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The authors have declared that no competing interests exist.

Conceived and designed the experiments: GJ. Analyzed the data: TS ESG. Contributed reagents/materials/analysis tools: DG GJ. Wrote the paper: TS ESG DG GJ.

In the context of agent based modeling and network theory, we focus on the problem of recovering behavior-related choice information from origin-destination type data, a topic also known under the name of network tomography. As a basis for predicting agents' choices we emphasize the connection between adaptive intelligent behavior, causal entropy maximization, and self-organized behavior in an open dynamic system. We cast this problem in the form of binary and weighted networks and suggest information theoretic entropy-driven methods to recover estimates of the unknown behavioral flow parameters. Our objective is to recover the unknown behavioral values across the ensemble analytically, without explicitly sampling the configuration space. In order to do so, we consider the Cressie-Read family of entropic functionals, enlarging the set of estimators commonly employed to make optimal use of the available information. More specifically, we explicitly work out two cases of particular interest: Shannon functional and the likelihood functional. We then employ them for the analysis of both univariate and bivariate data sets, comparing their accuracy in reproducing the observed trends.

In this paper we focus on the problem of recovering behavior-related micro choice information from aggregate data. In particular, we consider origin-destination data, casting this problem as an inference problem concerning the prediction of flows on networks [

To go beyond traditional reductionist modeling and mathematical anomalies, we use a new paradigm that is developing under the name of Network Science (see, for example, [

We seek an expression for the probabilities that the origin and the destination nodes are connected along a specific pathway in the statistical ensemble of possible pathways, without explicitly sampling the configuration space. Given information about the origin-destination network structure in the form of a matrix _{ij} must be estimated from aggregate flow data that may be noisy in nature. The number of unknown pathway parameters of the protocol matrix

As we seek new ways to think about the causal adaptive behavior of complex and dynamic micro systems, we note that problems of this type may be re-formulated as problems of constrained entropy-maximization over the pathways. In other words, causal entropy maximization can be adopted as the systems status-measure and optimization criterion (following [

This permits us to recast a behavioral system in terms of path microstates where entropy reflects the number of ways a macrostate can evolve along a path of possible microstates: the more diverse the number of path microstates, the larger the causal path entropy. The result is a causal entropic force that captures self-organized equilibrium seeking behavior (see [

In the sections ahead we analyse systems within this framework, that permits the interpretation of adaptive economic behavior in terms of entropic functions: as a basis for solving micro-behavioral information recovery problems, we suggest an information theoretic family of entropic functions; to demonstrate applicability, we consider binary and weighted data sets and recover the optimum corresponding unknown probabilities.

In developing a basis for the use of information theoretic (IT) methods to infer origin-destination networks flows, we focus on a stochastic ill posed inverse problem and the corresponding regularization method it implies (the pure, without-noise inverse problem is just a special case). In this context the Cressie-Read (CR) family of entropic functions [

This permits the researcher to exploit the statistical machinery of information theory to gain insights on the underlying adaptive behavior of a dynamic process from a system that may not be in equilibrium. This approach contrasts with the traditional approach to micro information recovery that rests on reductionist economic and econometric functional analysis and observational agent behavior data: however, precisely because of the nonlinear and ordinal nature of dynamic micro systems, the traditional approach is cumbersome in terms of identifying and expressing adaptive behavior.

We start introducing the CR multi parametric convex family of entropic functional measures [

In _{c}’s represent the subject probabilities and the _{c}’s are interpreted as reference (or prior) probabilities (the reason for indexing our coefficients with _{c}, _{c} ∈ [0, 1], ∀_{c} _{c} = 1, ∑_{c} _{c} = 1 are assumed to hold. As

In other words, the CR family of power divergences is a class of additive convex functions that encompasses a broad family of test statistics, in turn representing a broad family of functional relationships within a moments-based estimation context. In addition, the CR measure exhibits proper convexity in

In what follows we consider the two values _{c} = 1/_{c} = 1/

We stress that while the Shannon functional has been already employed for the analysis of univariate and bivariate data sets, the likelihood functional case has not been explicitly worked out yet, thus representing the major contribution of this paper to the analysis of behavioral networks.

To demonstrate the applicability of our approach in the binary network area, an example may be useful. Consider the problem of determining least-time, point-to-point traffic flows between sub-networks, when only aggregate origin-destinations volumes are known (see

Blue dots represent the origin and the destination nodes. Connections between them represent the ensemble of pathways described by the probability distribution _{c}, ∀

If we indicate by _{tot} ≡ ∑_{c} _{c}:
_{c} _{c} = 1. We have thus rewritten

In particular, since the functional

A similar problem is faced whenever a whole matrix of probability coefficients (and not a simple vector),

These are just the solutions to a standard problem when a function must be inferred from insufficient sample-data information. Thus network inference and monitoring problems have a strong resemblance to an inverse problem in which key aspects of a system are not directly observable (for details on the use of information theoretic entropic methods for this type of network information flow problems see also [

To test the effectiveness of our method, in what follows we analyze two aggregate data sets (for which origin-destination traffic volumes were collected), the first one concerning traffic on a local area network and the second one concerning consumers’ choices of complementary products.

