Dominating Sets, Multiple Egocentric Networks and Modularity Maximizing Clustering of Social Networks. By M.A. Boudourides & S.T. Lenis

1. Dominating Sets, Multiple Egocentric Networks
and Modularity Maximizing Clustering
of Social Networks
Moses A. Boudourides1 & Sergios T. Lenis2
Department of Mathematics
University of Patras, Greece
1Moses.Boudourides@gmail.com
2sergioslenis@gmail.com
1st European Conference on Social Networks
Barcelona, July 1-4, 2014
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

2. Dominating Sets in a Graph
Definition
Let G = (V , E) be a (simple undirected) graph. A set S ⊆ V of
vertices is called a dominating set (or externally stable) if every
vertex v ∈ V is either an element of S or is adjacent to an element
of S.
Remark
A set S ⊆ V is a dominating set if and only if:
for every v ∈ V S, |N(v) ∩ S| ≥ 1, i.e., V S is enclaveless;
N[S] = V ;
for every v ∈ V 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 closed
neighborhood of vertex v, i.e., N[v] = N(v) ∪ {v}, d(v, x) is the geodesic distance between vertices v and x and
d(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.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

3. Definition
Let S be a set of vertices in graph G.
S is called independent (or internally stable) if no two
vertices in S are adjacent.
If S is a dominating set, then
S is called minimal dominating set if no proper subset
S ⊂ S is a dominating set;
S is called a minimum dominating set if the cardinality of S
is minimum among the cardinalities of any other dominating
set;
The cardinality of a minimum dominating set is called the
domination number of graph G and is denoted by γ(G);
S is called independent dominating set if S is both an
independent and a dominating set.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

4. 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 all
minimal dominating sets. However, only the first is a minimum
dominating set. Apparently γ = 3 for this graph.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

5. Theorem (Ore, 1962)
A dominating set S is a minimal dominating set if and only if, for
each vertex u ∈ S, one of the following two conditions holds:
1 u is such that N(u) ⊆ V S, i.e., u is an isolate of S,
2 there exists a vertex v ∈ V 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.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

6. Structural Equivalence and Domination
Definition
Let S be a set of vertices in a graph G such that |S| ≥ 2. The set
S is called a status (in G) if, for any u, v ∈ S, N(u) ∩ (V S) =
N(v) ∩ (V S), i.e., the set of vertices in V S dominated by u
equals the set of vertices in V S dominated by v.
Definition
Let S be a set of vertices in a graph G. The set S is called a
clique (in G) if the subgraph induced by S (in G) is complete or
an 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].
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

7. Theorem (Kelleher & Cozzens, 1988)
Let S be a minimum dominating set in a connected graph G. If S
is 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 with
n vertices. If S is structurally equivalent (in G), then S is an
independent dominating set such that S = {v1, v2} (i.e.,
γ(G) = 2), where both v1, v2 have degree n − 1.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

8. 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 are
certain efficient algorithms (typically based on linear and
integer programming) for its approximate solution.
Here, we are algorithms that have been already implemented
in Python’s Sage. However, these implementations return only
one of the minimum or independing (minimimum) dominating
sets.
To be able to obtain all possible (minimum and independent)
dominating sets for rather small graphs, we have written a
Python script implementing the brute force algorithm for
dominating sets which is described in the sequel.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

9. 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, where
u1, . . . , uk are all vertices adjacent to v.
In this way, Poly(G) = v∈V N[v] becomes a polynomial in
graph vertices.
Notice that each monomial of Poly(G) can be simplified by
removing numerical coefficients and repeated (more than
once) vertices in it.
Therefore, each simplified monomial of Poly(G) including r
vertices is a minimal dominating set of cardinality r and a
minimum dominating set corresponds to the minimal r.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

10. 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.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

11. Therefore,
{a, b}, {a, e}, {b, d} is the collection of minimum
dominating sets (each one of cardinality 2 equal to the
domination 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.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

12. 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 dominating
egocentric (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|-egocentric
graph corresponding to the dominating set S.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

13. Remark
Given a (simple undirected) graph, typically, there is a
multiplicity of dominating sets for that graph.
Therefore, any egocentric decomposition of a graph depends
on the dominating set that was considered in the generation
of the constituent egocentric subgraphs.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

14. Private and Public Alters
Definition
Let S be a dominating set in graph G = (V , E) (|S| ≥ 2) and let
v ∈ S an ego.
An alter w ∈ N(v) (w /∈ S) is called private alter if
N(w) ⊂ N[v].
An alter w ∈ N(v) (w /∈ S) is called public alter if
N(w) N[v] = ∅.
Remark
Two adjacent egos u, v are not considered alters to each
other! 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 a
single ego and to another alter which is adjacent to a different
ego.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

15. Figure: Five private alters (blue circles) adjacent to an ego (red square).
Figure: Left: one public alter (green circle) shared by two egos (red
squares). Right: two public alters (green circles), each one adjacent to a
different ego (red square).
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

