# Filtering Statistics on Networks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Filtering Statistics on a Ring

#### 2.1. Degeneracy Distribution

#### 2.2. Effect of Filter Length

## 3. Filtering on Graphs

#### 3.1. Degeneracy Distributions

#### 3.2. Behaviour of Important Quantities

^{th}root of $M\left(n\right)$, which tends to ${z}_{g}$ for large n, approaches 2 for large q. By the same token, most outputs have a degeneracy of one, so the number of outputs of degeneracy one, $N(1,n)$ also approaches ${2}^{n}$ (${z}_{a}$ approaching 2) for large q (panel (f)), with while the largest degeneracy ${d}_{D}\left(n\right)$ (whose asymptotic behaviour is given by ${z}_{d}$) only grows slowly with n, (panel (d)). The resolution $H\left[y\right]$ measures how well the filter distinguishes different inputs, and as we see in panel (b) of Figure 6, and in agreement with the above observations, the resolution for the weak filter is high. The maximum possible value of $H\left[y\right]$ is $nln2$, which corresponds to a value of $H\left[y\right]/n=0.693...$ in the figure. We see that the resolution is already close to this value at $q=5$.

## 4. Discussion

## 5. Materials and Methods

#### 5.1. Calculation of Degeneracy Distributions

#### 5.2. Asymptotics of the Degeneracy Distribution on Rings

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Further Results for the Weak Rule Filter

**Figure A1.**Degeneracy distributions (

**left**) and cumulative degeneracy distributions (

**right**) for outputs of the WR filter on selected deterministic graphs of degree 2 (

**a**,

**b**) 3 (

**c**,

**d**) and 4 (

**e**,

**f**).

**Figure A2.**Degeneracy distributions and cumulative degeneracy distributions for outputs of the WR filter on random regular graphs of degree 2 (

**a**,

**b**) 3 (

**c**,

**d**) and 4 (

**e**,

**f**).

**Table A1.**Important values for the degeneracy distribution resulting from applying the weak rule (WR) filter to various graphs. The numbers $\sqrt[{\scriptstyle n}]{M\left(n\right)}$, $\sqrt[{\scriptstyle n}]{{d}_{D}\left(n\right)}$ and $\sqrt[{\scriptstyle n}]{N(1,n)}$ approximate ${z}_{g}$, ${z}_{d}$ and ${z}_{a}$ respectively. We also give the relevance per node $H\left[d\right]/n$ and the resolution per node $H\left[y\right]/n$. Numbers for RRG(q) and SW(q) were obtained by averaging over 10 random realizations.

Graph | n | $\sqrt[{\scriptstyle \mathit{n}}]{\mathit{M}\left(\mathit{n}\right)}$ | $\sqrt[{\scriptstyle \mathit{n}}]{{\mathit{d}}_{\mathit{D}}\left(\mathit{n}\right)}$ | $\sqrt[{\scriptstyle \mathit{n}}]{\mathit{N}(1,\mathit{n})}$ | $\mathit{H}\left[\mathit{d}\right]/\mathit{n}$ | $\mathit{H}\left[\mathit{y}\right]/\mathit{n}$ |
---|---|---|---|---|---|---|

