Modeling and learning network structure and dynamics
Nino Antulov-Fantulin
Laboratory for Information Systems,
Division of Electronics,
Rud̄er Bošković Institute,
Zagreb, Croatia
[email protected]
ABSTRACT
In this review paper, we give a brief introduction to the
complex network theory, where we explain the basic mathematics of networks, measures, metrics and the topology
properties of real networks. After a brief introduction, we
give an overview of the state of the art algorithms for modeling the network structure. Modeling the network structure
is divided into three main categories: modeling the global
network structure, local network structure and the network
structure at the level of groups or communities. In the final
part of this paper, we describe processes that take place on
the network structure like random failures, attacks, cascading failures, random walks, disease spreading and diffusion
process.
Keywords
complex networks, machine learning
1.
FUNDAMENTALS OF NETWORK THEORY
A network is a set of nodes interconnected with a set of
links. Network, nodes and links are also called graph, vertices and edges, respectively, in the graph theory literature
[1],[2]. In this paper we will mostly use the terms: network,
vertices and edges. We will also study only simple networks
- a network that contains no multiedge nor a self-edge.
Mathematics of networks
An adjacency matrix is one way of representing a network
structure. The adjacency matrix A of a simple network is the
matrix which contains non-zero elements Aij if there exists
an edge between vertices i and j. The non-zero element Aij
for an unweighted network is 1 and for a weighted network
is arbitrary number called a weight. The degree of a vertex
in undirected network is the number of edges connected to
it. In a directed networks edges have property of direction,
therefore an adjacency matrix contains the non-zero element
.
Aij only if there exists an edge from j to i. Note that the
adjacency matrix of directed graph is asymmetric in general
and symmetric for the undirected networks. In the directed
networks we define the in-degree and the out-degree as the
number of ingoing and outgoing edges, respectively.
It is sometimes more convenient to convert a directed network to an undirected network. One simple approach is to
make all the edges a symmetric ones. Second approach is
to construct cocitation network C. The cocitation of two
vertices i and j is the number of other vertices that both
have outgoing edges to i and j. The adjacency matrix of
the cocitation network C = AAT is calculated from the adjacency matrix of the directed network and putting all the
diagonal elements to zero.
A tree is a connected, undirected network without cycles.
A directed network without any cycles is called an acyclic
directed network. A citation network is an example of an
acyclic directed network. For every acyclic directed network
there exists labeling of the vertices such that the adjacency
matrix is strictly upper triangular. The eigenvalues of the
adjacency matrix are zeros if and only if the network is an
acyclic directed network.
In a bipartite network (e.g. movie recommendation network) there exist two types of vertices (users - type1 and
movies - type2 ) and edges which connect only vertices of different types. Structure of bipartite network is represented
with a rectangular incidence matrix. Incidence matrix B has
non-zero elements Bij if a vertex of type1 is connected to a
vertex of type2 . Adjacency matrices P1 and P2 of one-mode
projections of the bipartite network to a type1 or a type2 network are calculated from the incidence matrix P1 = BB T ,
P2 = B T B and setting all the diagonal elements to zero.
Another very important matrix that is used to represent a
network structure is the Laplacian matrix L of the network.
The Laplacian matrix is calculated as L = D −A, where A is
the adjacency matrix and D is the diagonal matrix with the
degrees ki of nodes on the diagonal. The Laplacian matrix is
a symmetric matrix, therefore it’s eigenvalues are real. Furthermore, the Laplacian matrix is a positive semi-definite
matrix because it can be decomposed as L = B T B, where
B is the edge incidence matrix. Note that the Laplacian
matrix is a singular matrix because the Laplacian matrix
always has at least one zero eigenvalue with the corresponding eigenvector 1 .
X
m
Lim × 1 =
X
m
(Dim − Aim ) × 1 =
X
m
(δim km − Aim ) × 1
= km −
X
Aim = km − km = 0
m
A network with n components has n zero eigenvalues and n
different corresponding eigenvectors ei . The eigenvector ei
of the i-th component contains ones on all the places k where
node vk is in the i-th component and zeros elsewhere. The
number of zero eigenvalues is equal to the number of components in the network. Therefore, if the network contains only
one component, the second eigenvalue (the algebraic connectivity) is positive. The Perron-Frobenius theorem tell us
that each real non-negative matrix (adjacency matrix) has
unique largest eigenvalue whose corresponding eigenvector
contains non-negative values.
