Graph Dynamical Systems - Network Dynamics and Simulation

MATHEMATICAL & COMPUTATIONAL THEORY OF
GRAPH DYNAMICAL SYSTEMS
Points of contact: Henning S. Mortveit, V. S. Anil Kumar
Graph Dynamical Systems as a Framework for Analysis and Simulation of Complex Systems:
Complex Systems and their models typically share the
following common structure:
• A collection of entities with states and local decision rules
interact with each other in some manner.
• The interaction is generally local.
• The global system dynamics result through composition
of local dynamics.
Examples: socio-technical systems with epidemics
on social contact networks, markets, power networks,
communication networks, biological systems, and in
computational paradigms such as numerical analysis and
functional annotation inference methods.
Graph Dynamical Systems (GDS)
These represent a mathematical abstraction of complex
systems and are constructed from:
• A graph X with vertices representing entities and edges
representing the possibility of interaction between
entities.
• An vertex state and vertex functions for each vertex
representing the configurations and decision mechanisms.
• An update scheme, such as a word w update order,
specifying the order in which entities act.
The GDS map is the map obtained through composition of
the vertex functions as specified by the update scheme.
5
2
• Allows for precise system specifications and designs.
• Directly supports HPC-based implementations
of scaleable simulation systems that map well to
hardware and software and that are amenable to
verification.
• Supports analysis and derivation of general (as
opposed to system specific) results, properties and
engineering and design principles.
• Makes it easier to use of existing theory from e.g.
mathematics, statistics, computer science, and physics.
• Ensures that models and their implementations
become amenable to rigorous analysis.
ase
Dise
l
idua
Indiv
Composed Map:
m
Syste ation
iz
l
Rea
ion
ract
Inte
DistributedComputing
• How does the graph, vertex functions and/or update scheme
affect the global dynamics?
• Is the system robust with respect to perturbations such as edge
alterations or changes to vertex states?
• For a given level of resolution (i.e. equivalence measure), how
can the neutral networks be characterized?
Since most system properties are only known locally, the approach
typically follows the local-to-global paradigm.
Example:
• Circle graph on 4 vertices, X = Circle4.
• Local Boolean functions nor3(x1, x2, x3) = ¬ (x1 ∨ x2 ∨ x3).
• F1(x1,x2,x3,x4) = (nor3(x4, x1, x2), x2, x3, x4).
• Update sequence π = (1,2,3,4).
State transition for the SDS map: Fπ(0,0,1,0) = (1,0,0,0).
Base graph
1
F1
1000
(1234)
F4
2
4
F2
F3
Update sequence (1234)
0011
0010
0111
0100
0000
1011
1101
1111
3
1010
0110
0011
0110
0001
0100
1001
1110
0101
0111
0010
1011
1000
1101
1111
yoffs
l pa
idua
Indiv
er
Play
tra
on/S
Acti
e
Gam ome
Outc
tegy
e
Stat
ge
han
eC
Stat
y
Entit
Asymptotic behavior of complex systems:
• Fixed points of permutation SDS are independent of the
update schedule.
• SDS induced by the Boolean function Nor never have fixed
points.
• SDS induced by threshold functions only have fixed points as
attractors.
Clasification of limit cycle structure in asynchronous systems:
The structural diversity of long-term behavior in asynchronous
systems is governed by k-equivalence. We have constructed a
bound for the number of long term behaviors that is possible
for a fixed graph and fixed functions when the permutation
composition sequence is varied.
Example: For the circle graph on n vertices, there are only n-1
possible long-term behaviors.
(3214) (2143)
(1432)
(1243)
(1423) (3241)
(3421)
(2134)
(2314) (4132)
(4312)
(1324)
(3124) (2413)
(4213)
(1342)
(3142) (2431)
(4231)
• Barrett C., Hunt III H.B., Marathe M., Ravi S., Rosenkrantz D. and Stearns R.
