# Graph Coloring via Locally-Active Memristor Oscillatory Networks

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. A Physics-Based Model for the Threshold Switching Dynamics of a Nano-Scale Locally-Active Memristor Device Stack

## 3. Memristive Computing Engine for Solving the Vertex Coloring Problem

#### 3.1. Operating Principles of the Capacitively-Coupled Networks

**Remark**

**1.**

#### 3.2. Compensation for an Imbalance in the Number of Couplings per Oscillator

#### 3.3. Compensation for the Memristor Device-to-Device Variability

**Remark**

**2.**

## 4. A Rigorous Strategy for Coloring a Graph via the Network Phase Dynamics

^{th}-placed vertex if the graph features no edge between these two vertices, otherwise the ${j}^{th}$-ranked vertex is defined as the first element of a new color group. After coloring vertex at row N in the table, a final check needs to be carried out to verify if the vertices in the last color group may be merged with those in the first color group. This may be done if and only if no pair of vertices in these two groups is connected by means of an edge in the undirected graph. In cycle $i\in \{2,\dots ,N\}$ of the iterative procedure, the color assignment step is repeated in a similar fashion, analysing progressively the vertices at positions i, $i+1$, …, N, 1, 2, …, $i-1$ in the table. With such iterative procedure, at least one of the cycles will allow to determine the minimum number of color groups, identifiable by the network, given the phase shift ordering it outputs at the end of a certain simulation. In other words, indicating the ${k}^{\mathrm{th}}$ color group, which, on the basis of the prediction of the ${i}^{th}$ cycle of the iterative procedure, is identifiable by the network, as ${\mathcal{C}}_{k}^{\left(i\right)}$ ($k\in \{1,\dots ,{m}_{i}\}$, where ${m}_{i}\in [n,N]$ denotes the total number of colors assigned to the N nodes of the associated graph in the ${i}^{\mathrm{th}}$ iteration, and n represents the chromatic number of the graph itself, the proposed strategy will output the particular group classification obtained from the ${q}^{\mathrm{th}}$ iteration, whereby ${m}_{q}={min}_{i=1}^{i=N}\left\{{m}_{i}\right\}$.

**Remark**

**3.**

## 5. Control Paradigms to Resolve Local Minima-Based Impasse Conditions

#### 5.1. Crossover Strategy

- (1)
- For each value of k in the set $\{0,\dots ,N-1\}$, the vertex k is removed from the original N-node graph, and the iterative vertex coloring strategy is applied to the resulting graph of $(N-1)$ nodes, using a modified version of the vertex ranking, which is tabulated beforehand, after a simulation of the oscillatory network, under a generic sub-optimal initialisation setting, attains the steady state. Specifically, the label of the vertex k, taken out of the original graph, is removed from the original vertex ranking, resulting in a new table with $N-1$ entries. For each value of k, a $N\times N$ matrix, denoted as ${\mathbf{A}}^{\left(k\right)}$, and obtained from the original adjacency matrix $\mathbf{A}$ by setting to 0 all the elements at row k and at column k, may still be used to define the connectivity of the respective $(N-1)$-node graph. Coloring the $N-1$ vertices of N distinct graphs, at least one of the N problems will be found to admit the best solution, allowing to categorise the $N-1$ nodes of the relative graph through the lowest number of color groups. The particular node k, which, extracted out of the original graph, allows the resulting network to identify the least number of colors according to our iterative vertex coloring procedure, may then be chosen as first vertex i to involve in the crossover20.
- (2)
- Assigning, one at a time, any integer from the set $\{0,\dots ,i-1,i+1,\dots ,N-1\}$ to k, the iterative vertex coloring strategy is then applied to a new vertex ranking, obtained from the original table by interchanging the positions of vertices i and k. Note that the original N-node graph, with connectivity defined by the adjacency matrix $\mathbf{A}$, should be considered in each of the $N-1$ applications of the iterative vertex coloring strategy, since nothing else, except for the correspondence between oscillators and vertices, is affected in a crossover operation21. Solving the resulting $N-1$ vertex coloring problems, the solution, assigning the least number of colors to the N nodes of the original graph, will be determined. It may happen that, on the basis of the proposed iterative procedure, for two or more values of k, the exchange between the positions of oscillators k and i in the original vertex ranking results in a common lowest number of color groups for the N nodes of the original graph. In this case, the choice of the second oscillator j to involve in the crossover falls for the particular candidate k, whose relative phase ${\phi}_{k}$ features the largest distance from the relative phase ${\phi}_{i}$ of oscillator i in the steady-state phase shift vector $\mathbf{\phi}$ obtained through the simulation preceding the application of the two-step strategy.

