Techniques for Analyzing Random Graph Dynamics and Their Applications

Graph theory is birth in 1736 with the publication of the work of the Swiss mathematician Leonhard Euler on the problem of finding a round trip path that would cross all the seven bridges of the city of Konigsberg exactly once (Euler, 1736). Since then, this theory has known many important developments and has answered to a lot of practical issues. Today, the graph theory is considered as an essential component of discrete mathematics. It aims at analyzing the structure induced by interactions between a set of elements and to study the resulting fundamental properties. Graph theory occurs as a fundamental and theoretical framework for analyzing a wide range of the so-called real-world networks in biology, computer sciences, multi-agent systems, chemistry, physics, economy, knowledge management, and sociology. In many works, graph models are employed as constructive descriptions to represent and understand the behavior of different complexe systems (Molloy and Reed, 1998; Mieghem et al., 2000; Newman, 2003; Kawahigashi et al. 2005; Jurdak, 2007). In such models, the graph vertices stand for the components (nodes) of the network that encode information about the values of the state variables of the dynamical system and the edges represent the mutual relationships between the correspondent endnodes. In practice, random graph theory has become increasingly important for modeling networks whose behaviors exhibit nondeterministic looks. In recent years, many significant results have used random graph models to explain, replicate and simulate the behavior of dynamic real-world networks (Hekmat and Van Mieghem, 2003; Kawahigashi et al., 2005; Durrett, 2006; Onat et al., 2008; Hewer et al., 2009; Hamlili, 2010; Trullols et al., 2010).


Introduction
Graph theory is birth in 1736 with the publication of the work of the Swiss mathematician Leonhard Euler on the problem of finding a round trip path that would cross all the seven bridges of the city of Königsberg exactly once (Euler, 1736). Since then, this theory has known many important developments and has answered to a lot of practical issues. Today, the graph theory is considered as an essential component of discrete mathematics. It aims at analyzing the structure induced by interactions between a set of elements and to study the resulting fundamental properties. Graph theory occurs as a fundamental and theoretical framework for analyzing a wide range of the so-called real-world networks in biology, computer sciences, multi-agent systems, chemistry, physics, economy, knowledge management, and sociology. In many works, graph models are employed as constructive descriptions to represent and understand the behavior of different complexe systems (Molloy and Reed, 1998;Mieghem et al., 2000;Newman, 2003;Kawahigashi et al. 2005;Jurdak, 2007). In such models, the graph vertices stand for the components (nodes) of the network that encode information about the values of the state variables of the dynamical system and the edges represent the mutual relationships between the correspondent endnodes. In practice, random graph theory has become increasingly important for modeling networks whose behaviors exhibit nondeterministic looks. In recent years, many significant results have used random graph models to explain, replicate and simulate the behavior of dynamic real-world networks (Hekmat and Van Mieghem, 2003;Kawahigashi et al., 2005;Durrett, 2006;Onat et al., 2008;Hewer et al., 2009;Hamlili, 2010;Trullols et al., 2010).
To provide a convenient way to represent and analyze dynamic networks by dynamic random graphs, it is very important to clarify how the model of random graphs should explain the behavior of change in the topology of the network. Thus, we introduce some stochastic processes (times of graph change, graph configurations, degree number at a chosen vertex …) in order to attempt to account for the observed statistical properties in graph dynamics. Therefore, we will try to highlight the basic mathematical operations that transform a graph into other one to make possible describing the dynamic change of graph configurations. In this objective, different concepts and notation are introduced in the preliminary sections and will be used throughout the chapter. A reader familiar with the common topics in general graph theory may skip ahead. However, he may use it as necessary to refer to unknown definitions or unusual notations. Also, a particular attention is agreed to Erdös-Rényi's random graph model (Erdös and Rényi, 1960;Bollobàs, 2001).

Preliminary concepts
This section is a short introduction on graph theory. It will review the basic definitions and notation used throughout all this chapter.

Graphs
A classical graph is a static structure of a set of objects where some pairs of these objects are connected by one or several directed or undirected links. In this chapter, we assume that there is no multiple links between a pair of objects and the orientation of the links doesn't play a decisive role. Since the graph is undirected,   , uv and   , vu designate the same edge which we write simply uv . Furthermore, the assumption "simple" states the fact that between two given vertices, we cannot pass more than a single edge.