Apollonian 2 | 7 | 1.95461 | 1.21901 | 1.91660 | 0.11519 | 0.66045 |

Apollonian 3 | 16 | 1.94788 | 1.43435 | 1.91189 | 0.09594 | 0.64711 |

(3,5)-cage | 10 | 1.91202 | 1.21481 | 1.84295 | 0.13705 | 0.62974 |

(3,6)-cage | 14 | 1.91394 | 1.34590 | 1.83757 | 0.12271 | 0.63149 |

(3,7)-cage | 24 | 1.91348 | 1.25055 | 1.83511 | 0.11462 | 0.63259 |

(3,8)-cage | 30 | 1.91330 | 1.34897 | 1.83337 | 0.10559 | 0.63275 |

(4,5)-cage | 19 | 1.95248 | 1.21101 | 1.91027 | 0.08217 | 0.65878 |

(4,6)-cage | 26 | 1.95322 | 1.37995 | 1.91085 | 0.07188 | 0.65902 |

(5,5)-cage 1 | 30 | 1.97461 | 1.16392 | 1.95220 | 0.04494 | 0.67453 |

(5,5)-cage 2 | 30 | 1.97461 | 1.18854 | 1.95219 | 0.04495 | 0.67453 |

(5,5)-cage 3 | 30 | 1.97461 | 1.21540 | 1.95220 | 0.04496 | 0.67453 |

(5,5)-cage 4 | 30 | 1.97461 | 1.17585 | 1.95220 | 0.04495 | 0.67453 |

torus $3\times 3$ | 9 | 1.95698 | 1.16653 | 1.92324 | 0.10088 | 0.66192 |

torus $4\times 4$ | 16 | 1.95546 | 1.38485 | 1.92191 | 0.09024 | 0.65777 |

torus $5\times 5$ | 25 | 1.95475 | 1.21993 | 1.91904 | 0.08076 | 0.65828 |

torus $6\times 5$ | 30 | 1.95475 | 1.28517 | 1.91898 | 0.07568 | 0.65831 |

torus $10\times 3$ | 30 | 1.95626 | 1.22522 | 1.91924 | 0.07072 | 0.66127 |

torus $8\times 4$ | 32 | 1.95510 | 1.38392 | 1.91932 | 0.07475 | 0.65813 |

torus $6\times 6$ | 36 | 1.95475 | 1.38400 | 1.91883 | 0.07034 | 0.65833 |

SWB(3) | 10 | 1.91492 | 1.35588 | 1.85212 | 0.14568 | 0.62873 |

SWB(3) | 20 | 1.91523 | 1.30100 | 1.84849 | 0.12731 | 0.62956 |

SWB(3) | 30 | 1.91523 | 1.35620 | 1.84851 | 0.11281 | 0.62956 |

SWB(4) | 12 | 1.95603 | 1.17605 | 1.91983 | 0.09152 | 0.66087 |

SWB(4) | 21 | 1.95626 | 1.21231 | 1.91923 | 0.08079 | 0.66127 |

SWB(4) | 30 | 1.95626 | 1.20790 | 1.91924 | 0.07071 | 0.66127 |

SWB(5) | 12 | 1.97929 | 1.17605 | 1.96131 | 0.05848 | 0.67803 |

SWB(5) | 20 | 1.97927 | 1.16442 | 1.96025 | 0.04881 | 0.67840 |

SWB(5) | 32 | 1.97927 | 1.14893 | 1.96021 | 0.04144 | 0.67842 |

RRG(2) | 10 | 1.76075 | 1.33214 | 1.62827 | 0.16029 | 0.53010 |

RRG(2) | 20 | 1.81141 | 1.29593 | 1.65746 | 0.14345 | 0.56553 |

RRG(2) | 30 | 1.83115 | 1.27447 | 1.67090 | 0.11728 | 0.58305 |

RRG(3) | 10 | 1.86350 | 1.31014 | 1.73760 | 0.18497 | 0.59914 |

RRG(3) | 20 | 1.88987 | 1.28690 | 1.78608 | 0.13766 | 0.61721 |

RRG(3) | 30 | 1.89895 | 1.27941 | 1.80483 | 0.11537 | 0.62310 |

RRG(4) | 10 | 1.92754 | 1.24293 | 1.86647 | 0.13602 | 0.64106 |

RRG(4) | 20 | 1.93917 | 1.25076 | 1.88272 | 0.09610 | 0.65019 |

RRG(4) | 30 | 1.94507 | 1.25247 | 1.89654 | 0.07819 | 0.65347 |

RRG(5) | 10 | 1.93764 | 1.26982 | 1.88340 | 0.12318 | 0.64808 |

RRG(5) | 20 | 1.95952 | 1.23790 | 1.92369 | 0.07520 | 0.66371 |

RRG(5) | 30 | 1.96616 | 1.22872 | 1.93641 | 0.05720 | 0.66839 |

SW(3) | 10 | 1.90692 | 1.25638 | 1.82855 | 0.15381 | 0.62737 |

SW(3) | 20 | 1.90025 | 1.27244 | 1.80865 | 0.13264 | 0.62404 |

SW(3) | 30 | 1.91093 | 1.27260 | 1.82941 | 0.10905 | 0.63105 |

SW(4) | 10 | 1.91972 | 1.26865 | 1.85154 | 0.14500 | 0.63556 |

SW(4) | 20 | 1.93992 | 1.25544 | 1.88853 | 0.09904 | 0.64926 |

SW(4) | 30 | 1.94584 | 1.26608 | 1.89776 | 0.07874 | 0.65403 |

SW(5) | 10 | 1.94942 | 1.23419 | 1.90650 | 0.11008 | 0.65650 |

SW(5) | 20 | 1.96150 | 1.23542 | 1.92709 | 0.07365 | 0.66525 |

SW(5) | 30 | 1.96889 | 1.23140 | 1.94184 | 0.05471 | 0.67024 |

