# Single-Threshold Model Resource Network and Its Double-Threshold Modifications

## Abstract

**:**

## 1. Introduction

## 2. Resource Network, the Standard Model

#### 2.1. Basic Definitions

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Remark**

**1.**

#### 2.2. Rules of Resource Distribution

**Definition**

**7.**

**Rule one**is applied when a vertex contains more resource than it can send to all adjacent vertices through outgoing edges; in this case, each edge transmits the resource amount equal to its throughput: ${f}_{ij}\left(t\right)={r}_{ij}$, and totally, the vertex gives away resource amount

**rule two**, a vertex sends out its entire resource. It distributes the resource to all outgoing edges in proportion to their throughputs.

**Remark**

**2.**

#### 2.3. Network Operation

**Definition**

**8.**

**Definition**

**9.**

**Definition**

**10.**

**Definition**

**11.**

**Definition**

**12.**

**Definition**

**13.**

#### 2.4. Classification of Networks by Topology

- Ergodic networks
- Regular networks;
- Cyclic networks;

- Non-ergodic networks
- Absorbing networks;
- Mixed networks.

**Definition**

**14.**

**Definition**

**15.**

**Definition**

**16.**

**Definition**

**17.**

**Definition**

**18.**

**Definition**

**19.**

**Definition**

**20.**

**Definition**

**21.**

#### 2.5. Classification of Networks by Total Throughputs of Vertices

**Definition**

**22.**

**Definition**

**23.**

- Receiver-vertices, ${r}_{i}^{in}-{r}_{i}^{out}>0$;
- Source-vertices, ${r}_{i}^{in}-{r}_{i}^{out}<0$;
- Neutral vertices, ${r}_{i}^{in}-{r}_{i}^{out}=0$.

**Definition**

**24.**

**Definition**

**25.**

#### 2.6. General Classification of Networks

## 3. Materials and Methods: The Standard Model

#### 3.1. Regular Resource Networks and Homogeneous Markov Chains

**Proposition**

**1.**

- The limit of degrees of matrix ${R}^{\prime}$ exists:$$\underset{t\to \infty}{lim}{\left({R}^{\prime}\right)}^{t}={R}^{\prime *};$$
- The limit state exists and is unique for any initial state.For an arbitrary state ${Q}^{1}\left(t\right)$ the equality holds$${Q}^{1}\left(t\right){R}^{\prime *}={Q}^{1*};$$
- The limit matrix ${R}^{\prime *}$ and limit vector ${Q}^{1*}$ are related by the following formula:$${R}^{\prime *}=\mathbf{1}\xb7{Q}^{1*},$$In other words, matrix ${R}^{\prime *}$ consists of n identical rows represented by vector ${Q}^{1*}$.
- Vector ${Q}^{1*}$ is a single left eigenvector of matrices ${R}^{\prime}$ and ${R}^{\prime *}$ corresponding to eigenvalue $\lambda =1$:$${Q}^{1*}{R}^{\prime}={Q}^{1*};$$$${Q}^{1*}{R}^{\prime *}={Q}^{1*}.$$

**Corollary**

**1.**

**Lemma**

**1.**

**Proof.**

- Consider an arbitrary source-vertex ${v}_{i}$. Let at $t=0$ its resource be more than its out-throughput: ${q}_{i}\left(0\right)>{r}_{i}^{out}$. It operates according to rule 1. At first time step it loses resource amount bounded from below by value $\Delta {q}_{i}={r}_{i}^{out}-{r}_{i}^{in}>0$. Then it will loose all the surplus ${s}_{i}={q}_{i}\left(0\right)-{r}_{i}^{out}$ in at most ${t}^{\prime}=\frac{{s}_{i}}{\Delta {q}_{i}}$ steps.
- Let there be neutral vertices in the network.Consider the time step ${t}^{\u2033}$ when all the source-vertices switched to rule 2.Consider a neutral vertex ${v}_{j}$ adjacent to any source vertex. As the network is regular, such a vertex always exists. Let at $t={t}^{\u2033}$ its resource be more than its out-throughput: ${q}_{j}\left({t}^{\u2033}\right)>{r}_{j}^{out}$. Then, it loses resource amount bounded from below by value $\Delta {q}_{j}={r}_{ji}-{f}_{ji}\left({t}^{\u2033}\right)>0$, as the source vertex cannot increase its resource. Then, it will loose all the surplus ${s}_{j}={q}_{j}\left({t}^{\u2033}\right)-{r}_{j}^{out}$ in at most ${t}^{\u2034}=\frac{{s}_{j}}{\Delta {q}_{j}}$ steps.Consider the next neutral vertex adjacent either to the source vertex or to ${v}_{j}$.Repeating these arguments, we get that in a finite number of steps, all neutral vertices will switch to rule 2.