#### 5.2. Pulse Destabilisation Strategy

- (1)
- The most suitable oscillator $i\in \{0,\dots ,N-1\}$ to target in the pulse destabilisation action is determined in the same way as was done for the selection of the cell $i\in \{0,\dots ,N-1\}$ to involve in the crossover process (see the first step in the procedure aimed to choose the right cell pair $(i,j)$ to involve in the coupling interchange strategy).
- (2)
- The second step is aimed to determine the appropriate shift $\mathsf{\Delta}\phantom{\rule{-0.166667em}{0ex}}\phi $ to be added to the steady-state relative phase ${\phi}_{i}^{\left(s\right)}$ of oscillator i for pulling the network out of the local minimum state, facilitating its convergence to an oscillatory solution, which would ideally correspond to the least number of color groups for the N vertices of the associated graph. To accomplish this task, for each value of k within the set $\{1,\dots ,M-1\}$, with M a predefined positive integer, the offset $\mathsf{\Delta}\phantom{\rule{-0.166667em}{0ex}}{\phi}_{k}=k\xb7\frac{{360}^{\circ}}{M}$ is added to the phase shift ${\phi}_{i}^{\left(s\right)}$ of cell i in the steady-state relative phase shift vector ${\mathbf{\phi}}^{\left(s\right)}$ recorded before the application of the pulse destabilisation process, and the iterative graph coloring procedure is applied to the resulting vertex ranking for the original graph. The choice for the most appropriate offset $\mathsf{\Delta}\phantom{\rule{-0.166667em}{0ex}}\phi $, within the specified set of k-dependent uniformly-spaced values, goes for the $\mathsf{\Delta}\phantom{\rule{-0.166667em}{0ex}}{\phi}_{k}$-candidate, which, according to our graph coloring strategy, allows the network to classify the nodes of the associated graph in the lowest number of color groups. If, for two or more k-values, the application of the iterative graph coloring procedure to the vertex ranking, resulting from the phase shift ordering, obtained by adding up the relevant offset $\mathsf{\Delta}\phantom{\rule{-0.166667em}{0ex}}{\phi}_{k}$ to the phase shift ${\phi}_{i}^{\left(s\right)}$ of oscillator i in the steady-state relative phase shift vector ${\phi}^{\left(s\right)}$, leads to the identification of the same lowest number of colors, the selection goes for the $\mathsf{\Delta}\phantom{\rule{-0.166667em}{0ex}}{\phi}_{k}$-candidate featuring the largest modulus. Finally, the pulse amplitude of the ${T}_{p}$-long stimulus to be applied to oscillator i is obtained from Equation (33).

#### 5.3. Discussion

^{nd}algorithm implementation challenge for NP-hard problems in Discrete Mathematics and Theoretical Computer Science (DIMACS) [51], derived from the application of various techniques, namely an algorithmic approach known as Brélaz heuristic [52], methods based upon the analysis of the phase dynamics of capacitively-coupled arrays of locally-active memristor oscillators without a control strategy for bypassing local minima solutions, as respectively presented in [42], and in Section 4, and, finally, paradigms including either a reconfigurability of the oscillators’ couplings, as discussed in Section 5.1, or a perturbation of the memristive array, as presented in Section 5.2, to enable the dynamical system to exit an impasse state, and to resume the calculations of the problem solution, thereafter.

**Remark**

**4.**

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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1 | The time it takes for a von Neumann computing machine to find the optimal solution to a NP-hard problem, which involves n elements, scales exponentially with n. |

2 | |

3 | Importantly, depending upon the graph under focus, the proposed Memristor CNN (M-CNN) may feature either local or non-local capacitive couplings. The solution of the vertex coloring problem through the proposed M-CNNs depends upon the phase differences among the oscillations developing in the constitutive units of the array at steady state. In order to highlight the steady-state oscillatory behaviours of the cells during operation, the bio-inspired arrays are also referred to as Memristor Oscillatory Networks (MONs) in the remainder of the manuscript. |

4 | |

5 | It is instructive to observe that there exist memristor physical realisations, whose models feature an even more general input- and state- dependent Ohm law than what is admissible for extended memristors. For these two-terminal devices, including the TiO${}_{2}$ memristor from HP Labs [39], the mathematical description includes an implicit Ohm law of the form $h(x,\phantom{\rule{0.166667em}{0ex}}{v}_{m},\phantom{\rule{0.166667em}{0ex}}{i}_{m})=0$, with x, ${v}_{m}$, and ${i}_{m}$ denoting the device state, voltage, and current, respectively. |

