A Game Theoretic Approach Proceedings of the 2015 ACM SIGMOD International Conference on Management of Data pp 16171628 20160512 M1 kosaka Author Nikos Armenatzoglou PhD at the Department of Computer Science and Engineering Hong Kong University of Science and ID: 784431
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Slide1
Real-Time Multi-Criteria Social Graph Partitioning: A Game Theoretic Approach
Proceedings of the 2015 ACM SIGMOD International Conference on Management of Data (pp. 1617-1628).
2016/05/12 M1
kosaka
Slide2AuthorNikos
ArmenatzoglouPh.D. at the Department of Computer Science and Engineering, Hong Kong University of Science and Technology
Research
Geo
-Social NetworksSpatial and spatio-temporal databasesInformation retrieval systems
Slide3IntroductionGraph partitioning is
particularly relevant to social networks because it enables the grouping of users into communities for market analysis and advertising purposes. RMGP
a novel framework for Real- time Multi-criteria Graph Partitioning
this framework
partitions a social network into a set of input classes
Slide4IntroductionSocial
GrouphG = (V, E, W ) V ... the set of users
E ... the set of edges (i.e., social connections)
W ... the set of edge weights (denoting the strength of social connections)
the edges may be directed , and the weights may be binary.
Slide5Introduction
RMGP returns an assignment of each user v to a single class sv that minimizes the following function:
①
②P a set classesc
(
v,s
v
)
the
cost of assigning v
∈ V to sv ∈ P α ∈ (0,1)The preference parameter adjusts the relative importance of the two factors
First sum represents
the assignment
cost
second one corresponds to
the social cost
Slide6Graph AlgorithmsExisting work on graph partitioning can be grouped into
threeattribute-based connectivity-based
attribute and connectivity-
based
Even though the above techniques partition a graph using multiple objectives none of them can be used for RMGP they consider the similarity of each node to a set of input classes, instead of the similarity between two nodes based on their attribute
values
Slide7Graph AlgorithmsUniform
Metric Labeling (UML) RMGP is closely related to UML
Input
undirected
edge-weighted graph G = (V, E, W ) a set L of k labels function c(v, l) that denotes the cost of assigning label l ∈L to node v ∈ V uniform function d(l, l’)
that re- turns 1 if l
≠ l’
; 0 otherwise.
Slide8Game Theory In strategic games
, players compete with each other over the same resources in order to optimize their individual objective functions.Under this framework, a player chooses a
strategy
that
minimizes his own cost without taking into account the effect of his choice on other players’ objectives(non-cooperative game).
Slide9Game Theory Formally, a strategic game is a tuple
Sv represents all the possible decisions that player v can take during the game to optimize his function
C
v
The optimization of Cv depends on v’s own strategy, as well as the strategies of the other players.
Slide10Game Theory
pure Nash equilibriumA strategic game has a pure Nash equilibrium
if
there exist a specific choice of strategies
sv ∈ Sv such that the following condition is true for all v ∈ V : In other words, no player has incentive to deviate from his current strategy.
Slide11Game TheoryFigure 2 illustrates a common framework for studying the dynamic process of decision-making in a strategic game
.
Slide12Game TheoryThree measures have been widely used to evaluate the quality of
equilibria : social optimum
(
OPT
)price of stability (PoS)price of anarchy (PoA).
PoS
= best equilibrium / OPT
PoA
= worst equilibrium / OPT
Slide13Game Theory
Potential Gamea single function , called the potential function
, can be used to express the objective functions of all the
players
A potential game is exact, if there exist a potential function Φ, such that for all si and all possible combinations of ,the following condition holds:
Slide14RMGP: GAME THEORETIC METHOD
In order to model RMGP as a game they assume that
each user is a player
,
whose goal is to join a class with low assignment cost that contains many of his closest friends. The RMGP game is a tuple the total cost
of
v is the weighted sum of
the
assignment cost
c
(v,sv)the social cost half of the total weight of edges that connect v with the subset of his friends in adj(v), who are assigned to different classes
Slide15RMGP: GAME THEORETIC METHOD The goal of each player v
∈ V is to find the class sv that minimizes his own total cost as expressed by Equation 3.
Similar to Equation 1, parameter
α
adjusts the relative importance of the assignment and social costs. the RMGP objective function in Equation 1 is equal to the sum of all individual user costs
Slide16Baseline Algorithm Figure 3 depicts the baseline algorithm RMGP
b, which applies the framework of Figure 2 to RMGP.