**Corollary**

**2.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**1.**

- if $W\le T$ for any initial state $Q\left(0\right)=({q}_{1}\left(0\right),\cdots ,{q}_{n}\left(0\right))$, there is a time step ${t}^{\prime}$, such that if $t>{t}^{\prime}$ all vertices operate according to rule 2.
- if $W>T$ for any initial state $Q\left(0\right)=({q}_{1}\left(0\right),\cdots ,{q}_{n}\left(0\right))$, there will always be at least one vertex operating according to rule 1.

**Proof.**

**Corollary**

**3.**

**Theorem**

**2.**

**Proof.**

**Definition**

**26.**

**Remark**

**3.**

#### 3.1.1. Complete Uniform Resource Networks. Small Resource

#### 3.1.2. Eulerian Networks: Small Resource

#### 3.1.3. Non-Symmetric Networks: Small Resource

#### 3.2. Regular Networks. Large resource

#### 3.2.1. Non-Symmetric Networks

**Definition**

**27.**

#### 3.2.2. Complete Uniform and Eulerian Networks: Large Resource

**Proposition**

**2.**

## 4. Results: Standard Model

#### 4.1. Non-Symmetric Networks

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

- 1
- If $W<T$ then vector ${Q}^{*}=({q}_{1}^{*},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}{q}_{n}^{*})$ is unique and defined by the formula$${Q}^{*}=W\xb7{Q}^{1*},$$$${R}^{\prime}=\mathrm{diag}\left(\frac{1}{{r}_{1}^{out}},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}\frac{1}{{r}_{n}^{out}}\right)\xb7R.$$All components of ${Q}^{*}$ are positive.
- 2
- If $W=T$ then vector $\tilde{Q}$ is unique and has the form$$\tilde{Q}=T\xb7{Q}^{1*}=({r}_{1}^{out},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}{r}_{l}^{out},{\tilde{q}}_{l+1},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}{\tilde{q}}_{n}),$$$${\tilde{q}}_{i}<{r}_{i}^{out},\phantom{\rule{3.33333pt}{0ex}}i=\overline{l+1,\phantom{\rule{3.33333pt}{0ex}}n}.$$
- 3
- If $W>T$, then vector ${Q}^{*}$ has the form$$\tilde{Q}=T\xb7{Q}^{1*}=({r}_{1}^{out}+{s}_{1}^{*},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}{r}_{l}^{out}+{s}_{l}^{*},{\tilde{q}}_{l+1},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}{\tilde{q}}_{n}),$$$$\sum _{i=1}^{l}{s}_{i}^{*}=W-T.$$The values of ${s}_{i}^{*}$ depend on the initial distribution.

**Theorem**

**5.**

- 1
- If $W<T$ then$${F}^{in*}={\left({F}^{out*}\right)}^{\mathrm{T}}=W\xb7{Q}^{1*},$$All components of matrix ${F}^{*}$ are positive.$${f}_{sum}=W.$$
- 2
- If $W\ge T$ then$${\tilde{F}}^{in}={\left({\tilde{F}}^{out*}\right)}^{\mathrm{T}}=T\xb7{Q}^{1*}=\tilde{Q}=({r}_{1}^{out},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}{r}_{l}^{out},{\tilde{q}}_{l+1},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}{\tilde{q}}_{n}),$$$${\tilde{q}}_{i}<{r}_{i}^{out},\phantom{\rule{3.33333pt}{0ex}}i=\overline{l+1,\phantom{\rule{3.33333pt}{0ex}}n}.$$$${f}_{sum}=T.$$

