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On Distributed Computation of Information Potentials Andreas Loukas, Matthias Woehrle, Philipp Glatz, Koen Langendoen Embedded Software Group Delft University of Technology, The Netherlands

{a.loukas, m.woehrle, p.m.glatz, k.g.langendoen}@tudelft.nl

ABSTRACT

1.

A common task of mobile wireless ad-hoc networks is to distributedly extract information from a monitored process. We define process information as a measure that is sensed and computed by each mobile node in a network. For complex tasks, such as searching in a network and coordination of robotic swarms, we are typically interested in the spatial distribution of the process information. Spatial distributions can be thought of as information potentials that recursively consider the richness of information around each node. This paper describes a localized mechanism for determining the information potential on each node based on local process information and the potential of neighboring nodes. The mechanism allows us to distributedly generate a spectrum of possible information potentials between the extreme points of a local view and distributed averaging. In this work, we describe the mechanism, prove its exponential convergence, and characterize the spectrum of information potentials. Moreover, we use the mechanism to generate information potentials that are unimodal, i.e., feature a single extremum. Unimodality is a very valuable property for chemotactic search, which can be used in diverse application tasks such as directed search of information and rendezvous of mobile agents.

Mobile ad-hoc wireless networks are often used to monitor a process and extract information from it. Such a process can be a physical, environmental process, e.g., measuring climate conditions and sensing human activities, or a virtual, intrinsic process of the network, e.g., measuring congestion. Monitoring relies on locally sensing the process and computing a quantity that we call process information. While the process information provides us with an accurate description of the process in close proximity of an individual node, we are often interested in its spatial distribution over a given (larger) area. In fact, there is not a single spatial distribution, but rather a whole spectrum of spatial distributions. The extremes of this spectrum can be characterized as on the one end only considering the local process information and on the other end as averaging all process information in the network. We denote the set of all spatial distributions as information potentials. An information potential recursively considers the richness of information around each node, i.e., the process information on each node and the potential of neighboring nodes. It is in essence an information aggregate over a network area of variable size. Our main objective is to address the problem of attaining global feedback from information potentials based on local interactions alone. An adaptive approach is necessary that considers the time-varying nature of the underlying process as well as the volatile nature of mobile, ad-hoc networks. This paper introduces a localized mechanism for determining the information potential on each node. Our mechanism can distributedly generate a spectrum of information potentials between the two extreme points of a local view and distributed averaging. We accomplish this transition from local to global through local (1-hop) state exchange that follows the diffusion paradigm: nodes influence their neighbors by exchanging and fusing their values locally. The resulting mechanism is (i) adaptive, since the exchange continues indefinitely, (ii) localized with a 1-hop knowledge requirement, (iii) lightweight with minimal computation and state requirements, and (iv ) converges exponentially fast, which makes it (v ) inherently scalable. The mechanism is also (vi) generic because it relies on process information as an abstract concept that describes a wide variety of physical and virtual processes. In addition to revealing the characteristics of an underlying process, information potentials provide an (implicit) local ranking of importance among nodes and identify some areas of the network as more important in terms of the specific information process. We exploit this ranking to find short

Categories and Subject Descriptors C.2.4 [Computer-Communication Networks]: Distributed Systems

General Terms Algorithms, Theory

Keywords Mobile Ad-Hoc Networks, Diffusion, Information Potentials, Chemotactic Search, Local Algorithms

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. FOMC’12, July 19, 2012, Madeira, Portugal. Copyright 2012 ACM 978-1-4503-1537-1/12/07 ...$15.00.

INTRODUCTION

class I class II

application routing self-monitoring rendezvous foraging

information context QoS various sensed quantity

entity message healing agent robot robot

Table 1: Application classes for information potentials. paths towards areas of the network with rich information. By interpreting assigned values as a chemical concentration over the network, we can use the biological process of chemotaxis to provide directions. Since chemotactic search [1] relies on greedy decisions, it can get stuck in local extrema. To overcome this shortcoming we identify the spectrum of information potentials that feature a single, global extremum and no local extrema, i.e., that are unimodal. Unimodality guarantees that a greedy search will be globally optimal. Because of the generality of our approach, we can use information potentials as a building block for a wide range of possible applications. We have identified two classes: (i) first, a class that relates to the movement of software entities and (ii) a second class that relates to the movement of physical entities. Both classes in essence encompass applications in which an entity is directed towards areas of the network with large information potential. They differ on which type of information is of interest and on the nature of the directed entities. We proceed by giving two concrete example applications from each class: Examples of the first class include context-aware routing and network self-monitoring. Context-aware routing entails the routing of messages through a network such that the routing load is balanced with respect to the availability of resources or QoS (e.g., congestion awareness [9], topology awareness [10]). In network self-monitoring, the network should detect failures or performance anomalies and software healing agents are dispatched to the affected area. The second class deals with the movement and coordination of robotic swarms. The two exemplary problems are rendezvous, where all robots should meet at a single point, and foraging, where the robots should discover and ‘consume’ a sensed quantity. The proposed application classes are summarized in Table 1. Note that of the four applications only the rendezvous requires a unimodal information potential. The other three applications rely on information potentials that consider network areas of smaller, yet variable size. This motivates the need for generating the whole spectrum of information potentials. Summarizing, this work makes the following contributions: • We describe our parameterized mechanism based on diffusion, demonstrate how it can generate a wide spectrum of information potentials, and prove its exponential convergence. • We discuss how our mechanism generates information potentials that are unimodal. • We present two case studies where we use our mechanism with chemotactic search for the rendezvous of mobile agents. The case studies are chosen to demonstrate the unimodality property. Our paper is structured as follows: Section 2 discusses related work. Section 3 details on the mechanism and proves

convergence to all possible information potentials. Section 4 describes the unimodality property and identifies parameterizations of the mechanism that generate unimodal information potentials. In Section 5, we present an application of our mechanism for rendezvous of mobile agents based on chemotactic search. Section 6 concludes our work.