Measures and metrics
Although, a degree centrality is the most intuitive measure, it does not capture most influential vertices in the
network. The eigenvector centrality [4] is based on a simple concept, a vertex is more important if it has more important neighbours. If we denote xi as the importance of
vertex vi , then the eigenvector centrality
can be calculated
P
with the following expression: xi = k Aij xk . This expression assumes we already know importances of the neighbours
and so on recursively. We can calculate importances iteratively like this x(t) = At x(0) by setting a vector x(0) to
some arbitrary
P value. The initial vector can be expressed
as: x(0) = j aj vj , the linear combination of the eigenvectors of the adjacency matrix. Finally, we can calculate the
eigenvector centrality for all the vertices in the network by
using only leading eigenvector v1 and eigenvalue λ1 of the
adjacency matrix A.
x(t) = At
X
k
ak vk =
X
ak λtk vk = λt1
k
X
k
ak
λtk
vk
λt1
In the limit of the time, we express the eigenvector centralities with only one leading eigenvector (Ax = λ1 x), which
elements are all non-negative. Nevertheless, the eigenvector
centrality fails as a importance measure on directed networks, so we introduce the PageRank centrality.
The PageRank [5] centrality measure of the particular vertex is proportional to the neighbours PageRank centrality
divided by their out-degree. This can be written in matrix
terms like this:
x = αAD−1 x + β1,
where A is the adjacency matrix, D the diagonal matrix
with the elements Dii = max(kiout , 1) down the diagonal.
Note that the PageRank centrality on undirected networks
reduces to the vertex degree.
Kleinberg [6] introduced two different types of the centrality importance for the vertices in directed networks. Each
vertex in the network can have a hub centrality (contain
information about the best authorities) and an authority
centrality (contain useful information). The authority centrality xi of the vertex vi in the network is proportional to
the sum of the hub centralities yj of the vertices that have
out-going edge to the i-th vertex vi :
X
Aij yj .
xi = α
j
The hub centrality yi of a vertex vi in the network is proportional to the sum of the authority centralities xj of the
vertices that have in-going edge from the i-th vertex vi :
X
yi = β
Aji xj .
j
In matrix terms, we can write x = αAy and y = βAT x.
From there we can write AAT x = (αβ)−1 x and AT Ay =
(αβ)−1 y. Therefore, we conclude that the hub and the authority centrality are the leading eigenvectors from matrices
AAT and AT A from the same eigenvalue (αβ)−1 , respectively. This procedure for computing hubs and authorities
centrality is used in the HITS algorithm.
A path in the network is defined as an arbitrary sequence
of vertices. Number of paths, between the vertices i and j,
with the given length k, can be computed from the adjacency
matrix: Akij . The number of cycles of length k in the netP
k
work can be computed as a sum over all vertices:
m Amm ,
k
which is equivalent to the trace of the matrix A , which is
equivalent to the sum of eigenvalues of the matrix Ak . A
geodesic path is a shortest path between two vertices. Let
us denote dij as the length of the geodesic path from the vertex i to the vertex j. The closeness centrality Ci of vertex
vi is the harmonic mean between the distances of geodesic
paths from the vertex vi to all others.
1 X 1
Ci =
.
n−1
dij
j(6=i)
We denote σst as the number of the geodesic paths between
pairs of vertices vs and vt and the σst (vi ) as the number
of the geodesic paths σst which pass through the vertex vi .
Then the betweenness centrality is defined as:
X σst (vi )
.
C(vi ) =
σst
st
The degree distribution P (k) defines the probability of
choosing a vertex with the degree k by uniform sampling
from the set of all vertices. The n-th moment of P (k) is
calculated as:
X n
k P (k).
hkn i =
k
We can also define the average degree of the nearest neighbours of the nodes with the degree k [7] as:
X ′
k P (k′ |k).
knn (k) =
k′
In uncorrelated networks, knn (k) is independent of k. Correlated networks are called assortative if knn (k) is an increasing function of k. If knn (k) is an decreasing function of k
then the network is disassortative [8]. The local clustering
coefficient Ci [9] is defined as the ratio of the edges ei between first neighbours of vi and all possible edges between
them.
2ei
Ci =
ki (ki − 1)
Topology of real networks
Most of real networks in information, social and biological systems are characterized with the similar topological
properties: small average path length, high clustering coefficients, fat tailed scale-free degree distributions, degree
correlations and presence of communities.