**Remark**

**4.**

#### 4.2. Complete Uniform Networks

**Theorem**

**6.**

- 1
- If $W<T$ then$${Q}^{*}=\left(\frac{W}{n},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}\frac{W}{n}\right),$$
- 2
- If $W=T=r{n}^{2}$ then$${Q}^{*}=\left(rn,\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}rn\right),$$
- 3
- If $W>T$, then vector ${Q}^{*}$ depends on the initial state$$Q\left(0\right)=(rn+{s}_{1}\left(0\right),\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}rn+{s}_{l}\left(0\right),\phantom{\rule{3.33333pt}{0ex}}rn-{d}_{l+1}\left(0\right),\cdots ,\phantom{\rule{3.33333pt}{0ex}}rn-{d}_{n}\left(0\right))$$$${Q}^{*}=(rn+{s}_{1}^{*},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}rn+{s}_{m}^{*},\phantom{\rule{3.33333pt}{0ex}}rn,\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}rn),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}m\le l,$$$$\sum _{i=1}^{m}{s}_{i}^{*}=W-T.$$The values of ${s}_{i}^{*}$ depend on the initial distribution.
- If$${s}_{l}\left(0\right)\ge \frac{{d}_{sum}\left(0\right)}{l}$$$${Q}^{*}=\left(rn+{s}_{1}\left(0\right)-\frac{{d}_{sum}\left(0\right)}{l},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}rn+{s}_{l}\left(0\right)-\frac{{d}_{sum}\left(0\right)}{l},rn,\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}rn\right),$$
- If$${s}_{l}\left(0\right)<\frac{{d}_{sum}\left(0\right)}{l}$$$${s}_{m}\left(0\right)\ge \frac{1}{m}\left({d}_{sum}\left(0\right)-\sum _{i=m}^{l}i\xb7{s}_{i}\left(0\right)\right)$$$${Q}^{*}=\left(rn+{s}_{1}\left(0\right)-{w}_{m}\left(0\right),\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}rn+{s}_{m}\left(0\right)-{w}_{m}\left(0\right),rn,\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}rn\right)$$$${w}_{m}\left(0\right)=\frac{1}{m}\left({d}_{sum}\left(0\right)-\sum _{i=m}^{l}i\xb7{s}_{i}\left(0\right)\right).$$

**Proof.**

- If ${s}_{l}\left(0\right)\ge \frac{{d}_{sum}\left(0\right)}{l}$ then the limit state is described by the Formula (25).
- If ${s}_{l}\left(0\right)<\frac{{d}_{sum}\left(0\right)}{l}$ then perform a number of steps.
- Reduce the surplus in each of the first l vertices by the value ${c}_{l}\left(0\right)$.
- Distribute the resource $l\xb7{c}_{l}\left(0\right)$ equally between the vertices ${v}_{l+1},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}{v}_{n}$.
- Take the resulting vector$$\begin{array}{c}{Q}^{\prime}\left(0\right)=(rn+{s}_{1}\left(0\right)-{s}_{l}\left(0\right),\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}rn+{s}_{l-1}\left(0\right)-{s}_{l}\left(0\right),rn,\\ \phantom{\rule{3.33333pt}{0ex}}rn-{d}_{l+1}\left(0\right)+\frac{l\xb7{c}_{l}\left(0\right)}{n-l},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}rn-{d}_{n}\left(0\right)+\frac{l\xb7{c}_{l}\left(0\right)}{n-l})\end{array}$$
- Calculate the new value of the total deficit: ${d}_{sum}^{\prime}\left(0\right)={d}_{sum}\left(0\right)-l\xb7{c}_{l}\left(0\right).$
- Evaluate the new value ${s}_{l-1}^{\prime}\left(0\right)$.
- –
- If ${s}_{l-1}^{\prime}\left(0\right)\ge \frac{{d}_{sum}\left(0\right)-l\xb7{c}_{l}\left(0\right)}{l-1}\phantom{\rule{3.33333pt}{0ex}}$ then $m=l-1$ and the limit state is described by the Formula (26).
- –
- If ${s}_{l-1}^{\prime}\left(0\right)<\frac{{d}_{sum}\left(0\right)-l\xb7{c}_{l}\left(0\right)}{l-1}\phantom{\rule{3.33333pt}{0ex}}$ then all steps $1\xf75$ must be repeated.