6 | The mathematical description of the nonlinear resistor, formulated in Equation (3), is in fact equivalent to the model of a NbO${}_{\mathrm{x}}$ memristor, as originally presented in [34,35], which reveals the correspondence of the real parameters ${R}_{02}$, ${a}_{02}$ and ${a}_{12}$ to physical properties of the nanostructure, in the limit when changes occurring in the device state, defined as its body temperature, are negligible. |

7 | The nominal parameter setting is obtained from Table 1 for $\alpha =0.5$. As will be shown later, simulating the memristor model under a quasi-DC voltage stimulus and with the variability parameter stepped across its existence domain, the locus observed for $\alpha =0.5$ appears in the center of the distribution of characteristics emerging in the voltage-current plane. |

8 | The voltage across (current through) the memristor in the cell i is indicated via ${v}_{m,i}$ (${i}_{m,i}$). |

9 | The units of the steady-state relative phase of oscillator i, computed via ${\phi}_{i}^{\left(s\right)}={\omega}_{0}\xb7\mathsf{\Delta}{t}_{i}$, are radiants. In order to express ${\phi}_{i}^{\left(s\right)}$ in degrees, its formula needs to be scaled by the factor $\frac{{180}^{\circ}}{\pi}$ ($i\in \{0,\dots ,N-1\}$). |

10 | A new graphical tool—which we call phase diagram—is introduced in this research study [31] for visualising the phase dynamics of the network. Referring, for example, to the phase diagram of Figure 6(a), a specific trace visualises the time evolution of the phase of oscillator j relative to oscillator 0 ($j\in \{1,\dots ,N-1\}$). Reading the time flow along the radial direction, the angle between the segment, joining the origin to the point, where the trace is found to lie at time t, and the blue horizontal line, denoting the ${0}^{\circ}$-valued reference level, represents the phase shift ${\phi}_{j}\left(t\right)$ of oscillator j with respect to oscillator 0 at time t. |

11 | The natural frequency of an oscillator is the inverse of the period of the oscillations developing across its circuitry. |

12 | In order to reprogram appropriately the operating point of the memristor in the cell $j\in \{1,2\}$ of the 3-oscillator network under focus, the cell j itself is capacitively coupled only to the reference cell 0, and, as described earlier, $\mathsf{\Delta}{R}_{S,j}$ is tuned until anti-phase synchronisation emerges in the resulting two-cell network. This procedure is carried out separately for oscillators 1 and 2. |

13 | It is important to pinpoint that, while the choice of a reference cell for the preliminary compensation of the memristor device-to-device variability should fall for a specific oscillator, as specified here, no rule dictates the selection of a reference cell for the later computation of the relative phase pattern of the array, as discussed in Section 4. |

14 | It is important to observe that, while taking the proposed device-to-device compensation measure, care need to be taken so as to keep the natural oscillation frequency of each oscillator within a close range. In fact, a wide spread in this parameter, inevitably differing across the cellular medium, due to the ${R}_{S}$ tuning procedure, would jeopardize the convergence of the bio-inspired computing engine to some steady state. |

15 | The proposed strategy will determine the minimum possible number of color groups, which, under a given initialisation setting, the network is able to identify as it classifies the nodes of the associated graph. Importantly, as will be clarified later, this minimum number does not necessarily coincide with the chromatic number of graph, since the network may converge to a correct but suboptimal solution. Methods allowing the memristive array to overcome a suboptimal solution so as to approach the optimal one will be presented shortly. |

16 | If an (no) edge connects vertices i and j, then ${a}_{i,j}={a}_{j,i}=1\left(0\right)$. Note that $\mathbf{A}={\mathbf{A}}^{T}\in {\mathbb{R}}^{N\times N}$. |

17 | Since, here, oscillator i is associated to vertex i for each $i\in \{0,\dots ,N-1\}$, the oscillator sequence, corresponding to the phase shift ordering, may be indifferently referred to as oscillator ranking or vertex ranking. As will be clarified later on, this is not always the case, when perturbation actions are performed on the network to enhance its performance. |

18 | in this work the estimation of the common period T of the oscillations developing in a N-cell network, and the associated group ordering of the phase shifts of the cells 1, …, $N-1$ relative to the null phase of the reference cell 0 are carried out every cycle throughout the duration of any simulation. |

19 | The application of a crossover to pairs of oscillators implies the necessity to endow the network with reprogrammable connections, e.g. via transistor-based switches, which, however, would add on to the integrated circuit (IC) overhead in a future hardware implementation of the network. |