Table 1: Topology characteristics of real networks:
size-N, average degree-hki, average path length-L,
average clustering coefficient-C, exponent of powerlaw distribution-γ, correlation-ν
Network
AS2001 [13]
WWW [14]
Protein [15]
Metabolic [16]
Math1999 [17]
Actors [9]
N
11,174
2 × 108
2,115
778
57,516
225,226
hki
4.19
7.5
6.8
3.2
5.0
61
L
3.62
16
2.12
7.4
8.46
3.65
C
0.24
0.11
0.07
0.7
0.15
0.79
γ
2.38
2.1/2.7
2.4
2.2/2.1
2.47
2.3
ν
<0
<0
<0
>0
>0
If the average shortest path length in the network depends
logarithmically on the network size, the network is considered to have the small-world property. Most real networks
have power law degree distribution P (k) = Ak−γ , where γ is
in the range 2<γ<3. The networks with the power law distribution are called scale-free networks (have the same functional form at different scales) [10],[11]. Finite-size networks
exhibit cut-offs in the fat-tailed degree distributions [12].
Distributions whose tail are not exponentially bounded are
called the fat-tailed or the heavy-tailed distributions (does
not have finite second moment).
2.
MODELING NETWORK STRUCTURE
Modeling global network structure
Random graphs were first studied by Erdös and Rényi in
1959. Their first model generated Erdös and Rényi random
graphs [19] with N vertices and K edges from entire statistical ensemble of all possible realizations. Later, another
model for the ER random graphs was presented, which generates a random graph of N vertices where the probability of
an edge occurrence is p. The graphs with k edges will appear
in the ensemble with the probability pk (1 − p)N(N−1)/2−k
[19],[18]. The structural properties of the ER random graphs
exhibit a phase transition at the critical probability pc = N1 .
When p<pc the graph almost surely has no components of
size grater than O(ln(N )). Above that critical probability
the random graph has a component of O(N ). The degree
distribution follows the binomial distribution:
!
N −1 k
P (K = k) =
p (1 − p)N−1−k .
k
For a large value of N and fixed hki the degree distribution
can be represented with the Poisson distribution:
P (k) = e−k
hki
.
k!
Although the ER random graphs are well mathematically
explained they do not reproduce the topological properties
of real networks.
The configuration model [20] allows to sample random
graphs from the ensemble with the arbitrary degree distribution P (k) and N vertices. A generalized random graph
is constructed by assigning a ki half-edges to the vertex vi
and wiring half-edges with uniform probability.
The Watts and Strogatz model generates small-world networks with a high clustering coefficient [9]. The model start
from the N vertex ring where each vertex is connected to
its 2m nearest neighbours. Then, the process of rewiring
start, each edge is rewired to the randomly chosen vertex
with the probability p. The regular lattice occurs when p
equals to zero and a random graph occurs when p equals to
one. For intermediate values of p small-world networks with
a high clustering coefficient occurs. The small rewiring procedure has a huge nonlinear effect on decreasing an average
shortest path L and linear effect on decreasing a clustering
coefficient.
The Barabási-Albert model (BA) is the model of evolving a scale-free network which uses a preferential attachment
[11] property. Starting from the m0 isolated vertices, at each
time step new vertices with m edges are added to the network (m < m0 ). The new vertex will create an edge to the
existing node vi with the probability proportional to its degree ki . The BA model produces the power-law distribution
P (k) ∼ k−3 in the limit of the time. The average distance
increases logarithmically with the size of the network. The
clustering coefficient vanishes with the system size slower
than in ER random graphs, but still different from smallworld models where C is a constant. Various authors have
proposed modifications and generalizations of the standard
BA model in order to became more realistic.
Although, various network models have been constructed,
they fail to reproduce several properties like: the scree plot,
the network values, the densification law or the shrinking diameter property. The eigenvalues versus their corresponding
rank of the adjacency matrix are represented by the scree
plot, this plot also obeys a power-law [21]. The densification
power law tell us that the relation between the number of
edges over time E(t) and the number of vertices over time
V (t) in the evolving network is: E(t) = V (t)a (the densification exponent a is greater than 1) [22]. The effective
diameter of the network tends to shrink in an evolving network [22]. A Kronecker graph K1k is defined by a k recursive
kronecker product of an initiator graph K1 .
