Online Social Space Identification. A Computational Tool for Optimizing Social Recommendations
Abstract
:1. Introduction
2. Related Work
- To handle complex relationships among the networks’ elements [18];
3. The “Social Space Identification” Problem
3.1. Problem Formulation
3.2. Main Notation and Definitions
4. Approximate Solutions
4.1. Chromosome Representation
4.2. Fitness Function
4.2.1. Case 1
- From one side, Equation (1) promotes chromosomes that identify, around feasible vertices, different (types of) services operated by HSPs and different (types of) services required by targets, i.e., we promote uniform distributions of the supply and demand of services.
- From the other side, we promote chromosomes that identify, around feasible vertices, the participation of different types of entities (here, we focus on two types of entities only: HSPs and target users, respectively), i.e., we promote uniform user-/resource-type distributions around feasible .
4.2.2. Case 2
4.3. Mutation
- Mutation Operator 1:Let c be a chromosome observed during some step of the evolution process. Assume that, at such a step of evolution, , the current hypothesis conjectured by c, is not feasible for Problem 1. Moreover, let be the subgraph induced by . In order to provide the system’s compliance to get feasible solutions, to obtain feasible two-clubs sparingly, we randomly sampled the vertices , and for each , we checked whether the minimum length between and v was . If this test was negative, then was flipped to zero, thus orienting the system towards the feasibility.
- Mutation Operator 2:Through this operator, we aimed to increment the size of a feasible solution. Let and be defined as above. We randomly sampled from , and then, we checked if the shortest distance of from was . If this test was negative, then was flipped to one.
- Mutation Operator 3:In this case, a standard mutation procedure was applied: bits were randomly switched either “on” or “off”.
4.4. Cross-Over
- Logical AND/OR cross-over: New offspring was generated by applying AND/OR logic operations on parent chromosomes.
- Standard cross-over: This operation is typical and is often used in standard applications of genetic algorithms. Parent chromosomes are reported and mixed in new descendants.
5. Distributed Learning
A Genetic Cascade Model
6. Numerical Experiments
6.1. Centralized Learning
- All models except identified two-clubs correctly.
- We considered the average value of the entropy for the health service and vertex-type distributions around expert patients. It is worth emphasizing that the system returned solutions whose entropy could not be associated with distributions having the whole probability mass centered on any specific value. Notice that, since only RD patients and HSP could be found around EX users, then we had . Similarly, as we had , i.e., five types of services were considered in our experiments, then was such that .
- Moreover, notice that, at least of the output nodes were identified as feasible. Larger values are reported for large input networks, e.g., or . This was a compelling property, for example, in the case of large communities.
- System time seemed reasonable (T2 s) for the applied instances.
6.2. Distributed Learning
- Input/Output diameters: the input graph diameter and output graph diameter proposed by the best chromosome solutions.
- Output vertices: the number of final vertices obtained in the chromosome solution.
- Fitness value: as described in Section 4.
- Ratio between the number of input vertices and the number of vertices within the two-club represented by the chromosome solution.
- Average CPU user time: for a single processing unit, i.e., standard GA evolution, this quantity corresponded to the (GA) execution time; when applying distributed learning, the whole execution time was averaged over the number of (framework) levels.
- Early stopping: the number of consecutive generations without improvement of the fitness value; the GA execution was stopped after the “early stopping”.
- Max number of generations: the maximum number of iterations before the GA search was halted.
- Final generation number: the iteration number associated with the final solution. Notice that, in our case (i.e., distributed evolution), this number corresponded to the iterations of the lowest site.
- All models, except , returned correct two-clubs. This was an interesting result when large input graphs (e.g., more than 1000 vertices in our table) were considered.
- Similarly to previous experiments, as we could not confront the problem’s optimal solution, to evaluate the solution proposed by the genetic algorithm qualitatively, we report the ratio between the number of vertices of the input and the output graphs. Notice that the approximability of the problem was very hard to obtain (not approximable within a factor , for each [47]). Considering our results, in particular the ratio in Table 3, we concluded that the most interesting solutions were those for which the larger the input graph (number of vertices), the lower the difference between the ratio and the unit.
