Homogeneous complex networks

doi:10.1016/j.physa.2005.10.024

Physica A: Statistical Mechanics and its Applications

Volume 366, 1 July 2006, Pages 587-607

https://doi.org/10.1016/j.physa.2005.10.024 Get rights and content

Abstract

We discuss various ensembles of homogeneous complex networks and a Monte-Carlo method of generating graphs from these ensembles. The method is quite general and can be applied to simulate micro-canonical, canonical or grand-canonical ensembles for systems with various statistical weights. It can be used to construct homogeneous networks with desired properties, or to construct a non-trivial scoring function for problems of advanced motif searching.

Introduction

Complex networks is a new emerging branch of random graph theory. For a long time random graphs have been mainly studied by pure mathematics but recently due to the availability of empirical data on real-world networks they have attracted the attention of physics and natural sciences (see for review Refs. [1], [2], [3]). Methods of statistical physics, both empirical and theoretical, have thus begun to play an important role in this research area.

The empirical observations of real-networks has had a feedback on theoretical development which is now concentrated on the understanding of the observed features. For example fat tails in node degree distribution, small world effect, degree–degree correlations, or high clustering. Two complementary approaches have been developed: diachronic, known as growing networks [1], [2], [3], and synchronic being a sort of statistical mechanics of networks [4], [5], [6], [7], [8], [9], [10], [11].

We will discuss here the latter. This approach is a natural extension of Erdös and Rényi ideas [12], [13]. It is well suited both for growing (causal) networks for which nodes’ labels reflect the causal order of nodes’ attachment to the network [14], [15] and for homogeneous networks for which nodes’ labels can be permuted freely in an arbitrary way. Here, we shall discuss mainly homogeneous networks. We shall shortly comment on causal networks towards the end of the paper.

The main aim of the paper is to present a consistent picture of statistical mechanics of networks. Some ideas have already been introduced earlier. They are scattered in many papers and discussed in many different contexts. We put them together, add some new material and introduce a guideline to obtain a self-contained introduction to statistical mechanics of complex networks.

The basic concept in the statistical formulation is statistical ensemble. Statistical ensemble of networks is defined by ascribing a statistical weight to every graph in the given set [4], [7]. Physical quantities are measured as weighted averages over all graphs in the ensemble. The probability of the occurrence of a graph in random sampling is proportional to its statistical weight. If the statistical weight changes then also the probability of occurrence of randomly sampled graphs will change and in effect different random graphs will be observed. The concept of statistical weight is crucial, since it defines randomness in the system. Statistical weight is built out of two ingredients: configuration space weight and functional weight. The configuration space weight is proportional to the uniform probability measure on the configuration space which tells us how to uniformly choose graphs in the configuration space. To illustrate the meaning of the uniform measure consider an ensemble of Erdös–Rényi graphs with N nodes and L links [12], [13]. The configuration space consists of $(\binom{(\binom{N}{2})}{L})$ graphs with labeled nodes. All those graphs are equiprobable, and therefore the configuration space weight is the same for each graph. It is convenient to choose this weight to be $1 / N!$ since then it can be interpreted as a factor which takes care of $N!$ possible permutations of nodes’ labels. This factor has the same origin as the corresponding factor in quantum mechanics for indistinguishable particles and it is constant for all graphs in a finite N-ensemble.

We can calculate the entropy of random graphs as $S = \ln \frac{1}{N!} (\binom{(\binom{N}{2})}{L}) .$ In the limit of large sparse graphs: $N \to \infty$ and $2 L / N = α = const > 2$ , the entropy is an over-extensive function of the system size: $S = \frac{α - 2}{2} N \ln N + \dots,$ unlike in standard thermodynamics.

Let us move to weighted graphs. The idea is to modify the Erdös–Rényi ensemble by introducing a functional weight which explicitly depends on graph's topology. For example, if we choose the functional weight to be a function of the number of loops on the graph, we can suppress favor loops of typical graphs in the ensemble. In a similar way we can choose statistical weights to control the node degree distribution to produce homogeneous scale-free graphs [16] or to introduce correlations between degrees of neighboring nodes [17], [18], [19], [20].