20 | In fact, it is highly probable that this node mostly prevents the optimisation measure of Equation (16) from attaining the global minimum, which would provide as solution to the vertex coloring task the chromatic number of the original N-node graph, as desired. In case, for each of two or more values of k, the application of our vertex coloring strategy to the respective $(N-1)$-node graph, obtained by removing the vertex k from the original graph, results in a common lowest number of colors, any of these node i candidates may be finally considered for the crossover |

21 | The interchange between nodes i and k operated on the original vertex ranking is due to the fact that the application of a crossover between the corresponding oscillators in the network is equivalent to exchanging their associations to the respective pair of vertices in the original graph. The relative phases, inherent to the oscillators, maintain the same ordering, as established originally. As a result, the oscillator ranking remains unaltered, but the mapping from oscillator ranking to vertex ranking is subject to the earlier mentioned node interchange. |

22 | In this work ${T}_{p}$ was set to twice the common graph-dependent period T of the oscillatory waveforms of the capacitor voltages and of the memristor currents in the network before the application of the pulse destabilisation paradigm. |

23 | We acknowledge, however, that the most suitable formula, expressing the relationship between the amplitude $\mathsf{\Delta}\phantom{\rule{-0.166667em}{0ex}}{V}_{S}$ of a destabilising pulse of fixed width ${T}_{p}$ and the resulting sudden shift $\mathsf{\Delta}\phantom{\rule{-0.166667em}{0ex}}\phi $ in the phase of the perturbed oscillator, may depend upon network properties and parameters. A deeper study, aimed to optimise the shape of the destabilisation stimulus, will be carried out in the future. |

24 | In order to present a fair comparison between the beneficial effects of the crossover and pulse destabilisation control paradigms, we ensured that the simulations in Figure 11 and Figure 12 provided identical results for $t\in [0,5)\phantom{\rule{0.166667em}{0ex}}$ms by choosing the same initialisation setting, and assigning a common random set of $\alpha $-values to the memristors. |

25 | In some cases, after overcoming the impasse situation, the dynamical system could approach a new oscillatory solution associated to another local minimum of $G\left(\mathbf{\phi}\right)$. |

26 | The time separation ${T}_{int}$ between consecutive applications of the crossover or pulse destabilisation strategy is set to $2\phantom{\rule{0.166667em}{0ex}}$ms in the simulations discussed in this section. Furthermore, in this work the estimation of the common period T of the oscillations appearing in a N-cell network, and the associated group ordering of the phase shifts of the cells 1, …, $N-1$ relative to the null phase of the reference cell 0 are carried out every cycle throughout the duration of any simulation. Moreover, in the first (latter) control strategy, the application of a pulse to the same oscillator (a crossover involving either oscillator from the same pair) is not allowed until at least 5 iterations of the control strategy have elapsed first. As a result, each pulse destabilisation (crossover) manoeuvre targets a different oscillator (involves a different pair of oscillators). |

**Figure 1.**In-memory computing in a $N\times N$ memristive crossbar array (here $N=4$). In-memory computing in an $N\times N$ memristive crossbar array (here, $N=4$). Naturally obeying Kirchhoff’s Current Law (KCL), the bio-inspired network enables a time-efficient computation of VMMs. With $j\in \{1,\dots ,N\}$, the current flowing down the ${j}^{th}$ column of the array is simply given by ${i}_{j}={\sum}_{i=1}^{N}{G}_{i,j}\xb7{v}_{i}$, where ${G}_{i,j}$ denotes the conductance of the memory resistive switch located at the intersection between the conductive nanowires stretching along row i and column j [29]. The computation of the currents at the outputs of the crossbar columns assumes that the bottom terminals of all the memristors—refer to the thick black horizontal segments in their circuit-theoretic symbols—are at virtual ground.

**Figure 2.**(

**a**) Equivalent circuit of the physical model of a NbO${}_{\mathrm{x}}$ nanoscale memristor $\mathcal{M}$ from NaMLab. The linear resistor ${R}_{C}$ and the nonlinear resistor $\mathcal{R}$ respectively account for the effects of electrode contact resistance and parasitics. (

**b**) Memristor circuit-theoretic symbol.

**Figure 3.**(

**a**) A 6-node ring-shaped undirected graph (

**b**) Associated MON. The oscillator i of the network corresponds to the vertex i of the graph ($i\in \{0,1,2,3,4,5\}$).

