61 research outputs found
Bad Communities with High Modularity
In this paper we discuss some problematic aspects of Newman's modularity
function QN. Given a graph G, the modularity of G can be written as QN = Qf
-Q0, where Qf is the intracluster edge fraction of G and Q0 is the expected
intracluster edge fraction of the null model, i.e., a randomly connected graph
with same expected degree distribution as G. It follows that the maximization
of QN must accomodate two factors pulling in opposite directions: Qf favors a
small number of clusters and Q0 favors many balanced (i.e., with approximately
equal degrees) clusters. In certain cases the Q0 term can cause overestimation
of the true cluster number; this is the opposite of the well-known under
estimation effect caused by the "resolution limit" of modularity. We illustrate
the overestimation effect by constructing families of graphs with a "natural"
community structure which, however, does not maximize modularity. In fact, we
prove that we can always find a graph G with a "natural clustering" V of G and
another, balanced clustering U of G such that (i) the pair (G; U) has higher
modularity than (G; V) and (ii) V and U are arbitrarily different.Comment: Significantly improved version of the paper, with the help of L.
Pitsouli
On the Nash Equilibria of a Simple Discounted Duel
We formulate and study a two-player static duel game as a nonzero-sum
discounted stochastic game. Players are standing in place and, in
each turn, one or both may shoot at the other player. If shoots at
(), either he hits and kills him (with probability ) or
he misses him and is unaffected (with probability ). The
process continues until at least one player dies; if nobody ever dies, the game
lasts an infinite number of turns. Each player receives unit payoff for each
turn in which he remains alive; no payoff is assigned to killing the opponent.
We show that the the always-shooting strategy is a NE but, in addition, the
game also possesses cooperative (i.e., non-shooting) Nash equilibria in both
stationary and nonstationary strategies. A certain similarity to the repeated
Prisoner's Dilemma is also noted and discussed
Cops and Invisible Robbers: the Cost of Drunkenness
We examine a version of the Cops and Robber (CR) game in which the robber is
invisible, i.e., the cops do not know his location until they capture him.
Apparently this game (CiR) has received little attention in the CR literature.
We examine two variants: in the first the robber is adversarial (he actively
tries to avoid capture); in the second he is drunk (he performs a random walk).
Our goal in this paper is to study the invisible Cost of Drunkenness (iCOD),
which is defined as the ratio ct_i(G)/dct_i(G), with ct_i(G) and dct_i(G) being
the expected capture times in the adversarial and drunk CiR variants,
respectively. We show that these capture times are well defined, using game
theory for the adversarial case and partially observable Markov decision
processes (POMDP) for the drunk case. We give exact asymptotic values of iCOD
for several special graph families such as -regular trees, give some bounds
for grids, and provide general upper and lower bounds for general classes of
graphs. We also give an infinite family of graphs showing that iCOD can be
arbitrarily close to any value in [2,infinty). Finally, we briefly examine one
more CiR variant, in which the robber is invisible and "infinitely fast"; we
argue that this variant is significantly different from the Graph Search game,
despite several similarities between the two games
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