The document describes an algorithm for drawing large graphs using divisive hierarchical k-means clustering. It first discusses the idea behind the algorithm, which is to arrange the vertices of a graph G into partitions at different levels of hierarchy to minimize the within-cluster sum of squares. It then describes implementing the algorithm using k-means clustering in an iterative fashion, and proves that the value being minimized decreases monotonically at each iteration.
Algorithmic Aspects of Vertex Geo-dominating Sets and Geonumber in GraphsIJERA Editor
In this paper we study about x-geodominating set, geodetic set, geo-set, geo-number of a graph G. We study the
binary operation, link vectors and some required results to develop algorithms. First we design two algorithms
to check whether given set is an x-geodominating set and to find the minimum x-geodominating set of a graph.
Finally we present another two algorithms to check whether a given vertex is geo-vertex or not and to find the
geo-number of a graph.
A Note on Non Split Locating Equitable Dominationijdmtaiir
Let G = (V,E) be a simple, undirected, finite
nontrivial graph. A non empty set DV of vertices in a graph
G is a dominating set if every vertex in V-D is adjacent to
some vertex in D. The domination number (G) of G is the
minimum cardinality of a dominating set. A dominating set D
is called a non split locating equitable dominating set if for
any two vertices u,wV-D, N(u)D N(w)D,
N(u)D=N(w)D and the induced sub graph V-D is
connected.The minimum cardinality of a non split locating
equitable dominating set is called the non split locating
equitable domination number of G and is denoted by nsle(G).
In this paper, bounds for nsle(G) and exact values for some
particular classes of graphs were found.
It is a algorithm used to find a minimum cost spanning tree for connected weighted undirected graph.This algorithm first appeared in Proceedings of the American Mathematical Society in 1956, and was written by Joseph Kruskal.
The degree equitable connected cototal dominating graph 퐷푐 푒 (퐺) of a graph 퐺 = (푉, 퐸) is a graph with 푉 퐷푐 푒 퐺 = 푉(퐺) ∪ 퐹, where F is the set of all minimal degree equitable connected cototal dominating sets of G and with two vertices 푢, 푣 ∈ 푉(퐷푐 푒 (퐺)) are adjacent if 푢 = 푣 푎푛푑 푣 = 퐷푐 푒 is a minimal degree equitable connected dominating set of G containing u. In this paper we introduce this new graph valued function and obtained some results
Algorithmic Aspects of Vertex Geo-dominating Sets and Geonumber in GraphsIJERA Editor
In this paper we study about x-geodominating set, geodetic set, geo-set, geo-number of a graph G. We study the
binary operation, link vectors and some required results to develop algorithms. First we design two algorithms
to check whether given set is an x-geodominating set and to find the minimum x-geodominating set of a graph.
Finally we present another two algorithms to check whether a given vertex is geo-vertex or not and to find the
geo-number of a graph.
A Note on Non Split Locating Equitable Dominationijdmtaiir
Let G = (V,E) be a simple, undirected, finite
nontrivial graph. A non empty set DV of vertices in a graph
G is a dominating set if every vertex in V-D is adjacent to
some vertex in D. The domination number (G) of G is the
minimum cardinality of a dominating set. A dominating set D
is called a non split locating equitable dominating set if for
any two vertices u,wV-D, N(u)D N(w)D,
N(u)D=N(w)D and the induced sub graph V-D is
connected.The minimum cardinality of a non split locating
equitable dominating set is called the non split locating
equitable domination number of G and is denoted by nsle(G).
In this paper, bounds for nsle(G) and exact values for some
particular classes of graphs were found.
It is a algorithm used to find a minimum cost spanning tree for connected weighted undirected graph.This algorithm first appeared in Proceedings of the American Mathematical Society in 1956, and was written by Joseph Kruskal.
The degree equitable connected cototal dominating graph 퐷푐 푒 (퐺) of a graph 퐺 = (푉, 퐸) is a graph with 푉 퐷푐 푒 퐺 = 푉(퐺) ∪ 퐹, where F is the set of all minimal degree equitable connected cototal dominating sets of G and with two vertices 푢, 푣 ∈ 푉(퐷푐 푒 (퐺)) are adjacent if 푢 = 푣 푎푛푑 푣 = 퐷푐 푒 is a minimal degree equitable connected dominating set of G containing u. In this paper we introduce this new graph valued function and obtained some results
LADDER AND SUBDIVISION OF LADDER GRAPHS WITH PENDANT EDGES ARE ODD GRACEFULFransiskeran
The ladder graph plays an important role in many applications as Electronics, Electrical and Wireless
communication areas. The aim of this work is to present a new class of odd graceful labeling for the ladder
graph. In particular, we show that the ladder graph Ln with m-pendant Ln mk1 is odd graceful. We also
show that the subdivision of ladder graph Ln with m-pendant S(Ln) mk1 is odd graceful. Finally, we
prove that all the subdivision of triangular snakes ( k snake ) with pendant edges
1
( ) k S snake mk are odd graceful.
We study communication cost of computing functions when inputs are distributed among k processors, each of which is located at one vertex of a network/graph called a terminal. Every other node of the network also has a processor, with no input. The communication is point-to-point and the cost is the total number of bits exchanged by the protocol, in the worst case, on all edges. Our results show the effect of topology of the network on the total communication cost. We prove tight bounds for simple functions like Element-Distinctness (ED), which depend on the 1-median of the graph. On the other hand, we show that for a large class of natural functions like Set-Disjointness the communication cost is essentially n times the cost of the optimal Steiner tree connecting the terminals. Further, we show for natural composed functions like ED∘XOR and XOR∘ED, the naive protocols suggested by their definition is optimal for general networks. Interestingly, the bounds for these functions depend on more involved topological parameters that are a combination of Steiner tree and 1-median costs. To obtain our results, we use some tools like metric embeddings and linear programming whose use in the context of communication complexity is novel as far as we know. (Based on joint works with Jaikumar Radhakrishnan and Atri Rudra)
In [8] Liang and Bai have shown that the - 4 kC snake graph is an odd harmonious graph for each k ³ 1.