K1k = Kk = K1 ⊕ K1 ⊕ ... ⊕ K1 = Kk−1 ⊕ K1 .
|
{z
}
k
The Kronecker graphs have a multinomial distribution for in
and out degrees, eigenvalues, components of leading eigenvector and follow the densification law [23]. For some choice
of the initiator K1 , the multinomial distribution behaves like
a power-law distribution. Stochastic Kronecker Graphs have
also been introduced [23], where values of Kk are probabilities of edges. The Kronecker graphs can model the real
networks by tuning the parameters in the initiator matrix
K1 . For the given graph G with the N1k vertices and the initiator matrix K1 (N1 × N1 ) one can generate the Kronecker
graph Kk with the N1k vertices and calculate the likelihood
between the graph G and the Kronecker graph Kk like this:
Y
Y
(1 − Kk (u, v)).
Kk (u, v)
P (G|K1 ) =
(u,v)∈G
(u,v)∈G
/
Calculating the likelihood by this approach has two problems. The first problem is matching the corresponding vertices between the adjacency matrix of G and the adjacency
matrix of the Kronecker graph Kk (factorial problem). The
second problem is a complexity of calculating the likelihood
when the vertices has been matched O(N 2 ). By using a
Markov Chain Monte Carlo method (the Metropolis sampling algorithm) for a vertex matching and the Taylor approximation of likelihood, calculations of the likelihood can
be made in linear time O(E). But, we want to find the
initiator matrix K1 such that has the maximum likelihood
P (G|K1 ). For simplicity we will denote the initiator matrix
K1 as Θ and one possible matching of vertices as σ. The
log-likelihood can then be written as:
X
P (G|Θ, σ)P (σ, Θ).
l(Θ) = logP (G|θ) = log
σ
To maximize the likelihood P (G|Θ) the gradient method can
be employed:
ˆ = Θ̂t + λ
Θt+1
∂l(Θ)
.
∂Θ
where the gradient is:
X ∂logP (G|σ, Θ)
∂l(Θ)
=
P (σ|G, Θ).
∂Θ
∂Θ
σ
is summing over all permutaNote, that the gradient ∂l(Θ)
∂Θ
tions σ. But, this can be calculated more efficiently in O(E)
by employing the Metropolis sampling from P (σ|G, Θ). The
Kronecker graphs have a static and a temporal properties of
real networks. Furthermore, fitting the parameters of the
initiator matrix is very fast even for large networks.
Modeling local network structure
Given a snapshot of the network at the time t1 , we want to
infer new interactions among the vertices of existing network
at some future time t2 . By some score function f (u, v) we
map a score to the particular edge (u, v) in the network and
propose a ranked list of all the missing edges by this score
function f (u, v) in decreasing order [26]. Various functions
for an edge confidence has been proposed. Let us denote
the set of neighbours of the vertex x by Γ(x). The common
neighbours function [27] returns the number of elements in
the intersection of the set:
\
f (u, v) = Γ(u) Γ(v).
The Adamic and Adar [28] method calculates the score as:
X
1
f (u, v) =
.
log|Γ(z)|
T
z∈Γ(u)
Γ(v)
This measure weights rarer features more heavily by using
log function. The preferential attachment [29] coefficient
calculates the score as:
f (u, v) = |Γ(u)||Γ(v)|.
The Jacard’s coefficient [30] calculates the score as:
T
|Γ(u) Γ(v)|
S
.
f (u, v) =
|Γ(u) Γ(v)|
The Katz measure [31] is a weighted sum over all the possible
paths between the vertices (u, v):
X l
β |pathsl (u, v)|.
f (u, v) =
l
In matrix terms the Katz measure between all pairs of the
vertices is: (I − βA)−1 − I, where A is the adjacency matrix
of the network. The commute time Cu,v is sum of expected
number of steps required that the random walker start at u
and reaches v and comes back.
A Low-rank matrix approximation with the singular value
decomposition Ak = Uk Sk VkT of an adjacency matrix can
also be used as one approach for an edge prediction. Many
of this measures have outperformed the random predictor,
just by using the topology properties [26].
