Instability of Graph Dynamical Systems and Edge Shatteringmacaule/slides/china_mathbio_dec08.pdf · 2008. 12. 24. · AMS-SMS Joint Meeting Fudan University Shanghai, China December,

Preliminaries: Graph dynamical systemsFixed Point Reachability

Role of cycles in the graphSummary and Future Work

Instability of Graph Dynamical Systems and Edge Shattering

Matthew Macauley

Department of Mathematical SciencesClemson University

Clemson, South Carolina, USA 29634

Special session: Biomathematics: Newly developed applied mathematics and newmathematics arising from biosciences

AMS-SMS Joint MeetingFudan UniversityShanghai, ChinaDecember, 2008



Outline

1 Preliminaries: Graph dynamical systems

2 Fixed Point Reachability

Application – Functional Linkage Networks

Abstraction to threshold systems

3 Role of cycles in the graph

Rank functions

Equivalence of SDSs

4 Summary and Future Work

Stability via Attractor Basin Reachability

Stability via Cycle equivalence

Edge shattering



Definitions

I A graph dynamical system (GDS) is a triple consisting of:

A graph Y with vertex set v[Y ] = 1, 2, . . . , n.

For each vertex i a state yi ∈ K (e.g. F2 = 0, 1) and a Y -local functionFi : Kn −→ K n

Fi (y = (y1, y2, . . . , yn)) = (y1, . . . , yi−1, fi(y[i ])| z

vertex function

, yi+1, . . . , yn) .

An update scheme that determines how to assemble the functions to obtain theglobal map F : K n −→ K n .

I Two standard choices for the update scheme:

Parallel: Generalized cellular automata

F(x1, . . . , xn) = (f1(x[1]), . . . , fn(x[n])).

Sequential: Sequential dynamical system

[FY , π] = Fπn Fπn−1 · · · Fπ1 .



Example. Define the function nork : Fk2 −→ F2 by nork(x) =

kY

i=1

(1 + xi).

I [NorCirc4, π] for given update sequences:

1000

0010

0100

0001 1010

0000

0101

0011

1011

0111

1111

1101

(1234)

0110 1110

1001

1100 (1423)

0000

1100

0110

0010

1000

0101

10101010

1101

1110

1111

1011

0111

1001 0100 0001

0111

1111

1101

1010

0000

0101

0010 1000

1110

1100

0110 0011

1001

1011

0001 0100(1324)



Applications

Large complex networks.

Epidemiology. Disease propogation over social contact graphs.

Agent-based transportation simulations.

Packet flow in wireless networks.

Gene annotation (functional linkage networks)

Transport computation on irregular grids (e.g., heat, radiation).

Image processing and pattern recognition.

Discrete event simulations (e.g., chemical reaction networks).



Generic GDS research questions

I What does it mean for two GDSs to be “equivalent”?

I What is a good way to measure “stability” of the dynamics of a GDS with respect tochanges in the system (e.g., functions, states, update sequence)?

I How is update stability correlated with properties of the system, such as the structureof the base graph?

I What is a good characterization of graphs that is useful to people studying dynamical

systems over them?



Application – Functional Linkage NetworksAbstraction to threshold systems

Functional Linkage Networks

I Begin with a functional-linkage graph:

Corresponds to a Gene Ontology (GO) function, f .

Vertices of a graph represent proteins.

Two proteins are adjacent if we think they share the same function. The edgeweight wij is our level of certainty.

Each protein is assigned a state xi from +1,−1, 0, depending on whether it isannotated with f . (+1 = yes, −1 = no, 0 = mystery).

Goal : Assign a value of +1 or −1 to all “mystery proteins.”

Basic approach: Given “mystery protein” i , is it adjacent to more +1 or more −1 proteins?i.e., compute

si = sign

0

@X

j |i,j∈E

wijxj − θ

1

A .

I If si > θ, then assign xi = +1. Otherwise, set xi = −1.




Functional Linkage Networks (cont.)

In [2], the proteins are updated sequentially, given by an update order chosen at random.This is an SDS.

