dominating sets, multiple egocentric networks and modularity maximizing clustering of social...

Dominating Sets, Multiple Egocentric Networksand Modularity Maximizing Clustering

of Social Networks

Moses A. Boudourides1 & Sergios T. Lenis2

Department of Mathematics

University of Patras, Greece

[email protected]@gmail.com

1st European Conference on Social NetworksBarcelona, July 1-4, 2014

M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

mailto:[email protected]

mailto:[email protected]

Dominating Sets in a Graph

Definition

Let G = (V ,E ) be a (simple undirected) graph. A set S ⊆ V ofvertices is called a dominating set (or externally stable) if everyvertex v ∈ V is either an element of S or is adjacent to an elementof S .

RemarkA set S ⊆ V is a dominating set if and only if:

for every v ∈ V r S , |N(v) ∩ S | ≥ 1, i.e., V r S is enclaveless;

N[S] = V ;

for every v ∈ V r S , d(v , S) ≤ 1.Above N(v) is the open neighborhood of vertex v , i.e., N(v) = {w ∈ V : (u,w) ∈ E}, N[v ] is the closedneighborhood of vertex v , i.e., N[v ] = N(v) ∪ {v}, d(v, x) is the geodesic distance between vertices v and x andd(v, S) = min{d(v, s) : s ∈ S} is the distance between vertex v and the set of vertices S .

Standard Reference

Haynes, Teresa W., Hedetniemi, Stephen T., & Slater, Peter J.(1998). Fundamentals of Domination in Graphs. New York:Marcel Dekker.


Definition

Let S be a set of vertices in graph G .

S is called independent (or internally stable) if no twovertices in S are adjacent.

If S is a dominating set, then

S is called minimal dominating set if no proper subsetS ′ ⊂ S is a dominating set;S is called a minimum dominating set if the cardinality of Sis minimum among the cardinalities of any other dominatingset;The cardinality of a minimum dominating set is called thedomination number of graph G and is denoted by γ(G );S is called independent dominating set if S is both anindependent and a dominating set.


Example

1

8

2

3

7

4

5 6 1

8

2

3

7

4

5 6

1

8

2

3

7

4

5 6 1

8

2

3

7

4

5 6

The sets {1, 3, 5}, {3, 6, 7, 8} and {2, 4, 6, 7, 8} (red circles) are allminimal dominating sets. However, only the first is a minimumdominating set. Apparently γ = 3 for this graph.


Theorem (Ore, 1962)

A dominating set S is a minimal dominating set if and only if, foreach vertex u ∈ S , one of the following two conditions holds:

1 u is such that N(u) ⊆ V r S , i.e., u is an isolate of S ,

2 there exists a vertex v ∈ V r S such that N(v) ∩ S = {u},i.e., v is a private neighbor of u.

Theorem (Ore, 1962)

Every connected graph G of order n ≥ 2 has a dominating set S .


Structural Equivalence and Domination

Definition

Let S be a set of vertices in a graph G such that |S | ≥ 2. The setS is called a status (in G ) if, for any u, v ∈ S , N(u) ∩ (V r S) =N(v) ∩ (V r S), i.e., the set of vertices in V r S dominated by uequals the set of vertices in V r S dominated by v .

Definition

Let S be a set of vertices in a graph G . The set S is called aclique (in G ) if the subgraph induced by S (in G ) is complete oran independent set.

Definition

Two vertices u, v in a graph G are called structurally equivalent(in G ) if N(u) = N(v) or N[u] = N[v ].


Theorem (Kelleher & Cozzens, 1988)

Let S be a minimum dominating set in a connected graph G . If Sis a status (in G ), then S is an independent dominating set and|S | = 2, i.e., γ(G ) = 2.

Corollary (Kelleher & Cozzens, 1988)

Let S be a minimum dominating set in a connected graph G withn vertices. If S is structurally equivalent (in G ), then S is anindependent dominating set such that S = {v1, v2} (i.e.,γ(G ) = 2), where both v1, v2 have degree n − 1.