2.

RELATED WORK

This section first discusses related work for computing single-valued information aggregates: distributed consensus and fusion mechanisms as a special case. Second, the idea of information potentials is compared to potential fields with a focus on robotics applications and wireless sensor networks.

2.1

Distributed Consensus

Different approaches exist for solving consensus problems that are either based on (i) flooding, i.e., propagating data to all nodes, (ii) having all information routed along a minimum spanning tree towards a sink or (iii) diffusion, i.e., updating local estimates by means of iteratively averaging over a 1-hop neighborhood. The latter have been proved as very effective methods of computation for mobile ad-hoc networks [1, 3, 5, 7, 14]. In their seminal paper, Kempe et al. [14] provide theoretical foundation for gossip-based consensus algorithms. The proposed algorithms compute aggregates such as averages, sums, and quantiles of an underlying process. Many improvements of those algorithms have been made in recent years: the theory and scope of computed functions has been expanded, (Cortes [5]), the convergence and mixing times have been studied and improved under specific assumptions (Boyd et al. [3], Dimakis et al. [7]), and the role of consensus as a “design pattern” has been investigated (Babaoglu et al. [1]). Our approach on the other hand does not compute a single aggregate of a monitored process, but constructs distributions over the network that reveal the characteristics of the underlying information process. Recent approaches have also combined consensus with other methods of computation [19, 20]. Xiao et al. [20] studied the common problem of distributed sensor fusion in wireless sensor networks by means of a distributed average consensus and maximum likelihood estimation, while Spanos et al. [19] used distributed Kalman Filtering methods for a similar purpose. While sensor fusion approaches require a model of a specific process, our approach is modelfree and therefore generic. Our approach additionally allows the realization of a spectrum of spatial distributions of process information, considering also potentials with a local scope.

2.2

Potential Fields, Search and Coordination

Potential functions have been used as a means of common information mediums in various fields of distributed systems. In the field of robotics potential fields [4, 8, 15, 17, 18, 21] are used to achieve coordination of robots. Potential fields are virtual, objective-specific functions without an underlying information process. They are distributedly computed over a network of robots. Goals are pursued by following gradients over respective functions with only localized interaction [17]. Through the use of potential fields one can plan and control robot trajectories, enforce connectivity [8, 21],

(a) network

(b) local, ϕ = 1

(c) semi-global, ϕ = 0.05

(d) global, ϕ = 0.005

Figure 1: Information potentials (b–d) for a network of 1000 nodes (a). We use Voronoi diagrams with cells that correspond to nodes. Darker color signifies richer information, i.e., smaller information potential. and achieve object manipulation [18]. As exemplified by Zavlanos et al. [21], combinations of potential functions can be used for accomplishing multiple objectives: a group of robots should rendezvous at a single location or follow a leader as well as maintain a connected network and avoid robot collisions. What our approach and potential fields have in common is that they both construct a spatial distribution over networks on which they rely as a coordination medium. Different from potential fields that are agnostic to the information process, our mechanism is inherently tied to and interacts with an underlying process. As such, both approaches are complementary. Furthermore, while in robotics one usually positions in absolute coordinate systems, our approach models position as being relative to the network graph. In essence, no global coordinate system is needed but only a sense of direction [12] relative to the information process at hand. Closest to our work, Lin et al. [16] focus on information discovery in wireless sensor networks. They construct smooth gradients towards sources such that local forwarding decisions suffice to guarantee the sources being discovered. This is similar to chemotactic search in Section 5. Their information gradients are computed by assigning constant values to the source and boundary nodes, while averaging the values on all other nodes. These information gradients are in essence harmonic functions, which are known for their various properties, especially the absence of local extrema and maxima only in the source nodes. A major caveat of the work by Lin et al. [16] is that it requires fixing source and boundary nodes and is thus typically limited to static networks. In contrast, our work focuses on mobile ad-hoc networks, where the network, including source and boundary nodes, is dynamic and open, i.e., subject to churn. Due to this difference in domain our assumptions and goals are slightly different as (i) we assume that every node is a source with its own information content and maintain no concept of boundaries and (ii) we construct a spectrum of potentials, where we can achieve a single (unimodal potential) or multiple (multimodal potential) extrema irrespectively of the number of source nodes. Noteworthy is the large application space described in Lin et al. [16], e.g., greedy local routing, routing diversity and traffic balancing, that directly applies to the the information potentials we introduce in this paper.

3.

DIFFUSION MECHANISM

In this section we provide a formal presentation of the mechanism. We analyze assumptions in Section 3.2 and convergence properties in Section 3.3. We then proceed to examine its stable state in Section 3.4. Our analysis provides insight to the mechanism’s principle of operation and provides the necessary foundation for generating unimodal information potentials as discussed in Section 4.