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**Figure 1.**The application of different filters to a set of zeros and ones place on a graph. Each node of the input and output graphs is in one of two states, namely 0 (open circles) or 1 (closed circles). In the strong rule (SR) filter, an output node is one only when the corresponding input node is one and all of its neighbours are zero. In the weak rule (WR) filter, an output node is one when the corresponding input node is one and one or more of its neighbours are zero.

**Figure 2.**Degeneracy distribution (

**a**) and cumulative degeneracy distribution (

**b**) for the filter 010 on a ring, and for its generalization on a torus, which is a 1 with four neighboring 0’s (panels (

**c**,

**d**)). These illustrate the typically broad, but complex, degeneracy distribution. The cumulative distribution exhibits a complex staircase like structure in the tail.

**Figure 3.**Evolution of the degeneracy distribution (

**left**) and cumulative degeneracy distributions (

**right**) with network degree for outputs of the SR filter on random regular graphs. There is a dramatic difference in the density and complexity of the distributions between degree 2 (

**a**,

**b**) and degree 3 (

**c**,

**d**). For degree 4 (

**e**,

**f**), the distributions appear to be similar, but are less broad.

**Figure 4.**Degeneracy distributions and cumulative degeneracy distributions for outputs of the SR filter on selected deterministic graphs of degree 2 (

**a**,

**b**) 3 (

**c**,

**d**), and 4 (

**e**,

**f**). We do not see the same increase in complexity for degrees that are greater than two that we observe in random graphs.

**Figure 5.**Degeneracy distributions and cumulative degeneracy distributions for outputs of different filters on selected deterministic graphs, showing a variety of possible distribution shapes. (

**a**,

**b**) The filter ‘00’ on a ring, (

**c**,

**d**) ‘01’ on a ring, (

**e**,

**f**) SR filter on an Apollonian graph with degree 3, and (

**g**,

**h**) WR filter on the same Apollonian graph.

**Figure 6.**Dependence of key observables related to the degeneracy distribution on graph degree q. (

**a**) The relevance entropy $H\left[d\right]$ scaled by system size n. (

**b**) Resolution $H\left[y\right]$. (

**c**) Total number of degeneracies $D\left(n\right)$ for $n=30$. (

**d**) The n

^{th}root of the largest degeneracy ${d}_{D}\left(n\right)$, which tends to ${z}_{d}$. (

**e**) The n

^{th}root of the number of outputs $M\left(n\right)$, tending to ${z}_{g}$. (

**f**) The n

^{th}root of the number of outputs of degeneracy onem $N(1,n)$, tending to ${z}_{a}$. There is a clear difference in the relevance entropy between random graphs and deterministic graphs under the SR filter, supporting the conclusion that this measure of entropy is the most informative. The entropy increases with a decreasing degree, except for degree two, which shows a sharp drop due to the highly predictable structure of degree two networks. These features are also reflected in the number of degeneracies, underlining the importance of studying the degeneracy distribution. The other quantities are almost entirely dependent on only network degree, and on which filter is used.