Definition 2
If S denotes the cardinality of a set S, the number of vertices NV  and the number of

GV E
 define respectively the order and the size of this graph.
Furthermore, we assume in this chapter that graphs can be finite or infinite according to their order and such that the sets of vertices and edges can't be jointly or separately empty.
where the vertices u and v are called end-vertices of the path.
An elementary path is a path such that when all the vertices are sequentially visited, a same vertex is never met twice. A path such that the end-vertices coincide is called cycle. An elementary cycle is a cycle such that all the vertices have exactly two neighbors. The concept of path is behind the notion of connectivity. In the rest of this chapter, we note , uv P the set of all paths between the vertices u and v.

Definitions 6
Let G be a simple graph, To achieve a fully connected graph G, there must exist a path from any vertex to each other vertex in the graph.
The path between the source vertex and the destination vertex may consist of one hop when source and destination are neighbors or several hops if they aren't directly connected by an edge of G. The hopcount specifies the number of hops through a path between two vertices. This measure is meaningful only when there is a path between the source and the destination. The average hopcount of a graph is the average value of the hopcount between the end-vertices of all the possible paths.
www.intechopen.com Furthermore, in a non-connected graph there is no path between at least one sourcedestination pair of vertices. Hence, a non-connected graph consists of several disconnected clusters and/or vertices. Thus, routing is only possible between the different vertices of a same cluster.

Matrix representation of graphs and degree function of vertices
The topological structure of the graph such that each entry is either 0 or 1 where , 1 uv a  signifies that uv is an edge of G. i.e. uv E  .

Definition 8
The degree of a vertex u is the number of its direct neighbors   GG du Nu  (4)

Proposition 1
Consider an undirected graph   , GVE a n d The two last equations (4) and (5) are equivalent by definition of the matrix G A . They induce a function G d from V to N (the set of nonnegative integers) called the degree function. Particularly, a vertex of degree 0 is called an isolated vertex and a vertex of degree 1 is called a leaf.

Random graphs
Another theory of graphs began in the late 1950s. It was baptized random graph theory in several papers by Paul Erdös and Alfréd Rényi. As a real-world network model, the Erdöswww.intechopen.com Rényi's random graph model has a number of attractive properties (Bollobàs, 2001). This model is exceptionally quantifiable; it allows an easy calculation of average values of the graph characteristics (Janson et al., 2000;Hamlili, 2010).
In this section, we want introduce in first a generalization of the concepts of the theory of random graphs. This generalization is intended to describe the issues in applicative frameworks where the number of the graph vertices can vary randomly (number of communicating entities in a wireless ad hoc network, number of routers in the Internet, etc). Also, we will show that most classical models, such as those of Erdös-Rényi random graphs and geometric random graphs can be derived as special cases of the model that we put forward as a generalized alternative.

Generalized random graph model
Intuitively, a generalized random graph representation can be defined in a simple way using the fully weighted graphs.

Definition 9
Consider a non-empty set  , called the set of possible vertices and   This definition of random graph models is very general. We should note that, if G is a generalized random graph, As will be discussed later, this way of modeling a random graph will represent opportunities for characterizing complex situations where classical models such as Erdös-Rényi model are not satisfactory.

Definition 10
The extended adjacency matrix

Practical examples
Different particular cases can be identified and as stated above, in different contexts it may be useful to define the term random graph with different degrees of generality. Hence, the generalized model can describe random geometric graphs (Steele, 1997;Barabasi and Albert, 1999;Penrose, 1999). It suffices to consider G such that V   and the edge probabilities This model is very interesting and as such it can formalize the framework of mobile wireless ad hoc networks where the connectivity of the network depends on the geometric positions of the communicating nodes and a radio coverage range which is generally supposed the same for all the network nodes. In such networks, the random dynamics of the associated graph is induced by the mobility of nodes.
Another example is the random graph model initiated by Erdös and Rényi in the 1950s. This kind of graphs can be represented by a generalized model where the set of vertices is not random (it is constantly the same V   ) and all the graph pairs of vertices are connected with the same probability , This model was be used in most areas of science and human activities in biology, chemistry, sociology, computer networks, manufacturing, etc. In a random Erdös-Rényi graph with N vertices, the edges are independently and randomly built with a probability p between the possible edges of the full mesh graph. This definition builds the binomial model   p V G of random graphs, also referred as Erdös-Rényi model (Bollobàs, 2001).