(2003) On some special classes of sequential dynamical systems. Annals of
Combinatorics, 7(4): 381-408.
m
Syste ation
iz
l
Rea
ction
a
Inter
ge
han
C
m
Syste ation
iz
l
a
Re
• Barrett C., Hunt III H.B., Marathe M., Ravi S., Rosenkrantz D., Stearns R. and
Thakur M. (2007) Predecessor existence problems for finite discrete dynamical
systems. Theoretical Computer Science, 386(1-2): 3-37.
on
riteri
C
brium
li
n
ositio ns
p
m
Co
ctio
tera
n
I
f
o
• Barrett C., Hunt III H.B., Marathe M., Ravi R., Rosenkrantz D. and Stearns R.
(2003) Reachability problems for sequential dynamical systems with threshold
functions. Theoretical Computer Science, 295(1-3): 41-64.
Markets
Algorithmic and Computational Aspects and Theory
Research theme: to form a bridge between a mathematical
theory of simulation (using GDS) and HPC algorithm design and
implementation and provide a natural formal framework for formally
specifying simulations of large complex systems in terms of GDS
consisting of composed local interactions. Our goal is to develop
general computational methods for abstract GDS, which can lead to a
better understanding and efficient implementation of socio-technical
simulations.
Reachability: Starting from a configuration P can S reach
T in less than r steps? Starting from a given configuration,
will the SDS S ever reach a fixed point? What sort of fixed
points?
Application - Viral Marketing: choose a small subset
initially to introduce a product, so that the maximum
number of people adopt it, assuming the popularity spreads
by a diffusion process.
Expressive and Computational power of GDS: GDS provide a
universal computing model, i.e., they are powerful enough to
encompass a number of other natural formal systems such as
cellular automata, Hopfield networks and communicating finite state
machines.
Computational Complexity Results:
Results: Fundamental and common dynamical systems problems
on GDS: Reachability, Predecessor existence, characteristics of Fixed
Points.
• Efficient algorithms often possible if the underlying graph
has a tree-like structure.
Generalizations of standard threshold systems
• Computing dynamical properties of general GDS not
known in polynomial time.
The InterSim Framework
Applications: a large range of phenomena can be captured and
described as threshold systems (epidemics, belief propagation,
rumors, fads).
0000
1010
Classical, networked threshold models use binary states and a
common threshold k for the transitions from 0 to 1, and from 1 to
0. Although these models are well-understood mathematically,
they do always suffice for modeling.
We have generalized the class of threshold systems in several
Scheduling matters for complex systems: Nonequivalent Nor-SDS
important ways. Examples include all of the following:
for two different schedules:
• Bi-threshold systems are extension of classical threshold
Results:
systems where the two transitions 0 to 1 and 1 to 0 have
• Functional scheduling neutrality of complex systems:
separate thresholds thresholds k" (up threshold) and k# (down
The number of different SDS that can be obtained through
threshold).
rescheduling alone is at most equal to the number of acyclic
• Dynamic threshold systems have thresholds that change with
orientations of the base graph Y.
the dynamics and can capture phenomena with increased/
• Dynamical scheduling neutrality of complex system: Update
decreased susceptibility, tolerance or immunity.
schedules connected by graph automorphisms give rise to
dynamically equivalent sequential dynamical systems.
• Multi-state bi-threshold systems combines the first extension
above with a generalized state set K={0,1,2,...,r-1 }.
• Connections to Coxeter groups determine the possible long
term behaviors of sequential dynamical systems.
Theorem: Synchronous bi-threshold systems only have fixed
• General framework for neutrality and equivalence. The
points and 2-cycles as limit cycles regardless of the choice of k"
SDS neutrality theory based on the notion of update graph
and k#.
generalizes the neutrality theory of for example neutral networks
Theorem: Asynchronous threshold systems undergo a
which describes the folding of sequences into RNA secondary
bifurcation at ¢ := k# ¡ k" = 2 . For ¢ ¸ 2 these systems
structures.
U(Circ4 )
may have long limit cycles. When ¢ < 2, only fixed points are
possible.
(1234)
(2341) (3412)
(4123)
(4321)
• Bisset K., Chen J., Feng X., Kumar A. and Marathe M. (2009) EpiFast: A
fast algorithm for large scale realistic epidemic simulations on distributed
memory systems In Proceedings of 23rd ACM International Conference on
Supercomputing (ICS'09), Austin, Texas.
Any GDS can be modeled with InterSim.