Complexity of the Dominating Set Problem

Theorem (Johnson, 1974)

The dominating set problem is NP–complete.

Algorithmic Computation of Dominating Sets

As the dominating set problem is NP–complete, there arecertain efficient algorithms (typically based on linear andinteger programming) for its approximate solution.

Here, we are algorithms that have been already implementedin Python’s Sage. However, these implementations return onlyone of the minimum or independing (minimimum) dominatingsets.

To be able to obtain all possible (minimum and independent)dominating sets for rather small graphs, we have written aPython script implementing the brute force algorithm fordominating sets which is described in the sequel.


An Algebraic Computation of All Minimal Dominating Sets

A Brute Force Heuristics (Berge)

Given two elements x and y of a set, let

“x + y” denote logical summation of x and y (i.e., “x or y”)and “x · y” denote their logical multiplication (i.e., “x and y”).

Thus, for each vertex v , N[v ] = v + u1 + . . .+ uk , whereu1, . . . , uk are all vertices adjacent to v .

In this way, Poly(G ) =∏

v∈V N[v ] becomes a polynomial ingraph vertices.

Notice that each monomial of Poly(G ) can be simplified byremoving numerical coefficients and repeated (more thanonce) vertices in it.

Therefore, each simplified monomial of Poly(G ) including rvertices is a minimal dominating set of cardinality r and aminimum dominating set corresponds to the minimal r .


Example

a b

c

d e

As N[a] = a + b + c + d ,N[b] = a + b + c + e,N[c] = a + b + c ,N[d ] =a + d ,N[e] = b + e, we find:

Poly(G ) = N[a]N[b]N[c]N[d ]N[e] =

= (a + b + c + d)(a + b + c + e)(a + b + c)(a + d)(b + e) =

= (a + b + c)3(a + d)(b + e) +

+(a + b + c)2(d + e)(a + d)(b + e) =

= ab + ae + bd +

+abc + abd + abe + ace + ade + bcd + bde + cde +

+abcd + abde + abce + acde + bcde +

+abcde.


Therefore,{{a, b}, {a, e}, {b, d}

}is the collection of minimum

dominating sets (each one of cardinality 2 equal to thedomination number of this graph),{{a, b, c}, {a, b, d}, {a, b, e}, {a, c, e}, {a, d , e}, {b, c , d},

{b, d , e}, {c , d , e}}

is the collection of minimal dominating

sets of cardinality 3,{{a, b, c , d}, {a, b, d , e}, {a, b, c , e}, {a, c, d , e}, {a, d , e},

{b, c, d , e}}

is the collection of minimal dominating sets of

cardinality 4 and

{a, b, c , d , e} is the minimal dominating set of cardinality 5.


Egocentric Subgraphs Induced by a Dominating Set in a Graph

Definition

Let G = (V ,E ) be a (simple undirected) graph.

Let U ⊂ V be a set of vertices in a graph G and let u ∈ U.

The subgraph induced by U (in G ) is called an egocentric(sub)graph (in G ) if

U = N[u],

i.e., if all vertices of U are dominated by u;vertex u is the ego of the egocentric (sub)graph N[u];vertices w ∈ N(u) are called alters of ego u.

Let S ⊂ V be a dominating set in G and let v ∈ S .

The subgraph induced by N[v ] (in G ) is called a dominatingegocentric (sub)graph;vertex v is the dominating ego;vertices w ∈ N(v) are the dominated alters by v .Graph G is called multiple egocentric or |S |-egocentricgraph corresponding to the dominating set S .


Remark

Given a (simple undirected) graph, typically, there is amultiplicity of dominating sets for that graph.

Therefore, any egocentric decomposition of a graph dependson the dominating set that was considered in the generationof the constituent egocentric subgraphs.


