Statistical mechanics of multiedge networks

Abstract

Statistical properties of binary complex networks are well understood and recently many attempts have been made to extend this knowledge to weighted ones. There are, however, subtle yet important considerations to be made regarding the nature of the weights used in this generalization. Weights can be either continuous or discrete magnitudes, and in the latter case, they can additionally have undistinguishable or distinguishable nature. This fact has not been addressed in the literature insofar and has deep implications on the network statistics. In this work we face this problem introducing multiedge networks as graphs where multiple (distinguishable) connections between nodes are considered. We develop a statistical mechanics framework where it is possible to get information about the most relevant observables given a large spectrum of linear and nonlinear constraints including those depending both on the number of multiedges per link and their binary projection. The latter case is particularly interesting as we show that binary projections can be understood from multiedge processes. The implications of these results are important as many real-agent-based problems mapped onto graphs require this treatment for a proper characterization of their collective behavior.

Role of adjacency-matrix degeneracy in maximum-entropy-weighted network models

“…Furthermore, such methods provide an easy way of rapidly simulating (and averaging) network instances belonging to a given ensemble. So far in the literature successful development of this kind of methodology has been performed for different types of monolayered networks [13,[18][19][20], directed [14] and bipartite [21] structures, and stochastic block models [22] and some multiplex weighted networks [4].…”

The configuration multi-edge model: Assessing the effect of fixing node strengths on weighted network magnitudes

“…We have that the strengths s i = j t ij will also be Poisson distributed random variables, being sums of independent occupation numbers. Moreover, since the binary projection of occupation numbers Θ(t ij ) are Bernoulli distributed variables with parameter P (t ij > 0) = 1 − e − tij [11] one can can also compute the associated degrees k i of the nodes, which will be sums of independent Bernoulli random variables 2 . We have that,…”

“…In fact, the need to distinguish different types of so-called weighted networks according to the nature of the events being represented has been pointed out recently [11]. If nodes accept multiple distinguishable connections, then one can speak of multi-edge networks, and p-1 it is in this scenario where we propose a flexible, general theory for null-model generation.…”

“…We further consider a fixed strength sequence {ŝ i } ; i ∈ 1, N , so that we have a total ofT = iŝ i distinguishable events to be randomly allocated among L edges. Note that the difference between distinguishable and non-distinguishable events is crucial, as it happens usually in statistical mechanics, because the resulting statistics of the ensembles one can consider depend on this property (see [11,22] for an extended discussion). As for the notation, note also that in general we will usex to denote the value to which a variable x is being constrained To clarify the elements of our system, we can specify them for the case of an OD study as an illustrative example: Nodes correspond to different locations, events to individual trips between locations, the strengths of nodes to the fixed amount of travelers departing/entering each location and the weights correspond to the observed flows between locations.…”

“…This means that both the occupation numbers or weights t ij and the node strengths s i = j t ij are integer random variables, and that each network belonging to the ensemble corresponds to a different realization of such variables. However, relative fluctuations around the constraints s i =ŝ i vanish in the large event limit [11]. The GC ensemble yields independent Poisson statistics for the occupation numbers t ij with mean t ij = βx i x j , where {x i } can be considered as node-specific hidden variables [25][26][27].…”