Some general questions:

I Does this process always converge to a fixed point? (Answer: YES)

I How much does the fixed point reached depend on the update order used?

I How quickly does it take to reach a fixed point?

I How reliable is this algorithm? (i.e., false negative & false positive rate)

In fact, this method has been well-received, and the reliability is superior to prior models.But as with any model, there are some “red flags” that are worth investigating.




A Mathematical Abstraction with 2-Threshold SDSs

Motivation: Let’s explore one of the “red flags” of this approach.

I Given FY , define

ωπ(y) =∞\

n=1

[FY , π]m(y) | m ≥ n .

I For P ⊆ SY , define

ωP(y) =[

π∈P

ωπ(y) .

I For a sequence of functions FY , define

ω(FY ) = max˘˛

˛ωSY(y)

˛˛ | y ∈ K n

¯.

I Consider an SDS [FY , π], where FY = T2Y

, the 2-threshold local functions.

I All periodic points of a threshold SDS are fixed points.




Summary of Results

I A threshold SDS over Kn can have at most n +1 fixed points, and this bound is sharp.

I If Y = Kn and k ≤ n, then Fix[TkY, π] = 0,1.

I If Y is connected with minimial degree d > (1 − 1k)n, for k > 0, then Fix[Tk

Y, π] ⊆

0,1.

I If Y is a tree, then Ω(2n) fixed points can be reached by varying the update order ofan SDS [T2

Y, π].

Theorem ([4])Let 0 < ε < 1. Threshold systems over Gn,p , with p = o

“nε

n

”

, contain initial

configurations from which Ω(2n1−ε

) different fixed points can be reached by changing

the update order, with probability 1 − o`

1nε

´

Conclusion. In general, adding edges to the dependency graph of an SDS causes the

dynamics to become more stable with respect to update order.



Rank functionsEquivalence of SDSs

Hyperplane arrangements – a motivating example

Consider the problem of counting the number of chambers |C(H)| of a hyperplanearrangement H in R

n.

Example:

PSfrag replacements

3 cuts of S23 cuts of S2

6 chambers8 chambers

Figure: Cutting the sphere with hyperplanes

The number of chambers depends not only on the number of hyperplanes, but also onthe linear dependencies of the normal vectors. This is a problem handled by matroids.




Consider a hyperplane arrangement H = H1, . . . ,Hk, with corresponding normalvectors V = v1, . . . , vk.

If the normal vectors are linearly independent, then |C(H)| = 2k .

If the hyperplanes (normal vectors) are in general position, |C(H)| = 2

n−1X

i=0

“k − i

i

”

.

I Define the rank function of V by

r : P(V) −→ Z , r(vi1 , . . . , vik) = dim〈vi1 , . . . , vik

〉 .

|C(H)| depends only on the rank function of V .




Update graphs of SDSs

I Question: When does [FY , π] = [FY , σ], for distinct update orders π, σ ∈ SY ?

Definition. The update graph U(Y ) has vertex set SY . The edge π, σ is present iff:

π and σ different by exactly an adjacent transposition (i , i + 1),

πi , πi+1 6∈ e[Y ].

Example. Let Circ4 be the circular graph on 4 vertices.

1243

3241

2134

4132

1423

3421

2314

4312

1234

1432

3214

2341

4123

2143

4321

1324

3124

2413

4213

1342

3142

2431

4231

3412

Figure: The update graph U(Circ4).




Functional equivalence of SDSs

Define an equivalence relation ∼Y on SY by π ∼Y σ if π and σ are on the sameconnected component of U(Y ).

Prop. If π ∼Y σ, then [FY , π] = [FY , σ].

I An ordering π ∈ SY induces an acyclic orientation of Y , denoted OπY

.

I There is a bijection between

fY : SY/∼Y −→ Acyc(Y ), fY ([π]Y ) = OπY .

Thus, α(Y ) = |Acyc(Y )| is an upper bound for the number of functionally distinct SDSmaps [FY , π] for a fixed choice of FY . This bound is known to be sharp.