Private and Public Alters

Definition

Let S be a dominating set in graph G = (V ,E ) (|S | ≥ 2) and letv ∈ S an ego.

An alter w ∈ N(v) (w /∈ S) is called private alter ifN(w) ⊂ N[v ].

An alter w ∈ N(v) (w /∈ S) is called public alter ifN(w) r N[v ] 6= ∅.

Remark

Two adjacent egos u, v are not considered alters to eachother! Nevertheless, they define an ego–to–ego edge (bridge).

A private alter is always adjacent to a single ego.

A public alter is adjacent either to at least two egos or to asingle ego and to another alter which is adjacent to a differentego.


Figure: Five private alters (blue circles) adjacent to an ego (red square).

Figure: Left: one public alter (green circle) shared by two egos (redsquares). Right: two public alters (green circles), each one adjacent to adifferent ego (red square).


Dominating and Dominated Bridges

Definition

Let S be a dominating set in a graph G = (V ,E ) (|S | ≥ 2).

If two egos v1, v2 ∈ S are adjacent, then edge (v1, v2) ∈ E iscalled dominating edge or bridge of egos.

If two private alters w1 ∈ N(v1),w2 ∈ N(v2) are adjacent(possibly v1 = v2), then edge (w1,w2) ∈ E is calleddominated edge among private alters or bridge of privatealters.

If two public alters w1 ∈ N(v1),w2 ∈ N(v2) are adjacent(possibly v1 = v2), then edge (w1,w2) ∈ E is calleddominated edge among public alters or bridge of publicalters.

If a private alter w1 ∈ N(v1) and a public alter w2 ∈ N(v2)are adjacent (possibly v1 = v2), then edge (w1,w2) is calleddominated edge among private–public alters or bridge ofprivate–to–public alters.


Figure: One edge (bridge) among private alters (yellow line), two edges(bridges) among public alters (magenta lines) and one edge (bridge)among private–to–public alters (brown lines).


Notation

The set of all private alters in G is denoted as:

Vprivate = {w ∈ V r S: ∃v ∈ S s.t. w ∈ N(v) r S and N(w) ⊂ N[v ]}.

The set of all public alters in G is denoted as:

Vpublic = {w ∈ V r S : N(w) r N[v ] 6= ∅, ∀v ∈ S}.

The set of all dominating bridges (among egos) in G is denoted as:

Eego = {(v1, v2) ∈ E: v1, v2 ∈ S}.

The set of all dominated bridges among private alters in G is denoted as:

Eprivate = {(v1, v2) ∈ E: v1, v2 ∈ Vprivate}.

The set of all dominated bridges among public alters in G is denoted as:

Epublic = {(v1, v2) ∈ E: v1, v2 ∈ Vpublic}.

The set of all dominated bridges among private–public alters in G is denoted as:

Eprivate− public = {(v1, v2) ∈ E: v1 ∈ Vprivate, v2 ∈ Vpublic}.


Definition

Let G r S be the graph remaining from G = (V ,E ) after removinga dominating set S (together with all edges which are incident toegos in S). Sometimes, G r S is called alter (sub)graph (oralter–to–alter (sub)graph) (of G ). In other words, G r S is thesubgraph induced by V rS and the set of edges of G rS is the set

Eprivate ∪ Epublic ∪ Eprivate− public.

Proposition

If S is a dominating set of a graph G = (V ,E ), then

|S |+ |Vprivate|+ |Vpublic| = |V |,

∑v∈S

degree(v) + |Eprivate|+ |Epublic|+ |Eprivate− public| − |Eego| = |E |.


Proposition

Let S be a dominating set of a graph G , A ⊂ Vprivate andB ⊂ Vpublic. Then

A ∪ B is a status if and only if all alters in A ∪ B aredominated by the same ego in S ;

B is a status if and only if all public alters in B areco–dominated by all the egos of a subset T ⊂ S .