**Theorem**

**7.**

- 1
- If $W<T$ then$${F}^{in*}={\left({F}^{out*}\right)}^{\mathrm{T}}=\left(\frac{W}{n},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}\frac{W}{n}\right),$$$${f}_{sum}=W.$$
- 2
- If $W\ge T=r{n}^{2}$ then$${\tilde{F}}^{in}={\left({\tilde{F}}^{out}\right)}^{\mathrm{T}}=\tilde{Q}=\left(rn,\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}rn\right),$$$${f}_{sum}=T=r{n}^{2}.$$

#### 4.3. Eulerian Networks

**Theorem**

**8.**

- 1
- If $W<T$ then$${Q}^{*}=W\left(\frac{{r}_{1}^{out}}{{r}_{sum}},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}\frac{{r}_{n}^{out}}{{r}_{sum}}\right),$$
- 2
- If $W=T={r}_{sum}$ then$${Q}^{*}=\left({r}_{1}^{out},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}{r}_{n}^{out}\right).$$
- 3
- If $W>T$, then vector ${Q}^{*}$ depends on the initial state $Q\left(0\right)=({r}_{1}^{out}+{s}_{1}\left(0\right),\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}{r}_{l}^{out}+{s}_{l}\left(0\right),\phantom{\rule{3.33333pt}{0ex}}{r}_{l+1}^{out}-{d}_{l+1}\left(0\right),\cdots ,\phantom{\rule{3.33333pt}{0ex}}{r}_{n}^{out}-{d}_{n}\left(0\right))$ and has the form$${Q}^{*}=({r}_{1}^{out}+{s}_{1}^{*},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}{r}_{m}^{out}+{s}_{m}^{*},\phantom{\rule{3.33333pt}{0ex}}{r}_{m+1}^{out},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}{r}_{n}^{out}),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}m\le l,$$$$\sum _{i=1}^{m}{s}_{i}^{*}=W-T.$$The values of ${s}_{i}^{*}$ depend on the initial distribution.

**Remark**

**5.**

#### 4.3.1. Initial State Analysis

**Proposition**

**3.**

**Proof.**

- If $\forall \phantom{\rule{3.33333pt}{0ex}}j=\overline{l+1,\phantom{\rule{3.33333pt}{0ex}}n}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{q}_{j}\left(0\right)=0$, the deficits in the corresponding vertices are ${r}_{j}^{out},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\frac{{d}_{j}\left(0\right)}{{r}_{j}^{out}}=1$, and the proof follows from Formula (32).
- If $\exists \phantom{\rule{3.33333pt}{0ex}}j=\overline{l+1,\phantom{\rule{3.33333pt}{0ex}}n}:\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{q}_{j}\left(0\right)>0$, then the j-th term in sum (33) will be reduced so that the right side of formula (32) has vector $({d}_{l+1}\left(0\right),\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}{d}_{n}\left(0\right))$ instead of vector $({r}_{l+1}^{out},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}{r}_{n}^{out})$.

#### 4.3.2. Operation Analysis

**Theorem**

**9.**

- 1
- If $W<T$, then$${F}^{in*}={\left({F}^{out*}\right)}^{\mathrm{T}}=W\left(\frac{{r}_{1}^{out}}{{r}_{sum}},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}\frac{{r}_{n}^{out}}{{r}_{sum}}\right),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{f}_{sum}=W.$$
- 2
- If $W\ge T={r}_{sum}$, then$${\tilde{F}}^{in}={\left({\tilde{F}}^{out}\right)}^{\mathrm{T}}=\tilde{Q}=\left({r}_{1}^{out},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}{r}_{n}^{out}\right),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{f}_{sum}=T={r}_{sum}.$$

**Proof.**

## 5. Resource Network with Limited Capacity of Attractor Vertices

**Proposition**

**4.**

- if $W\le {T}_{2}+lp$, the network operates as a network without capacity limitations;
- if $W>{T}_{2}+lp$, the dynamics of the network changes,

**Proposition**

**5.**

- For $W\in (0,T]$, from some moment ${t}^{\prime}\ge 0$ the network operation is described by a homogeneous Markov chain. The limit state and flow vectors exist, are unique and coincide. The total limit flow is equal to W and increases with the growth of W.
- For $W\in \left(T,T+lp\right]$, the limit state and flow exist. The limit flow is unique, the limit state is unique for $l=1$; for $l>1$, the limit state is unique at all vertices, except attractors. The resource in attractors is not less than their output throughputs. The surpluses in attractors depend on the initial distribution of the resource, but the sum of these surpluses does not depend on the initial state and is equal to $W-T\le lp$. The total limit flow is equal to T and does not change with increasing W.
- For $W\in (T+lp,{T}_{2}+lp]$, all attractors reach a capacity limitation. An excess resource begins to accumulate in the remaining vertices, but none of them is still able to exceed the value ${r}_{i}^{out}$, that is, switch to rule 1 operation. The total limit flow is equal to $W-lp$ and increases with the growth of W.
- For $W\in ({T}_{2}+lp,\infty )$, the new vertices are saturated to the total output ${r}_{i}^{out}$ and the flow is stabilized at the value ${T}_{2}$. For any arbitrarily large value of W, the total limit flow is equal to ${T}_{2}$.