Permutahedra

The n-permutahedron Πn is the convex hull of all permutations of the points(1, 2, . . . , n) ∈ R

n. It is an (n − 1)-dimensional polytope.

The vertices and edges of Πn can be labeled as follows:

Two vertices are adjacent if they differ by swapping two coordinates in adjacentposition.

An edge is labeled with a transposition (xi , xj) of the values of the two entries thatare swapped.

Note: This labeling scheme does not agree with the geometric coordinates of the vertices!




123

132 213

231312

321

(2 3)

(2 3)

(1 2)

(1 2)

(1 3) (1 3)

(a) Π3

2143

1243

1324

2413

2314

1342

1423

1432

3142

3124

3214

2431

4231

4321

3412

3241

4123

2134

2341

1234

3421

4132

4312

4213

(b) Π4

Figure: Permutahedra, for n = 3 and n = 4.




Constructing U(Y ) from Πn

I Πn is the update graph of En.

I Each transposition (i j) ∈ Sn corresponds with a complete set of parallel edges of Πn .

I The update graph U(Y ) can be constructed by “cutting” Πn with a hyperplane Hni,j

for every edge i , j ∈ e[Y ].




An example

2

3 4

1

(a) Y

2143

1243

2314 1423

1432

2431

4231

4321

3241

4123

2134

2341

1234

3421

4132

4312

4213

2413

(1 3)

(2 3)

( 1 2)

(b) Constructing U(Y )

Figure: Hyperplanes cuts corresponding with the edges 1, 2, 2, 3, and 1, 3 in Y < K4.




The rank function of a graph

Definition. The rank function of a graph Y is the function

rY : P(Y) −→ Z , rY (Z) = r(vni,j | i , j ∈ e[Z ]) ,

where vni,j

is the normal vector of the hyperplane Hni,j

from the constructing of U(Y ).

Definition. Let f1, . . . , fn be a basis for the dual space of Rn. For a graph G = (V ,E),let H(G) be the arrangement defined by

H(G) = ker(fi − fj) | i , j ∈ E .

H(G) is called the graphic arrangement of G .

Prop. If Z < Y , then rY (Z) is the number of edges in a spanning forest of Z , i.e.,

rY (Z) = |v[Z ]| − n(Z) .




Cycle equivalence of SDSs

Definition Two finite dynamical systems φ,ψ : K n → K n are cycle equivalent if thereexists a bijection h : Per(φ) −→ Per(ψ) such that

ψ|Per(ψ) h = h φ|Per(φ) .

I Let σ, τ ∈ Sn be

σ = (n, n − 1, . . . , 2, 1) , τ = (1, n)(2, n − 1) · · · (d n2e, b n

2c + 1) ,

and let Cn and Dn be the groups

Cn = 〈σ〉 Dn = 〈σ, τ〉 .

These groups act on update orders π = π1π2 · · ·πn by shift and reflection as follows:

σ(π) := σ · π = π2π3 · · ·πnπ1 , ρ(π) := τ · π = πnπn−1 · · ·π2π1 .




Cycle equivalence of SDSs

TheoremFor any π ∈ SY , the SDS maps [FY , π] and [FY ,σ(π)] are cycle equivalent. Moreover,

if K = F2, then these are cycle equivalent to [FY ,ρ(π)] as well.

Example. [NorCirc4, π] for given update sequences:

1000

0010

0100

0001 1010

0000

0101

0011

1011

0111

1111

1101

(1234)

0110 1110

1001

1100 (1423)

0000

1100

0110

0010

1000

0101

10101010

1101

1110

1111

1011

0111

1001 0100 0001

0111

1111

1101

1010

0000

0101

0010 1000

1110

1100

0110 0011

1001

1011

0001 0100(1324)




Enumeration of equivalence classes of SDS maps

α(Y ) := Acyc(Y ) = TY (2, 0) satisfies

α(Y ) = α(Y /e) + α(Y \ e) for any edge e

and is an upper bound for the number of SDS maps [FY , π] for a fixed choice ofFY up to functional equivalence.