Moreover, A,B are structurally equivalent (classes) if and onlyif they are cliques.


The Network of the 15 Florentine Families: All Dominating Egocentric

Subgraphs

Minimum Dominating Set (S) IDS1 |Vprivate| |Vpublic|∑

v∈S degree(v) |Eprivate| |Epublic| |Eprivate− public| |Eego|{A,C ,K , J,M} 3 7 14 0 5 2 1

{C ,E ,F , I , J} 2 8 15 0 5 0 0

{C ,F , I , J,M} 1 9 12 0 8 0 0

{C , I ,K , J,M} 2 8 13 0 8 0 1

{C ,D,F , I ,M} 2 8 14 0 7 0 1

{C ,E ,D,F , L} 3 7 17 0 5 0 2

{A,C ,E ,K , J} 4 6 17 0 2 2 1

{A,C ,F , J,M} 2 8 13 0 5 2 0

{C ,E , I ,K , J} 3 7 16 0 5 0 1

{A,C ,D,F ,M} 3 7 15 0 4 2 1

{A,C ,E ,D,K} 5 5 19 0 2 2 3

{A,C ,D,K ,M} 4 6 16 0 4 2 2

{C ,E ,K , J, L} 3 7 16 0 5 0 1

{C ,E ,F , J, L} 2 8 15 0 5 0 0

{C ,E ,D,F , I} 3 7 17 0 5 0 2

{A,C ,E ,D,F} 4 6 18 0 2 2 2

{C ,D, I ,K ,M} 3 7 15 0 7 0 2

{C ,E ,D, I ,K} 4 6 18 0 5 0 3

{A,C ,E ,F , J} 3 7 16 0 2 2 0

{C ,E ,D,K , L} 4 6 18 0 5 0 3

The 15 Florentine Families: A = Strozzi, B = Tornabuoni, C = Medici, D = Albizzi, E = Guadagni, F = Pazzi,G = Acciaiuoli, H = Bischeri, I = Peruzzi, J = Ginori, K = Salviati, L = Castellani, M = Lamberteschi, N =Ridolfi, O = Barbadori.

1Independent Dominating Set (IDS).


Example (The Voyaging Network among 14 Western CarolinesIslands, Hage & Harary, 1991)

|S| = 3 (3 egos, red big circles);|Vprivate| = 5 (5 private alters, blue circles);|Vpublic| = 6 (6 public alters, green circles);Eprivate = 3 (3 edges among private alters, blue lines);Epublic = 6 (6 edges among public alters, magenta lines);Eprivate− public = 4 (4 edges among private–to–public alters, brown lines).(Three communities enclosed in dotted rectangles.)

Minimum Dominating Set (S) IDS |Vprivate| |Vpublic|∑

v∈S degree(v) |Eprivate| |Epublic| |Eprivate− public| |Eego|{G , I ,M} 5 6 11 3 6 4 0

{C , I ,M} 5 6 11 3 6 4 0

The Dominant Western Carolines Islands: C = Puluwat, G = Pulap, I = Fais, M = Elato.


Community Partitions in a GraphDefinition (Newman & Girvan, 2004)

Let G = (V ,E ) be a graph.A clustering of vertices of G is a partition of the set of verticies Vinto a (finite) family C ⊂ 2V of subsets of vertices, often calledmodules, such that

⋃C∈C C = V and C ∩ C ′ = ∅, for each

C ,C ′ ∈ C,C 6= C ′.

A clustering C may be assessed by a quality function Q = Q(C),called modularity, which is defined as:

Q =∑C∈C

[|E (C )||E |

−(

2|E (C )|+∑

C ′∈C,C 6=C ′ |E (C ,C ′)|2|E |

)2]

=∑C∈C

[|E (C )||E |

−(∑

v∈C degree(v)

2|E |

)2],

where E(C) is the number of edges inside module C and E(C ,C ′) is thenumber of edges among modules C and C ′. Essentially, modularity comparesthe number of edges inside a given module with the expected value for arandomized graph of the same size and same degree sequence.