In this paper we generalize this result on cycles by showing that the - n kC snake with string 1,1,…,1 when
n º 0 (mod 4) are odd harmonious graph. Also we show that the - 4 kC snake with m-pendant edges for
each k,m ³ 1 , (for linear case and for general case). Moreover, we show that, all subdivision of 2 k mD -
snake are odd harmonious for each k,m ³ 1 . Finally we present some examples to illustrate the proposed
theories.
In [1] Abdel-Aal has introduced the notions of m-shadow graphs and n-splitting graphs, for all m, n ³ 1.
In this paper, we prove that, the m-shadow graphs for paths and complete bipartite graphs are odd
harmonious graphs for allm³ 1. Also, we prove the n-splitting graphs for paths, stars and symmetric
product between paths and null graphs are odd harmonious graphs for all n³ 1. In addition, we present
some examples to illustrate the proposed theories. Moreover, we show that some families of graphs admit
odd harmonious libeling.
On the Odd Gracefulness of Cyclic Snakes With Pendant EdgesGiselleginaGloria
Graceful and odd gracefulness of a graph are two entirely different concepts. A graph may posses one or both of these or neither. We present four new families of odd graceful graphs. In particular we show an odd graceful labeling of the linear 4 1 kC snake mK − e and therefore we introduce the odd graceful labeling of 4 1 kC snake mK − e ( for the general case ). We prove that the subdivision of linear 3 kC snake − is odd graceful. We also prove that the subdivision of linear 3 kC snake − with m-pendant edges is odd graceful. Finally, we present an odd graceful labeling of the crown graph P mK n 1 e .
LADDER AND SUBDIVISION OF LADDER GRAPHS WITH PENDANT EDGES ARE ODD GRACEFULFransiskeran
The ladder graph plays an important role in many applications as Electronics, Electrical and Wireless
communication areas. The aim of this work is to present a new class of odd graceful labeling for the ladder
graph. In particular, we show that the ladder graph Ln with m-pendant Ln mk1 is odd graceful. We also
show that the subdivision of ladder graph Ln with m-pendant S(Ln) mk1 is odd graceful. Finally, we
prove that all the subdivision of triangular snakes ( k snake ) with pendant edges
1
( ) k S snake mk are odd graceful.
We study communication cost of computing functions when inputs are distributed among k processors, each of which is located at one vertex of a network/graph called a terminal. Every other node of the network also has a processor, with no input. The communication is point-to-point and the cost is the total number of bits exchanged by the protocol, in the worst case, on all edges. Our results show the effect of topology of the network on the total communication cost. We prove tight bounds for simple functions like Element-Distinctness (ED), which depend on the 1-median of the graph. On the other hand, we show that for a large class of natural functions like Set-Disjointness the communication cost is essentially n times the cost of the optimal Steiner tree connecting the terminals. Further, we show for natural composed functions like ED∘XOR and XOR∘ED, the naive protocols suggested by their definition is optimal for general networks. Interestingly, the bounds for these functions depend on more involved topological parameters that are a combination of Steiner tree and 1-median costs. To obtain our results, we use some tools like metric embeddings and linear programming whose use in the context of communication complexity is novel as far as we know. (Based on joint works with Jaikumar Radhakrishnan and Atri Rudra)
In [8] Liang and Bai have shown that the - 4 kC snake graph is an odd harmonious graph for each k ³ 1.
In this paper we generalize this result on cycles by showing that the - n kC snake with string 1,1,…,1 when
n º 0 (mod 4) are odd harmonious graph. Also we show that the - 4 kC snake with m-pendant edges for
each k,m ³ 1 , (for linear case and for general case). Moreover, we show that, all subdivision of 2 k mD -
snake are odd harmonious for each k,m ³ 1 . Finally we present some examples to illustrate the proposed
theories.
In [1] Abdel-Aal has introduced the notions of m-shadow graphs and n-splitting graphs, for all m, n ³ 1.
In this paper, we prove that, the m-shadow graphs for paths and complete bipartite graphs are odd
harmonious graphs for allm³ 1. Also, we prove the n-splitting graphs for paths, stars and symmetric
product between paths and null graphs are odd harmonious graphs for all n³ 1. In addition, we present
some examples to illustrate the proposed theories. Moreover, we show that some families of graphs admit
odd harmonious libeling.
On the Odd Gracefulness of Cyclic Snakes With Pendant EdgesGiselleginaGloria
Graceful and odd gracefulness of a graph are two entirely different concepts. A graph may posses one or both of these or neither. We present four new families of odd graceful graphs. In particular we show an odd graceful labeling of the linear 4 1 kC snake mK − e and therefore we introduce the odd graceful labeling of 4 1 kC snake mK − e ( for the general case ). We prove that the subdivision of linear 3 kC snake − is odd graceful. We also prove that the subdivision of linear 3 kC snake − with m-pendant edges is odd graceful. Finally, we present an odd graceful labeling of the crown graph P mK n 1 e .
A review of one of the most popular methods of clustering, a part of what is know as unsupervised learning, K-Means. Here, we go from the basic heuristic used to solve the NP-Hard problem to an approximation algorithm K-Centers. Additionally, we look at variations coming from the Fuzzy Set ideas. In the future, we will add more about On-Line algorithms in the line of Stochastic Gradient Ideas...
E-Cordial Labeling of Some Mirror GraphsWaqas Tariq
Let G be a bipartite graph with a partite sets V1 and V2 and G\' be the copy of G with corresponding partite sets V1\' and V2\' . The mirror graph M(G) of G is obtained from G and G\' by joining each vertex of V2 to its corresponding vertex in V2\' by an edge. Here we investigate E-cordial labeling of some mirror graphs. We prove that the mirror graphs of even cycle Cn, even path Pn and hypercube Qk are E-cordial graphs.