**Definition**

**28.**

#### 5.1. Network Operation with Resource $W\in (T,{T}_{2})$

**Proposition**

**6.**

**Proof.**

**Proposition**

**7.**

**Proof.**

**Proposition**

**8.**

**Remark**

**6.**

**Definition**

**29.**

**1**is a column vector of n units, π is a certain probability vector.

**Proposition**

**9.**

**Proof.**

**Theorem**

**10.**

#### 5.2. Network Operation with Resource $W={T}_{2}$

**Theorem**

**11.**

**Proposition**

**10.**

**1**is a column vector of n units.

**Corollary**

**4.**

- ${Q}^{{1}^{\prime \prime}*}=\frac{1}{{T}_{2}}\widehat{Q}$;
- the threshold value ${T}^{\u2033}={T}_{2}$;
- the limit state vector at $W={T}^{\u2033}$ is ${\tilde{Q}}^{\u2033}=\widehat{Q}$;
- the limit of powers of the stochastic matrix is$${R}^{\u2033*}={\left({R}^{\prime}{\widehat{{S}^{\prime}}}^{-1}\right)}^{*}=\frac{1}{{T}_{2}}\xb7\left(\mathbf{1}\xb7\widehat{Q}\right).$$

**Proposition**

**11.**

**Theorem**

**12**

**Theorem**

**13**

**Theorem**

**14**

- For $W\in (0,T]$, the limit state is unique and is found by the formula ${Q}^{*}=W\xb7{Q}^{1*}$;
- For $W\in \left(T,T+lp\right]$, the limit state is unique for $l=1$; for $l>1$, the limit state is unique at all vertices, except for attractors:$${Q}^{*}=({r}_{1}^{out}+{s}_{1}^{*},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}{r}_{l}^{out}+{s}_{l}^{*},{\tilde{q}}_{l+1},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}{\tilde{q}}_{n})$$
- For $W\in (T+lp,{T}_{2}+lp]$, the limit state is unique. It consists of the sum of two vectors ${Q}^{*}=P+{Q}_{W}^{*}$. Here $P=(p,\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}p,\phantom{\rule{3.33333pt}{0ex}}0,\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}0)$; ${Q}_{W}^{*}$ is the limit state of an induced network with matrix$${R}^{new}\left(W\right)=\mathrm{diag}\left({r}_{1}^{out},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}{r}_{n}^{out}\right){R}^{\prime}{{S}^{\prime}}^{*-1}\left(W\right).$$If $W={T}_{2}+lp$, then this equality transforms into formula (40) and the limit state is$${Q}^{*}=\widehat{Q}+P=\left({r}_{1}^{out}+p,\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}{r}_{l}^{out}+p,\phantom{\rule{3.33333pt}{0ex}}{r}_{l+1}^{out},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}{r}_{m}^{out},\phantom{\rule{3.33333pt}{0ex}}{\widehat{q}}_{m+1},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\widehat{q}}_{n}\right),$$
- For $W\in ({T}_{2}+lp,\infty )$, the limit state is unique up to surpluses in the secondary attractors. It is found by the formula$${Q}^{*}=\left({r}_{1}^{out}+p,\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}{r}_{l}^{out}+p,\phantom{\rule{3.33333pt}{0ex}}{r}_{l+1}^{out}+{s}_{l+1}^{*},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}{r}_{m}^{out}+{s}_{m}^{*},\phantom{\rule{3.33333pt}{0ex}}{\widehat{q}}_{m+1},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\widehat{q}}_{n}\right).$$