3.1

Network Model and Notation

We model a network as a connected graph G = (V, E), where V is the node set of cardinality n and the edge set E that contains one edge for each pair of nodes that reside inside each other’s transmission range, i.e., we use a unitdisc model. We denote a node pair as neighbors if (i, j) ∈ E and we write i ∼ j. The node cell is a set that contains a node and its neighbors, Ci = {j : i ∼ j or i = j} and its cardinality is the node density di = |Ci | = deg(i) + 1. Let A be the n×n symmetric adjacency matrix, such that Aij = 1 if i ∼ j or i = j, and Aij = 0 otherwise. Contrary to common definition, A has ones on its diagonal. Let D and D−1 be the n × n diagonal density and inverse density −1 matrices with Dii = di and Dii = 1/di , ∀ i ∈ V and zero otherwise. 1 is a vector with all elements equal to one. I is the identity matrix with ones on the main diagonal and zeros elsewhere. Unless otherwise specified, all variables are n × 1 vectors and vector elements are denoted by indexes. All matrices are n × n. Table 2 outlines the notations used in the presentation of our mechanism. Process information is a value that is computed/sensed by each node and that represents the quantity or quality of a physical/virtual process at the vicinity of the node. We define it as a real-valued function over V, x : V → R+ that assigns a real value to each node. In order to facilitate presentation, we additionally introduce the process entropy vector z, defined as the inverse information process zi = 1/xi . Our mechanism computes the information potential y ∈ R+ , which is again a real-valued function over V that diffuses x over the network. Since x, z and y are in essence vectors indexed by the vertices of G, we use a vector representation in the following.

x z y ϕ

process information process entropy information potential inhibiting factor

n×1 n×1 n×1 scalar

xi ∈ R+ zi ∈ R + yi ∈ R+ ϕ ∈ (0, 1]

Algorithm 1 Mechanism running on node i 1: input 2: inhibiting factor ϕ

Table 2: Notation for the analysis of the mechanism.

3: initialize 4: round t := 0 5: yi := random

3.2

6: event at each slot 7: broadcast current yi (t) 8: end event

Mechanism

The mechanism with which a node i computes information potential yi (t + 1) at time t + 1 is given by the following simple equation: yi (t + 1) =

X yj (t) X ϕ + (zi − yj (t)) di di j∈C j∈C i

(1)

i

The inhibiting factor ϕ is a parameter that inhibits or stimulates diffusion and controls the transition from global to local. It is easy to see that when ϕ = 0 the equation simplifies to non-mass conserving distributed averaging [1], i.e., a pure diffusive process that always results in uniform y and thus realizes a global view. In contrast, when ϕ = 1 the information potential only relies on a local one-hop view: yi = zi/di . We show how non-extremal values of the inhibiting factor influence our mechanism in Section 3.4. Please note that we normalize both the information process and the process information of neighboring nodes by the density, i.e., the information potential is dependent on both process information and density. This is useful for several applications such as network monitoring where the density characteristics provide us with additional information as well as when higher (sampling) density in an area indicates richness of process information. Nevertheless, if we want to remove any weighting effect of the density on the process information, we can do so by choosing the process entropy as zi = di /xi . This does neither affect the theoretical analysis nor the operation of the mechanism in practice. As we will see in the following, the choice of z is not crucial for the properties of the mechanism. Instead it alters the semantics of the potential. Here, we choose to assign to z the inverse value of information process. By doing so, the mechanism creates potentials where the largest information coincides with the minimum value. This however is purely a convention and can be changed as one sees fit. In Figure 1(b)–1(d) we display the mechanism’s operation on the static network shown in Figure 1(a) by coloring the area around each node, in this case using Voronoi cells centered at each node, corresponding to a node’s information potential. Nodes with smaller y and thus richer information are drawn with darker color. For ease of presentation we choose a constant process information x = z = 1, such that the potential only depends on the network (and therefore information) density. For smaller values of the inhibiting factor we observe that the potential becomes smoother as it represents the information of a wider area. The case of Figure 1(d) is particularly interesting because the potential features a single minimum, i.e., it is unimodal. Since the information is constant across nodes, the minimum is positioned on the most connected node, i.e., the node with the most neighbors, that have the most neighbors, and so on. We discuss the property of unimodality is detail in Section 4 and show its usefulness in Section 5. Additionally, we can see the influence of ϕ on modality: With a small ϕ

9: event at each round 10: t=t+1 11: update information process xi and zi = 1/xi 12: compute yi (t) based on Equation (1) and received yj ∈ C i 13: end event

we achieve unimodality as described above across the whole network. When increasing ϕ to 0.05 in Figure 1(c), we see that one additional minimum emerges on the lower left. Furthermore, the two clusters of nodes on the middle left of the network appear as separate regions, each region having its own minimum. While the resulting information potential is not unimodal over the complete network, it is in fact unimodal over the three different regions that are indicated by the dividing white lines in the figure. Our mechanism is based on Equation (1) and Algorithm 1. The algorithm is simple and relies on few assumptions: (i) a synchronous slotted execution model and (ii) a round length that allows nodes to gather complete 1-hop knowledge in each round. The first assumption is only needed for the following theoretical analysis of the mechanism; we use an asynchronous execution model in the simulations of our mechanism in the case study presented in Section 5. Equation (1) assumes that the changes of x are on a longer time scale than the execution of the algorithm, which is usually true for process information such as physical phenomena, e.g., climate, and in applications such as rendezvous studied in Section 5.1.