- Although the size of the identified communities did not seem to differ, the (average) number of iterations of the last level site (distributed execution) was much lower than the iterations reported by the standard, centralized evolution. This was also evident from the decrease in the average execution time per level, reported by the distributed process. Since GAs use the same parameters for the standard and the distributed evolution, this behavior could be traced back to the initialization that each lower level site received from the higher levels.
- The computational cost seemed to depend on both the edge numbers (i.e., high expected connectivity of random models ) and the number of input vertices. As the fitness computational complexity was related to the diameter computation, this assertion could be justified considering the diameter computational cost, which is known to be bounded by .
7. Conclusions
- We formulated a computational optimization problem for the future suggestions of the SENIOR recommender system.
- Algorithmic solutions were proposed as well, based on evolutionary heuristics, both for centralized and parallel processing environments.
- As reported in recent studies, using additional user and item relationships could improve the recommendation quality [48,49]. In this perspective, the optimization objective formulated in Equation (1) will be further constrained by taking into account relationships between target users. These relationships could be interpretable as either “friendships between target users” or “targets sharing similar experiences”. The feasible solutions (i.e., two-clubs) optimized with such a type of connection would allow the future recommendation system to induce even more compelling communication among users, perhaps looking for similar reliable resources.Furthermore, edges among “experts ” (within the identified two-clubs) could be used as an endorsement, thus allowing target users to evaluate new different experiences or the consistency of the information of other expert users. Such links could also induce more effective communication between experts to facilitate final patient support. Similarly, a constrained optimization accounting for “resource to resource” relationships, within the identified community, could offer the user equivalent available resources.
- In Section 4.2, we presented a general fitness formulation for two different evaluations of a free parameter (). A more context-based formulation able to manage real needs will certainly provide more effective results in this sense. As an example, we can consider the first reported case (Number 1), where is related to the health service and vertex-type distributions. In this situation, a linear combination of entropies is used to optimize (and balance) an induced two-club community. Direct comparisons (e.g., using cross-entropy evaluations) of user requirement distribution vs. service distribution could be similarly evaluated for future numerical experiments.
- Finally, we assumed so far that each provider could supply one service only. In fact, this was “coded” by the labeling function h discussed in Section 3.2. It is straightforward to extend this labeling to multiple services for a more realistic implementation.