Classical thermodynamics describes systems in equilibrium for which the functional weight is given by the Gibbs measure: $\sim \exp (- β E)$ , where E is the energy of the system. When discussing complex networks it is convenient to abandon the concept of energy and Gibbs measure and consider a more general form of statistical weights because many networks are not in equilibrium. Indeed, many networks emerging as a result of a dynamical process like growth are far from equilibrium [1], [2], [3]. It does not mean though that one cannot introduce a statistical ensemble of growing networks. On the contrary, one can for example consider an ensemble of networks which result of many independent repetitions of the growth process terminated when the network reaches a certain size. Such a collection of networks does not describe a thermodynamic equilibrium. The functional weight can be deduced from the parameters of the growth process but of course it has nothing to do with the Gibbs measure.

In fact, many real-world networks result from a combination of a growth process and some thermalization processes. For example, the Internet grows but at the same time it continuously rearranges. The latter process introduces a sort of thermalization. Today the growth has probably still larger influence on the topology of the underlying network but in the future the growth may slow down due to saturation and then equilibration processes resulting from continuous rewirings will take over. Similarly all evolutionary networks emerge from a growth mixed with a sort of thermalization related to the continuous network rearrangement. Therefore, it is convenient to have a formalism which can extrapolate between the two regimes in a flexible way. The approach which we propose here is capable of modeling functional properties of networks by choosing an appropriate functional weight.

Let us return to the configuration space weight. As we mentioned this weight is equivalent to the uniform probability measure on the configuration space for which all graphs are equiprobable. It is a very crucial part of the construction of the ensemble to carefully specify what one means by equiprobable graphs. Consider first graphs with N nodes. There are at least two natural candidates for the uniform measure in such a set of graphs. Since one is interested in shape (topology) of graphs one can define all shapes to be equiprobable. Alternatively one can introduce labels for nodes of each graph to obtain a set of labeled graphs and then one can define all labeled graphs to be equiprobable. The two definitions give two different probability measures since the number of ways in which one can label graph nodes depends on graph's topology and thus the probability of occurrence of a given graph will depend on its topology too. It turns out that the latter definition is more natural. As we have seen above this definition leads to Erdös–Rényi graphs. So we stick to this definition and from here on we shall ascribe to each labeled graph the configuration space weight $1 / N!$ which is constant in the set of graphs of size N.

The situation is more complex if one considers pseudographs that is graphs which have multiple connections (more than one link between two nodes) or self-connections (a link having the same node at its endpoints). In this case one can also label links and ascribe the same statistical weight to each fully labeled graph. For this choice the statistical weight of each graph is equal to the symmetry factor of Feynman diagrams generated in the Gaussian perturbation field theory [4].

The paper is organized as follows. In the next section we will recall some basic definitions. Then we will discuss Erdös–Rényi graphs in the context of constructing statistical ensemble and later we will generalize the construction to weighted homogeneous graphs. After this we will describe Monte-Carlo algorithms to generate graphs for canonical, grand-canonical and micro-canonical ensembles and discuss their representation in terms of adjacency matrices. Next we will discuss pseudographs. In the last section we will briefly summarize the paper.

Section snippets

Definitions

Let us first introduce some terminology. Graph is a set of N nodes (vertices) connected by L edges (links). A graph need not be connected. It may have many disconnected components including empty nodes (without any link). If a graph has no multiple or self-connected links we shall call it simple graph or graph. An example is illustrated in Fig. 1. Later we shall also discuss graphs with multiple- and self-connections. To distinguish them from simple graphs we shall call them degenerate graphs

Statistical ensemble for Erdös–Rényi random graphs

For simplicity, we start from a well-known model of Erdös–Rényi's graphs [12], [13]. In this classical model one considers simple graphs with N labeled nodes and L links¹ chosen at random from all $(\binom{N}{2})$ possibilities. All possibilities are equiprobable and so are the corresponding graphs—understood as graphs whose vertices are labeled. Usually one is interested in unlabeled graphs that is

Weighted homogeneous graphs

In the previous section we described ensembles for which all labeled graphs had the same statistical weight. Random graphs in such ensembles have well known properties. It turns out, however, that most of these properties do not correspond to those observed for real-world networks. One needs a more general setup to define an ensemble of complex random networks. Such a set-up can be introduced as follows. One considers the same set of graphs as in Erdös–Rényi model but one ascribes to each graph

Monte-Carlo generator of homogeneous networks

Erdös–Rényi graphs are exceptional in the sense that one can calculate for them almost all quantities of interest analytically. This is not the case for weighted networks. Various methods have been proposed for generating random graphs [28]. In this section we will describe a Monte-Carlo method which allows one to study a wide class of random weighted graphs experimentally by a sort of numerical experiments. The basic idea behind this type of experiments is to sample the configuration space of