A clustering that maximizes modularity is usually called communitypartition and the corresponding modules are called communities.


Theorem (Brandes, Delling, Gaertler, Gorke, Hoefer, Nikoloski &Wagner, 2008)

Modularity is strongly NP–complete.

Algorithmic Computation of Dominating Sets

Here, we are using the community detection algorithm (throughmodularity maximization) of the Louvain method (Blondel,Guillaume, Lambiotte & Lefebvre, 2008) as implemented in Pythonby Thomas Aynaud.


Communities in 1–Egocentric Networks (γ = 1)

Proposition

Let G = (V ,E ) be 1-egocentric network, i.e., G is a graph in which thereis a single ego a and all other vertices are alters of a. In other wordes,S = {a} is a (minimum) dominating set or γ = 1. We denote by

Vleaves, the set of private alters having degree 1 (leaves) and, by

Gjoint alter, the subgraph induced by Vjoint alter = V r (Vleaves ∪ {a}).

Then, if C is a (modularity maximizing) community partition of G , one ofthe following cases holds:

1 G is a single community and a necessary and sufficient condition forthis is that the alter subgraph Galter = G r {a} is a proper clique;

2 {a} ∪ Vleaves is a community in C;

3 there is a collection Hjoint alter of connected components of Gjoint alter

such that {a} ∪ Vleaves ∪Hjoint alter is a community in C;

4 there is a subset Ujoint alter ⊂ Vjoint alter such that{a} ∪ Vjoint leaves ∪ Ujoint alter is a community in C.


Ego–Free Communities

Definition

Let S be a domination set in a graph G and C be a communitypartition of G . A community C ∈ C is called ego–free ifC ∩ S = ∅, i.e., if no ego is included among the vertices whichbelong to that community.

Corollary

Any community in a 1–egocentric network is either an ego–freecommunity or a (proper) clique (i.e., a structurally equivalentclass).


The four types of communities in a 1–egocentric network.


Counting Ego–Free Communities in γ–EgocentricErdos–Renyi Networks (for 1 ≤ γ ≤ 10)

1,000 simulations of γ–egocentric Erdos–Renyi networks with 100vertices and (uniformly) randomly selected link probability.


The Florentine Families Network: All Dominating Egocentric Subgraphs and

their Community Partitions

Minimum Dominating Set (S) γ |C| Community Partition (C)

{A,C ,K , J,M} 5 4 {A(−),C (−), J + M,K (−)}{C ,E ,F , I , J} 5 4 {I ,C (−),E (−) + J,F}{C ,F , I , J,M} 5 4 {I ,C (−), J + M,F}{C , I ,K , J,M} 5 4 {I ,C (−), J + M,K (−)}{C ,D,F , I ,M} 5 4 {I ,C (−),D(−) + M,F}{C ,E ,D,F , L} 5 4 {L(−),C (−),E (−) + D(−),F}{A,C ,E ,K , J} 5 4 {A(−),C (−),E (−) + J,K (−)}{A,C ,F , J,M} 5 4 {A(−),C (−), J + M,F}{C ,E , I ,K , J} 5 4 {I ,C (−),E (−) + J,K (−)}{A,C ,D,F ,M} 5 4 {A(−),C (−),D(−) + M,F}{A,C ,E ,D,K} 5 4 {A(−),C (−),E (−) + D(−),K (−)}{A,C ,D,K ,M} 5 4 {A(−),C (−),D(−) + M,K (−)}{C ,E ,K , J, L} 5 4 {L(−),C (−),E (−) + J,K (−)}{C ,E ,F , J, L} 5 4 {L(−),C (−),E (−) + J,F}{C ,E ,D,F , I} 5 4 {I ,C (−),E (−) + D(−),F}{A,C ,E ,D,F} 5 4 {A(−),C (−),E (−) + D(−),F}{C ,D, I ,K ,M} 5 4 {I ,C (−),D(−) + M,K (−)}{C ,E ,D, I ,K} 5 4 {I ,C (−),E (−) + D(−),K (−)}{A,C ,E ,F , J} 5 4 {A(−),C (−),E (−) + J,F}{C ,E ,D,K , L} 5 4 {L(−),C (−),E (−) + D(−),K (−)}