Hierarchical Random Graphs (HRG) [24] are the general
method for inferring a hierarchical network structure. The
hierarchical structure is represented by a tree or a dendogram in which the lowest common ancestor represents the
probability pr of the edge between pair of vertices in the network. The number of leaves in the dendogram is equal to the
number of vertices in the network. We are interested in fitting the hierarchical model (D, {pr }) with the real network
G. This is accomplished by using a maximum likelihood
method with the Monte Carlo sampling algorithm on the
space off all possible dendograms (D, {pr }). The likelihood
between real network G and the HRG is:
Y E
L(D, {pr }) =
pr r (1 − pr )Lr Rr −Er ,
r∈D
where Er is the number of edges in G whose vertices have
r as the lowest common ancestor, Lr and Rr represent the
number of leafs in the left and the right subtrees at the
lowest common ancestor r. By this method we can sample dendograms proportional to their likelihood to generate
real network. The final result of this method is an ensemble
of dendograms which are merged to the consensus dendogram. The hierarchical structure can be used for prediction
of missing edges in near future. We just output the ranked
list of the edges that are missing in the original network G
according to the corresponding edge probabilities.
Network structures at the level of groups
A community or a cluster or a subgroup is a subgraph whose
vertices are connected more cohesive or dense than the outside vertices. Different definitions of communities (clique,
n-clique, k-plex, etc.) are possible. A clique is a subgraph
where all vertices are connected with each other. A n-clique
is a subgraph where all pairs of vertices have a geodesic distance less or equal to n. A k-plex is a maximal subgraph
with m nodes where each vertex has m − k neighbours in
the subgraph.
The Kernighan-Lin algorithm [32] is a heuristic algorithm
used for the graph bisection problem (division of vertices
into two cohesive groups). This algorithm starts with an
arbitrary division into two groups and searches over all pairs
of vertices whose interchange would minimize the cut size.
Spectral graph partitioning uses the second eigenvector v2
(Fiedler eigenvector) associated to the second lowest eigenvalue λ2 . The positive components in Fiedler’s eigenvector
represent vertices in the first subgraph, while other components represent vertices in the second subgraph [33],[34].
A hierarchical clustering is used when the number of clusters is not known in advance. The aim is to divide vertices
into clusters, such that vertices within the cluster are more
closely related. This agglomerative hierarchical clustering
[35] starts by assigning each vertex its own cluster and iteratively merges the closest (similar) pairs of clusters into a
single cluster. The hierarchical random graph model [24] is
also one example of this technique.
The algorithm by Girvan and Newman [36] for commu-
nity detection is based on iterative pruning the edges with
the highest betweenness, until the network breaks into components. Other very important community detection algorithms can be found in the review paper by Fortunato [37].
3.
PROCESSES ON NETWORKS
We will now concentrate on the processes that take place
on the complex network structures like errors, attacks, cascading failures or spreading processes (diseases, opinions,
rumors, etc.).
A robustness is the ability of the network to overcome
the absence of certain network elements (vertices or edges).
Absence of certain network elements can changes the network structure profoundly, which can change behaviour of
the process dramatically. The robustness will be observed
with respect to the random failures or errors and the intentional attacks on vertices and edges.
A static robustness or a tolerance or a resilience is the
ability of the network to maintain its connectivity when the
random failure or the intentional attack happens to the fraction of its vertices or edges. The analytical results can be
obtained by using the percolation theory [38] (site or bond).
Site percolation assumes that the vertices are erased from
the network with the probability f , while bond percolation
assumes that edges are erased with the probability f from
the network. It is interesting to observe the size of the giant component and the average shortest path length with
respect to the probability f . The giant component persists
for high fraction of random failures of nodes in the scale-free
networks [39]. But, the giant component fragments after
only small fraction of the intentional attacks on nodes [39].
Random graphs display a threshold for intentional attacks
which is in general higher than in scale-free networks.
A dynamic robustness refers to the case of modelling dynamics of a flow of the particular quantities over the network structure when the random failure or the attack happens. Here we will mention only the problem of avalanches
of node breaking and modelling the cascading failures. Simple model for modelling cascading failures was introduced by
Motter and Lai [40]. This model assumes that the quantity
is transmitted along the shortest paths between all pairs of
vertices. Each vertex vi in the network posses finite capacity Ci which determines the maximum load that can hold.
The load for the particular vertex at the time t is equal to
its betweenness bi (t). The vertex vi can operate only if the
load is less or equal to its capacity: bi (t) ≤ Ci , otherwise
the vertex vi breaks and is removed from the network. A removal of the certain vertex lead to the redistribution of flows
(shortest paths) to his neighbours and may even trigger a
cascade of overload failures in the network. One example of
this scenario happened on the 10th August 1996 when the
initial failure of 1300 MW electrical line in southern Oregon
triggered cascades and the blackout in 11 US states [41].