κ(Y ) := Acyc(Y ) = TY (1, 0) satisfies

κ(Y ) = κ(Y /e) + κ(Y \ e) for any cycle edge e

and is an upper bound for the number of SDS maps [FY , π] for a fixed choice ofFY up to cycle equivalence [7].

Remark: κ(Y ) is also a sharp upper bound for the number of conjugacy classes of Coxeterelements of a simply-laced Coxeter system with Coxeter graph Y (see [8, 9]).



Stability via Attractor Basin ReachabilityStability via Cycle equivalenceEdge shattering

Reachable attractor basins as a measure of stability

Consider a sequential dynamical system over a graph Y , with 2-threshold functions.

I If Y is a tree, then there are states that can reach 2n − n = O(2n) fixed points byvarying the update order.

I If Y = Kn, then any given state can reach at most n + 1 fixed points by varying theupdate order.

I This can be extended to Gn,p for various values of p (see [4]).

Conclusion. In general, adding edges to the dependency graph of an SDS causes thedynamics to become more stable with respect to update order.




Cycle equivalence as a measure of stability

Prop. If Y is a tree, then κ(Y ) = 1.

Corollary. If Y is a tree, then for a fixed choice of functions FY , all SDS maps [FY , π]are cycle equivalent.

The functions α(Y ) and κ(Y ) are a measure of system complexity, and are Tutte-Grothendieck invariants.

I Adding more edges to Y can only increase α(Y ) and κ(Y ), and thus the number ofpossible SDS maps, and cycle structures of the maps.

Conclusion. In general, adding edges to the dependency graph of an SDS causes thedynamics to become less stable with respect to update order.




What is going on???

We are discussing different notions of update order instability!

By update order stability, one can measure:

How many different possible cycles structures (long term behaviors) are there. . .

How many different attractor basins can be reached from a particular state . . .

Or something else! (e.g., which states can arise as fixed points [5, 6]) . . .

. . . as the update order is perturbed.

Moral : Be careful when making general statements about the update order stability ofa dynamical system!

Question: These ideas have only been studied independently. Is there a way to tie themtogether to paint a clearer picture?




Application – Role of the graph structure in complex systemsWhen studying complex systems, it is natural to ask the following questions:

Question 1. What role does the structure of the dependency graph play in the dynamics

of the a system defined over it?

Question 2. Does there exist a good measure, or classification of graphs, useful for

people studying graph dynamical systems?

Key idea: Such a measure should be able to capture the cycle structure of the graph.

Measures such as the degree distribution or clustering coefficient can’t detect the presenceof large cycles.

PSfrag replacements

LC14,4 LC1,17

Figure: Two graphs with the same degree distribution and clustering coefficient, but “dynamicallydifferent.”




Approach: Edge shattering

I Recall that the rank function rY captures the cycle structure of a graph, and isdetermined by how many components remain upon removal of a certain subset of edges.

I For a graph with m edges, the rank function is of size Θ(2m), thus it is uncomputablefor most graphs.

I However, we can extract useful edge shattering properties from the following functions[0, 1] → N:

µY (k): average number of components when k% of edges are removed from Y .

MY (k): maximum number of components when k% of edges are removed from Y .

λY (k): average size of largest component when k% of edges are removed from Y .

σY (k): average size of a component when k% of edges are removed from Y .




Questions and Future ResearchI How well do these edge shattering functions distinguish commonly studied classes of graphs?For example:

Classical random graphs, such as Gn,p and Gn,M .Small-world networks (Watts & Strogatz, 1998)Scale-free networks (Barabasi and Albert, 1999)Real-world biological, epidemiological, and social networks.

I Can one re-construct a network of a particular size that satisfies certain edge shatteringproperties (hard!).

(a) Gn,p for n = 104 , p = 10−3 (b) A 10000-vertex tree

Figure: Edge shattering plots of σY for two graphs




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Instability of Graph Dynamical Systems and Edge Shatteringmacaule/slides/china_mathbio_dec08.pdf · 2008. 12. 24. · AMS-SMS Joint Meeting Fudan University Shanghai, China December,