Introduction:
RNA interference (RNAi) or Post-Transcriptional Gene Silencing (PTGS) is an important biological process for modulating eukaryotic gene expression.
It is highly conserved process of posttranscriptional gene silencing by which double stranded RNA (dsRNA) causes sequence-specific degradation of mRNA sequences.
dsRNA-induced gene silencing (RNAi) is reported in a wide range of eukaryotes ranging from worms, insects, mammals and plants.
This process mediates resistance to both endogenous parasitic and exogenous pathogenic nucleic acids, and regulates the expression of protein-coding genes.
What are small ncRNAs?
micro RNA (miRNA)
short interfering RNA (siRNA)
Properties of small non-coding RNA:
Involved in silencing mRNA transcripts.
Called “small” because they are usually only about 21-24 nucleotides long.
Synthesized by first cutting up longer precursor sequences (like the 61nt one that Lee discovered).
Silence an mRNA by base pairing with some sequence on the mRNA.
Discovery of siRNA?
The first small RNA:
In 1993 Rosalind Lee (Victor Ambros lab) was studying a non- coding gene in C. elegans, lin-4, that was involved in silencing of another gene, lin-14, at the appropriate time in the
development of the worm C. elegans.
Two small transcripts of lin-4 (22nt and 61nt) were found to be complementary to a sequence in the 3' UTR of lin-14.
Because lin-4 encoded no protein, she deduced that it must be these transcripts that are causing the silencing by RNA-RNA interactions.
Types of RNAi ( non coding RNA)
MiRNA
Length (23-25 nt)
Trans acting
Binds with target MRNA in mismatch
Translation inhibition
Si RNA
Length 21 nt.
Cis acting
Bind with target Mrna in perfect complementary sequence
Piwi-RNA
Length ; 25 to 36 nt.
Expressed in Germ Cells
Regulates trnasposomes activity
MECHANISM OF RNAI:
First the double-stranded RNA teams up with a protein complex named Dicer, which cuts the long RNA into short pieces.
Then another protein complex called RISC (RNA-induced silencing complex) discards one of the two RNA strands.
The RISC-docked, single-stranded RNA then pairs with the homologous mRNA and destroys it.
THE RISC COMPLEX:
RISC is large(>500kD) RNA multi- protein Binding complex which triggers MRNA degradation in response to MRNA
Unwinding of double stranded Si RNA by ATP independent Helicase
Active component of RISC is Ago proteins( ENDONUCLEASE) which cleave target MRNA.
DICER: endonuclease (RNase Family III)
Argonaute: Central Component of the RNA-Induced Silencing Complex (RISC)
One strand of the dsRNA produced by Dicer is retained in the RISC complex in association with Argonaute
ARGONAUTE PROTEIN :
1.PAZ(PIWI/Argonaute/ Zwille)- Recognition of target MRNA
2.PIWI (p-element induced wimpy Testis)- breaks Phosphodiester bond of mRNA.)RNAse H activity.
MiRNA:
The Double-stranded RNAs are naturally produced in eukaryotic cells during development, and they have a key role in regulating gene expression .
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
Slide 1: Title Slide
Extrachromosomal Inheritance
Slide 2: Introduction to Extrachromosomal Inheritance
Definition: Extrachromosomal inheritance refers to the transmission of genetic material that is not found within the nucleus.
Key Components: Involves genes located in mitochondria, chloroplasts, and plasmids.
Slide 3: Mitochondrial Inheritance
Mitochondria: Organelles responsible for energy production.
Mitochondrial DNA (mtDNA): Circular DNA molecule found in mitochondria.
Inheritance Pattern: Maternally inherited, meaning it is passed from mothers to all their offspring.
Diseases: Examples include Leber’s hereditary optic neuropathy (LHON) and mitochondrial myopathy.
Slide 4: Chloroplast Inheritance
Chloroplasts: Organelles responsible for photosynthesis in plants.
Chloroplast DNA (cpDNA): Circular DNA molecule found in chloroplasts.
Inheritance Pattern: Often maternally inherited in most plants, but can vary in some species.
Examples: Variegation in plants, where leaf color patterns are determined by chloroplast DNA.
Slide 5: Plasmid Inheritance
Plasmids: Small, circular DNA molecules found in bacteria and some eukaryotes.
Features: Can carry antibiotic resistance genes and can be transferred between cells through processes like conjugation.
Significance: Important in biotechnology for gene cloning and genetic engineering.
Slide 6: Mechanisms of Extrachromosomal Inheritance
Non-Mendelian Patterns: Do not follow Mendel’s laws of inheritance.
Cytoplasmic Segregation: During cell division, organelles like mitochondria and chloroplasts are randomly distributed to daughter cells.
Heteroplasmy: Presence of more than one type of organellar genome within a cell, leading to variation in expression.
Slide 7: Examples of Extrachromosomal Inheritance
Four O’clock Plant (Mirabilis jalapa): Shows variegated leaves due to different cpDNA in leaf cells.
Petite Mutants in Yeast: Result from mutations in mitochondrial DNA affecting respiration.
Slide 8: Importance of Extrachromosomal Inheritance
Evolution: Provides insight into the evolution of eukaryotic cells.
Medicine: Understanding mitochondrial inheritance helps in diagnosing and treating mitochondrial diseases.
Agriculture: Chloroplast inheritance can be used in plant breeding and genetic modification.
Slide 9: Recent Research and Advances
Gene Editing: Techniques like CRISPR-Cas9 are being used to edit mitochondrial and chloroplast DNA.
Therapies: Development of mitochondrial replacement therapy (MRT) for preventing mitochondrial diseases.