**Figure 5.**(

**a**) A 2-node 1-edge graph. Its chromatic number is 2. (

**b**) Oscillatory network corresponding to the graph in (

**a**). (

**c**) A 3-vertex 2-edge graph. Its chromatic number is once again 2. Interestingly, the number of edges departing from vertex 0 (from either vertex 1 or vertex 2) is 2 (1). (

**d**) Oscillatory network corresponding to the graph in (

**c**).

**Figure 6.**(

**a**) Phase diagram visualising the time evolution of the phases of oscillators 1 (in orange) and 2 (in green) relative to oscillator 0, sitting on the ${0}^{\circ}$ phase state throughout the simulation (blue horizontal line) for the original unbalanced network of Figure 5(c) (see the dashed traces) and for the compensated network in plot (

**b**) of this figure (refer to the solid traces). In the first (latter) case, the phases of oscillators 1 and 2 are found to cluster together, and to distance themselves from the reference ${0}^{\circ}$ phase, associated to oscillator 0, by approximately ${50}^{\circ}$ (${180}^{\circ}$) at steady state. (

**b**) Complete circuitry of the memristive oscillatory network of Figure 5(d) after compensation for the imbalance in the number of couplings per oscillator. Here ${C}_{\mathrm{comp},1}={C}_{\mathrm{comp},2}={C}_{C}\Vert C$.

**Figure 7.**(

**a**) Spread in the distribution of quasi-DC memristor current-voltage loci, as emerging from numerical simulations of the model Equations (1), (2), and (3) for all values of the variability parameter $\alpha $ in the set $\{0,0.2,0.4,0.6,0.8,1.0\}$. (

**b**) Phase diagram showing the time evolution of the phase shift of oscillator 1 relative to the ${0}^{\circ}$-valued phase of the reference oscillator 0 for each of the values of the series resistance increment $\mathsf{\Delta}{R}_{S,1}$ in the set $\{50,100,125,151\}\phantom{\rule{0.166667em}{0ex}}\Omega $ (refer in turn to the orange, green, purple, and red traces). The two capacitively-coupled oscillators achieve anti-phase synchronisation for the largest $\mathsf{\Delta}{R}_{S,1}$-value in this set. (

**c**) Phase diagram illustrating the phase dynamics of oscillators 1 (in orange) and 2 (in green) for the balanced network of Figure 6(b) for the case where specific parameters in the model of the memristor in the cells 0, 1, and 2 are respectively controlled by the first, second, and third $\alpha $-value within the set $\{0.5,0,1\}$ (see the dashed traces), and after the negative effects on the network performance associated to the memristor device-to-device variability have been compensated by incrementing (decrementing) the series resistance ${R}_{S}$ by $\mathsf{\Delta}{R}_{S,1}=-134\phantom{\rule{0.166667em}{0ex}}\Omega $ ($\mathsf{\Delta}{R}_{S,2}=+151\phantom{\rule{0.166667em}{0ex}}\Omega $) (refer to the solid lines).

**Figure 8.**(

**a**) ((

**c**)) Phase dynamics of the balanced network of Figure 3(b) with compensation for the memristor device-to-device variability and under a suboptimal (the optimal) initialisation setting. (

**b**) ((

**d**)) Time evolution of the optimisation goal function toward a local (the global) minimum in the simulation scenario illustrated in plot (

**a**) ((

**c**)). In the first (latter) case the application of the vertex coloring strategy to the respective vertex ranking divides the 6 nodes of the graph of Figure 3(a) into 3 (2) colors. In the first (latter) case the composition of each of the 3 (2) color groups is made clear by the colors assigned in plot (

**a**) ((

**c**)) to the arcs of the circular sectors, which host the final destinations of the traces associated to phase shifts clustering together, as well as by the colors assigned in the inset of plot (

**b**) ((

**d**)) to the respective nodes of the graph. Since the chromatic number of the graph is 2, the relative phases among the oscillators of the network converge to a suboptimal (the optimal) pattern in the simulation of plot (

**a**) ((

**c**)).

**Figure 9.**(

**a**) Steady-state time evolution of the memristor current ${i}_{m,i}$ of oscillator $i\in \{0,1,2,3,4,5\}$ over the time interval $[9.9,10]\phantom{\rule{0.166667em}{0ex}}$ms for the simulation of the balanced network of Figure 3(b) with compensation for the memristor device-to-device variability and under the optimal initialisation setting (see also Figure 8(c),(d) for more results obtained from this simulation). The differences in the peak values of the waveforms originate from the variability in the static and dynamic properties of the samples, as reproduced by our memristor model. As was already shown in relation to their relative phases in Figure 8(c), the oscillator triplets $(1,5,3)$ and $(0,2,4)$ group together, as respectively indicated over the first and second half of the first observable cycle, where the order of appearance of the nearby peaks of the memristor currents follows in turn the patterns 1-5-3 and 0-2-4. The same color coding map, as established in Figure 8(c) and reused in the inset of Figure 8(d), is adopted here to differentiate between the traces pertaining to distinct oscillators. (

