Enhanced reconstruction of weighted networks from strengths and degrees

Abstract

Network topology plays a key role in many phenomena, from the spreading of diseases to that of financial crises. Whenever the whole structure of a network is unknown, one must resort to reconstruction methods that identify the least biased ensemble of networks consistent with the partial information available. A challenging case, frequently encountered due to privacy issues in the analysis of interbank flows and Big Data, is when there is only local (node-specific) aggregate information available. For binary networks, the relevant ensemble is one where the degree (number of links) of each node is constrained to its observed value. However, for weighted networks the problem is much more complicated. While the naïve approach prescribes to constrain the strengths (total link weights) of all nodes, recent counter-intuitive results suggest that in weighted networks the degrees are often more informative than the strengths. This implies that the reconstruction of weighted networks would be significantly enhanced by the specification of both strengths and degrees, a computationally hard and bias-prone procedure. Here we solve this problem by introducing an analytical and unbiased maximum-entropy method that works in the shortest possible time and does not require the explicit generation of reconstructed samples. We consider several real-world examples and show that, while the strengths alone give poor results, the additional knowledge of the degrees yields Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. accurately reconstructed networks. Information-theoretic criteria rigorously confirm that the degree sequence, as soon as it is non-trivial, is irreducible to the strength sequence. Our results have strong implications for the analysis of motifs and communities and whenever the reconstructed ensemble is required as a null model to detect higher-order patterns.A range of phenomena of critical importance, from the spread of infectious diseases to the diffusion of opinions and the propagation of financial crises, is highly sensitive to the topology of the underlying network that mediates the interactions [1]. This sensitivity implies that, whenever it is not possible to have a complete empirical knowledge of the network, one should make an optimal use of the partial information available and try to reconstruct the most likely network, or rather an ensemble of likely networks, in the least biased way. In the Big Data era, this kind of problem is becoming more and more important given the ever-increasing availability of data that, for privacy issues, are often of aggregate nature [2, 3].Among the possible types of incomplete topological information (e.g. missing links, missing nodes, etc), one of the most frequently encountered situations is when only a local knowledge of the network is available [6][7][8][9][10][11]. For instance, in binary n...

Assessing Systemic Risk Due to Fire Sales Spillover Through Maximum Entropy Network Reconstruction

“…The performances of this newly proposed approach are shown to outperform those obtained with standard maximum entropy approaches (Saracco et al, 2015), i.e. the bipartite extensions of the models proposed by Mastrandrea et al (2014) for unipartite networks.…”

“…Since other specifications of maximum entropy are quite popular in the literature of network reconstruction, only for comparison purposes we take into considerations two other ensembles, mainly inspired by the paper by Mastrandrea et al (2014) and Saracco et al (2015). Each of them is characterized by different constraints imposed on the maximization of the Shannon's entropy.…”

“…Finally, we consider another (richer) statistical ensemble whose probability mass function, derived in Appendix B.3, corresponds in our bipartite framework to the enhanced configuration model of Mastrandrea et al (2014). This newly defined ensemble, that we address as Bipartite Enhanced Configuration Model (BIPECM), is obtained via Maximum Entropy imposing both the mean value of strengths (as in BIPWCM) and the mean value of degrees, that is the number of edges incident in each vertex.…”

The role of weighted and topological network information to understand animal social networks: a null model approach

“…Outside of behavioural ecology, there is a growing suite of random-graph algorithms (Watts & Strogatz, 1998;Serrano et al, 2006;Leskovec et al, 2010;Ansmann & Lehnertz, 2011;Prettejohn et al, 2011) which emphasise core properties such as the degree-sequence, strength-sequence, network size and density; they have shown that unless such properties are held constant across random-graphs, then any conclusions about network properties will just reect variation in the degreesequence, strength-sequence, network-size, etc. There is a near consensus about the need to condition on the degree-sequence for binary-networks, but the matter is more controversial for weighted-networks, and one's conclusions are sensitive to such conditioning (Garlaschelli & Loredo, 2009;Mastrandrea et al, 2014). This paper follows in the spirit of Garlaschelli & Loredo (2008), to calculate metric expectations based on null-models that assume only basic individual-level properties, and to do so by generating an ensemble of random networks based on the Exponential Random Graph formulation.…”

“…For example, Garlaschelli & Loredo (2009) discovered that some weighted measures "inherit" trivially from topological features and called for "a systematic redenition of weighted network properties", while Mastrandrea et al (2014) noted that "the strength sequence is in general uninformative about the higher-order properties of the network". The implications for behavioural ecologists are that: 1) many weighted-network metrics may not depend on weights per se and actually depend on the underlying binary, topological patterns; and 2) that many metrics of higher-order structure are not signicantly dierent from (and often highly corrected with) the values one would expect from networks with only individual-level constraints (degree and/or strength).…”