3.3

State Evolution

To facilitate the analysis of the state evolution of our mechanism, we rewrite Equation (1) in matrix form. y(t + 1) =D−1 Ay(t) + ϕD−1 (z − Ay(t)) =(1 − ϕ)P y(t) + ϕD−1 z

(2)

where P = D−1 A. Solving iteratively we express the state at step t based only on the initial state y(0). state evolution

z }| { initial state t−1 }| { z X (1 − ϕ)k P k D−1 z y(t) = (1 − ϕ)t P t y(0) +ϕ

(3)

k=0

Equation (3) is a closed-form solution of the stable state. The equation converges exponentially with convergence rate of log(1 − ϕ). In more detail, the geometric sequence, with common ratio (1−ϕ) P and scale factor 1, converges to zero, i.e., limt→∞ (1 − ϕ)t P t = 0. Due to the fact that P is row stochastic and as such P t 1 = 1 for t ∈ Z+ , where 1 is a

10

2

10

0

random walk we are considering is defined by the transition matrix P = D−1 A and describes a random transition in a graph where the transition probability from node i to a neighboring node j is inversely proportional to i’s density, Pij = 1/di if i ∼ j or i = j, and Pij = 0 otherwise. Note that the walk considers density instead of degree; thus there is a non-zero probability of the walk staying at the same node at each step.

variance

φ = 10-1

φ = 10-2 −2

10

φ = 10-3

In a connected network, the Markov chain based on P is:

−4

time-homogeneous, the probability of a transition is independent of t,

10

φ = 10-4

1

5

10

time

15

20

25

Figure 2: Average node variance per simulation step for varying values of ϕ in a random network of 1000 nodes. vector with all elements equal to one, both the initial state y(0) as well as state evolution diminish with rate log(1 − ϕ). Nevertheless, the graph structure does play a role on the convergence. It is well known that the convergence of P k depends on the second eigenvalue of P , λP,2 : ||P k − π|| ≤ √ n|λkP,2 |. Therefore, more connected graphs exhibit faster converge. Attention has to be given to Equation (3) for ϕ = 1. In this particular case, the equation simplifies to y(t) = 00 D−1 z. The interpretation of 00 is actually an issue of debate by mathematicians. In cases that involve exponent continuity, 00 is undefined. When however the exponent is discrete (e.g., power series, binomial theorem) it is generally accepted that 00 = 1. Here, the exponent k is discrete and we interpret 00 as 1. This simplifies our analysis as it eliminates the need for a special case. Figure 2 demonstrates the evolution of state in a random network of 1000 nodes with an average degree of 8 for different values of the inhibiting factor ϕ. Instead of showing the state of all nodes individually, the figure displays the average state variance of y for each simulated time step t. First, we can see that even though the convergence is exponential to ϕ and therefore takes long when ϕ is very small, node variance and hence relative node potentials stabilize after a few steps. Even though it is not apparent in the figure, the relative ordering of node potentials also stabilizes fast. This is a useful property of our mechanism that allows us to rank nodes based on their potentials even before the mechanism convergences. The property in fact appears in most cases where diffusion is involved. A second observation from the figure is that the variance is a function of the inhibiting factor. As described in Section 3.2, larger values of ϕ result in a more local view and as such in a larger variance. Smaller values take a larger part of the network into account and decrease the variance.

3.4

Stable State

The information potential is the stable state our mechanism converges to. In the following we analyze this stable state. The analysis relies on modeling the operation of the mechanism as random walks on graphs. In particular, the

irreducible, any state can be reached from any other state with positive probability, aperiodic, the period of each state is one because we explicitly model self-loops, and ergodic, any state can be reached from any other state in a finite number of steps equal to the network diameter. For irreducible, aperiodic, time-homogeneous chains, it is well known that a unique stationary distribution state exists and that the chain converges to this distribution from any initial state. According to the Perron-Frobenius theorem, the transition matrix P has a unique largest eigenvalue that is equal to 1 and all other eigenvalues are real and have smaller magnitude. 1 = λP,1 > λP,2 ≥ · · · ≥ λP,s > −1 In our notation λP,i denotes the ith eigenvalue of matrix P and s is the number of distinct eigenvalues with s ≤ n. Since 0 ≤ 1 − ϕ < 1, the largest eigenvalue of the matrix (1 − ϕ) P is always smaller than 1 and we can rewrite Equation (3) as: ∞ X (1 − ϕ)t P t = (I − (1 − ϕ) P )−1 = F −1

(4)

t=0

From its definition, we can show that matrix F has the same eigenvectors u as P and that its eigenvalues λF relate to ϕ and λP . F u = (I − (1 − ϕ) P ) u = λF u ⇔ 1 − λF u = λP u Pu= 1−ϕ

(5)

Therefore, λF = 1 − (1 − ϕ) λP and the (left and right respectively) eigenvectors of P and F are the same. Applying eigenvalue decomposition we can decompose F into −1 F −1 = U Λ−1 F U

(6)

where U is the matrix of eigenvectors of P , Λ−1 F is the inverse of the diagonal eigenvector matrix, and U −1 is the matrix whose rows equal to the left eigenvectors of P . Using Equation (3) and (6) the stable state becomes −1 y = lim y(t) = U (ϕ Λ−1 D−1 z F )U

(7)

t→∞

    ϕ Λ−1 F =  

1 0 .. . 0

0 ϕ 1−(1−ϕ) λP,2

.. . 0

··· ··· .. . ···

0 0 .. . ϕ 1−(1−ϕ) λP,s

     

Let us interpret the information potential y through the spectral expansion of Equation (7) in Equation (8). The potential is equal to the sum of the eigenvectors of P weighted by the corresponding eigenvalues. The multiplication with −1 D−1 z acts as a normalization mapping of U (ϕ Λ−1 to F )U an n × 1 vector. We distinguish between right (column vector) and left (row vector) eigenvectors by their dimension. As such ui is the ith n × 1 right eigenvector and u> i is the ith 1 × n left eigenvector. y=

s X > −1 (ϕΛ−1 z F )ii ui ui D

(8)

i=1

According to the value of the inhibiting factor, we distinguish three cases: (i) ϕ  0+ : Only the first element of ϕΛ−1 remains nonF zero. This means that the equation is considering the principal left and right eigenvectors alone, y = 1 π > D−1 z, where π > is the transpose of the stationary distribution of the random walk. If one ignores D−1 z the result is identical to the stationary distribution of the random walk; it is the probability of ending up in a certain node after a sufficiently long random walk. By multiplying with D−1 z however, the information potential of all nodes becomes equal to the normalized inverse global network information: y|ϕ→0+ = (

n X

dj )−1

n X

j=1

x−1 j 1

(9)

j=1

(ii) ϕ = 1: ϕΛ−1 F = I and the information potential becomes equal to the normalized inverse local information: y|ϕ=1 = D−1 z = [(x1 d1 )−1 . . . (xn dn )−1 ]>