Slide 10: Conclusion
Summary: Extrachromosomal inheritance involves the transmission of genetic material outside the nucleus and plays a crucial role in genetics, medicine, and biotechnology.
Future Directions: Continued research and technological advancements hold promise for new treatments and applications.
Slide 11: Questions and Discussion
Invite Audience: Open the floor for any questions or further discussion on the topic.
Cancer cell metabolism: special Reference to Lactate PathwayAADYARAJPANDEY1
Normal Cell Metabolism:
Cellular respiration describes the series of steps that cells use to break down sugar and other chemicals to get the energy we need to function.
Energy is stored in the bonds of glucose and when glucose is broken down, much of that energy is released.
Cell utilize energy in the form of ATP.
The first step of respiration is called glycolysis. In a series of steps, glycolysis breaks glucose into two smaller molecules - a chemical called pyruvate. A small amount of ATP is formed during this process.
Most healthy cells continue the breakdown in a second process, called the Kreb's cycle. The Kreb's cycle allows cells to “burn” the pyruvates made in glycolysis to get more ATP.
The last step in the breakdown of glucose is called oxidative phosphorylation (Ox-Phos).
It takes place in specialized cell structures called mitochondria. This process produces a large amount of ATP. Importantly, cells need oxygen to complete oxidative phosphorylation.
If a cell completes only glycolysis, only 2 molecules of ATP are made per glucose. However, if the cell completes the entire respiration process (glycolysis - Kreb's - oxidative phosphorylation), about 36 molecules of ATP are created, giving it much more energy to use.
IN CANCER CELL:
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
introduction to WARBERG PHENOMENA:
WARBURG EFFECT Usually, cancer cells are highly glycolytic (glucose addiction) and take up more glucose than do normal cells from outside.
Otto Heinrich Warburg (; 8 October 1883 – 1 August 1970) In 1931 was awarded the Nobel Prize in Physiology for his "discovery of the nature and mode of action of the respiratory enzyme.
WARNBURG EFFECT : cancer cells under aerobic (well-oxygenated) conditions to metabolize glucose to lactate (aerobic glycolysis) is known as the Warburg effect. Warburg made the observation that tumor slices consume glucose and secrete lactate at a higher rate than normal tissues.
(May 29th, 2024) Advancements in Intravital Microscopy- Insights for Preclini...Scintica Instrumentation
Intravital microscopy (IVM) is a powerful tool utilized to study cellular behavior over time and space in vivo. Much of our understanding of cell biology has been accomplished using various in vitro and ex vivo methods; however, these studies do not necessarily reflect the natural dynamics of biological processes. Unlike traditional cell culture or fixed tissue imaging, IVM allows for the ultra-fast high-resolution imaging of cellular processes over time and space and were studied in its natural environment. Real-time visualization of biological processes in the context of an intact organism helps maintain physiological relevance and provide insights into the progression of disease, response to treatments or developmental processes.
In this webinar we give an overview of advanced applications of the IVM system in preclinical research. IVIM technology is a provider of all-in-one intravital microscopy systems and solutions optimized for in vivo imaging of live animal models at sub-micron resolution. The system’s unique features and user-friendly software enables researchers to probe fast dynamic biological processes such as immune cell tracking, cell-cell interaction as well as vascularization and tumor metastasis with exceptional detail. This webinar will also give an overview of IVM being utilized in drug development, offering a view into the intricate interaction between drugs/nanoparticles and tissues in vivo and allows for the evaluation of therapeutic intervention in a variety of tissues and organs. This interdisciplinary collaboration continues to drive the advancements of novel therapeutic strategies.
1. Drawing Large Graphs Using Divisive Hierarchical
k-means
Barbara Ikica
22. 11. 2012
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
2. Abstract
Idea behind the algorithm
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
3. Abstract
Idea behind the algorithm
Implementation
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
4. Abstract
Idea behind the algorithm
Implementation
Hierarchical clustering (Divisive Hierarchical k-means)
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
5. Abstract
Idea behind the algorithm
Implementation
Hierarchical clustering (Divisive Hierarchical k-means)
Graph representation
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
6. Abstract
Idea behind the algorithm
Implementation
Hierarchical clustering (Divisive Hierarchical k-means)
Graph representation
Identifying connected components
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
7. Abstract
Idea behind the algorithm
Implementation
Hierarchical clustering (Divisive Hierarchical k-means)
Graph representation
Identifying connected components
Diffusion kernels on graphs
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
8. Abstract
Idea behind the algorithm
Implementation
Hierarchical clustering (Divisive Hierarchical k-means)
Graph representation
Identifying connected components
Diffusion kernels on graphs
Drawing
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
9. Abstract
Idea behind the algorithm
Implementation
Hierarchical clustering (Divisive Hierarchical k-means)
Graph representation
Identifying connected components
Diffusion kernels on graphs
Drawing
Time complexity
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
10. Abstract
Idea behind the algorithm
Implementation
Hierarchical clustering (Divisive Hierarchical k-means)
Graph representation
Identifying connected components
Diffusion kernels on graphs
Drawing
Time complexity
Random projections
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
11. Idea behind the algorithm
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5 6
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Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
12. Idea behind the algorithm
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Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
13. Idea behind the algorithm
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012 3456
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Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
14. Idea behind the algorithm
Objective: Arrangement of the (sets of the) vertices of a graph
G = V (G), E(G) at the level of hierarchy n
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
15. Idea behind the algorithm
Objective: Arrangement of the (sets of the) vertices of a graph
G = V (G), E(G) at the level of hierarchy n
1. Determining the partition of V (G) : Pn = {Mi}m
i=1
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
16. Idea behind the algorithm
Objective: Arrangement of the (sets of the) vertices of a graph
G = V (G), E(G) at the level of hierarchy n
1. Determining the partition of V (G) : Pn = {Mi}m
i=1
2. Determining the drawing area for each Mi ∈ Pn
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
18. Implementation – Hierarchical clustering
k-means clustering
M := {vi}n
i=1, vi ∈ Rd ∀i, 1 ≤ i ≤ n.