The network topology we consider yields 7 observed aggregate traffic volumes and 16 origin-destination traffic volumes.

Eggs | ||||||
---|---|---|---|---|---|---|

Bacon | 0 | 1 | 2 | 3 | 4 | Total |

0 | 254 | 115 | 42 | 13 | 6 | 430 |

1 | 34 | 29 | 16 | 6 | 1 | 86 |

2 | 8 | 8 | 3 | 3 | 1 | 23 |

3 | 0 | 0 | 4 | 1 | 1 | 6 |

4 | 1 | 1 | 1 | 0 | 0 | 3 |

Total | 297 | 153 | 66 | 23 | 9 | 548 |

The analysis of Bell Labs data is illustrated in Figs

The number of the channel is reported on the x-axis. Observed and estimated

The number of the channel is reported on the x-axis. Observed and estimated

As a general comment, the predictions of both functionals reproduce the majority of the observed trends satisfactorily, with the likelihood functional performing slightly better than Shannon functional whose estimates, in some cases, show larger discrepancies. Moreover, the performance of both functionals improves when single peaks are registered on a single channel, accompanied by small traffic volumes on the others. However, at night, whenever the latter are exactly zero the agreement between our estimates and observations seems to deteriorate: as shown in the left panel of

A solution to improve the predictions accuracy is to explicitly exclude zero values from our dataset. This can be achieved by considering a reduced _{1} = _{16} = 0. The right panel of

The result of the application of our information recovery method to the “eggs and bacon” data set is shown in

Shannon functional | ||||||
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Eggs | ||||||

Bacon | 0 | 1 | 2 | 3 | 4 | |

0 | 262.378 | 122.478 | 40.468 | 4.65702 | 0.0191661 | 0.999453 |

1 | 27.3702 | 23.502 | 18.8328 | 12.2212 | 4.0738 | 0.970398 |

2 | 5.38417 | 5.16918 | 4.87188 | 4.33981 | 3.23497 | 0.86233 |

3 | 1.25404 | 1.24078 | 1.22175 | 1.18545 | 1.09798 | −0.0718339 |

4 | 0.613532 | 0.61028 | 0.605583 | 0.596516 | 0.574089 | 0.847078 |

Likelihood functional | ||||||

Eggs | ||||||

Bacon | 0 | 1 | 2 | 3 | 4 | |

0 | 258.603 | 118.489 | 40.1875 | 9.81096 | 2.90897 | 0.99991 |

1 | 30.4192 | 26.7046 | 18.5562 | 7.63744 | 2.68261 | 0.993516 |

2 | 6.02486 | 5.86333 | 5.34772 | 3.78732 | 1.97677 | 0.850168 |

3 | 1.32087 | 1.31294 | 1.28519 | 1.1694 | 0.911598 | 0.019223 |

4 | 0.631723 | 0.629903 | 0.623446 | 0.594872 | 0.520056 | 0.824691 |

A closer inspection of

This paper represents a contribution to the study of behavioral information recovery for self-organizing systems. The approach we proposed questions the use of traditional information recovery methods (see [

The class of entropic functionals employed in this work is known as Cressie-Read family, which not only constitutes the analytical basis of our analysis but also represents a solution to the issue of solving ill-posed inverse problems by formally treating them as inference problems. Our results indicate that the performance of functionals constituting the CR family may vary significantly: in some cases, the likelihood functional (to the best of our knowledge, explicitly worked out here for the first time) provides the best performance; in others, it is outperformed by the Shannon functional. This indicates these two functionals are the ones making the best possible use of the available information, predicting the closest values to the observed ones.

In order to suggest applicability of our procedure, we have considered behavioral problems within the framework of network theory. The results we obtained not only indicate the effectiveness of our algorithm (applicable to univariate as well as bivariate data sets and for both

Given the importance of recovering dynamic economic behavioral information, a natural question arises about the continued use of traditional regularization information recovery methods as a solution basis for traditional pure and stochastic inverse type problems. For this reason, the next step is to extend the concept of adaptive-optimizing behavior and apply it (within the information theoretic framework) in the context of a range of micro economic settings, thus opening the promising perspective of turning the descriptive character of behavioral disciplines into a more quantitative one.

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TS acknowledges support from the Italian PNR project CRISIS-Lab.

ESG acknowledges support from the European Commission Marie-Curie ITN program (FP7-320 PEOPLE-2011-ITN) through the LINC project (no. 289447).

DG acknowledges support from the Dutch Econophysics Foundation (Stichting Econophysics, Leiden, the Netherlands). This work was also supported by the project MULTIPLEX (contract 317532) and the Netherlands Organization for Scientific Research (NWO/OCW).

The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.