Registry values: N10218) (NGC ID# 24ZU, PCGS# 6985)
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DirectFB Updates GTK3 Support, Working Towards DirectFB 1.8
Published on 21 August 2014 04:26 AM EDT
Written by
Michael Larabel
Free Software
2 Comments
While we haven't heard much out of the
camp in recent months, it turns out their code is still churning and they continue making progress for this library that provides a plethora of features while running off the Linux frame-buffer.
Some of the DirectFB news we missed was that GTK+ 3.12 has been ported to DirectFB and uses a brand new GDK back-end. Back in the GTK2 days there was DirectFB support but that was
dropped prior to GTK3
. This current DirectFB GTK+ 3.12 port lacks drag and drop, selection support, cursor shapes, full-screen support, geometry hints, properties, and many other features, but according to the developers maintaining this out-of-tree GTK+ code it's "basically usable" for running modern GTK apps on DirectFB.
DirectFB 1.8 is also coming along to succeed the year-old DirectFB 1.7. There isn't yet any organized change-log but those interested can
browse the commits
. There's been more work on Wayland support, EGL, libdirect, etc. Most of the DirectFB development continues to be led by Denis Oliver Kropp.
Those wishing to learn about the other recent DirectFB advancements can visit
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How Pedro, 'Ellen,' 'Will & Grace' and other gay TV icons helped pave the way for gay marriage
The prominence of gay people on TV, real and fictional, helped normalize gay culture and relationships
Jun 26, 2015
Songs on Screen: 'Game of Thrones' theme is designed to make you feel ALL the emotions
And it works!
Jun 26, 2015
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'Scarface,' Big Boo, Cartman, 'Stand By Me' and the other swearing-est characters
Jun 26, 2015
Benedict Cumberbatch, Kevin Hart, Emma Stone and 319 others invited to vote for Oscar
Progress is being made but women still sadly underrepresented
Jun 26, 2015
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When Jon Davis got the assignment of his dreams, his problems had just begun.
Jun 26, 2015
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Did 'American Horror Story' or 'Hannibal' make our list?
Jun 26, 2015
Exclusive: 'Gone Girl's' Rosamund Pike is back with a vengeance in 'Return to Sender'
New thriller co-starring Nick Nolte and Shiloh Fernandez hits theaters and VOD on August 14
Jun 26, 2015
Exclusive: In 'Rectify' season 3 trailer, the law has more in mind for Daniel
A more plot-heavy preview than the great Sundance drama tends to be
Jun 26, 2015
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The trappings of childhood have a lasting effect in our lives and in movies.
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Songs on Screen: The Great Journey of Lonely Dreamers in America's Rainbow Trilogy
From Dorothy to Kermit, how rainbows brought us togeter, but do they still?
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Press release from Marketwire
Eldorado Gold Achieves Two Key Permitting Milestones in Turkey
Monday, June 13, 2011
Eldorado Gold Achieves Two Key Permitting Milestones in Turkey07:55 EDT Monday, June 13, 2011VANCOUVER, BRITISH COLUMBIA--(Marketwire - June 13, 2011) - Paul N. Wright, President and Chief Executive Officer of Eldorado Gold Corporation (TSX:ELD)(NYSE:EGO)(ASX:EAU) ("Eldorado" the "Company" or "we") is pleased to announce that Eldorado, through its 100% owned subsidiary Tuprag Metal Madencilik Sanayi ve Ticaret ("Tuprag"), has received its Trial Operating Permit ("Permit") at the Efemcukuru Gold Project in Izmir and the supplementary Environmental Impact Assessment Positive Certificate ("EIA") at Kisladag Gold Mine in Usak."We are extremely pleased with the timely receipt of our Permit at Efemcukuru and approval for the supplementary EIA to increase annual throughput to 12.5 million tonnes of ore at Kisladag," said Paul Wright President and CEO of Eldorado. "We continue to operate according to plan and look forward to start up of operations at Efemcukuru this month and expansion of the process facilities at Kisladag to treat the flotation concentrate from Efemcukuru. Our thanks to the Tuprag team who have worked diligently to achieve these two important permitting milestones."Efemcukuru Gold Project, Izmir ProvinceThe Permit is the final step in the process leading to start up of operations. It is issued at the completion of construction and will allow the Company to carry out commissioning of the process facilities and to commence production of gold. The Permit is valid for a period of one year, during which time the Company will apply for the mine Operating Permit. The mine Operating Permit will then be issued when the company demonstrates compliance with the operational, safety and environmental parameters set out in the project Environmental Impact Assessment documents. Commissioning of the concentrator plant with ore from the underground mine will begin immediately, with first gold production planned prior to the end of June.Kisladag Gold Mine, Usak ProvinceThe Company has also received approval for the supplementary EIA for the planned expansion of the Kisladag Gold Mine. Approval of this supplementary EIA allows the company to increase annual throughput by 25% to approximately 12.5 million tonnes of ore to be processed through the heap leach facility. The approval of the supplementary EIA also covers the expansion of the process facilities on site to include a treatment plant to carry out final gold extraction from the flotation concentrate generated at the Efemcukuru Project. This plant will recover approximately seventy percent