Simplicial Complexes as Complex Networks Statistical Mechanics of Simplicial Complexes

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Simplicial Complexes as Complex Networks
(Statistical Mechanics of Simplicial Complexes)

• Slobodan Maletic, Zoran Mihailovic and Milan
  Rajkovic
• Instute of Nuclear Sciences Vinca, Belgrade

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Why need for simplicial complexes?

• Use of simple or directed graphs to represent
  complex networks does not provide an adequate
  description
• Examples
• In a collaboration network represented as simple
  graph we only know whether 3 or more authors
  linked together in the network were couthors of
  the same paper or not.

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• Social networks necessary to consider
  coordinated action of more than 2 agents, such as
  a buyer, a seller and a broker. Network theory
  doesnt have flexibility to represent higher
  order aggregations, where several agents interact
  as a group, rather than as a collection of pairs.
• Not only agents taking part in the actions are
  important but time, place etc.
• Protein complex networks need information about
  proteins, regulation, localization, turnover etc.
• Reaction and metabolic networks

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Definition of a Simplex

• Def A simplicial complex K on a finite set V
  v0, v1, , vn of vertices is a nonempty subset
  of the power set of V with the property that K is
  closed under the formation of subsets, i.e. if s
  3 K and
• t 2 s, then t 2 K. The dimension of a simplex
  s is equal to one less than the number of
  vertices defining it. The dimension of K is the
  max of dim of all simplices in K.

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Example
Simplex convex hull of k1 affinely
independent points
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• Simplical Complex finite set of simplices

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Simplices and relations

• Any relation l between the elements of a set X
  and the elements of a set Y is associated with
  two simplicial complexes K(l) and the conjugate
  K(l-1).
• A simplex of K(l) is a finite set of elements of
  X related to a common element of Y, a simplex of
  K(l-1) is a finite set of elements of Y related
  to a common element of X.
• Dowker, Homology groups of relations (Annals of
  Math., 1952)

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Incidence matrix relation l between a set of
people and their intellectual interests
B
A
E
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K(l)
K(l-1)
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Why need for simplicial complexes (cont.)?

• To detect qualitative features of the network
  structure
• S.Cs are combinatorial versions of topological
  spaces that can be analyzed with combinatorial,
  topological, or algebraic methods

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Simplicial complex may be viewed as

• a combinatorial object (consider numerical
  invariants) Q-analysis of R. Atkin
• a combinatorial model of a topological space
  (consider algebraic topological measures, such as
  homotopy and homology groups).
• an algebraic model of the complex, its so called
  Stanley-Reisner ring (the quotient of a
  polynomial ring on variables corresponding to
  vertices, divided by the ideal generated by the
  non-faces of the complex).

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Statistical mechanics of S.C.?

• Can we gauge topological objects with statistical
  mechanical tools?
• Are there power laws connected to the measures of
  the combinatorial, topological or algebraic
  aspect of the simplicial complex obtained from
  scale-free networks?

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Simplicial complexes from digraphs

• 1
• Neighborhood complex N(G) of graph G. Its
  vertices are the vertices of G. For each vertex v
  of G there is a simplex containing the vertex v,
  together with all vertices w corresponding to
  directed edges v ?w. By including all faces of
  those simplices, the neighborhood simplex is
  obtained.

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Simplicial complexes from digraphs

• 2
• The complex C(G) has complete subgraphs of G as
  simplices. C(G) has as vertices again the
  vertices of G. The maximal simplices are given by
  collections of vertices that make up maximal
  (un)directed complete subgraphs, or cliques of G.
  
• In general, any property of G that is monotone
  (preserved under deletion of vertices or eges)
  may be used e.g. simplices consist of all
  subgraphs of G without (dir or undirected) cycles.

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Combinatorial aspect Q-analysisInvariants and
useful quantities

• Dimension of the complex K
• Vector valued quantities
• f-vector (second structure vector)
• (f0,.fD)
• fi number of i-dim simplices in K

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q-nearness and q-connectedness

• Two simplices K1 and K2 are q-near in the
  simplicial family iff they share at least q1
  vertices.
• Two simplices are q-connected by a chain of
  complexes of length r iff there is sequence of r
  pair-wise q-near simplices. Equivalence relation
  ? generates the partition of the simplicial
  family into q-connected components (q-chains).
  Enumeration of all q-connected components for
  each q is the essence of Q-analysis (R. Atkin).

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Structure vectors
First structure vector
QQN,QN-1,,Q0
i-th entry is equal to the number of
i-connectivity classes
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Q41, f41
Q33, f33
Q24, f25
Q14, f16
Q01, f06
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Q41, f41, P40
Q33, f33, P30
Q24, f25, P20.2
Q14, f16, P10.33
Q01, f06, P00.83
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Third structure vector
Q expresses the number of q-connectivity
components in a complex, and in general, each of
such components can join together several
complexes
Number of connectivity components per one
simplex of dimension greater or equal to q
The range of Pq is 0,1 increasing fq causes
increase (small) of connectivity degree, when Qe
is fixed.
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Obstruction vector

• QQN-1,QN-1-1,,Q0-1
• Measures structural limitations in simplicial
  connectivity at q. Gives some sense of the gaps
  in the complex.