**Table 1.**Values of the numbers ${z}_{g}$, ${z}_{d}$, and ${z}_{a}$ for different filters. Note that we also included filter patterns that consist of all zeroes. For each filter, we also give the relevance per node $H\left[d\right]/n$ (in nats) calculated from the degeneracy distribution and the resolution per node $H\left[y\right]/n$. For the sake of comparison, the standard entropy of the inputs of this size is $H/n=ln2=0.69315$. Finally, we include the number of distinct degeneracies D for each pattern. Inputs of size $n=36$ were used, except for filters 00 and 10, for which $n=34$, and 000, for which $n=35$. Values for D for these three filters were extrapolated to $n=36$ for comparison with other results.

Pattern | ${\mathit{z}}_{\mathit{g}}$ | ${\mathit{z}}_{\mathit{d}}$ | ${\mathit{z}}_{\mathit{a}}$ | $\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\mathit{H}\left[\mathit{d}\right]/\mathit{n}$ | $\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\mathit{H}\left[\mathit{y}\right]/\mathit{n}$ | D | |
---|---|---|---|---|---|---|---|

0 (or 1) | 2 | 1 | 2 | 0 | 0.69315 | 1 | |

00 | 1.75488 | 1.61803 | 1.61803 | 0.18261 | 0.48468 | 924(1) | |

10 | 1.61803 | 1.31951 | 1 | 0.13954 | 0.46986 | 513(1) | |

010 | 1.61803 | 1.75488 | 1.46557 | 0.17248 | 0.35187 | 777 | |

000 | 1.61803 | 1.83929 | 1.49710 | 0.1453(1) | 0.30105 | 554(2) | |

0110 | $\}$ | 1.46557 | 1.86676 | 1.22074 | 0.11881 | 0.22387 | 698 |

0100 | |||||||

0000 | 1.52895 | 1.92756 | 1.41963(2) | 0.08856 | 0.17673 | 311 | |

00100 | 1.46557 | 1.9417 | 1.38028 | 0.06434 | 0.13371 | 291 | |

01110 | $\}$ | 1.38028 | 1.93318 | 1.16730 | 0.06312 | 0.13562 | 255 |

01100 | |||||||

01000 | |||||||

01010 | 1.44327 | 1.94789 | 1.32472 | 0.06117 | 0.12584 | 301 | |

00000 | 1.46557(2) | 1.96595 | 1.3652(2) | 0.05108 | 0.10052 | 190 | |

001100 | $\}$ | 1.38028 | 1.96931 | 1.2499(2) | 0.03606 | 0.07899 | 197 |

001000 | |||||||

010010 | 1.37108(1) | 1.97113 | 1.1938(5) | 0.03586 | 0.07692 | 218 | |

011110 | $\}$ | 1.32472 | 1.96717 | 1.13472 | 0.03448 | 0.07939 | 123 |

011100 | |||||||

011010 | |||||||

011000 | |||||||

010100 | |||||||

010000 | |||||||

000000 | 1.4176(2) | 1.98358 | 1.32486 | 0.02968 | 0.05606 | 123 | |

0110110 | 1.32472 | 1.98574 | 1.158(2) | 0.02084 | 0.04353 | 129 | |

0111110 | $\}$ | 1.28520 | 1.98386 | 1.11278 | 0.02016 | 0.04535 | 64 |

0111010 | |||||||

01111110 | 1.25542 | 1.99203 | 1.09698 | 0.01213 | 0.02546 | 36 | |

011111110 | 1.23205 | 1.99605 | 1.08507 | 0.00727 | 0.01411 | 25 | |

0111111110 | 1.21315 | 1.99803 | 1.07577 | 0.00427 | 0.00774 | 16 |

**Table 2.**Important values for the degeneracy distribution that results from applying the strong rule (SR) filter to various graphs. The numbers $\sqrt[{\scriptstyle n}]{M\left(n\right)}$, $\sqrt[{\scriptstyle n}]{{d}_{D}\left(n\right)}$ and $\sqrt[{\scriptstyle n}]{N(1,n)}$ approximate ${z}_{g}$, ${z}_{d}$, and ${z}_{a}$ respectively. We also give the relevance per node $H\left[d\right]/n$ and the resolution per node $H\left[y\right]/n$. Numbers for RRG(q) and SW(q) were obtained by averaging over 10 random realizations.