**b**) Zoom-in view of the time behaviour of each memristor current in the network across the time span $[9.97,10]\phantom{\rule{0.166667em}{0ex}}$ms. The common period T of the oscillatory waveforms is found to be equal to $19.21\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}$s. As an example, the time instant ${t}_{0}$ (${t}_{3}$), at which the memristor current ${i}_{m,0}$ (${i}_{m,5}$) of oscillator 0 (3) crosses the threshold ${I}_{th}=0.5\phantom{\rule{0.166667em}{0ex}}$mA in its ascending phase over the first observable cycle, gives $9973.8457$$\mathsf{\mu}$s ($9984.2351$$\mathsf{\mu}$s), allowing to compute the steady-state phase of oscillator 3 relative to the reference oscillator 0 via ${\phi}_{3}^{\left(s\right)}=\mathsf{\Delta}{t}_{3}\xb7{\omega}_{0}$, with ${\omega}_{0}=\frac{2\xb7\pi}{T}$. Performing a calculation of this kind for each of the remaining 5 oscillators results in the steady-state phase shift vector ${\mathbf{\phi}}^{\left(s\right)}$ reported in Equation (25).

**Figure 10.**(

**a**) Original 6-node ring-based graph. (

**b**) ((

**c**) or, equivalently, (

**d**)) Coupling arrangement in the network associated to the graph in (

**a**), before (after) a crossover between cells 1 and 2, which swaps the correspondence between these cells and the respective nodes in the graph.

**Figure 11.**(

**a**) Time evolution of the relative phases of the oscillators of the balanced network of Figure 3(b), with compensation for the memristor device-to-device variability, and under the same sub-optimal initialisation setting as in the simulation illustrated in Figure 8(a),(b), for the case where the connections of oscillators 1 and 2 are interchanged at $t=5\phantom{\rule{0.166667em}{0ex}}$ms. Right before the application of the crossover to the two cells, the network is found to sit on a stable oscillatory solution associated to a local minimum of the optimisation goal function (see also the three phase clusters emerging in plot (

**a**) right before the crossover procedure). Despite the phase shift vector, measured much earlier than it was done in the simulation of Figure 8(a),(b), was found to be slightly different from the one reported in Equation (17), specifically ${\mathbf{\phi}}^{\left(s\right)}={[{0}^{\circ},{118}^{\circ},{238}^{\circ},{359}^{\circ},{119}^{\circ},{240}^{\circ}]}^{\mathrm{T}}$, the resulting vertex ordering remains defined by Equation (18). (

**b**) Evolution of the optimisation goal function over time. (

**c**) Minimum number of color groups assigned through the iterative vertex coloring procedure of Section 4 to the nodes of the graph in Figure 3(a) over time (the procedure is applied once every T-long cycle, with the common period T of the oscillatory waveforms of the currents through the memristors, measured over the time interval $[4.965,4.984]\phantom{\rule{0.166667em}{0ex}}$ms, found to be equal to $19.24$$\mathsf{\mu}$s). After the crossover these nodes are classified into 2 groups (

**c**). The interchange between the couplings of oscillators 1 and 2 was thus found to resolve the impasse, allowing the memristive array to approach the optimal solution associated to the global minimum of $G\left(\mathbf{\phi}\right)$, and to identify the chromatic number $n=2$ of the associated graph (see also the two phase clusters emerging in plot (

**a**) at the end of the simulation).

**Figure 12.**(

**a**) Phase dynamics of the balanced network of Figure 3(b), with compensation for the memristor device-to-device variability, and under the same sub-optimal initialisation setting as in the simulation illustrated in Figure 8(a),(b), for the case where the voltage ${V}_{S}$ of the DC source in oscillator 1 is offset by $\mathsf{\Delta}\phantom{\rule{-0.166667em}{0ex}}{V}_{s}=-0.23\phantom{\rule{0.166667em}{0ex}}$V from the time instant $t=5\phantom{\rule{0.166667em}{0ex}}$ms for a temporal window of duration ${T}_{p}=38.48\phantom{\rule{0.166667em}{0ex}}$$\mathsf{\mu}$s (the common period T of the oscillatory waveforms of the currents through the memristors, measured over the time interval $[4.965,4.984]\phantom{\rule{0.166667em}{0ex}}$ms, was found to be equal to $19.24$$\mathsf{\mu}$s). As discussed earlier, right before the application of the pulse to cell 1, which triggers a sudden shift in its relative phase by approximately ${180}^{\circ}$, the network is found to sit on a stable oscillatory solution associated to a local minimum of the optimisation goal function (see also the three phase clusters emerging in plot (