We aim to partition the set M into k clusters P = {S1, S2, . . . , Sk}
so as to minimize the within-cluster sum of squares
min
P
k
i=1 vj∈Si
vj − µi
2
,
where µi := 1
|Si|( vj∈Si
vj) is the centroid vector of the cluster Si.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
19. Implementation – Hierarchical clustering
k-means clustering
NP-hard problem
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
20. Implementation – Hierarchical clustering
k-means clustering
NP-hard problem
If k and d are fixed, the problem can be exactly solved in
O(ndk+1 log n) steps.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
21. Implementation – Hierarchical clustering
k-means clustering
Recall that:
We aim to partition the set M into k clusters P = {S1, S2, . . . , Sk} so as to minimize
the within-cluster sum of squares
min
P
k
i=1 vj ∈Si
vj − µi
2
.
Corollary 1
min
z∈Rd
v∈S
v − z 2
=
v∈S
v − µ 2
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
22. Implementation – Hierarchical clustering
k-means clustering
Lemma 1
Choose arbitrary S ⊂ Rd and z ∈ Rd. Then
v∈S
v − z 2
=
v∈S
v − µ 2
+ |S| z − µ 2
,
where µ is the centroid vector of the cluster S, i.e. µ = 1
|S| v∈S
v.
Corollary 1
min
z∈Rd
v∈S
v − z 2
=
v∈S
v − µ 2
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
23. Implementation – Hierarchical clustering
k-means clustering
Lemma 1
Choose arbitrary S ⊂ Rd and z ∈ Rd. Then
v∈S
v − z 2
=
v∈S
v − µ 2
+ |S| z − µ 2
,
where µ is the centroid vector of the cluster S, i.e. µ = 1
|S| v∈S
v.
Lemma 2
Let X denote an arbitrary random variable with values in Rd. For any z ∈ Rd the
following holds:
E( X − z 2
) = E( X − E(X) 2
) + z − E(X) 2
.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
24. Implementation – Hierarchical clustering
k-means clustering – iterative algorithm
Randomly pick k vectors from M = {v1, v2, . . . , vn} as the centroids µ
(0)
i for
each i = 1, 2, . . . , k.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
25. Implementation – Hierarchical clustering
k-means clustering – iterative algorithm
Randomly pick k vectors from M = {v1, v2, . . . , vn} as the centroids µ
(0)
i for
each i = 1, 2, . . . , k.
For t = 0, 1, 2, . . . repeat (for each i = 1, 2, . . . , k)
1. S
(t)
i = vp : vp − µ
(t)
i
2
≤ vp − µ
(t)
j
2
∀ j : 1 ≤ j ≤ k
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
26. Implementation – Hierarchical clustering
k-means clustering – iterative algorithm
Randomly pick k vectors from M = {v1, v2, . . . , vn} as the centroids µ
(0)
i for
each i = 1, 2, . . . , k.
For t = 0, 1, 2, . . . repeat (for each i = 1, 2, . . . , k)
1. S
(t)
i = vp : vp − µ
(t)
i
2
≤ vp − µ
(t)
j
2
∀ j : 1 ≤ j ≤ k and
2. µ
(t+1)
i = 1
S
(t)
i
vj ∈S
(t)
i
vj
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
27. Implementation – Hierarchical clustering
k-means clustering – iterative algorithm
Randomly pick k vectors from M = {v1, v2, . . . , vn} as the centroids µ
(0)
i for
each i = 1, 2, . . . , k.
For t = 0, 1, 2, . . . repeat (for each i = 1, 2, . . . , k)
1. S
(t)
i = vp : vp − µ
(t)
i
2
≤ vp − µ
(t)
j
2
∀ j : 1 ≤ j ≤ k and
2. µ
(t+1)
i = 1
S
(t)
i
vj ∈S
(t)
i
vj
until there is no further change in the assignments of the vectors to the clusters
S
(t)
i (for each i) in the partition V.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
28. Implementation – Hierarchical clustering
k-means clustering – iterative algorithm
Lemma 3
The value of the expression
k
i=1 vj ∈S
(t)
i
vj − µ
(t)
i
2
decreases monotonically
during iteration.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
29. Implementation – Hierarchical clustering
k-means clustering – iterative algorithm
Lemma 3
The value of the expression
k
i=1 vj ∈S
(t)
i
vj − µ
(t)
i
2
decreases monotonically
during iteration.
Proof.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
30. Implementation – Hierarchical clustering
k-means clustering – iterative algorithm
Lemma 3
The value of the expression
k
i=1 vj ∈S
(t)
i
vj − µ
(t)
i
2
decreases monotonically
during iteration.
Proof.
k
i=1 vj ∈S
(t)
i
vj − µ
(t)
i
2
≤
k
i=1 vj ∈S
(t−1)
i
vj − µ
(t)
i
2
(1)
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
31. Implementation – Hierarchical clustering
k-means clustering – iterative algorithm
Lemma 3
The value of the expression
k
i=1 vj ∈S
(t)
i
vj − µ
(t)
i
2
decreases monotonically
during iteration.
Proof.
k
i=1 vj ∈S
(t)
i
vj − µ
(t)
i
2
≤
k
i=1 vj ∈S
(t−1)
i
vj − µ
(t)
i
2
, since (1)
S
(t)
i = vp : vp − µ
(t)
i
2
≤ vp − µ
(t)
j
2
∀ j : 1 ≤ j ≤ k .
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
32. Implementation – Hierarchical clustering
k-means clustering – iterative algorithm
Lemma 3
The value of the expression
k
i=1 vj ∈S
(t)
i
vj − µ
(t)
i
2
decreases monotonically
during iteration.
