Abstract
In this chapter we describe the core method that will be used throughout the rest of the book, i.e. the construction of a constrained maximum-entropy ensemble of networks. This procedure requires the definition of the entropy of a network ensemble, the specification of structural properties to be enforced as constraints, the calculation of the resulting maximum-entropy probability of network configurations, and the maximization of the likelihood, given the empirical values of the enforced constraints. We describe this procedure explicitly, after giving some general motivations. In particular, we discuss the crucial importance of enforcing local constraints that preserve the (empirical) heterogeneity of node properties. The maximum-entropy method not only generates the exact probabilities of occurrence of any graph in the ensemble, but also the expectation values and the higher moments of any quantity of interest. Moreover, unlike most alternative approaches, it is applicable to networks that are either binary or weighted, either undirected or directed, either sparse or dense, either tree-like or clustered, either small or large. We also discuss various likelihood-based statistical criteria to rank competing models resulting from different choices of the constraints. These criteria are useful to assess the informativeness of different network properties.
Whereof one cannot speak, thereof one must be silent.
—Ludwig Josef Johann Wittgenstein, Logisch-Philosophische Abhandlung
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Notes
- 1.
A topological property f, where \(f(\mathbf {G})\) is the value of the property in graph \(\mathbf {G}\), is said to evaluate to a graphic (or graphical) value \(\tilde{f}\) if there exist at least one graph \(\tilde{\mathbf {G}}\) that realizes such value, i.e. for which \(f(\tilde{\mathbf {G}})=\tilde{f}\).
- 2.
An undirected graph (or network) is a graph where no direction is specified for the edges. An undirected graph is binary or simple if each pair of nodes i and j (with \(i\ne j\)) is connected by at most one edge, i.e. if there are no multiple edges between the same two nodes. We will also assume the absence of self-loops (edges starting and ending at the same node) throughout the book.
- 3.
A weighted graph (or network) is a graph where links may carry different intensities. When dealing with weighted networks, throughout the book we will assume non-negative integer link weights (i.e. \(w_{ij}=0,1,2\dots +\infty \)) for simplicity. This corresponds to the assumption that an indivisible unit of measure of link weights has been preliminary specified. Under this assumption, a weighted network can also be regarded as a graph that is in general not simple, i.e. where multiple links of unit weight are allowed between the same two nodes. We will still exclude the possibility of self-loops. Ideally, one may think of link weights becoming continuous as the unit of measure is chosen to be vanishingly small.
- 4.
A directed graph is a graph where a direction is specified for each edge (self-loops are not allowed in this case as well). A directed graph is binary (or simple) if any two nodes i and j are connected in one of the following four mutually-exclusive ways: via only a directed link from i to j, via only a directed link from j to i, via both such links, or via no link at all. A directed graph is weighted if links can carry different intensities, including when they are pointing in opposite direction between the same two nodes. Again, we will assume non-negative integer weights.
- 5.
The empirical degree distribution is defined, for a given network, as the fraction P(k) of nodes that have degree k.
- 6.
The empirical strength distribution is defined, for a given network, as the fraction P(s) of nodes that have strength s.
- 7.
Throughout the book, by expected value (or expectation) of a topological property we mean the average of that property over the ensemble of random graphs under consideration. We denote expectation values with angular brackets \(\langle \cdot \rangle \). The rigorous definition is given later in Eq. (2.7).
- 8.
The average degree in a simple undirected graph with N nodes is defined as \(\bar{k}=N^{-1}\sum _{i=1}^N k_i\) and necessarily equals 2L / N, where L is the total number of links.
- 9.
The average strength in a weighted undirected graph with N nodes is defined as \(\bar{s}=N^{-1}\sum _{i=1}^N s_i\) and necessarily equals 2W / N, where W is the total weight of all links in the network.
- 10.
The maximum value of the entropy S(P) depends on the total number of configurations over which the sum in Eq. (2.8) runs. This number can be rescaled to one for all probability distributions, upon normalizing S(P) by the maximum value itself.
- 11.
In statistical physics, the thermodynamic limit is defined as the limit where the number of fundamental units that describe the microscopic configurations of the system diverges. In our graph ensembles, we regard the nodes as the units and their connections as the interactions.
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Squartini, T., Garlaschelli, D. (2017). Maximum-Entropy Ensembles of Graphs. In: Maximum-Entropy Networks. SpringerBriefs in Complexity. Springer, Cham. https://doi.org/10.1007/978-3-319-69438-2_2
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