(10)

(iii) ϕ ∈ (0, 1): For intermediate values of ϕ the diagonal elements of ϕΛ−1 F are positive, smaller than one – with the exception of the first element that is always one – and sorted with decreasing magnitude. The inhibiting factor ϕ affects the system by decreasing the importance of secondary eigenvectors relative to the principal one. Its effect becomes clear if we consider that larger eigenvectors (i.e., that correspond to larger eigenvalues) reveal more global properties than smaller ones. As the inhibiting factor becomes smaller, the influence of secondary eigenvectors decreases and the potential becomes closer to the inverse global information. For larger values of ϕ on the other hand the potential reveals the information of a smaller area. In short, through ϕ the potential becomes more or less sensitive to local information effects as compared to the global information view.

4.

THE UNIMODALITY PROPERTY

A key property of our mechanism is that it can generate unimodal information potentials. Unimodality guarantees a single extremum and allows us to combine information potentials with chemotactic search, such that a greedy search can not get stuck in local minima. Instead it always finds a short path to the largest information potential.

4.1

Unimodality Definition

Unimodality is classically defined for continuous functions:

Definition 1 (Unimodality). A function f : R → R is unimodal if and only if it has a single extremum and is monotonic on either side of the extremum. Even though unimodality is often associated to strict monotony, we define it in a non-strict meaning. It follows that the first derivative of f can be zero more than once (i.e., not solely at the extremum). Nevertheless, there is only one (global) extremum. We extend unimodality to functions defined on graphs: Definition 2 (Unimodality of Graph Functions). A function f : V → R defined on a graph is unimodal if and only if it has a single extremum at a node i and for each node at least one path to node i exists on which the values of f are monotonic. Similarly, we conjecture that our mechanism can generate unimodal information potentials by choosing appropriate ϕ. Conjecture 1 (Unimodality Conjecture). Let y be the information potential of a graph G for a given process information x. A sufficiently small ϕ exists such that y is unimodal. Actually, given our convention of using information entropy, the global extremum of our mechanism is a minimum and it coincides with the largest information potential. In the following we outline a proof idea of the unimodality property. We start by relating the transition matrix P of our mechanism to the normalized Laplacian L of a graph. Hence, we can use results from spectral analysis to prove the unimodality of the eigenvectors of L. From the unimodality of eigenvectors, we can reason about the unimodality of the information potential by considering that it is a sum of unimodal eigenvector nodal domains multiplied with density and process information.

4.2

Proof Idea

Given that our adjacency matrix has ones in its diagonal, we define the normalized Laplacian as  1   1− if i = j   di  1 Lij = −p if i ∼ j    d i dj   0 otherwise where we add self-loops on the diagonal. We have that P = D− /2 (I − L)D 1

1/2

(11)

The eigenvalues and eigenvectors of P and thus F can be related to those of L as shown below. (D

1/2

P uP = λP uP 1 uP = λp D /2 uP

P D− /2 )D 1

1/2

(I − L)D

1/2

and by setting uL = D

1/2

uP = λp D

1/2

⇔ ⇔

uP

uP , it follows that

LuL = (1 − λP )uL

(12)

Combining Equation (5) and (12) we determine the eigenvectors of P and F as: λP = 1 − λL

and

λF = λL + ϕ(1 − λL )

(13)

Furthermore, the eigenvectors of F and P are the same and they relate to the eigenvector of the normalized Laplacian: uF = uP = D− /2 uL 1

1

process entropy z

(14)

As is customary, the eigenvalues and corresponding eigenvectors of the graph Laplacian are sorted by increasing magnitude. This is contrary to the typical ordering of P . In the following we re-order eigenvalues of P to match the ordering of the Laplacian. Also note that when the eigenvector index is dropped, e.g., ui , it is implied that the eigenvector belongs to matrix P (and F ). We follow with some results from spectral graph theory [2] that helps us understand the nature of the eigenvectors of the Laplacian. In particular we rely on the concept of a nodal domain (also called sign graph). A nodal domain of a function f defined on a graph is a maximal connected induced subgraph consisting only of nodes on which the function takes either positive or negative values. Strong positive (negative) nodal domains of f consist entirely of nodes with positive (negative) values fi > 0 (fi < 0), whereas in weak positive (negative) nodal domains fi ≥ 0 (fi ≤ 0). Theorem 1 (Discrete Nodal Domain Theorem [6]). Let L be a generalized Laplacian of a connected graph with n vertices. Then any eigenvector uP,k of P corresponding to the kth eigenvalue λk with multiplicity r has at most k weak nodal domains and at most k + r − 1 strong nodal domains. The nodal domain theorem provides an upper bound on the number of nodal domains of the eigenvectors of the Laplacian. It states that the kth eigenvector passes through zero at most k − 1 times. In the following we use terms domain and nodal domain interchangeably. To understand the shape of the eigenvectors at the nodal domains we use the following theorem. Theorem 2 ([11, 13]). An eigenvector of a graph Laplacian L cannot have a nonnegative local minimum or a nonpositive local maximum. If a minimum (maximum) of an eigenvector exists, it resides in a negative (positive) nodal domain. We follow with a simple extension of Theorem 2. Lemma 1. The nodal domains of the eigenvectors of a graph Laplacian L are unimodal. Proof. We prove Lemma 1 by method of contradiction. Without loss of generality, let us assume a negative nodal domain and suppose there are two local minima residing in the domain. This necessitates the existence of a local maximum between the two minima, a contradiction to Theorem 2. Therefore, a single minimum (maximum) must exist at every negative (positive) nodal domain and the nodal domain is unimodal. Next, we have to distinguish the special case that a nodal domain features no extremum. As an example, let us consider the primary eigenvector of any graph laplacian. According to Theorem 1, the primary eigenvector has exactly one nodal domain. What is more, all its elements are constant. It may seem like a contradiction that a constant nodal domain is unimodal. However, in our notion of unimodality the monotonicity is not strict. Constant nodal domains are also unimodal.