Proof.
k
i=1 vj ∈S
(t)
i
vj − µ
(t+1)
i
2
≤
k
i=1 vj ∈S
(t)
i
vj − µ
(t)
i
2
(2)
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
33. Implementation – Hierarchical clustering
k-means clustering – iterative algorithm
Lemma 3
The value of the expression
k
i=1 vj ∈S
(t)
i
vj − µ
(t)
i
2
decreases monotonically
during iteration.
Proof.
k
i=1 vj ∈S
(t)
i
vj − µ
(t+1)
i
2
≤
k
i=1 vj ∈S
(t)
i
vj − µ
(t)
i
2
, since (2)
µ
(t+1)
i =
1
S
(t)
i
vj ∈S
(t)
i
vj and min
z∈Rd
v∈S
v − z 2
=
v∈S
v − µ 2
.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
34. Implementation – Hierarchical clustering
k-means clustering – iterative algorithm
Lemma 3
The value of the expression
k
i=1 vj ∈S
(t)
i
vj − µ
(t)
i
2
decreases monotonically
during iteration.
Proof. Thus
k
i=1 vj ∈S
(t)
i
vj − µ
(t)
i
2
≤
k
i=1 vj ∈S
(t−1)
i
vj − µ
(t)
i
2
(1)
k
i=1 vj ∈S
(t)
i
vj − µ
(t+1)
i
2
≤
k
i=1 vj ∈S
(t)
i
vj − µ
(t)
i
2
(2)
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
35. Implementation – Hierarchical clustering
k-means clustering – iterative algorithm
Lemma 3
The value of the expression
k
i=1 vj ∈S
(t)
i
vj − µ
(t)
i
2
decreases monotonically
during iteration.
Proof. Thus
k
i=1 vj ∈S
(t+1)
i
vj − µ
(t+1)
i
2
≤
k
i=1 vj ∈S
(t)
i
vj − µ
(t+1)
i
2
(1)
k
i=1 vj ∈S
(t)
i
vj − µ
(t+1)
i
2
≤
k
i=1 vj ∈S
(t)
i
vj − µ
(t)
i
2
(2)
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
36. Implementation – Hierarchical clustering
k-means clustering – iterative algorithm
Lemma 3
The value of the expression
k
i=1 vj ∈S
(t)
i
vj − µ
(t)
i
2
decreases monotonically
during iteration.
Proof. Thus
k
i=1 vj ∈S
(t+1)
i
vj − µ
(t+1)
i
2
≤
k
i=1 vj ∈S
(t)
i
vj − µ
(t)
i
2
.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
37. Implementation – Hierarchical clustering
Hierarchical clustering
M
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Figure: Schematic representation of the hierarchical partition of the set
M.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
38. Implementation – Hierarchical clustering
Hierarchical clustering
M
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Figure: Schematic representation of the hierarchical partition of the set
M.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
39. Implementation – Hierarchical clustering
Hierarchical clustering
M
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Figure: Schematic representation of the hierarchical partition of the set
M.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
40. Implementation – Hierarchical clustering
Hierarchical clustering
M
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Figure: Schematic representation of the hierarchical partition of the set
M.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
41. Implementation – Hierarchical clustering
Hierarchical clustering
M
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Figure: Schematic representation of the hierarchical partition of the set
M.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
42. Implementation – Hierarchical clustering
Hierarchical clustering
M
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Figure: Schematic representation of the hierarchical partition of the set
M.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
43. Implementation – Hierarchical clustering
Hierarchical clustering
M
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Figure: Schematic representation of the hierarchical partition of the set
M.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
44. Implementation – Hierarchical clustering
Hierarchical clustering
M
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Figure: Schematic representation of the hierarchical partition of the set
M.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
45. Implementation – Hierarchical clustering
Hierarchical clustering
M
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Figure: Schematic representation of the hierarchical partition of the set
M.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
46. Implementation – Hierarchical clustering
Hierarchical clustering
M
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Figure: Schematic representation of the hierarchical partition of the set
M.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
47. Implementation – Hierarchical clustering
Hierarchical clustering
M
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Figure: Schematic representation of the hierarchical partition of the set
M.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
48. Implementation – Hierarchical clustering
Hierarchical clustering
M
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Figure: Schematic representation of the hierarchical partition of the set
M.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
49. Implementation – Hierarchical clustering
Hierarchical clustering
M
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Figure: Schematic representation of the hierarchical partition of the set
M.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
50. Implementation – Hierarchical clustering
Hierarchical clustering
M
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Figure: Schematic representation of the hierarchical partition of the set
M.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
78. Implementation – Diffusion kernels on graphs
0 1
2 3
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
79. Implementation – Diffusion kernels on graphs
0 1
2 3
0
3
1
2 Hierarchical clustering on the row
vectors of the adjacency matrix
MG:
S1 = {1, 2}
S2 = {0, 3}
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
80. Implementation – Diffusion kernels on graphs
0 1
2 3
0
3
1
2 Hierarchical clustering on the row
vectors of ?:
S1 = {0, 1, 2}
S2 = {3}
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
81. Implementation – Diffusion kernels on graphs
The adjacency matrix MG has the following property:
[(MG
)k
]ij = # of all paths of length up to k between vertices i and j
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
82. Implementation – Diffusion kernels on graphs
The adjacency matrix MG has the following property:
[(MG
)k
]ij = # of all paths of length up to k between vertices i and j
We modify the algorithm by replacing MG with the kernel matrix
KG:
KG
:=
∞
k=0
αk(MG)k
k!
= exp(αMG
) .