**a**) right before the pulse destabilisation action). Despite the phase shift vector, measured much earlier than it was done in the simulation of Figure 8(a)-(b), was found to be slightly different from the one reported in Equation (17), specifically ${\mathbf{\phi}}^{\left(s\right)}={[{0}^{\circ},{118}^{\circ},{238}^{\circ},{359}^{\circ},{119}^{\circ},{240}^{\circ}]}^{\mathrm{T}}$, the resulting vertex ordering remains defined by Equation (18). (

**b**) Time evolution of the optimisation goal function. (

**c**) Minimum number of color groups assigned through the iterative vertex coloring procedure of Section 4 to the nodes of the graph in Figure 3(a) over time (the procedure is applied once every T-long cycle). After the pulse destabilisation 2 colors are assigned to these nodes. The pulse-based perturbation of oscillator 1 was thus found to resolve the impasse, allowing the memristive array to approach the optimal solution associated to the global minimum of $G\left(\mathbf{\phi}\right)$, and to identify the chromatic number $n=2$ of the associated graph (see also the two phase clusters emerging in plot (

**a**) at the end of the simulation). Despite, at the time instant $t=5\phantom{\rule{0.166667em}{0ex}}$ms, when the pulse perturbation commences, $G\left(\mathbf{\phi}\right)$ undergoes a sudden increase from the local minimum value of $-3$, it descends steeply straight away, decreasing monotonically toward the global minimum value of $-6$ thereafter.

**Figure 13.**Evidence for the capability of a 25-cell network, preliminarily compensated for the unbalance in the number of connections per oscillator, and for the inter-device variability inherent to memristors, to converge toward the optimal solution of the vertex coloring problem for the graph queen5_5 [50]. The cyclic application of a pulse stimulus of fixed length and appropriate amplitude to an ad-hoc oscillator of the network guides the phase dynamics toward the global minimum solution. (

**a**) Phase diagram visualising the time evolution of the phase of each oscillator $j\in \{1,\dots ,24\}$ relative to the phase of the reference oscillator 0. (

**b**) Time waveform of the optimisation goal function $G\left(\mathbf{\phi}\right)$. (

**c**) Progression of the outcome of the iterative vertex coloring procedure of Section 4 over time. Throughout the second half of the simulation the minimum number of color groups, assigned to the vertices of the graph queen5_5, visualised in the inset of plot (

**b**), is fixed to the chromatic number $n=5$ of the graph itself, despite the network is subject to further pulse-based perturbations.

**Figure 14.**(

**a**) Phase dynamics of the network associated to the graph queen6_6 [50], after its preliminary compensation for the unbalance in the number of connections per oscillator, and for the inter-device variability inherent to memristors, upon the periodic interchange between the couplings of two specific oscillators. In this case the phase dynamics of the network keep in a transient state throughout the $100\phantom{\rule{0.166667em}{0ex}}$ms-long simulation. (

**b**) Evolution of the optimisation goal function $G\left(\mathbf{\phi}\right)$ over time. (

**c**) Minimum number of colors, assigned to the nodes of the graph queen6_6 through the iterative vertex coloring procedure of Section 4, applied once every cycle, versus time. Half way through the simulation the nodes of the graph, illustrated in the inset of plot (

**b**), are classified into 8 color groups, one more than the correct number (refer to Table 3), but this solution proves to be unstable, when the network, thereafter, is subject to further crossover-based perturbations.

**Table 1.**Parameter setting for NbO${}_{\mathrm{x}}$ nanoscale threshold switch from NaMLab. The effects of the memristor-to-memristor variability are accounted through the assignment of a distinct value, chosen randomly within the closed set $[0,1]$ to the variable $\alpha $, controlling specific coefficients of Equations (1), (2), and (3).