## 6. Resource Network with Greedy Vertices

**Definition**

**30.**

**Remark**

**7.**

**Proposition**

**12.**

**Definition**

**31.**

**Definition**

**32.**

**Proposition**

**13.**

#### 6.1. Two Threshold Values

**Definition**

**33.**

**Proposition**

**14.**

**Theorem**

**15.**

**Proof.**

**Theorem**

**16.**

**Theorem**

**17.**

#### 6.2. Insufficient Resource $W<{T}_{0}$

**Proposition**

**15.**

**Proof.**

**Theorem**

**18.**

**Proof.**

**Theorem**

**19.**

**Proof.**

#### 6.3. Greedy Vertices and Cyclic Networks

**Definition**

**34.**

**Theorem**

**20.**

## 7. Resource Networks with the Limited Capacity and Greedy Vertices

## 8. Discussion

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Kuznetsov, O.P. Uniform Resource Networks. I. Complete Graphs. Autom. Remote Control
**2009**, 70, 1767–1775. [Google Scholar] [CrossRef] - Zhilyakova, L.Y. Asymmetrical Resource Networks. I. Stabilization Processes for Low Resources. Autom. Remote Control
**2011**, 72, 798–807. [Google Scholar] [CrossRef] - Zhilyakova, L.Y. Asymmetric resource networks. II. Flows for large resources and their stabilization. Autom. Remote Control
**2012**, 73, 1016–1028. [Google Scholar] [CrossRef] - Zhilyakova, L.Y. Asymmetric resource networks. III. A study of limit states. Autom. Remote Control
**2012**, 73, 1165–1172. [Google Scholar] [CrossRef] - Zhilyakova, L. Resource Allocation among Attractor Vertices in Asymmetric Regular Resource Networks. Autom. Remote Control
**2019**, 80, 1519–1540. [Google Scholar] [CrossRef] - Zhilyakova, L.Y. A study of Euler resource networks. Autom. Remote Control
**2014**, 75, 2248–2261. [Google Scholar] [CrossRef] - Zhilyakova, L.Y. Resource Network with Limited Capacity of Attractor Vertices. Autom. Remote Control
**2019**, 80, 543–555. [Google Scholar] [CrossRef] - Zhilyakova, L.; Chaplinskaya, N. Research of complete homogeneous “greedy-vertices” resource networks. UBS
**2021**, 89. (In Russian) [Google Scholar] [CrossRef] - Ford, L.R., Jr.; Fulkerson, D.R. Flows in Networks; Princeton Univ. Press: Princeton, NJ, USA, 1962. [Google Scholar]
- Ahuja, R.K.; Magnati, T.L.; Orlin, J.B. Network Flows: Theory, Algorithms and Applications; Prentice Hall: Hoboken, NJ, USA, 1993. [Google Scholar]
- Blanchard, P.; Volchenkov, D. Random Walks and Diffusions on Graphs and Data-Bases: An Introduction; Springer Series in Synergetics; Springer: Berlin/Heisenberg, Germany, 2011. [Google Scholar]
- Volchenkov, D. Infinite Ergodic Walks in Finite Connected Undirected Graphs. Entropy
**2021**, 23, 205. [Google Scholar] [CrossRef] - Oliveira, R.I.; Peres, Y. Random walks on graphs: New bounds on hitting, meeting, coalescing and returning. In Proceedings of the Meeting on Analytic Algorithmics and Combinatorics (ANALCO), San Diego, CA, USA, 6–7 January 2019. [Google Scholar] [CrossRef] [Green Version]
- Erusalimskii, Y.M. 2–3 Paths in a Lattice Graph: Random Walks. Math Notes
**2018**, 104, 395–403. [Google Scholar] [CrossRef] - Jin, C. Simulating Random Walks on Graphs in the Streaming Model. arXiv
**2018**, arXiv:1811.08205. [Google Scholar] - Björner, A.; Lovász, L.; Shor, P. Chip-firing games on graphs. Europ. J. Comb.
**1991**, 12, 283–291. [Google Scholar] [CrossRef] [Green Version] - Björner, A.; Lovász, L. Chip-firing games on directed graphs. J. Algebr. Comb.
**1992**, 1, 305–328. [Google Scholar] [CrossRef] - Biggs, N.L. Chip-Firing and the Critical Group of a Graph. J. Algebr. Comb.
**1999**, 9, 25–45. [Google Scholar] [CrossRef] - Liscio, P. Lattices in Chip-Firing. arXiv
**2020**, arXiv:2010.15650. [Google Scholar] - Glass, D.; Kaplan, N. Chip-Firing Games and Critical Groups. In A Project-Based Guide to Undergraduate Research in Mathematics. Foundations for Undergraduate Research in Mathematics; Harris, P., Insko, E., Wootton, A., Eds.; Birkhäuser: Cham, Switzerland, 2020; pp. 32–58. [Google Scholar]
- Dochtermann, A.; Meyers, E.; Samavedan, R.; Yi, A. Cycle and circuit chip-firing on graphs. arXiv
