Smartphones are shortly evolving and their buildings are straightforward to determine. Thus, the relationships between (anti-) motifs and the product structural robustness of 25 generations of smartphones starting from June 2007 to October 2020 are analysed to reveal the proposed methodology. First, the product information of 25 generations of smartphones are collected from the iFixit web site36. iFixit gives detailed details about the teardown of smartphones. Often, a smartphone consists of six fundamental parts (display screen meeting, digital camera, logic board, buttons, battery, and canopy), involving greater than 50 elements. (The connection between elements of every product is supplied within the Supplementary info file). Then, based mostly on the community mannequin proposed in “Building of product structural networks” Part, the elements are represented as nodes and the relationships between elements are represented as edges to type the product networks. As proven in Fig. 7, the dimensions of every node straight depends on the node diploma within the product networks. The precise variety of nodes and edges for every technology is listed in Desk 2.

Determine 7
figure 7

Product structural networks of the 25 generations of smartphones.

Desk 2 Statistics of nodes and edges of the 25 generations of smartphones.

Motifs and anti-motifs in product structural networks

The prevalence variety of every meta-structure in each product structural networks and random networks is countered via the enumeration algorithm proposed in “Identification of motifs based mostly on the enumeration algorithm” part. Subsequent, the Z rating of every motif in product structural networks is analysed based mostly on Eq. (2). In response to the worth of the Z rating (whether or not Z rating > 2 or not), the sorts of every meta-structure are distinguished. As proven in Fig. 8, the meta-structures of M1 and M7 are all the time anti-motifs within the 25 generations of product structural networks. In distinction, the meta-structure of M2 is all the time the motif. As well as, the meta-structures of M3, M4, M5, M6, and M8 all have a excessive chance of being motifs and a low chance of being anti-motifs. For instance, M3 has a 92% chance of being a motif within the 25 product structural networks. Thus, the meta-structures of M2, M3, M4, M5, M6, and M8 are considered motifs from the worldwide perspective. The meta-structures of M1 and M7 are considered the anti-motif.

Determine 8
figure 8

Motifs and anti-motifs of the 25 product structural networks.

As proven in Desk 3, the variety of every motif and anti-motif is countered. Then their frequencies are calculated in response to Eq. (2), as proven in Desk 4.

Desk 3 Variety of every (anti-) motif within the 25 product structural networks.
Desk 4 Frequencies of every (anti-) motif within the 25 product structural networks.

Robustness of product structural networks

As proven in Desk 5, the robustness for the 25 generations of product structural networks is obtained in response to Eq. (4). As proven in Fig. 9, product structural robustness underneath the random failure of elements is greater than 3 instances the robustness underneath FKC. As well as, the robustness underneath random assaults for the product generations fluctuates within the interval of [0.365, 0.390]. Nevertheless, the robustness underneath FKC is smaller than 0.12. With the evolution of the product construction, it regularly decreases. That is brought on by the growing diploma of inside integration of smartphones. With the failure of the important thing elements, many extra elements related with them have a excessive chance of failure. Remarkably, as a result of progress and maturity of expertise, the efficiency of elements is enhancing, and the chance of failures of the elements can be lowered. This examine solely considers the scenario after FKC however doesn’t goal at whether or not the important thing elements will fail.

Desk 5 Robustness of the 25 generations of product structural networks.
Determine 9
figure 9

Product structural robustness underneath random assaults and FKC.

Evaluation of relationships between motifs and product structural robustness based mostly on principal part regression

As proven in Fig. 10, the correlations between some impartial variables are greater than these between the dependent variable and a few impartial variables. For instance, the correlation between f2 and φ is 0.808, and the correlation between f2 and f3 is 0.902. To cut back multicollinearity, PCR is carried out.

Determine 10
figure 10

Correlation evaluation matrix.

As proven in Desk 6, the eigenvalues of C1, C2, and C3 are all bigger than 1, and their cumulative variance accounts for greater than 92%. Subsequently, three principal elements (C1, C2, and C3) are chosen to characterize the 8 impartial variables. Then the regression mannequin is established in response to Eq. (10). The detailed parameters of the regression equation are listed in Desk 7. The p for the regression mannequin is the same as 0, which suggests the regression mannequin is important. The p values for the coefficients of C1, C2, and C3 are all smaller than 0.01, which implies that C1, C2, and C3 all have a major impact on φ inside a 99% confidence interval. The variance inflation issue (VIF) for the coefficients of C1, C2, and C3 are all equal to 1, which is smaller than 5. Subsequently, the multicollinearity on this mannequin is comparatively weak.