The 15 Florentine Families: A = Strozzi, B = Tornabuoni, C = Medici, D = Albizzi, E = Guadagni, F = Pazzi,G = Acciaiuoli, H = Bischeri, I = Peruzzi, J = Ginori, K = Salviati, L = Castellani, M = Lamberteschi, N =Ridolfi, O = Barbadori.


The Network of Florentine Families: Egos in Communities

Five egos distributed inside four communities.Egos are colored red, private alters blue and public alters green.Dotted rectangles embrace vertices lying inside the same communities.


Zachary’s Karate Club Network: Egos in Communities

Four egos distributed inside four communities.Egos are colored red, private alters blue and public alters green.Dotted rectangles embrace vertices lying inside the same communities.


Network of Les Miserables: Egos in Communities

Ten egos distributed inside six communities with one of them being ego–free.Egos are colored red, private alters blue and public alters green.Dotted rectangles embrace vertices lying inside the same communities.


American College Football Network: Egos in Communities

Twelve egos distributed inside ten communities.Egos are colored red, private alters blue and public alters green.Dotted rectangles embrace vertices lying inside the same communities.


Food–Web Network: Egos in Communities

Fifteen egos distributed inside four communities.Egos are colored red, private alters blue and public alters green.Dotted rectangles embrace vertices lying inside the same communities.


C. elegans Network: Egos in Communities

Sixteen egos distributed inside five communities.Egos are colored red, private alters blue and public alters green.Dotted rectangles embrace vertices lying inside the same communities.


Barabasi–Albert Network (m = 4): Minimum Dominating Egocentric Subgraphs

and their Community Partitions

Network |V | |E | |Vprivate| |Vpublic| |Eprivate| |Epublic| |Eprivate− public| |Eego| γ |C| Ego–freeCommunities

Simulation 1 100 384 1 82 0 203 3 14 17 6

Simulation 2 100 384 0 86 0 214 0 12 14 7

Simulation 3 100 384 0 83 0 214 0 14 17 6

Simulation 4 100 384 1 85 0 210 3 13 14 7

Simulation 5 100 384 0 84 0 208 0 19 16 7

Simulation 6 100 384 1 83 0 211 3 9 16 7

Simulation 7 100 384 0 85 0 216 0 15 15 7

Simulation 8 100 384 1 79 0 200 3 12 20 6

Simulation 9 100 384 0 85 0 216 0 14 15 7

Simulation 10 100 384 1 81 0 208 3 12 18 8

Simulation 11 100 384 1 81 0 200 4 18 18 7

Simulation 12 100 384 0 86 0 220 0 13 14 7

Simulation 13 100 384 0 83 0 218 0 15 17 7

Simulation 14 100 384 0 85 0 215 0 17 15 6

Simulation 15 100 384 1 81 0 204 3 12 18 9

Simulation 16 100 384 4 82 0 202 13 10 14 6

Simulation 17 100 384 0 85 0 215 0 12 15 6

Simulation 18 100 384 0 82 0 212 0 14 18 7

Simulation 19 100 384 0 83 0 206 0 12 17 8

Simulation 20 100 384 0 86 0 233 0 13 14 7

Simulation 21 100 384 0 85 0 215 0 17 15 7

Simulation 22 100 384 1 85 0 205 3 15 14 8


dominating sets, multiple egocentric networks and modularity maximizing clustering of social...

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dominating set s

set s v of vertices

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vertex v v s

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minimum dominating set

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cardinality of s