of the recoverable gold from the Efemcukuru ore, the balance being recovered through gravity concentration at the Efemcukuru plant.The receipt of these two important milestone permitting documents is the culmination of many years of continued effort by the Company to work with the government authorities in Turkey to realize the value from the country's considerable mineral resources.Eldorado is a gold producing, exploration and development company actively growing businesses in Turkey, China, Brazil and Greece. With our international expertise in mining, finance and project development, together with highly skilled and dedicated staff, we believe that Eldorado is well positioned to grow in value as we create and pursue new opportunities.ON BEHALF OF ELDORADO GOLD CORPORATION"Paul N. Wright"Paul N. Wright President and Chief Executive OfficerCAUTIONARY CONCERNING FORWARD-LOOKING STATEMENTSCertain of the statements made herein may contain forward-looking statements or information within the meaning of the United States Private Securities Litigation Reform Act of 1995 and applicable Canadian securities laws. Often, but not always, forward-looking statements and forward-looking information can be identified by the use of words such as "plans", "expects", "is expected", "budget", "scheduled", "estimates", "forecasts", "intends", "anticipates", or "believes" or the negatives thereof or variations of such words and phrases or statements that certain actions, events or results "may", "could", "would", "might" or "will" be taken, occur or be achieved. Forward-looking statements or information herein include, but are not limited, to statements or information with respect to the permitting milestones in Turkey.Forward-looking statements and forward-looking information by their nature are based on assumptions and involve known and unknown risks, uncertainties and other factors which may cause the actual results, performance or achievements of the Company to be materially different from any future results, performance or achievements expressed or implied by such forward-looking statements or information. We have made certain assumptions about the forward-looking statements and information, including assumptions about the price of gold; anticipated costs and expenditures; estimated production, mineral reserves and metallurgical recoveries; the impact of the integration of acquired businesses on our operation, financial position, reserves and resources and gold production;and the ability to achieve our goals. Although our management believes that the assumptions made and the expectations represented by such statements or information are reasonable, there can be no assurance that the forward-looking statements or information will prove to be accurate. Furthermore, should one or more of the risks, uncertainties or other factors materialize, or should underlying assumptions prove incorrect, actual results may vary materially from those described in forward-looking statements or information. These risks, uncertainties and other factors include, among others, the following: gold price volatility; risks of not meeting production and cost targets; discrepancies between actual and estimated production, mineral reserves and resources and metallurgical recoveries; mining operational and development risk; litigation risks; regulatory restrictions, including environmental regulatory restrictions and liability; risks of sovereign investment and operating in foreign countries; currency fluctuations; speculative nature of gold exploration; global economic climate; dilution; share price volatility; the risk that the integration of acquired businesses taking longer than expected, the anticipated benefits of the integration may be less than estimated and the costs of acquisition higher than anticipated; ability to complete acquisitions; competition; loss of key employees; additional funding requirements; and defective title to mineral claims or property, as well as those factors discussed in the sections entitled "Forward-Looking Statements" and "Risk Factors" in the Company's Annual Information Form & Form 40-F dated March 31, 2011.There can be no assurance that forward-looking statements or information will prove to be accurate, as actual results and future events could differ materially from those anticipated in such statements. Accordingly, you should not place undue reliance on the forward-looking statements or information contained herein. Except as required by law, we do not expect to update forward-looking statements and information continually as conditions change and you are referred to the full discussion of the Company's business contained in the Company's reports filed with the securities regulatory authorities in Canada and the U.S.Eldorado Gold Corporation's shares trade on the Toronto Stock Exchange (TSX:ELD) and the New York Stock Exchange (NYSE:EGO). Our Chess Depositary Interests trade on the Australian Securities Exchange (ASX:EAU).Request for information packages: email@example.com FURTHER INFORMATION PLEASE CONTACT: Nancy E. WooEldorado Gold CorporationVice President Investor Relations604.601.6650 or 1.888.353.8166604.687.4026 (FAX)firstname.lastname@example.com


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text: Advertising trends and techniques tend to change all the time. Something that was popular last year might not necessarily be popular this year. Something that was once disregarded as ineffective may now be the 'in' thing everybody is using. It's important that you stay up to date with your advertising techniques, or you might find that your business dries up. With that in mind, here are the 6 most effective advertising techniques today: Continue reading