Desk 6 Cumulative variance of the principal elements.
Desk 7 Detailed parameters of the regression mannequin.

Then the coefficients for the eight sorts of (anti-) motifs will be obtained in response to Eq. (11). The connection between product structural robustness and the frequency of eight sorts of (anti-) motifs is

$$varphi = – 0.{17}f_{{1}}^{^{prime}} + 0.{78}f_{{2}}^{^{prime}} + 0.{77}f_{{3}}^{^{prime}} + 0.{73}f_{{4}}^{^{prime}} + 0.{7}0f_{{5}}^{^{prime}} + 0.{51}f_{{6}}^{^{prime}} – 0.{18}f_{{7}}^{^{prime}} – 0.{58}f_{{8}}^{^{prime}} + 0.0{63}.$$

If the coefficient is bigger than 0, then the corresponding (anti-) motif has a constructive impact on product structural robustness; in distinction, if the coefficient is smaller than 0, the corresponding (anti-) motif is unfavourable for product structural robustness. The bigger absolutely the worth of the coefficient is, the extra important the corresponding (anti-) motif’s impact on product structural robustness. For instance, the coefficients of f1, f7, and f8 are smaller than 0, so the corresponding (anti-) motifs of M1, M7, and M8 are all unfavourable on product structural robustness. In the meantime, absolutely the worth of the coefficient for f8 is 0.58, which is bigger than that of f1 and f7. Subsequently, the unfavourable impact on the product structural robustness of M8 is way more important. Equally, the coefficients of f2, f3, f4, f5, and f6 are all bigger than 0. Subsequently, the corresponding motifs of M2, M3, M4, M5, and M6 all play a constructive position in product structural robustness. Absolutely the worth of f5 is the smallest within the coefficients of M2, M3, M4, M5, and M6. Thus, it has a comparatively small constructive affect on product structural robustness.

As proven in Desk 8, 4 observations will be concluded. (1) All anti-motifs (M1 and M7) are unfavourable for product structural robustness. (2) Most motifs have a constructive impact on product structural robustness, besides M8. (3) Motifs that include a loop construction are constructive for product structural robustness. (For instance, motifs of M2, M3, M4, and M5 have a three-node loop construction, and M6 has a four-node loop construction. All of them have a constructive impact on product structural robustness. In distinction, M1, M7, and M8 should not have a loop construction and are unfavourable for product structural robustness.) (4) The constructive impact on product structural robustness of motifs with the three-node loop construction is greater than that of motifs with a four-node loop construction. Motifs (M2, M3, M4, and M5) with a three-node loop construction all have a coefficient bigger than 0.7, and the motif (M6) with a four-node loop construction solely has a coefficient of 0.51.

Desk 8 Impact of (anti-) motifs on product structural robustness.

Enhance product structural robustness based mostly on three motif-based methods

In response to the methods proposed in “Motif-based methods to enhance product structural robustness” part and the observations concluded in “Robustness of product structural networks” part, three detailed motif-based methods are analysed to enhance product structural robustness.

Technique 1: Enhance the frequencies of motifs with loop buildings in product structural networks

As proven in Fig. 11, the digital camera is related to the logic board and guarded by the digital camera ring on the unique product technology. The digital camera ring can solely restrict the liberty of the digital camera in 5 instructions, which ends up in the opportunity of longitudinal loosening of the digital camera. Then, the digital camera tends to be structurally unstable and it could not totally implement its features. In later generations, designers added a digital camera bracket to safe the digital camera. The digital camera bracket types a four-node loop construction with different elements to higher keep the soundness of the digital camera with out affecting the construction or operate of different elements. Subsequently, including motifs with loop buildings is useful for the development of product structural robustness.

Determine 11
figure 11

A four-node loop construction was added to maintain the digital camera secure. (Tailored in part with free permission from iFixit, https://www.ifixit.com/Teardown/iPhone/. Licenced underneath the CC BY open entry licence).

In the meantime, the simulation of accelerating the frequencies of motifs with the loop construction is carried out to analyse its impact on product structural robustness. As proven in Fig. 12, we improve the frequency of motifs with three-node and four-node loop buildings within the structural community of P1. (Remarkably, as a result of the correlation coefficients between M2, M3, M4, and M5 are excessive (see Fig. 10); the coefficients of the regression mannequin for them are very shut (see Desk 8); and they’re all shaped by a three-node loop construction, M2 is chosen to characterize M3, M4, and M5 on this part). As proven in Fig. 12a, because the frequency of M2 regularly will increase, product structural robustness additionally will increase adopted by a step sample. This step sample is brought on by the totally different results on the community connectivity of motifs which are shaped by nodes with totally different properties.