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Models | AvInD | AvOutD | OutN | T1 | T2 | HTS | HVS | TI | FeasA |
---|---|---|---|---|---|---|---|---|---|
ER(50,0.15) | 3.33 | 2 | 14 | 169 | 3.31 | 0.51 | 0.6 | 1 | 0.123 |
ER(100,0.15) | 3 | 2 | 26 | 401 | 6.53 | 0.62 | 0.757 | 1 | 0.283 |
ER(200,0.15) | 3 | 1 | 15 | 233 | 2.89 | 0.66 | 0.755 | 0.3 | 0.12 |
ER(300,0.15) | 3 | 2 | 249 | 3510 | 5.40 | 0.67 | 0.989 | 1 | 0.359 |
ER(50,0.3) | 3 | 2 | 39.7 | 684 | 6.87 | 0.65 | 0.93 | 1 | 0.42 |
ER(100,0.3) | 2 | 2 | 42.7 | 1553 | 7.2 | 0.67 | 0.92 | 1 | 0.30 |
ER(150,0.3) | 2 | 2 | 147 | 1750 | 4.17 | 0.67 | 0.95 | 1 | 0.32 |
Centralized Execution | ||||
---|---|---|---|---|
Model | InpDiam | OutDiam | OutN | OutP |
ER(150,0.1) | 3.3 (0.48) | 2 (0) | 18.5 (3.53) | 25.5 (21.20) |
ER(150,0.2) | 3 (0) | 2 (0) | 52.8 (38.30) | 387.4 (633.60) |
ER(500,0.1) | 3 (0) | 3.4 (3.30) | 16.9 (9.60) | 34.9 (36) |
ER(500,0.2) | 2 (0) | 2 (0) | 430.3 (68.50) | 20840.9 (6164) |
ER(1500,0.1) | 2.43 (0.51) | 2 (0) | 105.4 (52.60) | 1278.5 (983.30) |
Distributed Execution | ||||
ER(150,0.1) | 3.2 (0.42) | 2 (0) | 18.8 (3.12) | 28.9 (18.30) |
ER(150,0.2) | 3 (0) | 2 (0) | 53.3 (25.60) | 294.6 (298.90) |
ER(500,0.1) | 3 (0) | 2.6 (1.1) | 42.6 (49) | 183.5 (496.30) |
ER(500,0.2) | 2 (0) | 2 (0) | 346.6 (110.30) | 14519.9 (9588.90) |
ER(1500,0.1) | 2.25 (0.46) | 2 (0) | 104.6 (22.70) | 1180.8 (645.30) |
Centralized Execution | ||||
---|---|---|---|---|
Model | Fit | Iter | CPU Time | Ratio |
ER(150,0.1) | 18.5 (3.53) | 88.6 (14.90) | 43.47 (14.90) | 8.10 |
ER(150,0.2) | 52.8 (38.30) | 267.6 (147.30) | 263.8 (120.10) | 2.84 |
ER(500,0.1) | 16.9 (9.60) | 64.74 (27.32) | 295.83 (163.40) | 29.60 |
ER(500,0.2) | 430.3 (68.50) | 46.09 (70.86) | 910.37 (1580.29) | 1.16 |
ER(1500,0.1) | 105.4 (52.60) | 408.43 (235.08) | 1171.79 (1056.60) | 14.23 |
Distributed Execution | ||||
ER(150,0.1) | 18.8 (3.12) | 20.20 (5.2) | 47.50 (21.7) | 7.97 |
ER(150,0.2) | 53.3 (25.60) | 59.80 (32.4) | 77.56 (32.13) | 2.81 |
ER(500,0.1) | 42.6 (49.00) | 13.29 (10.57) | 119.80 (233.65) | 11.73 |
ER(500,0.2) | 346.6 (110.30) | 4.33 (6.40) | 107.89 (522.38) | 1.44 |
ER(1500,0.1) | 104.6 (22.70) | 112.50 (10.35) | 234.51 (247.24) | 14.34 |
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Zoppis, I.; Trentini, A.; Manzoni, S.; Micucci, D.; Mauri, G.; Pietrabissa, G.; Castelnuovo, G. Online Social Space Identification. A Computational Tool for Optimizing Social Recommendations. Appl. Sci. 2020, 10, 3024. https://doi.org/10.3390/app10093024
Zoppis I, Trentini A, Manzoni S, Micucci D, Mauri G, Pietrabissa G, Castelnuovo G. Online Social Space Identification. A Computational Tool for Optimizing Social Recommendations. Applied Sciences. 2020; 10(9):3024. https://doi.org/10.3390/app10093024
Chicago/Turabian StyleZoppis, Italo, Andrea Trentini, Sara Manzoni, Daniela Micucci, Giancarlo Mauri, Giada Pietrabissa, and Gianluca Castelnuovo. 2020. "Online Social Space Identification. A Computational Tool for Optimizing Social Recommendations" Applied Sciences 10, no. 9: 3024. https://doi.org/10.3390/app10093024
APA StyleZoppis, I., Trentini, A., Manzoni, S., Micucci, D., Mauri, G., Pietrabissa, G., & Castelnuovo, G. (2020). Online Social Space Identification. A Computational Tool for Optimizing Social Recommendations. Applied Sciences, 10(9), 3024. https://doi.org/10.3390/app10093024