One possible strategy of defence [42] is based on the selective removal of vertices after failure or attack happens in
order to reduce the size of the cascade failures. The similar strategy can be used in modelling the congestion in the
communication networks like the Internet where the traffic
is rerouted to bypass the broken routers, which may lead to
an avalanche of overloads on other routers.
The simple diffusion process on the network can be analytically explained [3]. Let us denote the amount of imaginary
substance at the vertex to be ψi , and consider a diffusion
process where the substance moves from the vertices with
high density to the vertices with low density proportional to
the diffusion constant C. Mathematically, we can express
the rate at which the ψi changes as:
X
dψi
Aij (ψj − ψi ).
=C
dt
j
By simple transformations, in matrix terms we can express
the rate for every vertex dψ
to be:
dt
dψ
dψ
= C(A − D)ψ =⇒
+ CLψ = 0,
dt
dt
where A is the adjacency matrix of the network, the D is
the diagonal matrix with the degrees down the diagonal and
L is the Laplacian matrix. The solution to this diffusion
equation can be written
P as linear combination of the eigenvectors ei : ψ(t) =
i ai (t)ei . Substituting this into the
diffusion equation we get:
X dai (t)
+ Cλi ai )ei .
(
dt
i
By taking the dot product with the eigenvector ej (eigenvectors of an symmetric matrix are orthogonal) we get:
dai (t)
+ Cλi ai = 0 =⇒ ai (t) = ai (0)e−Cλi t ,
dt
where ai (0) represents the initial condition of substance for
vertex vi .
A random walks [43] also represent one type of dynamic
processes on networks (discrete time Markov chains). The
random walk on the network starts at an arbitrary vertex,
vk . At each discrete time step n, random walker resides
at the vertex vm and randomly chooses an adjacent vertex
vl from a uniform distribution of adjacent vertices. Let us
denote the pi (n) probability that the random walker will be
at the vertex vi at discrete time n. This probability can be
expressed as:
X Aij
pj (n − 1),
pi (n) =
kj
j
where kj denotes the degree of a vertex vj . In matrix form
we can write p(n) = A×D−1 p(n−1) = T p(n−1), where D
is the diagonal matrix with the degrees of vertices ki down
the diagonal and T is a transition matrix. Stationary distributions p(∞) or just p can be expressed as:
p = A × D−1 p,
(I − A × D−1 )p = (D − A) × D−1 = LD−1 p = 0,
where matrix L is the Laplacian matrix of the network and
D−1 p is the eigenvector with the corresponding zero eigenvalue. Thus, we know that D−1 p = c1 for every
P connected
graph. And finally pi = cki , where c−1 =
j kj . Therefore we conclude that the stationary probability of a random
walker being at a vertex vi is proportional to its degree ki [3].
Memory biased random walks can be used in the domain of
the recommender systems for generation of the clickstreams
[44].
The study of epidemic spreading on complex networks
[45],[46] has very important applications in modelling the
disease spreading in human society or in modelling the computer virus propagation in computer networks. Small world
network property and scale-free network property have great
impact on epidemic spreading outcome. Scale-free complex
networks exhibit no epidemic threshold below which the infection cannot produce epidemic outbreak (endemic state)
in SIS model [47]. Different mathematical models have been
used to study epidemic spreading. Under the assumption of
homogeneous mixing among people inside different compartments (Kermack-McKendrick model) differential equations
can be applied to understand epidemic dynamics [48]. Contact network epidemiology applies bond percolation on random graphs (not on arbitrary structure) to model epidemic
spreading on heterogeneous population (epidemic dynamics
is neglected) [49] [50] [51].
Novel algorithm for epidemic simulation spreading with
SIR model on networks called the FastSIR algorithm [53]
was introduced. It reduces the average case running time
by constant factor by using probability distributions [52] of
the number of infected nodes:
! k
!
n X k
(1 − p)n−k+l
.
P (Xn = k) = q
(−1)l
k l=0 l
1 − (1 − q) (1 − p)n−k+l
Recently, the phase diagrams of epidemic spreading with
the SIR model on complex networks were introduced as a
useful tool for epidemic spreading analysis [52]. They allowed the definition of epidemic centrality [54], which is the
weighted average of the epidemic impact over all possible
disease spreading regimes.
4.
ACKNOWLEDGMENTS
Many thanks to my mentors: Tom Šmuc and Mile Šikić
and to: Alen Lančić, Hrvoje Štefančić and Vinko Zlatić for
collaboration in the research of the complex networks.
5.
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