Proposition 2
Consider a nonempty set V, a real p in  

Degree distribution
As defined above, the degree function G d on the graph vertices returns the number of vertices directly connected to the considered vertex. Thus, the degree distribution measures the local connectivity relevance of the graph vertices. In a generalized random graph model, the degree distribution can be set depending on the wished point of view. Its general form is defined by

Binomial model and Poisson approximation
In particular, in the Enrdös-Rényi representation   p V G , the theoretical degree distribution of any vertex u is defined by a binomial distribution (Bollobàs, 2001) when the number of vertices V is small. Otherwise, from the limit central theorem (LCT), the binomial distribution is approximately Gaussian with parameters t h e v a r i a n c e But the Gaussian distribution is continuous, while the binomial distribution is discrete. Thus, sometimes when V is large and under certain special assumptions we prefer the Poisson approximation to the LCT approximation.

Proposition 3
When the graph order is large, the degree distribution is in the order of where p is small and V is sufficiently large.

Proof
When the graph oder V is large, we can also write the degree density as where p is small and V is sufficiently large. This shows the equation (13).
More generally, this approximation is known in probability theory as the De Moivre's Poisson approximation to the Binomial distribution. Indeed, the Poisson distribution can be applied whenever it is dealing with systems with a large number of possible events such that each of which is rare.
Thus, it is in the order of things to note that the main advantage of the Enrdös-Rényi models comes from the traceability of calculations and the simplicity of parameter estimation. A form or another of this model will be applied as and when it's required.

Power-law models
The traditional model of Erdös and Rényi is not a universal representation for all the random graph behaviors. Sometimes, in many natural scale-free networks (such as World Wide Web pages and their links, Internet, grid computer networking, etc), despite the randomness of the resulting graphs, experimental studies have revealed that the degree distribution can have a pure power-law tail Barabàsi and Albert, 1999;Faloutsos et al. 1999) where 0   is a scaling constant and 0   is a constant scaling exponent. Scale-free Barabàsi-Albert random graphs are generally built through a growth process combined to a www.intechopen.com profile of preferential attachment to existing vertices . Although such graphs have a large number of vertices, the degree distribution deviates significantly from the Poisson law expected for classical random graphs . Other forms of degree distribution with a power-law tail have been studied (Amaral et al., 2000;Newman et al., 2001) where C is a constant fixed by the requirement of normalization,  is the constant scaling exponent, and  is a typical degree size from which the exponential adaptation becomes significant.
In scale-free graphs, the parameters estimation of the degree distribution has to be obviously adapted to each specific case of power-law. In general, the distribution coordinates have to be converted into the logarithmic scales and then apply a method such as least-square method.

Random Graph Dynamics (RGD)
In classic graph theory a graph is simply a collection of objects connected to each other in some manner. This description is very restrictive. In fact, the notions of random graph theory have been introduced in the objective to produce better models and more complete tools to represent non deterministic looks of configurations of a dynamic network. Here again, the language of random graphs is used simply to relate the graph structure of the different network situations. This language must be completed by defining a number of indispensable operations in order to introduce a basic mathematical framework for graph dynamics modeling.

Definition 11
A dynamic graph is a graph such that its configuration (or topology) is subject to dynamic changes with time.
Hence, it goes without saying that the topology changes induced by the random graph dynamics are made through a number of fundamental operations that affect only the set of edges E.

Random change process
Generally, we can define several kinds of graph dynamics. The following contexts are the basic ways of defining this kind of behavior:  Vertex-dynamic graph model: the set of vertices varies with time. In this context vertices may be added or deleted. However, we must be careful in this case to the edges such that an end-vertex is removed. They should just be deleted too.  Edge-dynamic graph model: the set of edges E varies with time. Thus, edges may be added or deleted from the graph. In this case, there are no consequences to fear on the set of vertices.
 Vertex-weighted dynamic graph model: the weights on the vertices vary with time.  Edge-weighted dynamic graph model: the weights on the edges vary with time.  Fully-weighted dynamic graph model: the weights on both vertices and edges vary with time.
Thus, we consider at first a model of dynamic graphs that combines all these aspects together. A such random dynamic graph G can be defined as a stochastic graph process i.e. a collection of independent random graphs where the parameter t is usually assumed to be time and which take values in a set I which can be continuous or countable (finite or infinite).
In the widest sense, each graph t G can belong to a different model The set of all possible states is called the state space. If the state space is discrete, we deal with a discrete state stochastic graph process, which is called a chain of graphs. The state space can also be continuous; we then deal with a continuous-state stochastic process. A similar classification can be made regarding whether the index set I is continuous leading to a continuous-time stochastic process or countable leading to a discrete-time stochastic process. Thus, a dynamic graph G is a representation which assumes that at any time t, there exists an instance To explain the changes in a dynamic random graph, we must often refer to events of presence, absence, addition, deletion, birth, death and structure of objects; where the term "object" refers to both vertices and edges of the graph. The following definitions clarify what do these events really mean and how can they be mathematically defined in the context of graph dynamics.