Slide17Assume
that all friends of v are assigned to a different class
When random assign result is “
p
1: v1,v4 p2:v2,v5,v6
p
3
:v
3
”, and α
is 0.5ex) Compute a maxSCv when v is v1maxSCv1 = 0.5 * 0.5 * (0.1+0.1) = 0.05
Slide18ex) First Round when v is v
1cost
v1
[p
1] = 0.5 * 0.48 + 0.05 = 0.29costv1[p2] = 0.5 * 0.6 + 0.05 = 0.35costv1[p
3
]
=
0.5 * 0.27 + 0.05 = 0.185
cost
v1
[sv2(p2)] = 0.35 – 0.5 * 0.5 * 0.1 = 0.325costv
1[sv4(p1
)] = 0.29 – 0.5 * 0.5 * 0.1 = 0.265
cost
v1
[p
1
] =
0.265
cost
v1
[
p
2
]
=
0.325
cost
v1
[
p
3
]
=
0.185
∴
minCost
=
cost
v1
[p
3
] ,
s
v
1
=
p
3
p
1
: v
1
,v
4
p
2
:v2,v5,v6 p3:v3
strategies
→
p1: v4 p2:v2,v5,v6 p3:v1,v3
LINE
7〜8
LINE9〜10
LINE 13
Slide19Baseline Algorithm All
Result
Slide20RMGP: Analysis
The running time of RMGPb
∵
In order to obtain the number of rounds,
they first show that RMGP is an exact potential game Potential function in RMGP is
Slide21RMGP: Analysis
Since RMGP is an exact potential gamethe set of strategic configurations S is finite, players need to change their strategies a finite number of times before a Nash equilibrium is reached.
In
order
to provide an upper bound for the number of rounds required for convergence of RMGPbassume an equivalent game with potential function , where d is a positive multiplicative factor chosen such that
Slide22RMGP: Analysis
Let ,and . is the (scaled) maximum assignment cost is
the (scaled) maximum social
cost
The following lemma describes an upper bound on the number of rounds required until RMGPb converges to a Nash equilibrium.
Slide23RMGP: Analysis
Some theoretical results on the quality of the solutions of RMGPb.
Slide24Normalization Issues
In several applications, the assignment and social costs may not be comparable. so this algorithm needs to have a way to normalize their costs.The objective function of conventional
RMGP (Equation 1) can be written in terms of the per user costs (Equation 3)
as
rewritten
Slide25Normalization Issues
In RMGPN(Normalized Version RMGP) for α
= 0.5, the average assignment and social
costs must be equal; i.e., it should hold that: To achieve this, the costs should be normalized, by re-adjusting the assignment cost or the edge weights. RMGPN aims at minimizing the function of Equation 7: (the
normalization
constant
)
Slide26Normalization IssuesHowever,
anybody cannot compute the exact value of cN because it requires AC
v
and
SCv, which can only be obtained after solving the problem.→ approximate cN using estimations of ACv and SC
v
based on two heuristic approaches.
optimistic
approach
assume that e
very user is assigned to the closest onepessimistic approach assume that each user is assigned to the event with the
median
distance
Slide27OPTIMIZATIONS They introduce three optimizations
that improve the performance of the baseline algorithm without compromising the convergence guarantee.
Pruning by Strategy Elimination
Parallelism with Independent Strategies Scheduling with Global Table
Slide28Pruning by Strategy Elimination RMGPb
computes the cost functions of all players for each strategy in every round. However, there are strategies that cannot be assigned to some users.
Consequently,
the authors
can reduce the strategic space, i.e., prune the events that are far from the user, and avoid redundant computations.
Slide29Pruning by Strategy Elimination aaa
aaaaaIn the worst case (i.e., assuming that none of his friends follows sv,min), the total cost of assigning v to
s
v,min
is: The reduced strategic space of v, denoted as Svr . The lowest cost for p is achieved when all friends of
v
follow p, i.e.,
α
·
c(v, p)+(1-α)· 0 = α· c(v, p).
Slide30Pruning by Strategy Elimination
Consequently, p belongs to Srv if:
The
valid region
VRv of player v is a bound such that no strategy with assignment cost higher than VRv can be assigned to v: In the running example of Figure 1, if
α
= 0.5, then
VR
v1
= 0.37. Srv１ contains only p3, since c(v1, p1) = 0.48, c(v1, p2) = 0.6, and c(v1 , p
3 ) = 0.27.