Any GDS can be modeled with InterSim. 1001
0001
Update sequence (1324)
(1324) 1100
1110
1100
0101
• Mortveit H. and Reidys C. (2007) An Introduction to Sequential Dynamical
Systems, Springer Verlag (Universitext) .
n
Mathematical Aspects and Theory
Research theme: Derive qualitative and quantitative information
about a complex system/model based on known properties (e.g.
graph, functions, update scheme) without exhaustive, brute-force
computations. Examples:
• Macauley M. and Mortveit H. (2008) On Enumeration of Conjugacy Classes
of Coxeter Elements. Proceedings of the American Mathematical Society, 136:
4157-4165.
tion
posi ions
m
o
C
ct
tera
n
I
f
o
Equi
Epidemiology
7
Stat
atio
n
ositio ns
p
m
io
Co
ract
e
t
n
I
of
Graph and
influence domain:
• Macauley M. and Mortveit H. (2009) Cycle equivalence of graph dynamical
systems. Nonlinearity, 22: 421-436.
ge
an
e Ch
y
Entit
n
State
• Macauley, M. and Mortveit, H.S. (2011) Posets from Admissible Coxeter
Sequences. The Electronic Journal of Combinatorics, 18(P197).
d
e an rleaving
g
a
s
Mes ute inte
p
com
m
Infor
emic
Epid
e
Curv
State
y
g
State
lutio
vo
se e
isea
Entit
Me
Selected Publications:
ram
Prog t
l
Resu
assin
ep
ssag
on
Con
n
tatio
pu
l com
Loca
pute
Com
e
Nod
gati
a
prop
s
tact
D
nd
es a e
u
l
a
stat
ory v
Mem program
l
loca
te
h Sta
lt
Hea
Int
8
6
Benefits of a mathematically based framework for
complex dynamical systems:
tion
erac
3 n[4]=(3,4,5,8)
4 f4(x3, x4, x5, x8)
1
About the research: The general research includes mathematical as well
as computational and algorithmic work. Members of the NDSSL and their
collaborators have developed extensive results during the fifteen years of this
program.
Update graph
Consequence: Theory gives insight into system behavior,
parameter interdependencies, scheduling sensitivity, and offers
support for system validation of corresponding interactionist
models.
InterSim is distributed, configurable, and extensible, with pluggable
InterSim is distributed, con,igurable, and extensible, with interaction models that represent vertex functions.
pluggable interaction models that represent vertex functions. domain-­‐speci,ic simulator = con,igurable InterSim unaffected
affected
0 1 unaffected
0 + unaffected
affected
affected’
negative
neutral
positive
0 1 2 -­‐1 0 1 negative
-­‐1 (f) mul1-­‐back-­‐and-­‐forth neutral
positive
0 1 Degrees of conviction
-­‐1 neutral
0 Compute Node InterSim Compute Node Compute Node Compute Node InterSim InterSim InterSim Partitioned nNetwork
Partitioned etwork Graph etween Graphedges edgesbbetween
partitions epresent partitionsrrepresent
interactions ommunicated interactionsccommunicated
through essages iin
n the throughmmessages
distributed InterSim the distributed
InterSim
framework. framework.
-­‐1 0 … 1 q (g) arbitrary cyclic positive
0 1 1 (d) Compe1ng contagions Cluster … -­‐q (c) mul1-­‐progressive compe1ng negative
InterSimscales scales
million
nodes
2.8 billion
InterSim to 1to
00 100
million nodes and 2and
.8 billion edges. edges.
1 (e) bi-­‐threshold (back-­‐and-­‐forth) (a) progressive (b) mul1-­‐progressive Interaction Model Interaction Model Interaction Model Interaction Model affected
… 2 4 6 3 5 7 (h) mul1-­‐level cyclic Deterministic
diffusion.
Deterministic versus
versus sstochastic
tochastic diffusion. 0 Deterministic bi-­‐threshold system: long-­‐term dynamics of 77000-­‐
node Slashdot network is a 2-­‐cycle, as predicted by theory. 1 Stochastic bi-­‐threshold system: long-­‐term dynamics of Slashdot network reaches a quasi-­‐steady state; 20% of nodes are transitioning at each time.