Monte-Carlo generator of canonical ensemble

Now, we want to apply this method to generate Erdös–Rényi graphs. Let us begin with the canonical ensemble with $N, L$ fixed. A good candidate for elementary transformation of graph is rewiring of a link as shown in Fig. 4, because it does not change N and L. As mentioned before it is convenient to introduce a representation in which each undirected link is represented by two directed links. The rewiring is done in two steps [4]. First, we choose a directed link ij and a vertex k at random. Then

Monte-Carlo generator of grand-canonical ensemble

The rewiring procedure described in the previous section does not change N and L. If we want to simulate graphs from a grand-canonical ensemble for which L is variable, we have to supplement the set of elementary transformations in the algorithm by transformations which change the number of links. We can introduce two mutually reciprocal transformations: adding and deleting a link. They both preserve the number of nodes N but change the number of links: $L \to L \pm 1$ . The two transformations must be

Monte-Carlo generator of micro-canonical ensemble

Another frequently encountered ensemble is an ensemble of graphs which have a given node degree sequence ${q_{1}, q_{2}, \dots, q_{N}}$ . The partition function Z has the form: $Z (N, {q_{i}}) = \sum_{α^{'} \in \lg (N, L)} (\prod_{i = 1}^{N} δ [q_{i} (α^{'}) - q_{i}]) 1 / N! W (α^{'}),$ where the product of delta functions allows one to include only those graphs which have a prescribed degree distribution $q_{i}$ . As before the factor $1 / N!$ is fixed in this ensemble and could in principle be skipped. The canonical partition function $Z (N, L)$ is related to the micro-canonical ones: $Z (N,$

Graph generator and adjacency matrices

All the elementary transformations: rewiring, adding or removing a link, and the X-move have a simple representation in terms of adjacency matrices. Rewiring relies on picking up at random an element $A_{ij} = 1$ of the adjacency matrix and flipping it with an element $A_{ik} = 0$ so that after the move $A_{ij} = 0$ and $A_{ik} = 1$ . For undirected links, adjacency matrices are symmetric and therefore at the same time one has to flip $A_{ji} = 1$ and $A_{ki} = 0$ . To add a link one chooses at random $A_{ij}$ and if $A_{ij} = 0$ and $i \neq j$ , one

Degenerated graphs (pseudographs)

In previous sections we described ensembles of simple graphs. Let us now discuss pseudographs that is graphs which may have multiple- and self-connections.

A degenerate undirected pseudograph can be represented by a symmetric adjacency matrix $A$ whose off diagonal entries $A_{ij}$ count the number of links between vertices i and j, and the diagonal ones $A_{ii}$ count twice the number of self-connecting links attached to vertex i. For example, the graph depicted in Fig. 7 has the following adjacency matrix:

Summary

We have discussed a statistical approach to homogeneous random graphs. This framework is a natural extension of the Erdös–Rényi theory to the case of weighted graphs: one considers the same set of graphs but with modified statistical weights. The statistical weights of homogeneous graphs depend only on graphs’ topology. In other words, if one assigns some labels to its nodes, they will have no physical meaning similarly as the numbers of indistinguishable particles in quantum mechanics. One can

Acknowledgements

We thank Piotr Bialas, Jerzy Jurkiewicz and Andrzej Krzywicki for stimulating discussions. This work was partially supported by the Polish State Committee for Scientific Research (KBN) Grant 2P03B-08225 (2003–2006) and Marie Curie Host Fellowship HPMD-CT-2001-00108 and by EU IST Center of Excellence “COPIRA”.

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View full text

Homogeneous complex networks

Abstract

Introduction

Section snippets

Definitions

Statistical ensemble for Erdös–Rényi random graphs

Weighted homogeneous graphs

Monte-Carlo generator of homogeneous networks

Monte-Carlo generator of canonical ensemble

Monte-Carlo generator of grand-canonical ensemble

Monte-Carlo generator of micro-canonical ensemble

Graph generator and adjacency matrices

Degenerated graphs (pseudographs)

Summary

Acknowledgements

Nucl. Phys. B

Physica A

Nucl. Phys. B

Phys. Rev. Lett.

Rev. Mod. Phys.

Adv. Phys.

SIAM Rev.

Phys. Rev. E

Phys. Rev. Lett.

Springer Lecture Notes Phys.

Phys. Rev. E

Publ. Math. Debrecen

Publ. Math. Inst. Hung. Acad. Sci.

Random Graphs

Phys. Rev. E

Phys. Rev. E