Definition 12
The weighted graphs   ; tt G P thus defined are called instantaneous configurations of the random dynamic graph G.
The random graph dynamics can be characterized by the ordered stochastic time process In this model, the diagonal entries of the matrix k P represent the probabilities to select, at time k t , the vertices of k V individually in  and the off-diagonal entries of this symmetric matrix represent the probabilities to activate the edges between two given vertices of k V . Both the sets of vertices and edges in the graph configuration process are chosen as temporary. Thus, between two existing end-vertices u and v an edge is selected with the temporary probability   , k uv p . Both the vertices and the edges of a dynamic random graph can be seen as subject to dynamic random changes.

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This type of model is very interesting for modeling dynamic networks where the parameters are all temporary. As examples, we can mention routers and links of the Internet, friends in web social networking, communicating nodes in mobile wireless mobile ad hoc networks that use essentially the methods of broadcast.

Change operations
In this section, the studied approaches are classified according to their own definition in the context of graph dynamics. The RGD is said low when only few elements (number of vertices, number of edges, building edges probability, vertices clustering …) change over time. Otherwise, the dynamics is strong. Furthermore, there are several operations that build new graphs from old ones. They might be characterized through a number of descriptor events and basic transforms.

Definition 13 (Presence)
In a dynamic graph G, an object is present at time t if it belongs to the instance G t of G.

In a dynamic graph G, an object is absent at time t, if it does not belong to the instance G t of G.
Operating dynamic changes on a random graph consists of a series of graph modifications (weight, vertex or edge adaptations). In contrast, the dynamicity analysis is more effective when all combination of additions and deletions of edges and vertices are taken into account.

Definition 15 (Addition)
An object appears in the dynamic graph G, if it transits from the state absent to the state present.
From an operational perspective, this definition should be clarified. Let u and v be two vertices of G such that uv is not an edge of G, the addition of the edge uv to the graph G is defined by the operation Now, there are two ways to add a new vertex to a graph. One way is to add an isolated vertex to the graph G, i.e.

  
and the other is to add a new vertex u which will be connected to an existing vertex v of the graph G, i.e.
From the algorithmic point of view this last operation (18) can be seen as a composition of the two previous ones (16) and (17) Thus, we conceive that the adjacency matrix of * G is obtained by completing the adjacency matrix of G by 0. This matrix will be called in the rest of the chapter extended adjacency matrix of G. The advantage of working with the host graph * G can be viewed rather as freeing from the assumption that the set of vertices of a random dynamic graph varies with time. But the downside of this alternative is that the graph model can exhibit needlessly an excessively large order or incomplete information.

Proposition 4
Let   V be two nonempty sets such that

Proof
This property results trivially from the definitions corresponding to the operations   and   .

Definition 16 (Deletion)
An object disappears (or is deleted) from the dynamic graph G, if it transits from the state present to the state absent.
In these terms, the edge deletion can be formalized as follow. Let u and v be two vertices of a graph such that uv is an edge of G, the deletion of the edge uv from the graph G is defined by the operation On another hand, the vertex deletion can be defined Indeed, the deletion of the vertex u induces the deletion of all the edges having u as endvertex.

Definition 17 (Birth)
The birth of an object is the date of its first appearance in the dynamic graph G.

A structure S of a graph consists in a dynamical set of elements that satisfies a given property.
A path, a click and a cluster are all examples of structures. Let us note that from the random viewpoint the succession of graph transforms and structure updates that convert a configuration of the dynamic graph in another one are not known in advance.

Graph topology changes in RGD
Remark that between two consecutive changes of the graph configuration recorded at k T and 1 k T  , the extended adjacency matrix  

Characterizing the change of the number of vertices
We can define for successive configurations of the graph, the number of new vertices, the number of lost vertices and the number of maintained vertices respectively at kk Tt  by    This shows the required result in (30).
Following this line of thinking, these results can be applied only if the number of the observed vertices varies with time. That is when the dynamicity affects the vertices of the dynamic graph. Otherwise, while the number observed vertices remain unchanged, we will show similar results in the next subsection under the assumption of dynamicity of edges. These two approaches are complementary.

Proof
The proof of (32) results directly from the definitions of   k u  and   k u  .
We can also define global graph indices which reflect the global configuration changes of the dynamic random graph (see figure 1) www.intechopen.com Under the condition of conservation of the number of vertices from a configuration to a successor one, we can establish a number of results for dynamic graphs under Erdös-Rényi model constrains.