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
83. Implementation – Diffusion kernels on graphs
k : Ω × Ω → R is a similarity measure if:
k(x, y) characterizes the similarities of x, y ∈ Ω.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
84. Implementation – Diffusion kernels on graphs
k : Ω × Ω → R is a similarity measure if:
k(x, y) characterizes the similarities of x, y ∈ Ω.
k : Ω × Ω → R is a kernel if:
1. k(x, y) = k(y, x), ∀x, y ∈ Ω,
2. k is positive semidefinite:
the kernel matrix K ∈ Rn×n, [K]ij = k(xi, xj), is positive
semidefinite for all x1, x2, . . . , xn ∈ Ω.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
85. Implementation – Diffusion kernels on graphs
k : Ω × Ω → R is a similarity measure if:
k(x, y) characterizes the similarities of x, y ∈ Ω.
k : Ω × Ω → R is a kernel if:
1. k(x, y) = k(y, x), ∀x, y ∈ Ω,
2. k is positive semidefinite:
the kernel matrix K ∈ Rn×n, [K]ij = k(xi, xj), is positive
semidefinite for all x1, x2, . . . , xn ∈ Ω.
Given a kernel k, there exist a Hilbert space Hk and a map φ : Ω → Hk such
that
φ(x), φ(y) Hk
= k(x, y) for all x, y ∈ Ω.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
86. Implementation – Diffusion kernels on graphs
KG
:=
∞
k=0
αk(MG)k
k!
= exp(αMG
) indeed is a kernel matrix,
since
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
87. Implementation – Diffusion kernels on graphs
KG
:=
∞
k=0
αk(MG)k
k!
= exp(αMG
) indeed is a kernel matrix,
since
KG
= exp(αMG
)
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
88. Implementation – Diffusion kernels on graphs
KG
:=
∞
k=0
αk(MG)k
k!
= exp(αMG
) indeed is a kernel matrix,
since
KG
= exp(αMG
) = exp(αUDUT
)
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
89. Implementation – Diffusion kernels on graphs
KG
:=
∞
k=0
αk(MG)k
k!
= exp(αMG
) indeed is a kernel matrix,
since
KG
= exp(αMG
) = exp(αUDUT
) = αU exp(D)UT
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
90. Implementation – Diffusion kernels on graphs
KG
:=
∞
k=0
αk(MG)k
k!
= exp(αMG
) indeed is a kernel matrix,
since
KG
= exp(αMG
) = exp(αUDUT
) = αU exp(D)UT
=⇒ KG is a symmetric positive semidefinite matrix
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
91. Implementation – Hierarchical clustering on KG
Example
"1#1"
"2#11"
"1#2"
"2#211"
"3#11"
"2#12"
"3#2"
"3#12"
"2#221"
"2#212"
"2#2221"
"2#2222"
0
2
4
6 7
1 3
5
8
9
10
11
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
92. Implementation – Hierarchical clustering on KG
Example
"1#1"
"2#11"
"1#2"
"2#211"
"3#11"
"2#12"
"3#2"
"3#12"
"2#221"
"2#212"
"2#2221"
"2#2222"
0
2
4
6 7
1 3
5
8
9
10
11
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
93. Implementation – Hierarchical clustering on KG
Example
"1#1"
"2#11"
"1#2"
"2#211"
"3#11"
"2#12"
"3#2"
"3#12"
"2#221"
"2#212"
"2#2221"
"2#2222"
0
2
4
6 7
1
5
3
8
9
10
11
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
94. Implementation – Hierarchical clustering on KG
Example
"1#1"
"2#11"
"1#2"
"2#211"
"3#11"
"2#12"
"3#2"
"3#12"
"2#221"
"2#212"
"2#2221"
"2#2222"
0
2
4
6 7
1
5
8
10
11
3
9
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
95. Implementation – Hierarchical clustering on KG
Example
"1#1"
"2#11"
"1#2"
"2#211"
"3#11"
"2#12"
"3#2"
"3#12"
"2#221"
"2#212"
"2#2221"
"2#2222"
0
2
4
6 7
1
5
8
10
11
3
9
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
96. Implementation – Drawing
A111A112
A12
A211 A212
A221 A222
Figure: Determining the
drawing area for the sets
in the partition P0.
V (G)
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
97. Implementation – Drawing
A111A112
A12
A211 A212
A221 A222
Figure: Determining the
drawing area for the sets
in the partition P1.
V (G)
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
98. Implementation – Drawing
A111A112
A12
A211 A212
A221 A222
Figure: Determining the
drawing area for the sets
in the partition P1.
V (G)
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
99. Implementation – Drawing
A111A112
A12
A211 A212
A221 A222
Figure: Determining the
drawing area for the sets
in the partition P1.
V (G)
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
100. Implementation – Drawing
A111A112
A12
A211 A212
A221 A222
Figure: Determining the
drawing area for the sets
in the partition P2.
V (G)
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
101. Implementation – Drawing
A111A112
A12
A211 A212
A221 A222
Figure: Determining the
drawing area for the sets
in the partition P2.
V (G)
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
102. Implementation – Drawing
A111A112
A12
A211 A212
A221 A222
Figure: Determining the
drawing area for the sets
in the partition P3.
V (G)
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
103. Implementation – Drawing
A111A112
A12
A211 A212
A221 A222
Figure: Determining the
drawing area for the sets
in the partition P3.
V (G)
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
104. Implementation – Drawing
A111A112
A12
A211 A212
A221 A222
Figure: Determining the
drawing area for the sets
in the partition P3.
V (G)
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
105. Implementation – Drawing
A111A112
A12
A211 A212
A221 A222
Figure: Determining the
drawing area for the sets
in the partition P3.
V (G)
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
106. Implementation – Drawing
A111A112
A12
A211 A212
A221 A222
Figure: Determining the
drawing area for the sets
in the partition P3.
V (G)
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
107. Implementation – Drawing
A111A112
A12
A211 A212
A221 A222
Figure: Determining the
drawing area for the sets
in the partition P3.
V (G)
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
108. Implementation – Drawing
A111A112
A12
A211 A212
A221 A222
Figure: Determining the
drawing area for the sets
in the partition P3.
V (G)
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
109. Implementation – Drawing
A111A112
A12
A211 A212
A221 A222
Figure: Determining the
drawing area for the sets
in the partition P3.