0

eigenvector u3



u3u3 D -1z .

1

10

20

chain graph

.

30

40

Figure 3: Nodal domain invariance of the 3rd eigenvector u3 of a chain graph. The multiplication of u3 with a randomly chosen process entropy vector z alters the amplitude but not the number or positions of the extrema. All shaded nodal domains are negative. The applicability of Lemma 1 is not restricted to L but it extends to matrix P . That can be primarily understood by Equation (12) and (14). An additional interpretation supports this claim: I − P is a generalized Laplacian of G with the same eigenvectors as P and eigenvalues λI−P = 1 − λP . In summary, we have proved that the nodal domains of the eigenvectors of our mechanism’s transition matrix P are unimodal. The invariance of nodal domain unimodality under multiplication with density and information is straightforward. We can rewrite the spectral expansion given in Equation (8) as follows. y=

s X > −1 (ϕΛ−1 z F )ii uP,i uP,i D i=1

=

s X

wi uP,i

(15)

i=1

where wi is a scalar given by > −1 z wi = (ϕΛ−1 F )ii uP,i D ϕ > = uP,i D−1 z 1 − (1 − ϕ)λP,i

(16)

In Equation (15), each eigenvector is weighted by wi , i.e., only its magnitude is altered. More importantly, neither the number of nodal domains of an eigenvector nor the unimodality is affected. By definition uP,i · u> P,i = 1, and the left eigenvectors u> exhibit the same nodal domains as uP,i . P,i −1 The inner product of u> with vector D z thus reveals the P,i importance of the information at the areas of the network that coincide with the right (and left) eigenvectors’ extrema. Therefore, z and x only alter the eigenvector magnitude and can take any arbitrary value. To demonstrate, we choose a graph that is embeddable in a single dimension, even though all concepts shown also apply to higher dimensions. We consider a chain graph of 40 nodes, i.e., a graph where each node has two neighbors with the exception of the boundary nodes that only have

y

u2 u1

u3

0

u4

1

10

20

chain graph

30

40

Figure 4: The four principal eigenvectors of P along with the approximated information potential 4 P > −1 y= (ϕΛ−1 1. The weighted sum y of F )ii ui ui D

Let us reconsider the chain graph of 40 nodes introduced above. In Figure 4 we plot for each node the corresponding element of the first four eigenvectors of P without considering the weights (ϕΛ−1 F )ii . Firstly, we notice that the eigenvectors domains are unimodal, which confirms Lemma 1. Secondly, we clearly see that for the first three eigenvectors the nodal domains result in a single extremum in the middle of the graph. Lastly, we also approximate the information potential y considering additional nodal domains in the figure, summing the first four terms of Equation (8) with ϕ = 0.1. Notice that the fourth domain actually has multiple extrema. However, due to the small weight on the local extrema of such eigenvectors relating to larger eigenvalues, this fourth domain has no effect on unimodality. As a result, we see in the figure the unimodality of the information potential.

5.

CHEMOTACTIC SEARCH

i=1

eigenvectors is unimodal. All shaded nodal domains are negative. one neighbor. Figure 3 focuses on the third eigenvector uP,3 and shows the invariance of nodal domain unimodality under multiplication with random vectors. The figure shading serves as a visual differentiation of the nodal domains to −1 negative and positive domains. We compute uP,3 u> z P,3 D for a randomly chosen process entropy vector z. We can see how the process entropy vector z influences the amplitude (and not the extrema) of uP,3 . We additionally need to give sufficient conditions on ϕ that guarantee unimodality of the sum of unimodal domains given the spectral characteristics of a graph. We provide an intuition in the following, and leave the proof for future work.

4.3

The Sum of Unimodal Domains

To understand the unimodality of the sum of unimodal domains given a specific ϕ, we focus on the weights of the spectral expansion (cf. Equation (16)). The weights play an important role on diminishing the effect of local extrema of eigenvectors relating to larger eigenvalues. As an example, if we sum only the first four terms of the spectral expansion but with equal weights, the resulting vector may have up to 5 unimodal nodal domains; this is the sum of the eigenvector domains while considering border effects only once (4 + 3 − 2 + 2 − 2). As we discussed in Section 3.4, the diagonal elements of ϕΛ−1 F are decreasing and ϕ determines how much larger the first element is relative to the rest. Diagonal elements therefore diminish the effect of multiple extrema of eigenvectors relating to larger eigenvalues. Since the left eigenvectors are bounded and D−1 z is the same for all eigenvectors, for sufficiently small ϕ, eigenvectors after the first three matter little and the number of unimodal domains is three. In this case, it is evident that there exists a single global extremum, following from the structure of the nodal domains. We can also observe that the primary eigenvector alters all node potentials equally and by a constant value. Therefore, it has no effect to the unimodality of the information potential.