16. 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 is
called 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 called
dominated edge among private alters or bridge of private
alters.
If two public alters w1 ∈ N(v1), w2 ∈ N(v2) are adjacent
(possibly v1 = v2), then edge (w1, w2) ∈ E is called
dominated edge among public alters or bridge of public
alters.
If a private alter w1 ∈ N(v1) and a public alter w2 ∈ N(v2)
are adjacent (possibly v1 = v2), then edge (w1, w2) is called
dominated edge among private–public alters or bridge of
private–to–public alters.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

17. 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).
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

18. Notation
The set of all private alters in G is denoted as:
Vprivate = {w ∈ V S: ∃v ∈ S s.t. w ∈ N(v) S and N(w) ⊂ N[v]}.
The set of all public alters in G is denoted as:
Vpublic = {w ∈ V S : N(w) N[v] = ∅, ∀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}.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

19. Definition
Let G S be the graph remaining from G = (V , E) after removing
a dominating set S (together with all edges which are incident to
egos in S). Sometimes, G S is called alter (sub)graph (or
alter–to–alter (sub)graph) (of G). In other words, G S is the
subgraph induced by V S and the set of edges of G S 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|.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

20. Proposition
Let S be a dominating set of a graph G, A ⊂ Vprivate and
B ⊂ Vpublic. Then
A ∪ B is a status if and only if all alters in A ∪ B are
dominated by the same ego in S;
B is a status if and only if all public alters in B are
co–dominated by all the egos of a subset T ⊂ S.
Moreover, A, B are structurally equivalent (classes) if and only
if they are cliques.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

21. 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.
1
Independent Dominating Set (IDS).
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

22. Example (Florentine Families Network)
|S| = 5 (5 egos, red big circles);
|Vprivate| = 5 (5 private alters, blue circles);
|Vpublic| = 5 (5 public alters, green circles);
|Eprivate| = 0 (no edges among private alters);
|Epublic| = 2 (2 edges among public alters, magenta lines);
|Eprivate − public| = 2 (2 edges among private–to–public alters, brown lines).
|Eego| = 3 (3 among egos, red lines).
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

23. Example (The Voyaging Network among 14 Western Carolines
Islands, 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.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

24. Community Partitions in a Graph
Definition (Newman & Girvan, 2004)
Let G = (V , E) be a graph.
A clustering of vertices of G is a partition of the set of verticies V
into a (finite) family C ⊂ 2V
of subsets of vertices, often called
modules, such that C∈C C = V and C ∩ C = ∅, for each
C, C ∈ C, C = 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=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 the
number of edges among modules C and C . Essentially, modularity compares
the number of edges inside a given module with the expected value for a
randomized graph of the same size and same degree sequence.
A clustering that maximizes modularity is usually called community
partition and the corresponding modules are called communities.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

25. Theorem (Brandes, Delling, Gaertler, G¨orke, Hoefer, Nikoloski &
Wagner, 2008)
Modularity is strongly NP–complete.
Algorithmic Computation of Dominating Sets
Here, we are using the community detection algorithm (through
modularity maximization) of the Louvain method (Blondel,
Guillaume, Lambiotte & Lefebvre, 2008) as implemented in Python
by Thomas Aynaud.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

26. Communities in 1–Egocentric Networks (γ = 1)
Proposition
Let G = (V , E) be 1-egocentric network, i.e., G is a graph in which there
is 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 (Vleaves ∪ {a}).
Then, if C is a (modularity maximizing) community partition of G, one of
the following cases holds:
1 G is a single community and a necessary and sufficient condition for
this is that the alter subgraph Galter = G {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.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

27. Ego–Free Communities
Definition
Let S be a domination set in a graph G and C be a community
partition of G. A community C ∈ C is called ego–free if
C ∩ S = ∅, i.e., if no ego is included among the vertices which
belong to that community.
Corollary
Any community in a 1–egocentric network is either an ego–free
community or a (proper) clique (i.e., a structurally equivalent
class).
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

28. The four types of communities in a 1–egocentric network.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

29. Counting Ego–Free Communities in γ–Egocentric
Erdos–Renyi Networks (for 1 ≤ γ ≤ 10)
1,000 simulations of γ–egocentric Erdos–Renyi networks with 100
vertices and (uniformly) randomly selected link probability.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

30. 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.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

31. 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.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

32. Five Empirical Networks: Minimum Dominating Egocentric Subgraphs and their
Community Partitions
Network |V | |E| |Vprivate| |Vpublic| |Eprivate| |Epublic| |Eprivate − public| |Eego| γ |C| Ego–free
Communities
Zachary’s
Karate Club
34 78 15 15 3 19 17 1 4 4
Les Miserables 77 254 38 29 28 70 40 10 10 6
American
College Football
115 613 0 103 0 487 0 3 12 10
Food–Web 161 592 30 116 0 230 1 13 15 4
C. elegans 297 2148 41 240 28 1471 132 12 16 5
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

33. 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.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

34. 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.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

35. 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.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

36. 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.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

37. 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.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities

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