**2020**, arXiv:2006.13397. [Google Scholar] - Bak, P.; Tang, C.; Wiesenfeld, K. Self-organized criticality. Phys. Rev. A
**1988**, 38, 364–374. [Google Scholar] [CrossRef] - Dhar, D. The abelian sandpile and related models. Phys. A Stat. Mech. Its Appl.
**1999**, 263, 4–25. [Google Scholar] [CrossRef] [Green Version] - Pegden, W.; Smart, C.K. Stability of Patterns in the Abelian Sandpile. Ann. Henri Poincaré
**2020**, 21, 1383–1399. [Google Scholar] [CrossRef] [Green Version] - Kim, S.; Wang, Y. A Stochastic Variant of the Abelian Sandpile Model. J. Stat. Phys.
**2020**, 178, 711–724. [Google Scholar] [CrossRef] [Green Version] - Dhar, D. Self-organized critical state of sandpile automaton models. Phys. Rev. Lett.
**1990**, 64, 1613–1616. [Google Scholar] [CrossRef] - Járai, A.A. The Sandpile Cellular Automaton. In Probabilistic Cellular Automata. Emergence, Complexity and Computation; Louis, P.Y., Nardi, F., Eds.; Springer: Cham, Switzerland, 2018; pp. 79–88. [Google Scholar]
- Lovász, L.; Winkler, P. Mixing of Random Walks and Other Diffusions on a Graph. In Surveys Combinat; Rowlinson, P., Ed.; London Math. Soc. Lecture Notes Ser.; Cambridge Univ. Press: Cambridge, UK, 1995; Volume 218, pp. 119–154. [Google Scholar]
- Duffy, C.; Lidbetter, T.F.; Messinger, M.E.; Nowakowski, R.J. A Variation on Chip-Firing: The diffusion game. Discret. Math. Theor. Comput. Sci.
**2018**, 20. [Google Scholar] [CrossRef] - Skorokhodov, V.A.; Chebotareva, A.S. The maximum flow problem in a network with special conditions of flow distribution. J. Appl. Ind. Math.
**2015**, 9, 435–446. [Google Scholar] [CrossRef] - Zhilyakova, L. Dynamic Graph Models and Their Properties. Autom. Remote Control
**2015**, 76, 1417–1435. [Google Scholar] [CrossRef] - Supplementary Materials for Article ‘Single-Threshold Model Resource Network and Its Double-Threshold Modifications’. Available online: https://www.researchgate.net/publication/352184919_Supplementary_Materials_for_article_%27Single-Threshold_Model_Resource_Network_and_its_Double-Threshold_Modifications%27 (accessed on 8 June 2021).
- Kemeny, J.G.; Snell, J.L. Finite Markov Chains; Van Nostrand Reinhold: New York, NY, USA, 1960. [Google Scholar]
- Zhilyakova, L. Resource networks with the capacity limitations on attractor-vertices. Formal characteristics. UBS
**2016**, 59, 72–119. (In Russian) [Google Scholar] - Hajnal, J. Weak ergodicity in non-homogeneous Markov chains. Proc. Cambridge Philos. Soc.
**1958**, 54, 233–246. [Google Scholar] [CrossRef] - Zhilyakova, L.Y. Ergodic cyclic resource networks. I. Oscillations and equilibrium at low resource. UBS
**2013**, 43, 34–54. (In Russian) [Google Scholar]

**Figure 3.**The general classification of networks. The focus of the study is the top row of the table. It lists the classes of networks under consideration.

**Figure 4.**Threshold scheme in a resource network with greedy vertices: ${T}_{0}$ separates insufficient and sufficient resource values, ${T}_{1}$ divides the sufficient resource into small and large values, ${T}_{2}$ determines the interval at which the limitations on the attractors begin to apply. Attractors of the next ranks are schematically shown.

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Zhilyakova, L.
Single-Threshold Model Resource Network and Its Double-Threshold Modifications. *Mathematics* **2021**, *9*, 1444.
https://doi.org/10.3390/math9121444

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Single-Threshold Model Resource Network and Its Double-Threshold Modifications. *Mathematics*. 2021; 9(12):1444.
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2021. "Single-Threshold Model Resource Network and Its Double-Threshold Modifications" *Mathematics* 9, no. 12: 1444.
https://doi.org/10.3390/math9121444