Chemotaxis is the phenomenon in which somatic cells, bacteria, and other single-cell or multicellular organisms direct their movements according to certain chemicals in their environment [1]. In our case, the chemical is a poison and, by following the path of least chemical concentration, one will always reach a concentration minimum that coincides with the largest information potential. We can thus find short paths to information hubs. We have to note that chemotactic search takes decisions greedily. As such, it cannot distinguish local from global extrema. The success of the search depends on the structure of the search space, or equivalently of the chemical concentration over the network. Smoothness of the search space results in more informed decisions that take a larger part of the space into account. In our mechanism the smoothness is controlled by the inhibiting factor, which is a parameter of our system that inhibits or stimulates the diffusive property. Large values hinder information diffusion, which makes the search space less smooth and more concerned with local effects. Smaller values on the other hand decrease the mechanism’s sensitivity to local effects, thus resulting in a more global view and a smoother search space. As we discussed in Section 3, the space becomes unimodal as the inhibiting factor decreases. Unimodality guarantees that while traversing the network, any encountered local extremum, is also a global extremum. To demonstrate, let us consider the example of two nearby areas of rich information separated by a poor information zone. Due to the unimodality principle, given sufficiently small values of the inhibiting factor, the search space has a single minimum. The minimum is positioned where the information potential is the highest, i.e., somewhere between the two areas. Additionally, a chemotactic search will reveal a short path to the minimum. Section 1 outlined that, by combining information potentials with chemotactic search, we can solve problems that relate to search and coordination. We proceed with two case studies. We have to note that the case studies are neither the only nor the best use cases of our mechanism. Instead, they are intended to pronounce the unimodality and multimodality properties presented in Section 4.

750

1000

meters

max distance in meters

1000

500

90

th

500

percentile

250 10th percentile

median

100 0

4

50

100

time in seconds

150

200

Figure 5: Rendezvous of 100 robots which are randomly placed in an area of 1 km by 1 km. The robots reach close (less than 100 m) to the rendezvous point in 100 seconds median.

5.1

Multi-Agent Rendezvous

We focus on the particular application of multi-agent rendezvous, where a group of or robots have to eventually meet at a single location on a plane. The coordination is achieved by robots exchanging information with their neighboring robots and autonomously deciding on the direction of their movement. We have to stress that we do not claim the superiority of our method to previous solutions, (cf. [21]). Instead, we wish to evaluate our mechanism, to show its unimodality property, and to demonstrate one of the ways that it can be employed to achieve coordination. For software entities, a chemotactic movement over a unimodal information potential always leads to rendezvous. In the case of robots however, the construction of the potential is coupled to the robot’s movement. As robots move, the potential itself changes to reflect the new topology. In line with Babaoglu et al. [1], we identify that when using chemotaxis the diffusion must be faster than topology changes in order to provide useful guidance to mobile agents. Hence, if the information potential is updated more frequently than the topology changes, the robots meet. It is usually the case that the semantics of the location where robots meet are not considered. In contrast, the rendezvous point here coincides with the area of the network that exhibits the richest information potential. This can be a useful feature in scenarios where the robots are affected by or influence the environment. A group of solar-powered robots for example benefit more from a rendezvous point where the sun is the strongest. By setting the process information to reflect sun intensity, our mechanism achieves exactly that. Given a unimodal information potential, robots can rendezvous by simply moving towards the direction of the richest information – in this case the potential minimum. To do so, they simply ask their neighbor with the smallest xi its location and move towards it with a speed that is proportional to their distance. Even though in this example we consider the locations of nearby robots known, in essence only a sense of direction is needed [12]. We demonstrate the effectiveness of this approach with a simulation. In the simulation, 100 robots are positioned

0

250

500

meters

750

1000

Figure 6: Multi-rendezvous of 100 randomly placed robots. The increase of the inhibiting factor ϕ introduces multiple rendezvous points and reduces the mean distance traveled by each robot.

randomly in a square area of 1 km by 1 km and execute our mechanism asynchronously with each round lasting 1 second. The robots can move with a speed that is equal to or smaller than 10 m/s. To avoid artifacts of initialization, we keep the robots stationary for a period of 4 seconds in the beginning of each simulation run. For simplicity, we assume that the process information is constant. The potential therefore is only influenced by the connectivity and nodes meet at the most dense area of the network. The inhibiting factor was chosen to be ϕ = 0.01. Figure 5.1 shows the maximum distance between any pair of nodes in each point in time. We summarize 30 independent simulation runs by the 10th , 50th , and 90th percentiles. Without considering the initialization phase, the maximum distance becomes less than 100 m in 96 seconds (for the median). We also observe a slow decrease of the maximum distance around 100 seconds for the worst cases (90th percentile). This phenomenon stems from an initially bimodal potential. Due to the latency of signal diffusion, the robots initially head towards two distinct directions. However, they merge once the signal is diffused properly.

5.2

Multi-Agent Multi-Rendezvous

By choosing larger values of ϕ, our mechanism constructs multimodal information potentials. We exploit such potentials to perform search local to an area of the network with variable size. An application for this is the multi-rendezvous problem, where robots that are within a certain area rendezvous. Since the network may contain multiple such areas, we have multiple rendezvous points and robots move to the closest one. We simulate this application, maintaining all parameters of the rendezvous simulation in Section 5.1, except from setting the inhibiting factor to ϕ = 0.6 to consider areas of smaller size. The value of the inhibiting factor was chosen by experimentation1 . Figure 5.1 shows the trajectories of the nodes towards two ending points, i.e., the rendezvous points, that are drawn with darker color. The robots initial 1 The derivation of bounds on the inhibiting factor is left for future work.

placement is indicated by white circles. We can see that robots choose the closer rendezvous point, thereby reducing travel distance and rendezvous time to approximately half of the single-rendezvous case. In larger networks, we can therefore trade off the number of rendezvous points to the rendezvous time.