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Q1
Q2
Q3
Q3
Q0
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Eccentricity
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• Eccentricity

Bottom q the q-level at which the it first
joins another simplex
Top q the q-level at which the simplex first
appears
ecc0 completely integrated (part of another
simplex) ecc1 completely disintegrated
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E
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Vertex significance
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Vertex significance

• ?(vi) - weight of vertex vi equal to the number
  of simplices which are formed by vertex vi.
• Any simplex sq in simplicial complex is formed by
  exactly q1 different vertices, calculate sum
  weight ? (sq ) of its vertices and define vertex
  significance

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Cohesiveness
Quantifies the ability of the simplex to form
q-chains. Simplex first appears at level q ?
dim sq q. At level q1 it first coalesces into
structure meaning that it shares with other
simplices face of dim q1 with mq1 being the
number of such smplices. At level q2 it forms
chain with simplices with faces of dim q1 and/or
q2 and the number of such simplices are mq1 and
mq2 respectively. The process continues to level
q0.
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Cohesiveness
3rd structure vector
Dimension of the simplex
Depends on Dimension of the simplex q Degree of
connectivity at each of q-levels Number of
simplices with which it shares face of dimension
f(mf))
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Clustering coefficient (1)
Characterizes property of the simplex based on
shared faces with neighbors
dimension of shared face
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Clustering coefficient (2)
Cnetworks
q ? 0 fij? 0
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Algebraic model of the complex

• x1,,xn - vertices of K k a field, kR.
• Consider polynomial ring kx1,,xnkx,
• containing all polynomials in x1,,xn (addition
  
• and multiplication of polynomials as ring
• operations).
• Each simplex xi1,,xir of K corresponds to a
• unique (square-free) monomial in kx. Let
• I ? kx be the ideal generated by all
  square-free
• monomials that correspond to non-faces in K,
  i.e.
• collection of vertices that do not represent
• simplices in K.

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Algebraic model of the complex (cont.)

• The Stanley-Reisner ring of K is the quotient
  ring
• RK kx/I
• Algebraic model of K and many combinatorial
  properties of K are contained in RK. Betti
  numbers measure of the relationships among
  monomials (among non-faces of K).

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Algebraic model of the complex (cont.)

• Betti vector
• B b01,b1,,br rmaxN (no. of vertices)
• simplest homological invariants
• The number of disjointed components that make
• up the simplicial complex at each level
  (dimension).
• What kind of information does it convey to .
• .social analyst?

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Algebraic model of the complex (cont.)

• Assume b31 ? somewhere in the complex a 3-dim
  subcomplex is missing (i.e. a 3-dim object
  assembled from simplices of dim at most 3),
  though all of its faces are already there.
• Conclusion the complex is weak in triadic
  relations. Depending on the statistics governing
  the complex, the group may complete by filling
  the whole at some later stage.

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Complex as a topological spaceSingular homology

• Singular homolgy groups of a topological space
  measure the existence of holes of various
  dimensions in the space
• HK H0 ,,Hd dmaxdim K
• Example K is a hollow tetrahedron, with a
  one-cycle graph attached to one if its vertices ?
  HK 1 ,1,1 .

Complex is connected
2-dim hole inside
Measures the attached graph which topologically
looks like a 1-dim sphere
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Random network2000 nodes p0.005
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random network
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random network
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Protein-protein interaction network in yeast S.
cerevisiae (2361 nodes)
g2.3
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Protein-protein interaction network in yeast S.
cerevisiae
g2.3
g2.0
g1.01
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Protein-protein interaction network in yeast S.
cerevisiae
g1.6
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Protein-protein interaction network in yeast S.
cerevisiae
g1.8
g1.7
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US Power grid 4941 nodes
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US Power grid 4941 nodes
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US Power grid 4941 nodes
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Computation geometry collaboration network
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Computation geometry collaboration network
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Conclusion

• Versatile method (combinatorial, topoloical and
  algebraic aspects)
• Distance measures
• Time series of graphs complexes
• Measures for time series of complexes
• Computational methods for high q

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Flatland (1884), Edwin Abbott
The more sides you have, the higher your social
standing. At the bottom are triangular laborers.
At the top are priestly circles
A. Square
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Then, one day, a sphere moves through the planar
world of Flatland. A. Square sees it as a dot
widening into a circle that then shrinks back to
a dot and disappears. The sphere takes A. Square
into the world of 3 dimensions and opens his eyes
to things beyond his imagining.
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Back in Flatland, A. Square tells his vision of
the third dimension. He is ridiculed, ignored,
and finally haled into prison where he writes his
book. Indeed, he was already in trouble during
his visit with the sphere. As the world of 3
dimensions opened up, he wondered if a fourth
dimension might lie beyond the sphere's
comprehension. The sphere scolded him for his
foolish speculation.