Graph | n | $\sqrt[{\scriptstyle \mathit{n}}]{\mathit{M}\left(\mathit{n}\right)}$ | $\sqrt[{\scriptstyle \mathit{n}}]{{\mathit{d}}_{\mathit{D}}\left(\mathit{n}\right)}$ | $\sqrt[{\scriptstyle \mathit{n}}]{\mathit{N}(1,\mathit{n})}$ | $\mathit{H}\left[\mathit{d}\right]/\mathit{n}$ | $\mathit{H}\left[\mathit{y}\right]/\mathit{n}$ |
---|---|---|---|---|---|---|

Apollonian 2 | 7 | 1.47236 | 1.94420 | 1.40854 | 0.08504 | 0.12919 |

Apollonian 3 | 16 | 1.52380 | 1.94596 | 1.49013 | 0.08148 | 0.13005 |

(3,5)-cage | 10 | 1.54199 | 1.88916 | 1.42694 | 0.10463 | 0.22185 |

(3,6)-cage | 14 | 1.54904 | 1.88549 | 1.46952 | 0.09741 | 0.22302 |

(3,7)-cage | 24 | 1.54516 | 1.88688 | 1.42191 | 0.12412 | 0.22268 |

(3,8)-cage | 30 | 1.54618 | 1.88722 | 1.44630 | 0.08763 | 0.22254 |

(4,5)-cage | 19 | 1.48991 | 1.94458 | 1.37494 | 0.08094 | 0.13458 |

(4,6)-cage | 26 | 1.50129 | 1.94386 | 1.44997 | 0.05243 | 0.13497 |

(5,5)-cage 1 | 30 | 1.44928 | 1.97192 | 1.34932 | 0.04164 | 0.07890 |

(5,5)-cage 2 | 30 | 1.44984 | 1.97191 | 1.35558 | 0.04602 | 0.07891 |

(5,5)-cage 3 | 30 | 1.44954 | 1.97192 | 1.35543 | 0.04201 | 0.07890 |

(5,5)-cage 4 | 30 | 1.44964 | 1.97191 | 1.35280 | 0.05264 | 0.07891 |

torus $3\times 3$ | 9 | 1.47967 | 1.94480 | 1.42350 | 0.07165 | 0.13112 |

torus $4\times 4$ | 16 | 1.51160 | 1.94843 | 1.46895 | 0.06205 | 0.13043 |

torus $5\times 5$ | 25 | 1.50066 | 1.94752 | 1.41779 | 0.05857 | 0.13132 |

torus $6\times 5$ | 30 | 1.50206 | 1.94754 | 1.42286 | 0.06159 | 0.13131 |

torus $10\times 3$ | 30 | 1.48922 | 1.94678 | 1.39796 | 0.05933 | 0.13100 |

torus $8\times 4$ | 32 | 1.50701 | 1.94785 | 1.44980 | 0.06251 | 0.13096 |

torus $6\times 6$ | 36 | 1.50405 | 1.94756 | 1.44490 | 0.05890 | 0.13130 |

SWB(3) | 10 | 1.55564 | 1.89336 | 1.48457 | 0.10987 | 0.21539 |

SWB(3) | 20 | 1.55376 | 1.89450 | 1.46394 | 0.09850 | 0.21540 |

SWB(3) | 30 | 1.55377 | 1.89450 | 1.46573 | 0.10256 | 0.21541 |

SWB(4) | 12 | 1.48818 | 1.94653 | 1.40063 | 0.07107 | 0.13103 |

SWB(4) | 21 | 1.48924 | 1.94678 | 1.39802 | 0.06401 | 0.13100 |

SWB(4) | 30 | 1.48922 | 1.94678 | 1.39797 | 0.06211 | 0.13100 |