V (G)
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
110. Implementation – Drawing
A111A112
A12
A211 A212
A221 A222
Figure: Determining the
drawing area for the sets
in the partition P3.
V (G)
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
111. Implementation – Drawing
A111A112
A12
A211 A212
A221 A222
Figure: Determining the
drawing area for the sets
in the partition P3.
V (G)
A1
A11
A111 A112
A12
A12
A2
A21
A211 A212
A22
A221 A222
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
112. Implementation – Time complexity
Efficient computation of KG
= exp(αMG
)
Computation of KG si needed in the algorithm:
to determine the centroids µj,
to minimize vi − µj
2
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
113. Implementation – Time complexity
Efficient computation of KG
= exp(αMG
)
Computation of KG si needed in the algorithm:
to determine the centroids µj,
to minimize vi − µj
2:
vi − µj
2
= vi − µj, vi − µj = vi, vi − 2 vi, µj + µj, µj .
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
114. Implementation – Time complexity
Efficient computation of KG
= exp(αMG
)
Computation of KG si needed in the algorithm:
to determine the centroids µj,
to minimize vi − µj
2:
vi − µj
2
= vi − µj, vi − µj = vi, vi − 2 vi, µj + µj, µj .
The question is thus:
How to efficiently multiply the matrix KG with an arbitrary vector and how to
efficiently compute the inner products vi, vi = vi
2?
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
115. Implementation – Time complexity
Multiplying the matrix KG
with an arbitrary vector v
KG
v = Iv +
α
1
MG
v +
α2
2!
(MG
)2
v +
α3
3!
(MG
)3
v + . . .
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
116. Implementation – Time complexity
Multiplying the matrix KG
with an arbitrary vector v
KG
v = Iv +
α
1
MG
v +
α2
2!
(MG
)2
v +
α3
3!
(MG
)3
v + . . . =
= v +
α
1
(MG
v) +
α
2
MG α
1
(MG
v) +
α
3
MG α
2
MG α
1
(MG
v) + . . .
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
117. Implementation – Time complexity
Multiplying the matrix KG
with an arbitrary vector v
KG
v = Iv +
α
1
MG
v +
α2
2!
(MG
)2
v +
α3
3!
(MG
)3
v + . . . =
= v +
α
1
(MG
v) +
α
2
MG α
1
(MG
v) +
α
3
MG α
2
MG α
1
(MG
v) + . . .
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
118. Implementation – Time complexity
Multiplying the matrix KG
with an arbitrary vector v
KG
v = Iv +
α
1
MG
v +
α2
2!
(MG
)2
v +
α3
3!
(MG
)3
v + . . . =
= v +
α
1
(MG
v) +
α
2
MG α
1
(MG
v) +
α
3
MG α
2
MG α
1
(MG
v) + . . .
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
119. Implementation – Time complexity
Multiplying the matrix KG
with an arbitrary vector v
KG
v = Iv +
α
1
MG
v +
α2
2!
(MG
)2
v +
α3
3!
(MG
)3
v + . . . =
= v +
α
1
(MG
v) +
α
2
MG α
1
(MG
v) +
α
3
MG α
2
MG α
1
(MG
v) + . . .
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
120. Implementation – Time complexity
Multiplying the matrix KG
with an arbitrary vector v
KG
v = Iv +
α
1
MG
v +
α2
2!
(MG
)2
v +
α3
3!
(MG
)3
v + . . . =
= v +
α
1
(MG
v) +
α
2
MG α
1
(MG
v) +
α
3
MG α
2
MG α
1
(MG
v) + . . .
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
121. Implementation – Time complexity
Multiplying the matrix KG
with an arbitrary vector v
KG
v = Iv +
α
1
MG
v +
α2
2!
(MG
)2
v +
α3
3!
(MG
)3
v + . . . =
= v +
α
1
(MG
v) +
α
2
MG α
1
(MG
v) +
α
3
MG α
2
MG α
1
(MG
v) + . . .
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
122. Implementation – Random projections
Random projections
Corollary of the Johnson–Lindenstrauss lemma:
Theorem 1
Let P be an arbitrary set of n points in Rd
, represented as an n × d matrix A ∈ Rn×d
. Given ε > 0, β > 0 let
k0 =
4+2β
ε2/2−ε3/3
log n. For integer k ≥ k0, let R ∈ Rd×k
be a random matrix with elements rij , where rij
are independent random variables from either one of the following two probability distributions:
rij :
1 −1
1
2
1
2
, rij :
1 0 −1
1
6
2
3
1
6
.
Let E = 1√
k
AR and let f : Rd
→ Rk
map the i-th row of A to the i-th row of E.
Then
(1 − ε) u − v
2
≤ f (u) − f (v)
2
≤ (1 + ε) u − v
2
holds for all u, v ∈ P with probability at least 1 − n−β
.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means
123. Implementation – Random projections
Random projections
Corollary of the Johnson–Lindenstrauss lemma:
Theorem 1
Let P be an arbitrary set of n points in Rd
, represented as an n × d matrix A ∈ Rn×d
. Given ε > 0, β > 0 let
k0 =
4+2β
ε2/2−ε3/3
log n. For integer k ≥ k0, let R ∈ Rd×k
be a random matrix with elements rij , where rij
are independent random variables from either one of the following two probability distributions:
rij :
1 −1
1
2
1
2
, rij :
1 0 −1
1
6
2
3
1
6
.
Let E =
1
√
k
AR and let f : Rd
→ Rk
map the i-th row of A to the i-th row of E.
Then
(1 − ε) u − v
2
≤ f (u) − f (v)
2
≤ (1 + ε) u − v
2
holds for all u, v ∈ P with probability at least 1 − n−β
.
Barbara Ikica
Drawing Large Graphs Using Divisive Hierarchical k-means