Determine 12
figure 12

Impact of motifs with the three-node and four-node loop construction on φ.

As proven in Fig. 13b, the added motif shaped by node 2 and its hanging nodes 1 and three doesn’t successfully improve the connectivity of the community. (Hanging nodes are these nodes that solely have one edge). Subsequently, the biggest related cluster doesn’t considerably improve when the community suffers from FKC, which ends up in the development of robustness being not apparent in contrast with the unique community (see Fig. 13a). As proven in Fig. 13c, if M2 is shaped by handing node 1 and different nodes (not the hanging nodes of node 2), the connectivity of the community is considerably improved. Then, the community robustness is improved too. As proven in Fig. 7, there are various hanging nodes within the product structural community. Subsequently, when the added M2 is shaped by the node and its two hanging nodes, product structural robustness solely will increase barely, as proven in Fig. 12a.

Determine 13
figure 13

Robustness of the community underneath totally different formations of M2.

Equally, as proven in Fig. 12b, with the rise within the frequency of the four-node loop construction (M6), product structural robustness additionally will increase. Though there nonetheless exists a step sample within the uptrend, the principle pattern is a rising line. It’s because the four-node loop construction makes it simpler for these hanging nodes to attach with different nodes. Subsequently, it’s simpler to enhance the general connectivity and the robustness of the community.

Technique 2: Scale back the frequencies of (anti-) motifs of M
1, M
7, and M
8 in product structural networks

Each occasion evaluation and simulation are carried out to analyse the impression of this technique on enhancing product structural robustness. As proven in Fig. 14, within the left product, the LCD connects to the digitizer they usually each transmit the knowledge to the logic board via the digitizer cable. The LCD, the digitizer (with digitizer cable), and the logic board type an anti-motif of M1. If the digitizer fails, the LCD can not work both. In distinction, as proven within the product in the best part of Fig. 14, the LCD transmits the knowledge to the logic board by the LCD cable. The construction of M1 modifications to M2 and the frequency of M1 is lowered. Then, the LCD and the digitizer can work independently and the failure of considered one of them doesn’t have an effect on the work of the opposite. Subsequently, product structural robustness is improved by decreasing the frequency of M2. Equally, the frequency of M7 and M8 may also be lowered by altering their construction to enhance product structural robustness.

Determine 14
figure 14

Scale back the frequency of M1 to enhance product structural robustness. (Tailored in part with free permission from iFixit, https://www.ifixit.com/Teardown/iPhone/. Licenced underneath the CC BY open entry licence).

As proven in Fig. 15, with the discount of the frequencies of those (anti-) motifs, product structural robustness regularly will increase. For instance, with the discount of the frequency of M1, product structural robustness is linearly growing. Product structural robustness sharply will increase when the frequency of M7 decreases from 0.007 to 0.004; thereafter, the expansion price turns into decrease. It’s because your complete community connectivity will increase shortly with the change within the frequency of M7; then, when the frequency of M7 reduces to a sure diploma, the community connectivity will increase slowly. Equally, because the frequency of M8 regularly decreases, product structural robustness will increase quickly first after which comparatively slowly.

Determine 15
figure 15

Change of φ with the discount of f1, f7, and f8.

Technique 3: Defending the important thing elements and fragile elements in (anti-) motifs

As proven in Desk 9, M1 consists of a key part (rear case) and two different elements (lightning connector and loudspeaker). The rear case is necessary in fixing varied elements. If it fails, each the lightning connector and loudspeaker can not stay secure. In contrast with M1, M8 additionally consists of a key part and lots of different elements with no relationship. As proven in Desk 9, many chips are related to the logic board to perform varied features. If the logic board fails, all of the chips can be disabled, and the smartphone can’t be used. The soundness of M8 has a major affect on product operate efficiency. The construction of M7 is a single hyperlink construction to carry out a selected operate. For instance, the operate movement in Desk 9 about M7 will be represented as follows: battery → logic board → lighting connector cable → lightning connector. If any part within the operate movement fails (particularly the lighting connector cable which is fragile), this operate can’t be carried out. To maintain the product way more sturdy, designers ought to deal with the important thing elements and fragile elements in these (anti-) motifs. In the meantime, designers have to develop a selected plan for the safety and common inspections.

Desk 9 Examples of eight sorts of (anti-) motifs (Tailored in part with free permission from iFixit, https://www.ifixit.com/Teardown/iPhone/. Licenced underneath the CC BY open entry licence).



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