∴directly remove v1 from the game, and directly assign
v1 to p3.
Slide31Parallelism with Independent Strategies In each round, RMGP
b computes the best response of all users sequentially, in a round-robin fashion. This is a fundamental requirement in best response dynamics:
if multiple players change strategies simultaneously their decisions may be based on “outdated” information and there is the chance that the overall potential function
increases
Slide32Parallelism with Independent Strategies However, in our setting,
if two users are not socially connected, the strategic deviations of one will not affect the best-response of the
other
;
i.e., these two users can select their best responses simultaneously. Based on this observation, the authors can partition the users in Ng groups such that no two users in the same group share an edge
.
Slide33Parallelism with Independent Strategies The best responses of the users in the same group are computed in parallel, either by the same machine through
multi-threading, or by different machines.
Slide34Parallelism with Independent Strategies As shown in Figure 4, RMGP
is examines each group in a round-robin manner.
Slide35Parallelism with Independent Strategies Groups can be computed by a graph coloring algorithm.
The problem of coloring a graph using the minimum number of colors is NP-Hard.
However
, there are polynomial greedy
approaches that use at most dmax + 1 colors, where dmax is the
maximum
degree.
Slide36Scheduling with Global Table Depending on the strategy deviations, numerous redundant computations can be avoided with proper book-keeping.
In particular, for a class/strategy p and a user v if the set of v’s friends, who followed p does not change, then the cost of assigning v to p remains the same
.
Even
if some of v’s friends have switched strategies, it is possible that v is not affected, and the costly examination of his strategies can be avoided.
Slide37Scheduling with Global Table Based on these observations we propose the
RMGPgt optimization. Figure 5 depicts the pseudocode of RMGPgt
.
Slide38DECENTRALIZED GAME Commercial social networks usually distribute the social graph across multiple servers in order to improve content delivery and accommodate the geographically diverse nature of data generation and demand
.Computation in decentralized storage systems is usually performed through a distributed data access interface (API) between the application and the corresponding servers.
Slide39DECENTRALIZED GAME However, for large problem instances, transferring the data to a server can be a major bottleneck in performance. Moreover, a single server may not be able to store the entire graph due to
memory limitations.To avoid these shortcomings, the authors
propose a framework for performing RMGP in a decentralized manner
.
Slide40DECENTRALIZED GAME Their
framework assumes that the graph is distributed over several slave nodes that have processing capabilities. A master node M acts as a game coordinator that manages the data exchange between the slave nodes.
Although
they
assume that the slaves can only communicate through M, DG can be easily extended to handle direct data exchange between slaves.
Slide41DECENTRALIZED GAME Figure 6 depicts the processing
method and communication steps between M and the slave nodes.
Slide42EXPERIMENTS They
evaluate the proposed approaches assuming a LAGP task, where the users of a geo-social network are assigned to events based on proximity and social aspects (i.e., the running example).
LAGP ...
a user is assigned to an event that is nearby and, at
the same time, it is recommended to several of his friends. The graph and the locations of users are stored in two mainmemory hash tables where the user IDs are used as keys.social hash table ... list of pairs (friend id, edge weight) location hash table ... list of pairs
(
user
id
,
the
coordinates of the last check-in )
Slide43EXPERIMENTS Conditions
All algorithms were implemented in C++, under Linux Ubuntu and executed on an Intel Xeon E5-2660 2.20GHz with 16GB RAM. DatasetsThey
use
two real datasets:
Gowalla and Foursquare. In both datasets, all edge weights equal to 1. For experiments, where they need to reduce the size of the social graph, they use the Forest Fire sampling technique.
Slide44Comparison with Other Approaches
Slide45Comparison with Other Approaches
Slide46Effect of Normalization
Slide47Centralized RMGP
Slide48Centralized RMGP
Slide49Centralized RMGP
Slide50Decentralized RMGP
Slide51Decentralized RMGP
Slide52CONCLUSION This paper studies a type of multi-criteria graph partitioning
To achieve efficiency, they model the problem as a game, develop a best-response algorithm, and propose several
optimizations
to enhance its performance.
They apply normalization to resolve issues that arise due to large differences in the assignment and social costs. They demonstrate the effectiveness of the proposed techniques with extensive experiments on real datasets.