Proposition 8
Let G be a random dynamic graph following the model

Proof
Since all trials are independent, each of the three probabilities can be decomposed such as follows

Parameter estimation of the random dynamic graph model
The ultimate objective in random graph dynamics analysis is the estimation of the graph model parameters. In some respects, nearly all types of dynamic changes can be interpreted in this way. In the empirical study, for reasons of traceability of the calculations, we restrict ourselves to an homogeneous model Furthermore, we assume that by some means a vertex may deduct its neighborhood set directly from information exchanged as part of edge sensing. This section aims to show how the model parameters of the random dynamic graph can be estimated.

Dynamic graphs with known number of vertices and unknown edge probability
The main goal here is to find the maximum-likelihood estimate (MLE) of the edge probability p when the number of the graph vertices NV  is known.

Proposition 11
Let G be a random dynamic graph following the model

Proof
If the graph order NV  is small, from equation (11), the log-likelihood function of the degree distribution in a given vertex u is for different configurations of the random dynamic graph, this quantity is estimated differently in each vertex (see figure 2).

Dynamic graphs with unknown number of vertices and edge probability
In the statistical common sense, when the graph order is unknown the method proposed in the previous paragraph cannot be applied. Another resourceful method of point estimation called method of moments (Lehmann E. L. and Casella G., 1998) can be used when the number N of the graph vertices and the edge density parameter p are both unknown. Likewise, these parameters should be estimated on the basis of an s-sample 12 ,,, s DD D  of the degree of a vertex u for s different instances of the random dynamic graph topology. www.intechopen.com

Proposition 12
Let G be a random dynamic graph following the model Depending on the neighbors met by each of the random dynamic graph vertices, the evaluation of the estimated parameters will be different.

Expected degree number of vertices in wide random dynamic graphs
Various random dynamic graph problems (Internet, out vehicle networks, wide ad hoc networks …) are analyzed and interpreted under the assumption that the order of the resulting graph may be relatively large (with some tens, hundreds or even thousands of vertices). Let  be the average degree number As a result of scaling, since we use an approximation of the degree distribution, it is very important to work with good estimates. This shows that the resulting computed values are most likely not due merely to chance.

Proposition 13
Let G be a random dynamic graph following the model which is the lower bound of the Fréchet-Darmois-Cramer-Rao (FDCR) inequality.
Then, since  is unbiased and verifies the lower bound of the FDCR inequality, it is an efficient estimate.

Proposition 14
Let G be a random dynamic graph following the model From the proposition 11 and the equation (45)

Giant component size estimation
Let us consider G a random dynamic graph following the model   p V G and suppose that the graph order V is large and known. From the theory of Erdös-Rényi graphs, we know (Molloy and Reed, 1998) that the graph will almost surely have a unique giant component containing a positive fraction of the graph vertices if 1 p V  (see figure 5) i.e. the average degree numbet 1   .

Proposition 15
Let G be a random dynamic graph following the model Elementary dissimilarities exist between random graph models and real-world graphs. Realworld graphs show strong clustering, but Erdös and Rényi's model does not. Many of the graphs, including Internet and World-Wide Web graphs, show a power-law degree distribution . This means that only a small part of the graph vertices can have a large degree. In fact, Erdös-Rényi assumptions can imply strong consequences on the behavior of the graph (Newman, 2003).

Conclusions
In this chapter we have illustrated several basic tools for representing and analyzing dynamic random graph in a general context and a practical approach to estimate the parameters of classical models of such behaviors. The generalized model that we have proposed can describe not only classical real-world networks models but also situations with more complex constraints. In this model, random graph dynamics is outlined by introducing a random time process where the dynamic graph topology changes are recorded. Among the advantages of this model is the possibility to generate successive independent random graphs with potentially different sets of vertices but all belonging to the same basic set of vertices. Otherwise, the fact that the generalized model allows to consider probabilities which are not necessarily all equal gives the possibility to favor the establishment of certain connections over others. This kind of behavior is often observed in www.intechopen.com wireless mobile networks and social networks. Here, there are ways to implement in the generalized model the ability to prioritize connections to closest vertices. These advantages of modeling are generally not possible through the traditional random graphs of Erdös-Rényi. Also, we have described the global behavior between two consecutive configurations by calculating the probability of change. This is done in the case where the dynamic change concerns only the graph edges, as well as in the case where the dynamic change affects also the graph vertices.
At the end of this chapter we have shown how the parameters of a dynamic random graph can be estimated under the assumptions of the Erdös-Rényi model. Thus, the estimation of these parameters has led to the estimates of the average degree of vertices, the average hopcount and the size of the giant component in large dynamic graphs.