${C}_{th}/\phantom{\rule{0.166667em}{0ex}}$J · K${}^{-1}$ | ${\Gamma}_{th,\alpha}/\phantom{\rule{0.166667em}{0ex}}$W · K${}^{-1}$ | ${T}_{amb}/\phantom{\rule{0.166667em}{0ex}}$K | ${R}_{01,\alpha}/\phantom{\rule{0.166667em}{0ex}}\Omega $ | ${a}_{01,\alpha}/\phantom{\rule{0.166667em}{0ex}}$K |

$1\xb7{10}^{-14}$ | $1.889\xb7{10}^{-6}\xb71.{064}^{\alpha}$ | 293 | $3.047\xb70.{831}^{\alpha}$ | $3620\xb71.{061}^{\alpha}$ |

${a}_{11,\alpha}/\phantom{\rule{0.166667em}{0ex}}$K· V${}^{-1}$ | ${R}_{c,\alpha}/\phantom{\rule{0.166667em}{0ex}}\Omega $ | ${R}_{02,\alpha}/\phantom{\rule{0.166667em}{0ex}}\Omega $ | ${a}_{02}/\phantom{\rule{0.166667em}{0ex}}$K | ${a}_{12,\alpha}/\phantom{\rule{0.166667em}{0ex}}$K· V${}^{-1/2}$ |

$820.4\xb71.{137}^{\alpha}$ | $173.8\xb71.{092}^{\alpha}$ | $565\xb71.{377}^{\alpha}$ | 1000 | $168.8\xb71.{083}^{\alpha}$ |

**Table 2.**Parameter setting for the non-memristive circuit elements in the oscillator of Figure 4(a).

${\mathit{V}}_{\mathit{S}}/\phantom{\rule{0.166667em}{0ex}}$V | ${\mathit{R}}_{\mathit{S}}/\phantom{\rule{0.166667em}{0ex}}{\Omega}$ | $\mathit{C}/\phantom{\rule{0.166667em}{0ex}}$F | ${\mathit{C}}_{\mathit{C}}/\phantom{\rule{0.166667em}{0ex}}$F |
---|---|---|---|

$2.5$ | 5525 | $10\xb7{10}^{-9}$ | $0.2\xb7{10}^{-9}$ |

**Table 3.**Comparison between the solutions of the vertex coloring problem for various graphs [50] for the 2

^{nd}algorithm implementation challenge for NP-hard problems in DIMACS [51], obtained through the application of a specific algorithm, known as Brélaz heuristic [52], by means of methods exploiting the phase dynamics of capacitively-coupled memristive networks without a control strategy for bypassing local minima solutions, namely the technique proposed in [42], and the iterative node coloring procedure, presented in Section 4, and via the iterative node coloring procedure augmented with strategies, based upon crossover and pulse destabilisation, presented in Section 5.1 and Section 5.2, respectively, and aimed to overcome local minima solutions [31]. The results tabulated in the last three columns were computed through the analysis of $100\phantom{\rule{0.166667em}{0ex}}$ms long numerical simulations.

Minimum Number of Color Groups for the Classification of the Vertices of the Associated Group | |||||||
---|---|---|---|---|---|---|---|

graph | vertices | n | Brélaz algorithm | [42] | iterative strategy | iterative strategy and crossover control | iterative strategy and pulse destabilisation control |

mycie13 | 11 | 4 | 4 | 4 | 4 | 4 | 4 |

mycie14 | 20 | 5 | 5 | 5 | 5 | 5 | 5 |

mycie15 | 47 | 6 | 6 | 6 | 7 | 6 | 6 |

queen5_5 | 25 | 5 | 7 | 6 | 7 | 5 | 5 |

queen6_6 | 36 | 7 | 10 | 12 | 11 | 8 | 8 |

queen7_7 | 49 | 7 | 12 | 12 | 14 | 10 | 10 |

queen8_8 | 64 | 9 | 15 | 14 | 15 | 13 | 13 |

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**MDPI and ACS Style**

Ascoli, A.; Weiher, M.; Herzig, M.; Slesazeck, S.; Mikolajick, T.; Tetzlaff, R.
Graph Coloring via Locally-Active Memristor Oscillatory Networks. *J. Low Power Electron. Appl.* **2022**, *12*, 22.
https://doi.org/10.3390/jlpea12020022

**AMA Style**

Ascoli A, Weiher M, Herzig M, Slesazeck S, Mikolajick T, Tetzlaff R.
Graph Coloring via Locally-Active Memristor Oscillatory Networks. *Journal of Low Power Electronics and Applications*. 2022; 12(2):22.
https://doi.org/10.3390/jlpea12020022

**Chicago/Turabian Style**

Ascoli, Alon, Martin Weiher, Melanie Herzig, Stefan Slesazeck, Thomas Mikolajick, and Ronald Tetzlaff.
2022. "Graph Coloring via Locally-Active Memristor Oscillatory Networks" *Journal of Low Power Electronics and Applications* 12, no. 2: 22.
https://doi.org/10.3390/jlpea12020022