6.

CONCLUSIONS

In this paper we introduced the concept of information potentials, aggregates of a process monitored by a network. Different from single-value, global aggregates, e.g., averageconsensus, information potentials are spatial distributions that aggregate over parts of the network. We proposed a distributed mechanism that computes a spectrum of information potentials. The mechanism is adaptive, generic, and scalable because it relies solely on local interactions. We proved that the mechanism converges exponentially, independently of network topology and initial conditions. We proceeded to analyze its stable state and showed that it generates potentials that feature a single extremum, i.e., that are unimodal. Due to this property, we can effectively use our mechanism in conjunction with chemotactic search in order to solve problems related to network search and robot coordination in a global scope. We evaluated an application of global search using a case study of multi-agent rendezvous. We additionally investigated the use of chemotaxis on multimodal information potentials in a second case study of multi-agent multi-rendezvous. These are only two examples out of the large application space that we identified in the paper. In the future we want to demonstrate that our mechanism is a building block with wide applicability. Acknowledgements: This work was supported by the Dutch Technology Foundation STW and the Technology Program of the Ministry of Economic Affairs, Agriculture and Innovation.

References [1] O. Babaoglu, G. Canright, A. Deutsch, G. A. D. Caro, F. Ducatelle, L. M. Gambardella, N. Ganguly, M. Jelasity, R. Montemanni, A. Montresor, and T. Urnes. Design patterns from biology for distributed computing. ACM Trans. Autonomous Adaptive Systems, 1:26–66, September 2006. [2] T. Biyikoglu, J. Leydold, and P. F. Stadler. Laplacian eigenvectors of graphs: Perron-Frobenius and Faber-Krahn Type Theorems. Springer, 2007. [3] S. Boyd, A. Ghosh, B. Prabhakar, and D. Shah. Randomized gossip algorithms. IEEE Transactions on Information Theory, 52:2508–2530, Jun. 2006. [4] J. Cort´ es. Achieving coordination tasks in finite time via nonsmooth gradient flows. In Proc. IEEE Conf. on Decision and Control, pages 6376–6381, Dec. 2005. [5] J. Cort´ es. Distributed algorithms for reaching consensus on general functions. Automatica, pages 726–737, 2008.

[6] E. B. Davies, G. M. L. Gladwell, J. Leydold, and P. F. Stadler. Discrete nodal domain theorems. Linear Algebra and its Applications, 336:51–60, 2001. [7] A. Dimakis, A. Sarwate, and M. Wainwright. Geographic gossip: Efficient averaging for sensor networks. IEEE Trans. on Signal Processing, 56(3):1205–1216, March 2008. [8] W. Fan, Y. Liu, F. Wang, and X. Cai. Multi-robot formation control using potential field for mobile ad-hoc networks. In Proc. IEEE Int’l Conf. on Robotics and Biomimetics, pages 133–138, July 2005. [9] A. Gaba, S. Voulgaris, and M. van Steen. Towards congestion-aware all-to-all information dissemination in mobile ad-hoc networks. In Proc. Workshop on Ubiquitous Computing and Networks, pages 1690–1695, Dec. 2010. [10] C. Gkantsidis, G. Goel, M. Mihail, and A. Saberi. Towards topology aware networks. In Proc. IEEE Int’l Conf. on Computer Communications, pages 2591–2595, May 2007. [11] G. M. L. Gladwell and H. Zhu. Courant’s nodal line theorem and its discrete counterparts. The Quarterly Journal of Mechanics and Applied Mathematics, 55:1–15, 2002. [12] M. A. Gonzalez, J. Gomez, M. Lopez-Guerrero, V. Rangel, and M. M. Oca. Guide-gradient: A guiding algorithm for mobile nodes in WLAN and ad-hoc networks. Wirel. Pers. Commun., 57(4):629–653, April 2011. [13] L. K. Grover. Local search and the local structure of NPcomplete problems. Operations Research Letters, 12:235– 243, 1992. [14] D. Kempe, A. Dobra, and J. Gehrke. Gossip-based computation of aggregate information. In Proc. of IEEE Symposium Foundations of Computer Science, pages 482–491, Oct. 2003. [15] O. Khatib. Real-time obstacle avoidance for manipulators and mobile robots. Int’l Journal of Robotics Research, 5:90– 98, Apr 1986. [16] H. Lin, M. Lu, N. Milosavljevic, J. Gao, and L. J. Guibas. Composable Information Gradients in Wireless Sensor Networks. In Proc. of IEEE International Conference on Information Processing in Sensor Networks, pages 121–132, Apr. 2008. [17] P. Ogren, E. Fiorelli, and N. Leonard. Cooperative control of mobile sensor networks: Adaptive gradient climbing in a distributed environment. IEEE Trans. on Automatic Control, 49:1292–1302, Aug. 2004. [18] P. Song and V. Kumar. A potential field based approach to multi-robot manipulation. In Proc. IEEE Int’l Conf. on Robotics and Automation, volume 2, pages 1217–1222, May 2002. [19] D. P. Spanos and R. M. Murray. Distributed sensor fusion using dynamic consensus. IFAC World Congress, 2005. [20] L. Xiao, S. Boyd, and S. Lall. A scheme for robust distributed sensor fusion based on average consensus. In Proc. Int’l Conf. on Information processing in sensor networks, Apr. 2005. [21] M. Zavlanos, M. Egerstedt, and G. Pappas. Graph-theoretic connectivity control of mobile robot networks. Proc. of the IEEE, 99:1525–1540, Sept. 2011.