SWB(5) | 12 | 1.43618 | 1.97359 | 1.32007 | 0.04433 | 0.07602 |

SWB(5) | 20 | 1.43469 | 1.97223 | 1.31634 | 0.03927 | 0.07765 |

SWB(5) | 32 | 1.43463 | 1.97225 | 1.31607 | 0.03597 | 0.07765 |

RRG(2) | 10 | 1.55934 | 1.77122 | 1.41900 | 0.15869 | 0.32044 |

RRG(2) | 20 | 1.60061 | 1.76297 | 1.45744 | 0.16195 | 0.33977 |

RRG(2) | 30 | 1.61251 | 1.75289 | 1.46125 | 0.16997 | 0.35053 |

RRG(3) | 10 | 1.49614 | 1.87903 | 1.30837 | 0.17514 | 0.21708 |

RRG(3) | 20 | 1.52503 | 1.87847 | 1.37793 | 0.20373 | 0.22357 |

RRG(3) | 30 | 1.54129 | 1.87706 | 1.41134 | 0.21442 | 0.22868 |

RRG(4) | 10 | 1.44023 | 1.93770 | 1.30201 | 0.11648 | 0.13463 |

RRG(4) | 20 | 1.48205 | 1.93399 | 1.36490 | 0.14077 | 0.14659 |

RRG(4) | 30 | 1.48439 | 1.93797 | 1.37705 | 0.13959 | 0.14166 |

RRG(5) | 10 | 1.41641 | 1.95111 | 1.27098 | 0.10038 | 0.11042 |

RRG(5) | 20 | 1.42488 | 1.96513 | 1.30706 | 0.08722 | 0.08896 |

RRG(5) | 30 | 1.43068 | 1.96825 | 1.31653 | 0.08344 | 0.08393 |

SW(3) | 10 | 1.55356 | 1.87107 | 1.43216 | 0.16610 | 0.23788 |

SW(3) | 20 | 1.55998 | 1.86334 | 1.43951 | 0.22050 | 0.24702 |

SW(3) | 30 | 1.54842 | 1.88077 | 1.43167 | 0.21643 | 0.22839 |

SW(4) | 10 | 1.47637 | 1.91449 | 1.33514 | 0.14321 | 0.16987 |

SW(4) | 20 | 1.49157 | 1.93385 | 1.38141 | 0.14235 | 0.14885 |

SW(4) | 30 | 1.49811 | 1.93505 | 1.39764 | 0.14519 | 0.14804 |

SW(5) | 10 | 1.43017 | 1.95045 | 1.29919 | 0.09802 | 0.11240 |

SW(5) | 20 | 1.44575 | 1.95995 | 1.33955 | 0.09722 | 0.09962 |

SW(5) | 30 | 1.45251 | 1.96562 | 1.35962 | 0.08951 | 0.09047 |

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**MDPI and ACS Style**

Baxter, G.J.; da Costa, R.A.; Dorogovtsev, S.N.; Mendes, J.F.F.
Filtering Statistics on Networks. *Entropy* **2020**, *22*, 1149.
https://doi.org/10.3390/e22101149

**AMA Style**

Baxter GJ, da Costa RA, Dorogovtsev SN, Mendes JFF.
Filtering Statistics on Networks. *Entropy*. 2020; 22(10):1149.
https://doi.org/10.3390/e22101149

**Chicago/Turabian Style**

Baxter, G. J., R. A. da Costa, S. N. Dorogovtsev, and J. F. F. Mendes.
2020. "Filtering Statistics on Networks" *Entropy* 22, no. 10: 1149.
https://doi.org/10.3390/e22101149